# Properties

 Label 1792.2.b.g.897.1 Level $1792$ Weight $2$ Character 1792.897 Analytic conductor $14.309$ 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] = [1792,2,Mod(897,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, 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("1792.897");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1792.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: $$14.3091920422$$ 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: $$2$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 897.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1792.897 Dual form 1792.2.b.g.897.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{3} +1.00000 q^{7} -1.00000 q^{9} +4.00000i q^{13} +6.00000 q^{17} +2.00000i q^{19} -2.00000i q^{21} +5.00000 q^{25} -4.00000i q^{27} +6.00000i q^{29} +4.00000 q^{31} +2.00000i q^{37} +8.00000 q^{39} -6.00000 q^{41} -8.00000i q^{43} +12.0000 q^{47} +1.00000 q^{49} -12.0000i q^{51} +6.00000i q^{53} +4.00000 q^{57} +6.00000i q^{59} -8.00000i q^{61} -1.00000 q^{63} -4.00000i q^{67} -2.00000 q^{73} -10.0000i q^{75} -8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} +12.0000 q^{87} +6.00000 q^{89} +4.00000i q^{91} -8.00000i q^{93} -10.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^7 - 2 * q^9 $$2 q + 2 q^{7} - 2 q^{9} + 12 q^{17} + 10 q^{25} + 8 q^{31} + 16 q^{39} - 12 q^{41} + 24 q^{47} + 2 q^{49} + 8 q^{57} - 2 q^{63} - 4 q^{73} - 16 q^{79} - 22 q^{81} + 24 q^{87} + 12 q^{89} - 20 q^{97}+O(q^{100})$$ 2 * q + 2 * q^7 - 2 * q^9 + 12 * q^17 + 10 * q^25 + 8 * q^31 + 16 * q^39 - 12 * q^41 + 24 * q^47 + 2 * q^49 + 8 * q^57 - 2 * q^63 - 4 * q^73 - 16 * q^79 - 22 * q^81 + 24 * q^87 + 12 * q^89 - 20 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 2.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ 0 0
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ 0 0
$$21$$ − 2.00000i − 0.436436i
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ − 4.00000i − 0.769800i
$$28$$ 0 0
$$29$$ 6.00000i 1.11417i 0.830455 + 0.557086i $$0.188081\pi$$
−0.830455 + 0.557086i $$0.811919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ 8.00000 1.28103
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ − 12.0000i − 1.68034i
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ 6.00000i 0.781133i 0.920575 + 0.390567i $$0.127721\pi$$
−0.920575 + 0.390567i $$0.872279\pi$$
$$60$$ 0 0
$$61$$ − 8.00000i − 1.02430i −0.858898 0.512148i $$-0.828850\pi$$
0.858898 0.512148i $$-0.171150\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ − 10.0000i − 1.15470i
$$76$$ 0 0
$$77$$ 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
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 12.0000 1.28654
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 4.00000i 0.419314i
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ 0 0
$$109$$ − 2.00000i − 0.191565i −0.995402 0.0957826i $$-0.969465\pi$$
0.995402 0.0957826i $$-0.0305354\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 4.00000i − 0.369800i
$$118$$ 0 0
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 12.0000i 1.08200i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 0 0
$$129$$ −16.0000 −1.40872
$$130$$ 0 0
$$131$$ 18.0000i 1.57267i 0.617802 + 0.786334i $$0.288023\pi$$
−0.617802 + 0.786334i $$0.711977\pi$$
$$132$$ 0 0
$$133$$ 2.00000i 0.173422i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ − 14.0000i − 1.18746i −0.804663 0.593732i $$-0.797654\pi$$
0.804663 0.593732i $$-0.202346\pi$$
$$140$$ 0 0
$$141$$ − 24.0000i − 2.02116i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 2.00000i − 0.164957i
$$148$$ 0 0
$$149$$ − 18.0000i − 1.47462i −0.675556 0.737309i $$-0.736096\pi$$
0.675556 0.737309i $$-0.263904\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.00000i 0.319235i 0.987179 + 0.159617i $$0.0510260\pi$$
−0.987179 + 0.159617i $$0.948974\pi$$
$$158$$ 0 0
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ − 2.00000i − 0.152944i
$$172$$ 0 0
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ 0 0
$$175$$ 5.00000 0.377964
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ − 12.0000i − 0.896922i −0.893802 0.448461i $$-0.851972\pi$$
0.893802 0.448461i $$-0.148028\pi$$
$$180$$ 0 0
$$181$$ 20.0000i 1.48659i 0.668965 + 0.743294i $$0.266738\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ −16.0000 −1.18275
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ − 4.00000i − 0.290957i
$$190$$ 0 0
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 0 0
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 0 0
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ − 4.00000i − 0.275371i −0.990476 0.137686i $$-0.956034\pi$$
0.990476 0.137686i $$-0.0439664\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 0 0
$$219$$ 4.00000i 0.270295i
$$220$$ 0 0
$$221$$ 24.0000i 1.61441i
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 0 0
$$225$$ −5.00000 −0.333333
$$226$$ 0 0
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ 0 0
$$229$$ − 4.00000i − 0.264327i −0.991228 0.132164i $$-0.957808\pi$$
0.991228 0.132164i $$-0.0421925\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 16.0000i 1.03931i
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ 10.0000i 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 18.0000i 1.13615i 0.822977 + 0.568075i $$0.192312\pi$$
−0.822977 + 0.568075i $$0.807688\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 2.00000i 0.124274i
$$260$$ 0 0
$$261$$ − 6.00000i − 0.371391i
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 12.0000i − 0.734388i
$$268$$ 0 0
$$269$$ 12.0000i 0.731653i 0.930683 + 0.365826i $$0.119214\pi$$
−0.930683 + 0.365826i $$0.880786\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ 8.00000 0.484182
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 22.0000i 1.30776i 0.756596 + 0.653882i $$0.226861\pi$$
−0.756596 + 0.653882i $$0.773139\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 20.0000i 1.17242i
$$292$$ 0 0
$$293$$ 24.0000i 1.40209i 0.713115 + 0.701047i $$0.247284\pi$$
−0.713115 + 0.701047i $$0.752716\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ − 8.00000i − 0.461112i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2.00000i 0.114146i 0.998370 + 0.0570730i $$0.0181768\pi$$
−0.998370 + 0.0570730i $$0.981823\pi$$
$$308$$ 0 0
$$309$$ 8.00000i 0.455104i
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ 0 0
$$323$$ 12.0000i 0.667698i
$$324$$ 0 0
$$325$$ 20.0000i 1.10940i
$$326$$ 0 0
$$327$$ −4.00000 −0.221201
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ − 8.00000i − 0.439720i −0.975531 0.219860i $$-0.929440\pi$$
0.975531 0.219860i $$-0.0705600\pi$$
$$332$$ 0 0
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ − 12.0000i − 0.651751i
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 24.0000i 1.28839i 0.764862 + 0.644194i $$0.222807\pi$$
−0.764862 + 0.644194i $$0.777193\pi$$
$$348$$ 0 0
$$349$$ 28.0000i 1.49881i 0.662114 + 0.749403i $$0.269659\pi$$
−0.662114 + 0.749403i $$0.730341\pi$$
$$350$$ 0 0
$$351$$ 16.0000 0.854017
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 12.0000i − 0.635107i
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ 15.0000 0.789474
$$362$$ 0 0
$$363$$ − 22.0000i − 1.15470i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 6.00000i 0.311504i
$$372$$ 0 0
$$373$$ 14.0000i 0.724893i 0.932005 + 0.362446i $$0.118058\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −24.0000 −1.23606
$$378$$ 0 0
$$379$$ 16.0000i 0.821865i 0.911666 + 0.410932i $$0.134797\pi$$
−0.911666 + 0.410932i $$0.865203\pi$$
$$380$$ 0 0
$$381$$ − 32.0000i − 1.63941i
$$382$$ 0 0
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.00000i 0.406663i
$$388$$ 0 0
$$389$$ 18.0000i 0.912636i 0.889817 + 0.456318i $$0.150832\pi$$
−0.889817 + 0.456318i $$0.849168\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 36.0000 1.81596
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 20.0000i − 1.00377i −0.864934 0.501886i $$-0.832640\pi$$
0.864934 0.501886i $$-0.167360\pi$$
$$398$$ 0 0
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 16.0000i 0.797017i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 36.0000i 1.77575i
$$412$$ 0 0
$$413$$ 6.00000i 0.295241i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −28.0000 −1.37117
$$418$$ 0 0
$$419$$ 6.00000i 0.293119i 0.989202 + 0.146560i $$0.0468200\pi$$
−0.989202 + 0.146560i $$0.953180\pi$$
$$420$$ 0 0
$$421$$ − 10.0000i − 0.487370i −0.969854 0.243685i $$-0.921644\pi$$
0.969854 0.243685i $$-0.0783563\pi$$
$$422$$ 0 0
$$423$$ −12.0000 −0.583460
$$424$$ 0 0
$$425$$ 30.0000 1.45521
$$426$$ 0 0
$$427$$ − 8.00000i − 0.387147i
$$428$$ 0 0
$$429$$ 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
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −1.00000 −0.0476190
$$442$$ 0 0
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −36.0000 −1.70274
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 16.0000i − 0.751746i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 0 0
$$459$$ − 24.0000i − 1.12022i
$$460$$ 0 0
$$461$$ − 12.0000i − 0.558896i −0.960161 0.279448i $$-0.909849\pi$$
0.960161 0.279448i $$-0.0901514\pi$$
$$462$$ 0 0
$$463$$ −32.0000 −1.48717 −0.743583 0.668644i $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 6.00000i − 0.277647i −0.990317 0.138823i $$-0.955668\pi$$
0.990317 0.138823i $$-0.0443321\pi$$
$$468$$ 0 0
$$469$$ − 4.00000i − 0.184703i
$$470$$ 0 0
$$471$$ 8.00000 0.368621
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 10.0000i 0.458831i
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 0 0
$$479$$ 36.0000 1.64488 0.822441 0.568850i $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ 0 0
$$489$$ −32.0000 −1.44709
$$490$$ 0 0
$$491$$ 12.0000i 0.541552i 0.962642 + 0.270776i $$0.0872803\pi$$
−0.962642 + 0.270776i $$0.912720\pi$$
$$492$$ 0 0
$$493$$ 36.0000i 1.62136i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 4.00000i − 0.179065i −0.995984 0.0895323i $$-0.971463\pi$$
0.995984 0.0895323i $$-0.0285372\pi$$
$$500$$ 0 0
$$501$$ 24.0000i 1.07224i
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 6.00000i 0.266469i
$$508$$ 0 0
$$509$$ − 36.0000i − 1.59567i −0.602875 0.797836i $$-0.705978\pi$$
0.602875 0.797836i $$-0.294022\pi$$
$$510$$ 0 0
$$511$$ −2.00000 −0.0884748
$$512$$ 0 0
$$513$$ 8.00000 0.353209
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 24.0000 1.05348
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ − 2.00000i − 0.0874539i −0.999044 0.0437269i $$-0.986077\pi$$
0.999044 0.0437269i $$-0.0139232\pi$$
$$524$$ 0 0
$$525$$ − 10.0000i − 0.436436i
$$526$$ 0 0
$$527$$ 24.0000 1.04546
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ − 6.00000i − 0.260378i
$$532$$ 0 0
$$533$$ − 24.0000i − 1.03956i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −24.0000 −1.03568
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 38.0000i − 1.63375i −0.576816 0.816874i $$-0.695705\pi$$
0.576816 0.816874i $$-0.304295\pi$$
$$542$$ 0 0
$$543$$ 40.0000 1.71656
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ 0 0
$$549$$ 8.00000i 0.341432i
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 6.00000i − 0.254228i −0.991888 0.127114i $$-0.959429\pi$$
0.991888 0.127114i $$-0.0405714\pi$$
$$558$$ 0 0
$$559$$ 32.0000 1.35346
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 30.0000i 1.26435i 0.774826 + 0.632175i $$0.217837\pi$$
−0.774826 + 0.632175i $$0.782163\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −11.0000 −0.461957
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ − 32.0000i − 1.33916i −0.742741 0.669579i $$-0.766474\pi$$
0.742741 0.669579i $$-0.233526\pi$$
$$572$$ 0 0
$$573$$ 48.0000i 2.00523i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ − 28.0000i − 1.16364i
$$580$$ 0 0
$$581$$ − 6.00000i − 0.248922i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 42.0000i 1.73353i 0.498721 + 0.866763i $$0.333803\pi$$
−0.498721 + 0.866763i $$0.666197\pi$$
$$588$$ 0 0
$$589$$ 8.00000i 0.329634i
$$590$$ 0 0
$$591$$ −36.0000 −1.48084
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 40.0000i − 1.63709i
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 0 0
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ 48.0000i 1.94187i
$$612$$ 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
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ − 26.0000i − 1.04503i −0.852631 0.522514i $$-0.824994\pi$$
0.852631 0.522514i $$-0.175006\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.0000i 0.478471i
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ −8.00000 −0.317971
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4.00000i 0.158486i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ 14.0000i 0.552106i 0.961142 + 0.276053i $$0.0890266\pi$$
−0.961142 + 0.276053i $$0.910973\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ − 8.00000i − 0.313545i
$$652$$ 0 0
$$653$$ − 18.0000i − 0.704394i −0.935926 0.352197i $$-0.885435\pi$$
0.935926 0.352197i $$-0.114565\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ − 24.0000i − 0.934907i −0.884018 0.467454i $$-0.845171\pi$$
0.884018 0.467454i $$-0.154829\pi$$
$$660$$ 0 0
$$661$$ − 40.0000i − 1.55582i −0.628376 0.777910i $$-0.716280\pi$$
0.628376 0.777910i $$-0.283720\pi$$
$$662$$ 0 0
$$663$$ 48.0000 1.86417
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 16.0000i 0.618596i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ 0 0
$$675$$ − 20.0000i − 0.769800i
$$676$$ 0 0
$$677$$ − 12.0000i − 0.461197i −0.973049 0.230599i $$-0.925932\pi$$
0.973049 0.230599i $$-0.0740685\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 36.0000 1.37952
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −8.00000 −0.305219
$$688$$ 0 0
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ − 46.0000i − 1.74992i −0.484193 0.874961i $$-0.660887\pi$$
0.484193 0.874961i $$-0.339113\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 0 0
$$699$$ − 12.0000i − 0.453882i
$$700$$ 0 0
$$701$$ − 18.0000i − 0.679851i −0.940452 0.339925i $$-0.889598\pi$$
0.940452 0.339925i $$-0.110402\pi$$
$$702$$ 0 0
$$703$$ −4.00000 −0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ − 46.0000i − 1.72757i −0.503864 0.863783i $$-0.668089\pi$$
0.503864 0.863783i $$-0.331911\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 48.0000i 1.79259i
$$718$$ 0 0
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ 0 0
$$723$$ 20.0000i 0.743808i
$$724$$ 0 0
$$725$$ 30.0000i 1.11417i
$$726$$ 0 0
$$727$$ 44.0000 1.63187 0.815935 0.578144i $$-0.196223\pi$$
0.815935 + 0.578144i $$0.196223\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ − 48.0000i − 1.77534i
$$732$$ 0 0
$$733$$ 40.0000i 1.47743i 0.674016 + 0.738717i $$0.264568\pi$$
−0.674016 + 0.738717i $$0.735432\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ − 16.0000i − 0.588570i −0.955718 0.294285i $$-0.904919\pi$$
0.955718 0.294285i $$-0.0950814\pi$$
$$740$$ 0 0
$$741$$ 16.0000i 0.587775i
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.00000i 0.219529i
$$748$$ 0 0
$$749$$ − 12.0000i − 0.438470i
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 36.0000 1.31191
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ 0 0
$$763$$ − 2.00000i − 0.0724049i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ − 36.0000i − 1.29651i
$$772$$ 0 0
$$773$$ 24.0000i 0.863220i 0.902060 + 0.431610i $$0.142054\pi$$
−0.902060 + 0.431610i $$0.857946\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ 0 0
$$777$$ 4.00000 0.143499
$$778$$ 0 0
$$779$$ − 12.0000i − 0.429945i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 24.0000 0.857690
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 22.0000i − 0.784215i −0.919919 0.392108i $$-0.871746\pi$$
0.919919 0.392108i $$-0.128254\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ 32.0000 1.13635
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 12.0000i 0.425062i 0.977154 + 0.212531i $$0.0681706\pi$$
−0.977154 + 0.212531i $$0.931829\pi$$
$$798$$ 0 0
$$799$$ 72.0000 2.54718
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 24.0000 0.844840
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ − 2.00000i − 0.0702295i −0.999383 0.0351147i $$-0.988820\pi$$
0.999383 0.0351147i $$-0.0111797\pi$$
$$812$$ 0 0
$$813$$ − 32.0000i − 1.12229i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000 0.559769
$$818$$ 0 0
$$819$$ − 4.00000i − 0.139771i
$$820$$ 0 0
$$821$$ 6.00000i 0.209401i 0.994504 + 0.104701i $$0.0333885\pi$$
−0.994504 + 0.104701i $$0.966612\pi$$
$$822$$ 0 0
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 0 0
$$829$$ − 56.0000i − 1.94496i −0.232986 0.972480i $$-0.574849\pi$$
0.232986 0.972480i $$-0.425151\pi$$
$$830$$ 0 0
$$831$$ −20.0000 −0.693792
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 16.0000i − 0.553041i
$$838$$ 0 0
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ − 12.0000i − 0.413302i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 11.0000 0.377964
$$848$$ 0 0
$$849$$ 44.0000 1.51008
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 44.0000i 1.50653i 0.657716 + 0.753266i $$0.271523\pi$$
−0.657716 + 0.753266i $$0.728477\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ − 14.0000i − 0.477674i −0.971060 0.238837i $$-0.923234\pi$$
0.971060 0.238837i $$-0.0767661\pi$$
$$860$$ 0 0
$$861$$ 12.0000i 0.408959i
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 38.0000i − 1.29055i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 0 0
$$873$$ 10.0000 0.338449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 0 0
$$879$$ 48.0000 1.61900
$$880$$ 0 0
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ 0 0
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 24.0000i 0.803129i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 24.0000i 0.800445i
$$900$$ 0 0
$$901$$ 36.0000i 1.19933i
$$902$$ 0 0
$$903$$ −16.0000 −0.532447
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 44.0000i − 1.46100i −0.682915 0.730498i $$-0.739288\pi$$
0.682915 0.730498i $$-0.260712\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 18.0000i 0.594412i
$$918$$ 0 0
$$919$$ 56.0000 1.84727 0.923635 0.383274i $$-0.125203\pi$$
0.923635 + 0.383274i $$0.125203\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 10.0000i 0.328798i
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 2.00000i 0.0655474i
$$932$$ 0 0
$$933$$ 48.0000i 1.57145i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −2.00000 −0.0653372 −0.0326686 0.999466i $$-0.510401\pi$$
−0.0326686 + 0.999466i $$0.510401\pi$$
$$938$$ 0 0
$$939$$ − 20.0000i − 0.652675i
$$940$$ 0 0
$$941$$ 24.0000i 0.782378i 0.920310 + 0.391189i $$0.127936\pi$$
−0.920310 + 0.391189i $$0.872064\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 24.0000i 0.779895i 0.920837 + 0.389948i $$0.127507\pi$$
−0.920837 + 0.389948i $$0.872493\pi$$
$$948$$ 0 0
$$949$$ − 8.00000i − 0.259691i
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 12.0000i 0.386695i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 0 0
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 6.00000i 0.192549i 0.995355 + 0.0962746i $$0.0306927\pi$$
−0.995355 + 0.0962746i $$0.969307\pi$$
$$972$$ 0 0
$$973$$ − 14.0000i − 0.448819i
$$974$$ 0 0
$$975$$ 40.0000 1.28103
$$976$$ 0 0
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 2.00000i 0.0638551i
$$982$$ 0 0
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 24.0000i − 0.763928i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ −16.0000 −0.507745
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 8.00000i 0.253363i 0.991943 + 0.126681i $$0.0404325\pi$$
−0.991943 + 0.126681i $$0.959567\pi$$
$$998$$ 0 0
$$999$$ 8.00000 0.253109
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 1792.2.b.g.897.1 2
4.3 odd 2 1792.2.b.c.897.2 2
8.3 odd 2 1792.2.b.c.897.1 2
8.5 even 2 inner 1792.2.b.g.897.2 2
16.3 odd 4 14.2.a.a.1.1 1
16.5 even 4 448.2.a.a.1.1 1
16.11 odd 4 448.2.a.g.1.1 1
16.13 even 4 112.2.a.c.1.1 1
48.5 odd 4 4032.2.a.r.1.1 1
48.11 even 4 4032.2.a.w.1.1 1
48.29 odd 4 1008.2.a.h.1.1 1
48.35 even 4 126.2.a.b.1.1 1
80.3 even 4 350.2.c.d.99.2 2
80.13 odd 4 2800.2.g.h.449.2 2
80.19 odd 4 350.2.a.f.1.1 1
80.29 even 4 2800.2.a.g.1.1 1
80.67 even 4 350.2.c.d.99.1 2
80.77 odd 4 2800.2.g.h.449.1 2
112.3 even 12 98.2.c.a.79.1 2
112.13 odd 4 784.2.a.b.1.1 1
112.19 even 12 98.2.c.a.67.1 2
112.27 even 4 3136.2.a.e.1.1 1
112.45 odd 12 784.2.i.i.177.1 2
112.51 odd 12 98.2.c.b.67.1 2
112.61 odd 12 784.2.i.i.753.1 2
112.67 odd 12 98.2.c.b.79.1 2
112.69 odd 4 3136.2.a.z.1.1 1
112.83 even 4 98.2.a.a.1.1 1
112.93 even 12 784.2.i.c.753.1 2
112.109 even 12 784.2.i.c.177.1 2
144.67 odd 12 1134.2.f.l.379.1 2
144.83 even 12 1134.2.f.f.757.1 2
144.115 odd 12 1134.2.f.l.757.1 2
144.131 even 12 1134.2.f.f.379.1 2
176.131 even 4 1694.2.a.e.1.1 1
208.51 odd 4 2366.2.a.j.1.1 1
208.83 even 4 2366.2.d.b.337.2 2
208.99 even 4 2366.2.d.b.337.1 2
240.83 odd 4 3150.2.g.j.2899.1 2
240.179 even 4 3150.2.a.i.1.1 1
240.227 odd 4 3150.2.g.j.2899.2 2
272.67 odd 4 4046.2.a.f.1.1 1
304.227 even 4 5054.2.a.c.1.1 1
336.83 odd 4 882.2.a.i.1.1 1
336.125 even 4 7056.2.a.bd.1.1 1
336.131 odd 12 882.2.g.d.361.1 2
336.179 even 12 882.2.g.c.667.1 2
336.227 odd 12 882.2.g.d.667.1 2
336.275 even 12 882.2.g.c.361.1 2
368.275 even 4 7406.2.a.a.1.1 1
560.83 odd 4 2450.2.c.c.99.2 2
560.307 odd 4 2450.2.c.c.99.1 2
560.419 even 4 2450.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 16.3 odd 4
98.2.a.a.1.1 1 112.83 even 4
98.2.c.a.67.1 2 112.19 even 12
98.2.c.a.79.1 2 112.3 even 12
98.2.c.b.67.1 2 112.51 odd 12
98.2.c.b.79.1 2 112.67 odd 12
112.2.a.c.1.1 1 16.13 even 4
126.2.a.b.1.1 1 48.35 even 4
350.2.a.f.1.1 1 80.19 odd 4
350.2.c.d.99.1 2 80.67 even 4
350.2.c.d.99.2 2 80.3 even 4
448.2.a.a.1.1 1 16.5 even 4
448.2.a.g.1.1 1 16.11 odd 4
784.2.a.b.1.1 1 112.13 odd 4
784.2.i.c.177.1 2 112.109 even 12
784.2.i.c.753.1 2 112.93 even 12
784.2.i.i.177.1 2 112.45 odd 12
784.2.i.i.753.1 2 112.61 odd 12
882.2.a.i.1.1 1 336.83 odd 4
882.2.g.c.361.1 2 336.275 even 12
882.2.g.c.667.1 2 336.179 even 12
882.2.g.d.361.1 2 336.131 odd 12
882.2.g.d.667.1 2 336.227 odd 12
1008.2.a.h.1.1 1 48.29 odd 4
1134.2.f.f.379.1 2 144.131 even 12
1134.2.f.f.757.1 2 144.83 even 12
1134.2.f.l.379.1 2 144.67 odd 12
1134.2.f.l.757.1 2 144.115 odd 12
1694.2.a.e.1.1 1 176.131 even 4
1792.2.b.c.897.1 2 8.3 odd 2
1792.2.b.c.897.2 2 4.3 odd 2
1792.2.b.g.897.1 2 1.1 even 1 trivial
1792.2.b.g.897.2 2 8.5 even 2 inner
2366.2.a.j.1.1 1 208.51 odd 4
2366.2.d.b.337.1 2 208.99 even 4
2366.2.d.b.337.2 2 208.83 even 4
2450.2.a.t.1.1 1 560.419 even 4
2450.2.c.c.99.1 2 560.307 odd 4
2450.2.c.c.99.2 2 560.83 odd 4
2800.2.a.g.1.1 1 80.29 even 4
2800.2.g.h.449.1 2 80.77 odd 4
2800.2.g.h.449.2 2 80.13 odd 4
3136.2.a.e.1.1 1 112.27 even 4
3136.2.a.z.1.1 1 112.69 odd 4
3150.2.a.i.1.1 1 240.179 even 4
3150.2.g.j.2899.1 2 240.83 odd 4
3150.2.g.j.2899.2 2 240.227 odd 4
4032.2.a.r.1.1 1 48.5 odd 4
4032.2.a.w.1.1 1 48.11 even 4
4046.2.a.f.1.1 1 272.67 odd 4
5054.2.a.c.1.1 1 304.227 even 4
7056.2.a.bd.1.1 1 336.125 even 4
7406.2.a.a.1.1 1 368.275 even 4