# Properties

 Label 1152.2.c.c.1151.4 Level $1152$ Weight $2$ Character 1152.1151 Analytic conductor $9.199$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(1151,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1152.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.c (of order $$2$$, degree $$1$$, 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: $$9.19876631285$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1151.4 Root $$-0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1152.1151 Dual form 1152.2.c.c.1151.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.41421i q^{5} +4.82843i q^{7} +O(q^{10})$$ $$q+3.41421i q^{5} +4.82843i q^{7} +2.82843 q^{11} -2.82843 q^{13} +5.41421i q^{17} -5.65685i q^{19} +1.17157 q^{23} -6.65685 q^{25} +0.585786i q^{29} -3.17157i q^{31} -16.4853 q^{35} -3.65685 q^{37} +2.58579i q^{41} -9.65685i q^{43} +12.4853 q^{47} -16.3137 q^{49} -5.07107i q^{53} +9.65685i q^{55} -2.34315 q^{59} -7.65685 q^{61} -9.65685i q^{65} +12.0000i q^{67} -4.48528 q^{71} +4.00000 q^{73} +13.6569i q^{77} +6.48528i q^{79} +5.17157 q^{83} -18.4853 q^{85} +12.2426i q^{89} -13.6569i q^{91} +19.3137 q^{95} +13.6569 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 16 q^{23} - 4 q^{25} - 32 q^{35} + 8 q^{37} + 16 q^{47} - 20 q^{49} - 32 q^{59} - 8 q^{61} + 16 q^{71} + 16 q^{73} + 32 q^{83} - 40 q^{85} + 32 q^{95} + 32 q^{97}+O(q^{100})$$ 4 * q + 16 * q^23 - 4 * q^25 - 32 * q^35 + 8 * q^37 + 16 * q^47 - 20 * q^49 - 32 * q^59 - 8 * q^61 + 16 * q^71 + 16 * q^73 + 32 * q^83 - 40 * q^85 + 32 * q^95 + 32 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 3.41421i 1.52688i 0.645877 + 0.763441i $$0.276492\pi$$
−0.645877 + 0.763441i $$0.723508\pi$$
$$6$$ 0 0
$$7$$ 4.82843i 1.82497i 0.409106 + 0.912487i $$0.365841\pi$$
−0.409106 + 0.912487i $$0.634159\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.82843 0.852803 0.426401 0.904534i $$-0.359781\pi$$
0.426401 + 0.904534i $$0.359781\pi$$
$$12$$ 0 0
$$13$$ −2.82843 −0.784465 −0.392232 0.919866i $$-0.628297\pi$$
−0.392232 + 0.919866i $$0.628297\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.41421i 1.31314i 0.754265 + 0.656570i $$0.227993\pi$$
−0.754265 + 0.656570i $$0.772007\pi$$
$$18$$ 0 0
$$19$$ − 5.65685i − 1.29777i −0.760886 0.648886i $$-0.775235\pi$$
0.760886 0.648886i $$-0.224765\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.17157 0.244290 0.122145 0.992512i $$-0.461023\pi$$
0.122145 + 0.992512i $$0.461023\pi$$
$$24$$ 0 0
$$25$$ −6.65685 −1.33137
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0.585786i 0.108778i 0.998520 + 0.0543889i $$0.0173211\pi$$
−0.998520 + 0.0543889i $$0.982679\pi$$
$$30$$ 0 0
$$31$$ − 3.17157i − 0.569631i −0.958582 0.284816i $$-0.908068\pi$$
0.958582 0.284816i $$-0.0919324\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −16.4853 −2.78652
$$36$$ 0 0
$$37$$ −3.65685 −0.601183 −0.300592 0.953753i $$-0.597184\pi$$
−0.300592 + 0.953753i $$0.597184\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.58579i 0.403832i 0.979403 + 0.201916i $$0.0647168\pi$$
−0.979403 + 0.201916i $$0.935283\pi$$
$$42$$ 0 0
$$43$$ − 9.65685i − 1.47266i −0.676625 0.736328i $$-0.736558\pi$$
0.676625 0.736328i $$-0.263442\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.4853 1.82117 0.910583 0.413327i $$-0.135633\pi$$
0.910583 + 0.413327i $$0.135633\pi$$
$$48$$ 0 0
$$49$$ −16.3137 −2.33053
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 5.07107i − 0.696565i −0.937390 0.348282i $$-0.886765\pi$$
0.937390 0.348282i $$-0.113235\pi$$
$$54$$ 0 0
$$55$$ 9.65685i 1.30213i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −2.34315 −0.305052 −0.152526 0.988299i $$-0.548741\pi$$
−0.152526 + 0.988299i $$0.548741\pi$$
$$60$$ 0 0
$$61$$ −7.65685 −0.980360 −0.490180 0.871621i $$-0.663069\pi$$
−0.490180 + 0.871621i $$0.663069\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 9.65685i − 1.19779i
$$66$$ 0 0
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.48528 −0.532305 −0.266152 0.963931i $$-0.585752\pi$$
−0.266152 + 0.963931i $$0.585752\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 13.6569i 1.55634i
$$78$$ 0 0
$$79$$ 6.48528i 0.729651i 0.931076 + 0.364826i $$0.118871\pi$$
−0.931076 + 0.364826i $$0.881129\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 5.17157 0.567654 0.283827 0.958876i $$-0.408396\pi$$
0.283827 + 0.958876i $$0.408396\pi$$
$$84$$ 0 0
$$85$$ −18.4853 −2.00501
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 12.2426i 1.29772i 0.760909 + 0.648859i $$0.224753\pi$$
−0.760909 + 0.648859i $$0.775247\pi$$
$$90$$ 0 0
$$91$$ − 13.6569i − 1.43163i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 19.3137 1.98154
$$96$$ 0 0
$$97$$ 13.6569 1.38664 0.693322 0.720628i $$-0.256146\pi$$
0.693322 + 0.720628i $$0.256146\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 10.7279i 1.06747i 0.845652 + 0.533734i $$0.179212\pi$$
−0.845652 + 0.533734i $$0.820788\pi$$
$$102$$ 0 0
$$103$$ 3.17157i 0.312504i 0.987717 + 0.156252i $$0.0499413\pi$$
−0.987717 + 0.156252i $$0.950059\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.3137 1.09374 0.546869 0.837218i $$-0.315820\pi$$
0.546869 + 0.837218i $$0.315820\pi$$
$$108$$ 0 0
$$109$$ 10.8284 1.03718 0.518588 0.855024i $$-0.326458\pi$$
0.518588 + 0.855024i $$0.326458\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.2426i 1.15169i 0.817559 + 0.575845i $$0.195327\pi$$
−0.817559 + 0.575845i $$0.804673\pi$$
$$114$$ 0 0
$$115$$ 4.00000i 0.373002i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −26.1421 −2.39645
$$120$$ 0 0
$$121$$ −3.00000 −0.272727
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 5.65685i − 0.505964i
$$126$$ 0 0
$$127$$ 4.82843i 0.428454i 0.976784 + 0.214227i $$0.0687232\pi$$
−0.976784 + 0.214227i $$0.931277\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −5.65685 −0.494242 −0.247121 0.968985i $$-0.579484\pi$$
−0.247121 + 0.968985i $$0.579484\pi$$
$$132$$ 0 0
$$133$$ 27.3137 2.36840
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 10.5858i − 0.904405i −0.891915 0.452202i $$-0.850638\pi$$
0.891915 0.452202i $$-0.149362\pi$$
$$138$$ 0 0
$$139$$ − 12.0000i − 1.01783i −0.860818 0.508913i $$-0.830047\pi$$
0.860818 0.508913i $$-0.169953\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 6.72792i − 0.551173i −0.961276 0.275586i $$-0.911128\pi$$
0.961276 0.275586i $$-0.0888720\pi$$
$$150$$ 0 0
$$151$$ − 6.48528i − 0.527765i −0.964555 0.263882i $$-0.914997\pi$$
0.964555 0.263882i $$-0.0850031\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 10.8284 0.869760
$$156$$ 0 0
$$157$$ −0.343146 −0.0273860 −0.0136930 0.999906i $$-0.504359\pi$$
−0.0136930 + 0.999906i $$0.504359\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 5.65685i 0.445823i
$$162$$ 0 0
$$163$$ 1.65685i 0.129775i 0.997893 + 0.0648874i $$0.0206688\pi$$
−0.997893 + 0.0648874i $$0.979331\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −5.65685 −0.437741 −0.218870 0.975754i $$-0.570237\pi$$
−0.218870 + 0.975754i $$0.570237\pi$$
$$168$$ 0 0
$$169$$ −5.00000 −0.384615
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 20.5858i − 1.56511i −0.622582 0.782554i $$-0.713916\pi$$
0.622582 0.782554i $$-0.286084\pi$$
$$174$$ 0 0
$$175$$ − 32.1421i − 2.42972i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −19.3137 −1.44357 −0.721787 0.692115i $$-0.756679\pi$$
−0.721787 + 0.692115i $$0.756679\pi$$
$$180$$ 0 0
$$181$$ 6.14214 0.456541 0.228271 0.973598i $$-0.426693\pi$$
0.228271 + 0.973598i $$0.426693\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 12.4853i − 0.917936i
$$186$$ 0 0
$$187$$ 15.3137i 1.11985i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.3431 −0.748404 −0.374202 0.927347i $$-0.622083\pi$$
−0.374202 + 0.927347i $$0.622083\pi$$
$$192$$ 0 0
$$193$$ 5.31371 0.382489 0.191245 0.981542i $$-0.438748\pi$$
0.191245 + 0.981542i $$0.438748\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.3848i 1.16737i 0.811981 + 0.583683i $$0.198389\pi$$
−0.811981 + 0.583683i $$0.801611\pi$$
$$198$$ 0 0
$$199$$ − 4.82843i − 0.342278i −0.985247 0.171139i $$-0.945255\pi$$
0.985247 0.171139i $$-0.0547447\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −2.82843 −0.198517
$$204$$ 0 0
$$205$$ −8.82843 −0.616604
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 16.0000i − 1.10674i
$$210$$ 0 0
$$211$$ 23.3137i 1.60498i 0.596664 + 0.802491i $$0.296492\pi$$
−0.596664 + 0.802491i $$0.703508\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 32.9706 2.24857
$$216$$ 0 0
$$217$$ 15.3137 1.03956
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 15.3137i − 1.03011i
$$222$$ 0 0
$$223$$ − 17.7990i − 1.19191i −0.803018 0.595954i $$-0.796774\pi$$
0.803018 0.595954i $$-0.203226\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 26.8284 1.78067 0.890333 0.455311i $$-0.150472\pi$$
0.890333 + 0.455311i $$0.150472\pi$$
$$228$$ 0 0
$$229$$ 5.17157 0.341747 0.170874 0.985293i $$-0.445341\pi$$
0.170874 + 0.985293i $$0.445341\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 17.8995i − 1.17263i −0.810081 0.586317i $$-0.800577\pi$$
0.810081 0.586317i $$-0.199423\pi$$
$$234$$ 0 0
$$235$$ 42.6274i 2.78071i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 20.9706 1.35083 0.675416 0.737437i $$-0.263964\pi$$
0.675416 + 0.737437i $$0.263964\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 55.6985i − 3.55845i
$$246$$ 0 0
$$247$$ 16.0000i 1.01806i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 24.4853 1.54550 0.772749 0.634712i $$-0.218881\pi$$
0.772749 + 0.634712i $$0.218881\pi$$
$$252$$ 0 0
$$253$$ 3.31371 0.208331
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 9.89949i − 0.617514i −0.951141 0.308757i $$-0.900087\pi$$
0.951141 0.308757i $$-0.0999129\pi$$
$$258$$ 0 0
$$259$$ − 17.6569i − 1.09714i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −18.3431 −1.13109 −0.565543 0.824719i $$-0.691334\pi$$
−0.565543 + 0.824719i $$0.691334\pi$$
$$264$$ 0 0
$$265$$ 17.3137 1.06357
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 32.3848i 1.97453i 0.159070 + 0.987267i $$0.449150\pi$$
−0.159070 + 0.987267i $$0.550850\pi$$
$$270$$ 0 0
$$271$$ 1.51472i 0.0920126i 0.998941 + 0.0460063i $$0.0146494\pi$$
−0.998941 + 0.0460063i $$0.985351\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −18.8284 −1.13540
$$276$$ 0 0
$$277$$ −21.1716 −1.27208 −0.636038 0.771658i $$-0.719428\pi$$
−0.636038 + 0.771658i $$0.719428\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 6.10051i − 0.363926i −0.983305 0.181963i $$-0.941755\pi$$
0.983305 0.181963i $$-0.0582450\pi$$
$$282$$ 0 0
$$283$$ 3.31371i 0.196980i 0.995138 + 0.0984898i $$0.0314012\pi$$
−0.995138 + 0.0984898i $$0.968599\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.4853 −0.736983
$$288$$ 0 0
$$289$$ −12.3137 −0.724336
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 28.8701i 1.68661i 0.537438 + 0.843303i $$0.319392\pi$$
−0.537438 + 0.843303i $$0.680608\pi$$
$$294$$ 0 0
$$295$$ − 8.00000i − 0.465778i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −3.31371 −0.191637
$$300$$ 0 0
$$301$$ 46.6274 2.68756
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 26.1421i − 1.49689i
$$306$$ 0 0
$$307$$ − 0.686292i − 0.0391687i −0.999808 0.0195844i $$-0.993766\pi$$
0.999808 0.0195844i $$-0.00623429\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −13.6569 −0.774409 −0.387205 0.921994i $$-0.626559\pi$$
−0.387205 + 0.921994i $$0.626559\pi$$
$$312$$ 0 0
$$313$$ 9.31371 0.526442 0.263221 0.964736i $$-0.415215\pi$$
0.263221 + 0.964736i $$0.415215\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 14.2426i − 0.799946i −0.916527 0.399973i $$-0.869019\pi$$
0.916527 0.399973i $$-0.130981\pi$$
$$318$$ 0 0
$$319$$ 1.65685i 0.0927660i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 30.6274 1.70416
$$324$$ 0 0
$$325$$ 18.8284 1.04441
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 60.2843i 3.32358i
$$330$$ 0 0
$$331$$ 4.00000i 0.219860i 0.993939 + 0.109930i $$0.0350627\pi$$
−0.993939 + 0.109930i $$0.964937\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −40.9706 −2.23846
$$336$$ 0 0
$$337$$ 23.3137 1.26998 0.634989 0.772521i $$-0.281004\pi$$
0.634989 + 0.772521i $$0.281004\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 8.97056i − 0.485783i
$$342$$ 0 0
$$343$$ − 44.9706i − 2.42818i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −16.4853 −0.884976 −0.442488 0.896774i $$-0.645904\pi$$
−0.442488 + 0.896774i $$0.645904\pi$$
$$348$$ 0 0
$$349$$ 11.6569 0.623977 0.311989 0.950086i $$-0.399005\pi$$
0.311989 + 0.950086i $$0.399005\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.8701i 1.00435i 0.864765 + 0.502176i $$0.167467\pi$$
−0.864765 + 0.502176i $$0.832533\pi$$
$$354$$ 0 0
$$355$$ − 15.3137i − 0.812767i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 20.4853 1.08117 0.540586 0.841289i $$-0.318203\pi$$
0.540586 + 0.841289i $$0.318203\pi$$
$$360$$ 0 0
$$361$$ −13.0000 −0.684211
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 13.6569i 0.714832i
$$366$$ 0 0
$$367$$ 20.8284i 1.08724i 0.839333 + 0.543618i $$0.182946\pi$$
−0.839333 + 0.543618i $$0.817054\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 24.4853 1.27121
$$372$$ 0 0
$$373$$ 14.9706 0.775146 0.387573 0.921839i $$-0.373313\pi$$
0.387573 + 0.921839i $$0.373313\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 1.65685i − 0.0853323i
$$378$$ 0 0
$$379$$ − 20.2843i − 1.04193i −0.853577 0.520967i $$-0.825572\pi$$
0.853577 0.520967i $$-0.174428\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −14.6274 −0.747426 −0.373713 0.927544i $$-0.621916\pi$$
−0.373713 + 0.927544i $$0.621916\pi$$
$$384$$ 0 0
$$385$$ −46.6274 −2.37635
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 2.44365i − 0.123898i −0.998079 0.0619490i $$-0.980268\pi$$
0.998079 0.0619490i $$-0.0197316\pi$$
$$390$$ 0 0
$$391$$ 6.34315i 0.320787i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −22.1421 −1.11409
$$396$$ 0 0
$$397$$ −18.9706 −0.952105 −0.476053 0.879417i $$-0.657933\pi$$
−0.476053 + 0.879417i $$0.657933\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 13.4142i 0.669874i 0.942241 + 0.334937i $$0.108715\pi$$
−0.942241 + 0.334937i $$0.891285\pi$$
$$402$$ 0 0
$$403$$ 8.97056i 0.446856i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.3431 −0.512691
$$408$$ 0 0
$$409$$ 10.3431 0.511436 0.255718 0.966751i $$-0.417688\pi$$
0.255718 + 0.966751i $$0.417688\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 11.3137i − 0.556711i
$$414$$ 0 0
$$415$$ 17.6569i 0.866741i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 11.7990 0.576418 0.288209 0.957567i $$-0.406940\pi$$
0.288209 + 0.957567i $$0.406940\pi$$
$$420$$ 0 0
$$421$$ 7.51472 0.366245 0.183122 0.983090i $$-0.441380\pi$$
0.183122 + 0.983090i $$0.441380\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 36.0416i − 1.74828i
$$426$$ 0 0
$$427$$ − 36.9706i − 1.78913i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 28.4853 1.37209 0.686044 0.727560i $$-0.259346\pi$$
0.686044 + 0.727560i $$0.259346\pi$$
$$432$$ 0 0
$$433$$ 28.6274 1.37575 0.687873 0.725831i $$-0.258545\pi$$
0.687873 + 0.725831i $$0.258545\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 6.62742i − 0.317032i
$$438$$ 0 0
$$439$$ − 7.85786i − 0.375035i −0.982261 0.187518i $$-0.939956\pi$$
0.982261 0.187518i $$-0.0600442\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −17.4558 −0.829352 −0.414676 0.909969i $$-0.636105\pi$$
−0.414676 + 0.909969i $$0.636105\pi$$
$$444$$ 0 0
$$445$$ −41.7990 −1.98146
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 15.2721i 0.720734i 0.932811 + 0.360367i $$0.117349\pi$$
−0.932811 + 0.360367i $$0.882651\pi$$
$$450$$ 0 0
$$451$$ 7.31371i 0.344389i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 46.6274 2.18593
$$456$$ 0 0
$$457$$ −8.97056 −0.419625 −0.209813 0.977742i $$-0.567285\pi$$
−0.209813 + 0.977742i $$0.567285\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 8.10051i − 0.377278i −0.982046 0.188639i $$-0.939592\pi$$
0.982046 0.188639i $$-0.0604076\pi$$
$$462$$ 0 0
$$463$$ 33.7990i 1.57077i 0.619006 + 0.785386i $$0.287536\pi$$
−0.619006 + 0.785386i $$0.712464\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.14214 0.284224 0.142112 0.989851i $$-0.454611\pi$$
0.142112 + 0.989851i $$0.454611\pi$$
$$468$$ 0 0
$$469$$ −57.9411 −2.67547
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 27.3137i − 1.25589i
$$474$$ 0 0
$$475$$ 37.6569i 1.72781i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 9.17157 0.419060 0.209530 0.977802i $$-0.432807\pi$$
0.209530 + 0.977802i $$0.432807\pi$$
$$480$$ 0 0
$$481$$ 10.3431 0.471607
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 46.6274i 2.11724i
$$486$$ 0 0
$$487$$ − 28.8284i − 1.30634i −0.757211 0.653170i $$-0.773439\pi$$
0.757211 0.653170i $$-0.226561\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −35.3137 −1.59369 −0.796843 0.604187i $$-0.793498\pi$$
−0.796843 + 0.604187i $$0.793498\pi$$
$$492$$ 0 0
$$493$$ −3.17157 −0.142840
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 21.6569i − 0.971443i
$$498$$ 0 0
$$499$$ − 6.62742i − 0.296684i −0.988936 0.148342i $$-0.952606\pi$$
0.988936 0.148342i $$-0.0473936\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 15.7990 0.704442 0.352221 0.935917i $$-0.385427\pi$$
0.352221 + 0.935917i $$0.385427\pi$$
$$504$$ 0 0
$$505$$ −36.6274 −1.62990
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 13.2721i 0.588275i 0.955763 + 0.294137i $$0.0950323\pi$$
−0.955763 + 0.294137i $$0.904968\pi$$
$$510$$ 0 0
$$511$$ 19.3137i 0.854388i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −10.8284 −0.477158
$$516$$ 0 0
$$517$$ 35.3137 1.55310
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3.07107i 0.134546i 0.997735 + 0.0672730i $$0.0214298\pi$$
−0.997735 + 0.0672730i $$0.978570\pi$$
$$522$$ 0 0
$$523$$ 45.6569i 1.99643i 0.0596823 + 0.998217i $$0.480991\pi$$
−0.0596823 + 0.998217i $$0.519009\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 17.1716 0.748005
$$528$$ 0 0
$$529$$ −21.6274 −0.940322
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 7.31371i − 0.316792i
$$534$$ 0 0
$$535$$ 38.6274i 1.67001i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −46.1421 −1.98748
$$540$$ 0 0
$$541$$ −35.7990 −1.53912 −0.769559 0.638575i $$-0.779524\pi$$
−0.769559 + 0.638575i $$0.779524\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 36.9706i 1.58364i
$$546$$ 0 0
$$547$$ − 9.65685i − 0.412897i −0.978457 0.206449i $$-0.933809\pi$$
0.978457 0.206449i $$-0.0661906\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.31371 0.141169
$$552$$ 0 0
$$553$$ −31.3137 −1.33159
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 9.07107i − 0.384353i −0.981360 0.192177i $$-0.938445\pi$$
0.981360 0.192177i $$-0.0615547\pi$$
$$558$$ 0 0
$$559$$ 27.3137i 1.15525i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 25.4558 1.07284 0.536418 0.843952i $$-0.319777\pi$$
0.536418 + 0.843952i $$0.319777\pi$$
$$564$$ 0 0
$$565$$ −41.7990 −1.75850
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 2.10051i 0.0880578i 0.999030 + 0.0440289i $$0.0140194\pi$$
−0.999030 + 0.0440289i $$0.985981\pi$$
$$570$$ 0 0
$$571$$ − 24.0000i − 1.00437i −0.864761 0.502184i $$-0.832530\pi$$
0.864761 0.502184i $$-0.167470\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −7.79899 −0.325240
$$576$$ 0 0
$$577$$ −29.3137 −1.22035 −0.610173 0.792268i $$-0.708900\pi$$
−0.610173 + 0.792268i $$0.708900\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.9706i 1.03595i
$$582$$ 0 0
$$583$$ − 14.3431i − 0.594032i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 29.6569 1.22407 0.612035 0.790831i $$-0.290351\pi$$
0.612035 + 0.790831i $$0.290351\pi$$
$$588$$ 0 0
$$589$$ −17.9411 −0.739251
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 15.0711i 0.618895i 0.950917 + 0.309447i $$0.100144\pi$$
−0.950917 + 0.309447i $$0.899856\pi$$
$$594$$ 0 0
$$595$$ − 89.2548i − 3.65909i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 44.4853 1.81762 0.908810 0.417211i $$-0.136992\pi$$
0.908810 + 0.417211i $$0.136992\pi$$
$$600$$ 0 0
$$601$$ 21.3137 0.869404 0.434702 0.900574i $$-0.356854\pi$$
0.434702 + 0.900574i $$0.356854\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 10.2426i − 0.416423i
$$606$$ 0 0
$$607$$ − 19.1716i − 0.778150i −0.921206 0.389075i $$-0.872795\pi$$
0.921206 0.389075i $$-0.127205\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −35.3137 −1.42864
$$612$$ 0 0
$$613$$ 11.6569 0.470816 0.235408 0.971897i $$-0.424357\pi$$
0.235408 + 0.971897i $$0.424357\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.5858i 1.23134i 0.788005 + 0.615669i $$0.211114\pi$$
−0.788005 + 0.615669i $$0.788886\pi$$
$$618$$ 0 0
$$619$$ − 33.9411i − 1.36421i −0.731255 0.682105i $$-0.761065\pi$$
0.731255 0.682105i $$-0.238935\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −59.1127 −2.36830
$$624$$ 0 0
$$625$$ −13.9706 −0.558823
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 19.7990i − 0.789437i
$$630$$ 0 0
$$631$$ − 1.51472i − 0.0603000i −0.999545 0.0301500i $$-0.990402\pi$$
0.999545 0.0301500i $$-0.00959850\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −16.4853 −0.654198
$$636$$ 0 0
$$637$$ 46.1421 1.82822
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 38.8701i − 1.53527i −0.640884 0.767637i $$-0.721432\pi$$
0.640884 0.767637i $$-0.278568\pi$$
$$642$$ 0 0
$$643$$ 1.65685i 0.0653400i 0.999466 + 0.0326700i $$0.0104010\pi$$
−0.999466 + 0.0326700i $$0.989599\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −23.7990 −0.935635 −0.467817 0.883825i $$-0.654960\pi$$
−0.467817 + 0.883825i $$0.654960\pi$$
$$648$$ 0 0
$$649$$ −6.62742 −0.260149
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 30.7279i 1.20248i 0.799070 + 0.601238i $$0.205326\pi$$
−0.799070 + 0.601238i $$0.794674\pi$$
$$654$$ 0 0
$$655$$ − 19.3137i − 0.754649i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 24.9706 0.972715 0.486358 0.873760i $$-0.338325\pi$$
0.486358 + 0.873760i $$0.338325\pi$$
$$660$$ 0 0
$$661$$ 7.65685 0.297817 0.148909 0.988851i $$-0.452424\pi$$
0.148909 + 0.988851i $$0.452424\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 93.2548i 3.61627i
$$666$$ 0 0
$$667$$ 0.686292i 0.0265733i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −21.6569 −0.836054
$$672$$ 0 0
$$673$$ 18.0000 0.693849 0.346925 0.937893i $$-0.387226\pi$$
0.346925 + 0.937893i $$0.387226\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.72792i 0.258575i 0.991607 + 0.129288i $$0.0412690\pi$$
−0.991607 + 0.129288i $$0.958731\pi$$
$$678$$ 0 0
$$679$$ 65.9411i 2.53059i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −6.14214 −0.235022 −0.117511 0.993072i $$-0.537492\pi$$
−0.117511 + 0.993072i $$0.537492\pi$$
$$684$$ 0 0
$$685$$ 36.1421 1.38092
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 14.3431i 0.546430i
$$690$$ 0 0
$$691$$ − 36.9706i − 1.40643i −0.710979 0.703213i $$-0.751748\pi$$
0.710979 0.703213i $$-0.248252\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 40.9706 1.55410
$$696$$ 0 0
$$697$$ −14.0000 −0.530288
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22.2426i 0.840093i 0.907503 + 0.420046i $$0.137986\pi$$
−0.907503 + 0.420046i $$0.862014\pi$$
$$702$$ 0 0
$$703$$ 20.6863i 0.780198i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −51.7990 −1.94810
$$708$$ 0 0
$$709$$ −48.0833 −1.80580 −0.902902 0.429846i $$-0.858568\pi$$
−0.902902 + 0.429846i $$0.858568\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 3.71573i − 0.139155i
$$714$$ 0 0
$$715$$ − 27.3137i − 1.02147i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 11.5147 0.429427 0.214713 0.976677i $$-0.431118\pi$$
0.214713 + 0.976677i $$0.431118\pi$$
$$720$$ 0 0
$$721$$ −15.3137 −0.570312
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 3.89949i − 0.144824i
$$726$$ 0 0
$$727$$ 3.17157i 0.117627i 0.998269 + 0.0588136i $$0.0187317\pi$$
−0.998269 + 0.0588136i $$0.981268\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 52.2843 1.93380
$$732$$ 0 0
$$733$$ −7.51472 −0.277562 −0.138781 0.990323i $$-0.544318\pi$$
−0.138781 + 0.990323i $$0.544318\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 33.9411i 1.25024i
$$738$$ 0 0
$$739$$ 8.00000i 0.294285i 0.989115 + 0.147142i $$0.0470076\pi$$
−0.989115 + 0.147142i $$0.952992\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16.9706 0.622590 0.311295 0.950313i $$-0.399237\pi$$
0.311295 + 0.950313i $$0.399237\pi$$
$$744$$ 0 0
$$745$$ 22.9706 0.841576
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 54.6274i 1.99604i
$$750$$ 0 0
$$751$$ − 35.1716i − 1.28343i −0.766944 0.641714i $$-0.778223\pi$$
0.766944 0.641714i $$-0.221777\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 22.1421 0.805835
$$756$$ 0 0
$$757$$ 16.4853 0.599168 0.299584 0.954070i $$-0.403152\pi$$
0.299584 + 0.954070i $$0.403152\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 33.6985i − 1.22157i −0.791797 0.610785i $$-0.790854\pi$$
0.791797 0.610785i $$-0.209146\pi$$
$$762$$ 0 0
$$763$$ 52.2843i 1.89282i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 6.62742 0.239302
$$768$$ 0 0
$$769$$ 1.31371 0.0473735 0.0236868 0.999719i $$-0.492460\pi$$
0.0236868 + 0.999719i $$0.492460\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 15.4142i − 0.554411i −0.960811 0.277205i $$-0.910592\pi$$
0.960811 0.277205i $$-0.0894082\pi$$
$$774$$ 0 0
$$775$$ 21.1127i 0.758391i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 14.6274 0.524082
$$780$$ 0 0
$$781$$ −12.6863 −0.453951
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ − 1.17157i − 0.0418152i
$$786$$ 0 0
$$787$$ 37.6569i 1.34232i 0.741312 + 0.671161i $$0.234204\pi$$
−0.741312 + 0.671161i $$0.765796\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −59.1127 −2.10181
$$792$$ 0 0
$$793$$ 21.6569 0.769057
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1.07107i 0.0379392i 0.999820 + 0.0189696i $$0.00603857\pi$$
−0.999820 + 0.0189696i $$0.993961\pi$$
$$798$$ 0 0
$$799$$ 67.5980i 2.39144i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 11.3137 0.399252
$$804$$ 0 0
$$805$$ −19.3137 −0.680719
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ − 50.6690i − 1.78143i −0.454563 0.890714i $$-0.650205\pi$$
0.454563 0.890714i $$-0.349795\pi$$
$$810$$ 0 0
$$811$$ 47.5980i 1.67139i 0.549193 + 0.835696i $$0.314935\pi$$
−0.549193 + 0.835696i $$0.685065\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −5.65685 −0.198151
$$816$$ 0 0
$$817$$ −54.6274 −1.91117
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 8.58579i − 0.299646i −0.988713 0.149823i $$-0.952130\pi$$
0.988713 0.149823i $$-0.0478704\pi$$
$$822$$ 0 0
$$823$$ − 32.4264i − 1.13031i −0.824984 0.565157i $$-0.808816\pi$$
0.824984 0.565157i $$-0.191184\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −48.9706 −1.70287 −0.851437 0.524457i $$-0.824268\pi$$
−0.851437 + 0.524457i $$0.824268\pi$$
$$828$$ 0 0
$$829$$ −43.7990 −1.52120 −0.760601 0.649220i $$-0.775096\pi$$
−0.760601 + 0.649220i $$0.775096\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 88.3259i − 3.06031i
$$834$$ 0 0
$$835$$ − 19.3137i − 0.668378i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 20.4853 0.707230 0.353615 0.935391i $$-0.384952\pi$$
0.353615 + 0.935391i $$0.384952\pi$$
$$840$$ 0 0
$$841$$ 28.6569 0.988167
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 17.0711i − 0.587263i
$$846$$ 0 0
$$847$$ − 14.4853i − 0.497720i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −4.28427 −0.146863
$$852$$ 0 0
$$853$$ 14.2843 0.489084 0.244542 0.969639i $$-0.421362\pi$$
0.244542 + 0.969639i $$0.421362\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 2.10051i − 0.0717519i −0.999356 0.0358759i $$-0.988578\pi$$
0.999356 0.0358759i $$-0.0114221\pi$$
$$858$$ 0 0
$$859$$ − 9.65685i − 0.329488i −0.986336 0.164744i $$-0.947320\pi$$
0.986336 0.164744i $$-0.0526797\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 32.9706 1.12233 0.561166 0.827704i $$-0.310353\pi$$
0.561166 + 0.827704i $$0.310353\pi$$
$$864$$ 0 0
$$865$$ 70.2843 2.38974
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 18.3431i 0.622249i
$$870$$ 0 0
$$871$$ − 33.9411i − 1.15005i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 27.3137 0.923372
$$876$$ 0 0
$$877$$ −26.9706 −0.910731 −0.455366 0.890305i $$-0.650491\pi$$
−0.455366 + 0.890305i $$0.650491\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 12.7279i 0.428815i 0.976744 + 0.214407i $$0.0687820\pi$$
−0.976744 + 0.214407i $$0.931218\pi$$
$$882$$ 0 0
$$883$$ 16.2843i 0.548009i 0.961728 + 0.274005i $$0.0883484\pi$$
−0.961728 + 0.274005i $$0.911652\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 13.6569 0.458552 0.229276 0.973361i $$-0.426364\pi$$
0.229276 + 0.973361i $$0.426364\pi$$
$$888$$ 0 0
$$889$$ −23.3137 −0.781917
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 70.6274i − 2.36346i
$$894$$ 0 0
$$895$$ − 65.9411i − 2.20417i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1.85786 0.0619632
$$900$$ 0 0
$$901$$ 27.4558 0.914687
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 20.9706i 0.697085i
$$906$$ 0 0
$$907$$ − 25.6569i − 0.851922i −0.904742 0.425961i $$-0.859936\pi$$
0.904742 0.425961i $$-0.140064\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 55.5980 1.84204 0.921022 0.389511i $$-0.127356\pi$$
0.921022 + 0.389511i $$0.127356\pi$$
$$912$$ 0 0
$$913$$ 14.6274 0.484097
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 27.3137i − 0.901978i
$$918$$ 0 0
$$919$$ 27.4558i 0.905685i 0.891591 + 0.452842i $$0.149590\pi$$
−0.891591 + 0.452842i $$0.850410\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 12.6863 0.417574
$$924$$ 0 0
$$925$$ 24.3431 0.800398
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 8.72792i 0.286354i 0.989697 + 0.143177i $$0.0457318\pi$$
−0.989697 + 0.143177i $$0.954268\pi$$
$$930$$ 0 0
$$931$$ 92.2843i 3.02449i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −52.2843 −1.70988
$$936$$ 0 0
$$937$$ −11.3726 −0.371526 −0.185763 0.982595i $$-0.559476\pi$$
−0.185763 + 0.982595i $$0.559476\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 46.7279i − 1.52329i −0.647996 0.761643i $$-0.724393\pi$$
0.647996 0.761643i $$-0.275607\pi$$
$$942$$ 0 0
$$943$$ 3.02944i 0.0986521i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −47.5980 −1.54673 −0.773363 0.633963i $$-0.781427\pi$$
−0.773363 + 0.633963i $$0.781427\pi$$
$$948$$ 0 0
$$949$$ −11.3137 −0.367259
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 35.5563i − 1.15178i −0.817526 0.575892i $$-0.804655\pi$$
0.817526 0.575892i $$-0.195345\pi$$
$$954$$ 0 0
$$955$$ − 35.3137i − 1.14272i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 51.1127 1.65052
$$960$$ 0 0
$$961$$ 20.9411 0.675520
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 18.1421i 0.584016i
$$966$$ 0 0
$$967$$ − 50.0833i − 1.61057i −0.592889 0.805285i $$-0.702013\pi$$
0.592889 0.805285i $$-0.297987\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −26.8284 −0.860965 −0.430483 0.902599i $$-0.641657\pi$$
−0.430483 + 0.902599i $$0.641657\pi$$
$$972$$ 0 0
$$973$$ 57.9411 1.85751
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 52.5269i 1.68048i 0.542211 + 0.840242i $$0.317587\pi$$
−0.542211 + 0.840242i $$0.682413\pi$$
$$978$$ 0 0
$$979$$ 34.6274i 1.10670i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −45.6569 −1.45623 −0.728114 0.685456i $$-0.759603\pi$$
−0.728114 + 0.685456i $$0.759603\pi$$
$$984$$ 0 0
$$985$$ −55.9411 −1.78243
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 11.3137i − 0.359755i
$$990$$ 0 0
$$991$$ − 25.7990i − 0.819532i −0.912191 0.409766i $$-0.865610\pi$$
0.912191 0.409766i $$-0.134390\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 16.4853 0.522619
$$996$$ 0 0
$$997$$ 61.5980 1.95083 0.975414 0.220381i $$-0.0707302\pi$$
0.975414 + 0.220381i $$0.0707302\pi$$
$$998$$ 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 1152.2.c.c.1151.4 yes 4
3.2 odd 2 1152.2.c.b.1151.1 yes 4
4.3 odd 2 1152.2.c.b.1151.4 yes 4
8.3 odd 2 1152.2.c.a.1151.1 4
8.5 even 2 1152.2.c.d.1151.1 yes 4
12.11 even 2 inner 1152.2.c.c.1151.1 yes 4
16.3 odd 4 2304.2.f.h.1151.4 4
16.5 even 4 2304.2.f.a.1151.1 4
16.11 odd 4 2304.2.f.b.1151.2 4
16.13 even 4 2304.2.f.g.1151.3 4
24.5 odd 2 1152.2.c.a.1151.4 yes 4
24.11 even 2 1152.2.c.d.1151.4 yes 4
48.5 odd 4 2304.2.f.h.1151.3 4
48.11 even 4 2304.2.f.g.1151.4 4
48.29 odd 4 2304.2.f.b.1151.1 4
48.35 even 4 2304.2.f.a.1151.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.c.a.1151.1 4 8.3 odd 2
1152.2.c.a.1151.4 yes 4 24.5 odd 2
1152.2.c.b.1151.1 yes 4 3.2 odd 2
1152.2.c.b.1151.4 yes 4 4.3 odd 2
1152.2.c.c.1151.1 yes 4 12.11 even 2 inner
1152.2.c.c.1151.4 yes 4 1.1 even 1 trivial
1152.2.c.d.1151.1 yes 4 8.5 even 2
1152.2.c.d.1151.4 yes 4 24.11 even 2
2304.2.f.a.1151.1 4 16.5 even 4
2304.2.f.a.1151.2 4 48.35 even 4
2304.2.f.b.1151.1 4 48.29 odd 4
2304.2.f.b.1151.2 4 16.11 odd 4
2304.2.f.g.1151.3 4 16.13 even 4
2304.2.f.g.1151.4 4 48.11 even 4
2304.2.f.h.1151.3 4 48.5 odd 4
2304.2.f.h.1151.4 4 16.3 odd 4