# Properties

 Label 2304.2.f.a.1151.2 Level $2304$ Weight $2$ Character 2304.1151 Analytic conductor $18.398$ 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] = [2304,2,Mod(1151,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2304.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.f (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: $$18.3975326257$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1151.2 Root $$-0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.1151 Dual form 2304.2.f.a.1151.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.41421 q^{5} +4.82843i q^{7} +O(q^{10})$$ $$q-3.41421 q^{5} +4.82843i q^{7} -2.82843i q^{11} -2.82843i q^{13} -5.41421i q^{17} -5.65685 q^{19} -1.17157 q^{23} +6.65685 q^{25} +0.585786 q^{29} +3.17157i q^{31} -16.4853i q^{35} +3.65685i q^{37} +2.58579i q^{41} +9.65685 q^{43} +12.4853 q^{47} -16.3137 q^{49} +5.07107 q^{53} +9.65685i q^{55} +2.34315i q^{59} -7.65685i q^{61} +9.65685i q^{65} +12.0000 q^{67} +4.48528 q^{71} -4.00000 q^{73} +13.6569 q^{77} -6.48528i q^{79} +5.17157i q^{83} +18.4853i q^{85} +12.2426i q^{89} +13.6569 q^{91} +19.3137 q^{95} +13.6569 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{5}+O(q^{10})$$ 4 * q - 8 * q^5 $$4 q - 8 q^{5} - 16 q^{23} + 4 q^{25} + 8 q^{29} + 16 q^{43} + 16 q^{47} - 20 q^{49} - 8 q^{53} + 48 q^{67} - 16 q^{71} - 16 q^{73} + 32 q^{77} + 32 q^{91} + 32 q^{95} + 32 q^{97}+O(q^{100})$$ 4 * q - 8 * q^5 - 16 * q^23 + 4 * q^25 + 8 * q^29 + 16 * q^43 + 16 * q^47 - 20 * q^49 - 8 * q^53 + 48 * q^67 - 16 * q^71 - 16 * q^73 + 32 * q^77 + 32 * q^91 + 32 * q^95 + 32 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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
$$5$$ −3.41421 −1.52688 −0.763441 0.645877i $$-0.776492\pi$$
−0.763441 + 0.645877i $$0.776492\pi$$
$$6$$ 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
$$11$$ − 2.82843i − 0.852803i −0.904534 0.426401i $$-0.859781\pi$$
0.904534 0.426401i $$-0.140219\pi$$
$$12$$ 0 0
$$13$$ − 2.82843i − 0.784465i −0.919866 0.392232i $$-0.871703\pi$$
0.919866 0.392232i $$-0.128297\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 5.41421i − 1.31314i −0.754265 0.656570i $$-0.772007\pi$$
0.754265 0.656570i $$-0.227993\pi$$
$$18$$ 0 0
$$19$$ −5.65685 −1.29777 −0.648886 0.760886i $$-0.724765\pi$$
−0.648886 + 0.760886i $$0.724765\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.17157 −0.244290 −0.122145 0.992512i $$-0.538977\pi$$
−0.122145 + 0.992512i $$0.538977\pi$$
$$24$$ 0 0
$$25$$ 6.65685 1.33137
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0.585786 0.108778 0.0543889 0.998520i $$-0.482679\pi$$
0.0543889 + 0.998520i $$0.482679\pi$$
$$30$$ 0 0
$$31$$ 3.17157i 0.569631i 0.958582 + 0.284816i $$0.0919324\pi$$
−0.958582 + 0.284816i $$0.908068\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 16.4853i − 2.78652i
$$36$$ 0 0
$$37$$ 3.65685i 0.601183i 0.953753 + 0.300592i $$0.0971841\pi$$
−0.953753 + 0.300592i $$0.902816\pi$$
$$38$$ 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
$$43$$ 9.65685 1.47266 0.736328 0.676625i $$-0.236558\pi$$
0.736328 + 0.676625i $$0.236558\pi$$
$$44$$ 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
$$53$$ 5.07107 0.696565 0.348282 0.937390i $$-0.386765\pi$$
0.348282 + 0.937390i $$0.386765\pi$$
$$54$$ 0 0
$$55$$ 9.65685i 1.30213i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 2.34315i 0.305052i 0.988299 + 0.152526i $$0.0487407\pi$$
−0.988299 + 0.152526i $$0.951259\pi$$
$$60$$ 0 0
$$61$$ − 7.65685i − 0.980360i −0.871621 0.490180i $$-0.836931\pi$$
0.871621 0.490180i $$-0.163069\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 9.65685i 1.19779i
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.48528 0.532305 0.266152 0.963931i $$-0.414248\pi$$
0.266152 + 0.963931i $$0.414248\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 13.6569 1.55634
$$78$$ 0 0
$$79$$ − 6.48528i − 0.729651i −0.931076 0.364826i $$-0.881129\pi$$
0.931076 0.364826i $$-0.118871\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 5.17157i 0.567654i 0.958876 + 0.283827i $$0.0916041\pi$$
−0.958876 + 0.283827i $$0.908396\pi$$
$$84$$ 0 0
$$85$$ 18.4853i 2.00501i
$$86$$ 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
$$91$$ 13.6569 1.43163
$$92$$ 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
$$101$$ −10.7279 −1.06747 −0.533734 0.845652i $$-0.679212\pi$$
−0.533734 + 0.845652i $$0.679212\pi$$
$$102$$ 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
$$107$$ − 11.3137i − 1.09374i −0.837218 0.546869i $$-0.815820\pi$$
0.837218 0.546869i $$-0.184180\pi$$
$$108$$ 0 0
$$109$$ 10.8284i 1.03718i 0.855024 + 0.518588i $$0.173542\pi$$
−0.855024 + 0.518588i $$0.826458\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 12.2426i − 1.15169i −0.817559 0.575845i $$-0.804673\pi$$
0.817559 0.575845i $$-0.195327\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 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
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ − 4.82843i − 0.428454i −0.976784 0.214227i $$-0.931277\pi$$
0.976784 0.214227i $$-0.0687232\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 5.65685i − 0.494242i −0.968985 0.247121i $$-0.920516\pi$$
0.968985 0.247121i $$-0.0794845\pi$$
$$132$$ 0 0
$$133$$ − 27.3137i − 2.36840i
$$134$$ 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
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 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
$$149$$ 6.72792 0.551173 0.275586 0.961276i $$-0.411128\pi$$
0.275586 + 0.961276i $$0.411128\pi$$
$$150$$ 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
$$155$$ − 10.8284i − 0.869760i
$$156$$ 0 0
$$157$$ − 0.343146i − 0.0273860i −0.999906 0.0136930i $$-0.995641\pi$$
0.999906 0.0136930i $$-0.00435876\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 5.65685i − 0.445823i
$$162$$ 0 0
$$163$$ 1.65685 0.129775 0.0648874 0.997893i $$-0.479331\pi$$
0.0648874 + 0.997893i $$0.479331\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 5.65685 0.437741 0.218870 0.975754i $$-0.429763\pi$$
0.218870 + 0.975754i $$0.429763\pi$$
$$168$$ 0 0
$$169$$ 5.00000 0.384615
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −20.5858 −1.56511 −0.782554 0.622582i $$-0.786084\pi$$
−0.782554 + 0.622582i $$0.786084\pi$$
$$174$$ 0 0
$$175$$ 32.1421i 2.42972i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 19.3137i − 1.44357i −0.692115 0.721787i $$-0.743321\pi$$
0.692115 0.721787i $$-0.256679\pi$$
$$180$$ 0 0
$$181$$ − 6.14214i − 0.456541i −0.973598 0.228271i $$-0.926693\pi$$
0.973598 0.228271i $$-0.0733071\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 12.4853i − 0.917936i
$$186$$ 0 0
$$187$$ −15.3137 −1.11985
$$188$$ 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
$$197$$ −16.3848 −1.16737 −0.583683 0.811981i $$-0.698389\pi$$
−0.583683 + 0.811981i $$0.698389\pi$$
$$198$$ 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
$$203$$ 2.82843i 0.198517i
$$204$$ 0 0
$$205$$ − 8.82843i − 0.616604i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 16.0000i 1.10674i
$$210$$ 0 0
$$211$$ 23.3137 1.60498 0.802491 0.596664i $$-0.203508\pi$$
0.802491 + 0.596664i $$0.203508\pi$$
$$212$$ 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
$$221$$ −15.3137 −1.03011
$$222$$ 0 0
$$223$$ 17.7990i 1.19191i 0.803018 + 0.595954i $$0.203226\pi$$
−0.803018 + 0.595954i $$0.796774\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 26.8284i 1.78067i 0.455311 + 0.890333i $$0.349528\pi$$
−0.455311 + 0.890333i $$0.650472\pi$$
$$228$$ 0 0
$$229$$ − 5.17157i − 0.341747i −0.985293 0.170874i $$-0.945341\pi$$
0.985293 0.170874i $$-0.0546590\pi$$
$$230$$ 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
$$235$$ −42.6274 −2.78071
$$236$$ 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
$$245$$ 55.6985 3.55845
$$246$$ 0 0
$$247$$ 16.0000i 1.01806i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 24.4853i − 1.54550i −0.634712 0.772749i $$-0.718881\pi$$
0.634712 0.772749i $$-0.281119\pi$$
$$252$$ 0 0
$$253$$ 3.31371i 0.208331i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 9.89949i 0.617514i 0.951141 + 0.308757i $$0.0999129\pi$$
−0.951141 + 0.308757i $$0.900087\pi$$
$$258$$ 0 0
$$259$$ −17.6569 −1.09714
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 18.3431 1.13109 0.565543 0.824719i $$-0.308666\pi$$
0.565543 + 0.824719i $$0.308666\pi$$
$$264$$ 0 0
$$265$$ −17.3137 −1.06357
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 32.3848 1.97453 0.987267 0.159070i $$-0.0508495\pi$$
0.987267 + 0.159070i $$0.0508495\pi$$
$$270$$ 0 0
$$271$$ − 1.51472i − 0.0920126i −0.998941 0.0460063i $$-0.985351\pi$$
0.998941 0.0460063i $$-0.0146494\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 18.8284i − 1.13540i
$$276$$ 0 0
$$277$$ 21.1716i 1.27208i 0.771658 + 0.636038i $$0.219428\pi$$
−0.771658 + 0.636038i $$0.780572\pi$$
$$278$$ 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
$$283$$ −3.31371 −0.196980 −0.0984898 0.995138i $$-0.531401\pi$$
−0.0984898 + 0.995138i $$0.531401\pi$$
$$284$$ 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
$$293$$ −28.8701 −1.68661 −0.843303 0.537438i $$-0.819392\pi$$
−0.843303 + 0.537438i $$0.819392\pi$$
$$294$$ 0 0
$$295$$ − 8.00000i − 0.465778i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 3.31371i 0.191637i
$$300$$ 0 0
$$301$$ 46.6274i 2.68756i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 26.1421i 1.49689i
$$306$$ 0 0
$$307$$ −0.686292 −0.0391687 −0.0195844 0.999808i $$-0.506234\pi$$
−0.0195844 + 0.999808i $$0.506234\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 13.6569 0.774409 0.387205 0.921994i $$-0.373441\pi$$
0.387205 + 0.921994i $$0.373441\pi$$
$$312$$ 0 0
$$313$$ −9.31371 −0.526442 −0.263221 0.964736i $$-0.584785\pi$$
−0.263221 + 0.964736i $$0.584785\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −14.2426 −0.799946 −0.399973 0.916527i $$-0.630981\pi$$
−0.399973 + 0.916527i $$0.630981\pi$$
$$318$$ 0 0
$$319$$ − 1.65685i − 0.0927660i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 30.6274i 1.70416i
$$324$$ 0 0
$$325$$ − 18.8284i − 1.04441i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 60.2843i 3.32358i
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 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
$$341$$ 8.97056 0.485783
$$342$$ 0 0
$$343$$ − 44.9706i − 2.42818i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 16.4853i 0.884976i 0.896774 + 0.442488i $$0.145904\pi$$
−0.896774 + 0.442488i $$0.854096\pi$$
$$348$$ 0 0
$$349$$ 11.6569i 0.623977i 0.950086 + 0.311989i $$0.100995\pi$$
−0.950086 + 0.311989i $$0.899005\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 18.8701i − 1.00435i −0.864765 0.502176i $$-0.832533\pi$$
0.864765 0.502176i $$-0.167467\pi$$
$$354$$ 0 0
$$355$$ −15.3137 −0.812767
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −20.4853 −1.08117 −0.540586 0.841289i $$-0.681797\pi$$
−0.540586 + 0.841289i $$0.681797\pi$$
$$360$$ 0 0
$$361$$ 13.0000 0.684211
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 13.6569 0.714832
$$366$$ 0 0
$$367$$ − 20.8284i − 1.08724i −0.839333 0.543618i $$-0.817054\pi$$
0.839333 0.543618i $$-0.182946\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 24.4853i 1.27121i
$$372$$ 0 0
$$373$$ − 14.9706i − 0.775146i −0.921839 0.387573i $$-0.873313\pi$$
0.921839 0.387573i $$-0.126687\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 1.65685i − 0.0853323i
$$378$$ 0 0
$$379$$ 20.2843 1.04193 0.520967 0.853577i $$-0.325572\pi$$
0.520967 + 0.853577i $$0.325572\pi$$
$$380$$ 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
$$389$$ 2.44365 0.123898 0.0619490 0.998079i $$-0.480268\pi$$
0.0619490 + 0.998079i $$0.480268\pi$$
$$390$$ 0 0
$$391$$ 6.34315i 0.320787i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 22.1421i 1.11409i
$$396$$ 0 0
$$397$$ − 18.9706i − 0.952105i −0.879417 0.476053i $$-0.842067\pi$$
0.879417 0.476053i $$-0.157933\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 13.4142i − 0.669874i −0.942241 0.334937i $$-0.891285\pi$$
0.942241 0.334937i $$-0.108715\pi$$
$$402$$ 0 0
$$403$$ 8.97056 0.446856
$$404$$ 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 $$-0.582312\pi$$
−0.255718 + 0.966751i $$0.582312\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −11.3137 −0.556711
$$414$$ 0 0
$$415$$ − 17.6569i − 0.866741i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 11.7990i 0.576418i 0.957567 + 0.288209i $$0.0930599\pi$$
−0.957567 + 0.288209i $$0.906940\pi$$
$$420$$ 0 0
$$421$$ − 7.51472i − 0.366245i −0.983090 0.183122i $$-0.941380\pi$$
0.983090 0.183122i $$-0.0586205\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 36.0416i − 1.74828i
$$426$$ 0 0
$$427$$ 36.9706 1.78913
$$428$$ 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
$$437$$ 6.62742 0.317032
$$438$$ 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
$$443$$ 17.4558i 0.829352i 0.909969 + 0.414676i $$0.136105\pi$$
−0.909969 + 0.414676i $$0.863895\pi$$
$$444$$ 0 0
$$445$$ − 41.7990i − 1.98146i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 15.2721i − 0.720734i −0.932811 0.360367i $$-0.882651\pi$$
0.932811 0.360367i $$-0.117349\pi$$
$$450$$ 0 0
$$451$$ 7.31371 0.344389
$$452$$ 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 $$-0.432715\pi$$
0.209813 + 0.977742i $$0.432715\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −8.10051 −0.377278 −0.188639 0.982046i $$-0.560408\pi$$
−0.188639 + 0.982046i $$0.560408\pi$$
$$462$$ 0 0
$$463$$ − 33.7990i − 1.57077i −0.619006 0.785386i $$-0.712464\pi$$
0.619006 0.785386i $$-0.287536\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.14214i 0.284224i 0.989851 + 0.142112i $$0.0453893\pi$$
−0.989851 + 0.142112i $$0.954611\pi$$
$$468$$ 0 0
$$469$$ 57.9411i 2.67547i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 27.3137i − 1.25589i
$$474$$ 0 0
$$475$$ −37.6569 −1.72781
$$476$$ 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
$$485$$ −46.6274 −2.11724
$$486$$ 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
$$491$$ 35.3137i 1.59369i 0.604187 + 0.796843i $$0.293498\pi$$
−0.604187 + 0.796843i $$0.706502\pi$$
$$492$$ 0 0
$$493$$ − 3.17157i − 0.142840i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 21.6569i 0.971443i
$$498$$ 0 0
$$499$$ −6.62742 −0.296684 −0.148342 0.988936i $$-0.547394\pi$$
−0.148342 + 0.988936i $$0.547394\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −15.7990 −0.704442 −0.352221 0.935917i $$-0.614573\pi$$
−0.352221 + 0.935917i $$0.614573\pi$$
$$504$$ 0 0
$$505$$ 36.6274 1.62990
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 13.2721 0.588275 0.294137 0.955763i $$-0.404968\pi$$
0.294137 + 0.955763i $$0.404968\pi$$
$$510$$ 0 0
$$511$$ − 19.3137i − 0.854388i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 10.8284i − 0.477158i
$$516$$ 0 0
$$517$$ − 35.3137i − 1.55310i
$$518$$ 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
$$523$$ −45.6569 −1.99643 −0.998217 0.0596823i $$-0.980991\pi$$
−0.998217 + 0.0596823i $$0.980991\pi$$
$$524$$ 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
$$533$$ 7.31371 0.316792
$$534$$ 0 0
$$535$$ 38.6274i 1.67001i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 46.1421i 1.98748i
$$540$$ 0 0
$$541$$ − 35.7990i − 1.53912i −0.638575 0.769559i $$-0.720476\pi$$
0.638575 0.769559i $$-0.279524\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ − 36.9706i − 1.58364i
$$546$$ 0 0
$$547$$ −9.65685 −0.412897 −0.206449 0.978457i $$-0.566191\pi$$
−0.206449 + 0.978457i $$0.566191\pi$$
$$548$$ 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
$$557$$ −9.07107 −0.384353 −0.192177 0.981360i $$-0.561555\pi$$
−0.192177 + 0.981360i $$0.561555\pi$$
$$558$$ 0 0
$$559$$ − 27.3137i − 1.15525i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 25.4558i 1.07284i 0.843952 + 0.536418i $$0.180223\pi$$
−0.843952 + 0.536418i $$0.819777\pi$$
$$564$$ 0 0
$$565$$ 41.7990i 1.75850i
$$566$$ 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
$$571$$ 24.0000 1.00437 0.502184 0.864761i $$-0.332530\pi$$
0.502184 + 0.864761i $$0.332530\pi$$
$$572$$ 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
$$581$$ −24.9706 −1.03595
$$582$$ 0 0
$$583$$ − 14.3431i − 0.594032i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 29.6569i − 1.22407i −0.790831 0.612035i $$-0.790351\pi$$
0.790831 0.612035i $$-0.209649\pi$$
$$588$$ 0 0
$$589$$ − 17.9411i − 0.739251i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 15.0711i − 0.618895i −0.950917 0.309447i $$-0.899856\pi$$
0.950917 0.309447i $$-0.100144\pi$$
$$594$$ 0 0
$$595$$ −89.2548 −3.65909
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −44.4853 −1.81762 −0.908810 0.417211i $$-0.863008\pi$$
−0.908810 + 0.417211i $$0.863008\pi$$
$$600$$ 0 0
$$601$$ −21.3137 −0.869404 −0.434702 0.900574i $$-0.643146\pi$$
−0.434702 + 0.900574i $$0.643146\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −10.2426 −0.416423
$$606$$ 0 0
$$607$$ 19.1716i 0.778150i 0.921206 + 0.389075i $$0.127205\pi$$
−0.921206 + 0.389075i $$0.872795\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 35.3137i − 1.42864i
$$612$$ 0 0
$$613$$ − 11.6569i − 0.470816i −0.971897 0.235408i $$-0.924357\pi$$
0.971897 0.235408i $$-0.0756426\pi$$
$$614$$ 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
$$619$$ 33.9411 1.36421 0.682105 0.731255i $$-0.261065\pi$$
0.682105 + 0.731255i $$0.261065\pi$$
$$620$$ 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
$$629$$ 19.7990 0.789437
$$630$$ 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
$$635$$ 16.4853i 0.654198i
$$636$$ 0 0
$$637$$ 46.1421i 1.82822i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 38.8701i 1.53527i 0.640884 + 0.767637i $$0.278568\pi$$
−0.640884 + 0.767637i $$0.721432\pi$$
$$642$$ 0 0
$$643$$ 1.65685 0.0653400 0.0326700 0.999466i $$-0.489599\pi$$
0.0326700 + 0.999466i $$0.489599\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 23.7990 0.935635 0.467817 0.883825i $$-0.345040\pi$$
0.467817 + 0.883825i $$0.345040\pi$$
$$648$$ 0 0
$$649$$ 6.62742 0.260149
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 30.7279 1.20248 0.601238 0.799070i $$-0.294674\pi$$
0.601238 + 0.799070i $$0.294674\pi$$
$$654$$ 0 0
$$655$$ 19.3137i 0.754649i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 24.9706i 0.972715i 0.873760 + 0.486358i $$0.161675\pi$$
−0.873760 + 0.486358i $$0.838325\pi$$
$$660$$ 0 0
$$661$$ − 7.65685i − 0.297817i −0.988851 0.148909i $$-0.952424\pi$$
0.988851 0.148909i $$-0.0475760\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 93.2548i 3.61627i
$$666$$ 0 0
$$667$$ −0.686292 −0.0265733
$$668$$ 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
$$677$$ −6.72792 −0.258575 −0.129288 0.991607i $$-0.541269\pi$$
−0.129288 + 0.991607i $$0.541269\pi$$
$$678$$ 0 0
$$679$$ 65.9411i 2.53059i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 6.14214i 0.235022i 0.993072 + 0.117511i $$0.0374916\pi$$
−0.993072 + 0.117511i $$0.962508\pi$$
$$684$$ 0 0
$$685$$ 36.1421i 1.38092i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 14.3431i − 0.546430i
$$690$$ 0 0
$$691$$ −36.9706 −1.40643 −0.703213 0.710979i $$-0.748252\pi$$
−0.703213 + 0.710979i $$0.748252\pi$$
$$692$$ 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
$$701$$ 22.2426 0.840093 0.420046 0.907503i $$-0.362014\pi$$
0.420046 + 0.907503i $$0.362014\pi$$
$$702$$ 0 0
$$703$$ − 20.6863i − 0.780198i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 51.7990i − 1.94810i
$$708$$ 0 0
$$709$$ 48.0833i 1.80580i 0.429846 + 0.902902i $$0.358568\pi$$
−0.429846 + 0.902902i $$0.641432\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 3.71573i − 0.139155i
$$714$$ 0 0
$$715$$ 27.3137 1.02147
$$716$$ 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
$$725$$ 3.89949 0.144824
$$726$$ 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
$$731$$ − 52.2843i − 1.93380i
$$732$$ 0 0
$$733$$ − 7.51472i − 0.277562i −0.990323 0.138781i $$-0.955682\pi$$
0.990323 0.138781i $$-0.0443185\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 33.9411i − 1.25024i
$$738$$ 0 0
$$739$$ 8.00000 0.294285 0.147142 0.989115i $$-0.452992\pi$$
0.147142 + 0.989115i $$0.452992\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16.9706 −0.622590 −0.311295 0.950313i $$-0.600763\pi$$
−0.311295 + 0.950313i $$0.600763\pi$$
$$744$$ 0 0
$$745$$ −22.9706 −0.841576
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 54.6274 1.99604
$$750$$ 0 0
$$751$$ 35.1716i 1.28343i 0.766944 + 0.641714i $$0.221777\pi$$
−0.766944 + 0.641714i $$0.778223\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 22.1421i 0.805835i
$$756$$ 0 0
$$757$$ − 16.4853i − 0.599168i −0.954070 0.299584i $$-0.903152\pi$$
0.954070 0.299584i $$-0.0968478\pi$$
$$758$$ 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
$$763$$ −52.2843 −1.89282
$$764$$ 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
$$773$$ 15.4142 0.554411 0.277205 0.960811i $$-0.410592\pi$$
0.277205 + 0.960811i $$0.410592\pi$$
$$774$$ 0 0
$$775$$ 21.1127i 0.758391i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 14.6274i − 0.524082i
$$780$$ 0 0
$$781$$ − 12.6863i − 0.453951i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.17157i 0.0418152i
$$786$$ 0 0
$$787$$ 37.6569 1.34232 0.671161 0.741312i $$-0.265796\pi$$
0.671161 + 0.741312i $$0.265796\pi$$
$$788$$ 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
$$797$$ 1.07107 0.0379392 0.0189696 0.999820i $$-0.493961\pi$$
0.0189696 + 0.999820i $$0.493961\pi$$
$$798$$ 0 0
$$799$$ − 67.5980i − 2.39144i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 11.3137i 0.399252i
$$804$$ 0 0
$$805$$ 19.3137i 0.680719i
$$806$$ 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
$$811$$ −47.5980 −1.67139 −0.835696 0.549193i $$-0.814935\pi$$
−0.835696 + 0.549193i $$0.814935\pi$$
$$812$$ 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
$$821$$ 8.58579 0.299646 0.149823 0.988713i $$-0.452130\pi$$
0.149823 + 0.988713i $$0.452130\pi$$
$$822$$ 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
$$827$$ 48.9706i 1.70287i 0.524457 + 0.851437i $$0.324268\pi$$
−0.524457 + 0.851437i $$0.675732\pi$$
$$828$$ 0 0
$$829$$ − 43.7990i − 1.52120i −0.649220 0.760601i $$-0.724904\pi$$
0.649220 0.760601i $$-0.275096\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 88.3259i 3.06031i
$$834$$ 0 0
$$835$$ −19.3137 −0.668378
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −20.4853 −0.707230 −0.353615 0.935391i $$-0.615048\pi$$
−0.353615 + 0.935391i $$0.615048\pi$$
$$840$$ 0 0
$$841$$ −28.6569 −0.988167
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −17.0711 −0.587263
$$846$$ 0 0
$$847$$ 14.4853i 0.497720i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 4.28427i − 0.146863i
$$852$$ 0 0
$$853$$ − 14.2843i − 0.489084i −0.969639 0.244542i $$-0.921362\pi$$
0.969639 0.244542i $$-0.0786376\pi$$
$$854$$ 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
$$859$$ 9.65685 0.329488 0.164744 0.986336i $$-0.447320\pi$$
0.164744 + 0.986336i $$0.447320\pi$$
$$860$$ 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
$$869$$ −18.3431 −0.622249
$$870$$ 0 0
$$871$$ − 33.9411i − 1.15005i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 27.3137i − 0.923372i
$$876$$ 0 0
$$877$$ − 26.9706i − 0.910731i −0.890305 0.455366i $$-0.849509\pi$$
0.890305 0.455366i $$-0.150491\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 12.7279i − 0.428815i −0.976744 0.214407i $$-0.931218\pi$$
0.976744 0.214407i $$-0.0687820\pi$$
$$882$$ 0 0
$$883$$ 16.2843 0.548009 0.274005 0.961728i $$-0.411652\pi$$
0.274005 + 0.961728i $$0.411652\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −13.6569 −0.458552 −0.229276 0.973361i $$-0.573636\pi$$
−0.229276 + 0.973361i $$0.573636\pi$$
$$888$$ 0 0
$$889$$ 23.3137 0.781917
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −70.6274 −2.36346
$$894$$ 0 0
$$895$$ 65.9411i 2.20417i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1.85786i 0.0619632i
$$900$$ 0 0
$$901$$ − 27.4558i − 0.914687i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 20.9706i 0.697085i
$$906$$ 0 0
$$907$$ 25.6569 0.851922 0.425961 0.904742i $$-0.359936\pi$$
0.425961 + 0.904742i $$0.359936\pi$$
$$908$$ 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
$$917$$ 27.3137 0.901978
$$918$$ 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
$$923$$ − 12.6863i − 0.417574i
$$924$$ 0 0
$$925$$ 24.3431i 0.800398i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ − 8.72792i − 0.286354i −0.989697 0.143177i $$-0.954268\pi$$
0.989697 0.143177i $$-0.0457318\pi$$
$$930$$ 0 0
$$931$$ 92.2843 3.02449
$$932$$ 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 $$-0.440524\pi$$
0.185763 + 0.982595i $$0.440524\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −46.7279 −1.52329 −0.761643 0.647996i $$-0.775607\pi$$
−0.761643 + 0.647996i $$0.775607\pi$$
$$942$$ 0 0
$$943$$ − 3.02944i − 0.0986521i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 47.5980i − 1.54673i −0.633963 0.773363i $$-0.718573\pi$$
0.633963 0.773363i $$-0.281427\pi$$
$$948$$ 0 0
$$949$$ 11.3137i 0.367259i
$$950$$ 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
$$955$$ 35.3137 1.14272
$$956$$ 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
$$965$$ −18.1421 −0.584016
$$966$$ 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
$$971$$ 26.8284i 0.860965i 0.902599 + 0.430483i $$0.141657\pi$$
−0.902599 + 0.430483i $$0.858343\pi$$
$$972$$ 0 0
$$973$$ 57.9411i 1.85751i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 52.5269i − 1.68048i −0.542211 0.840242i $$-0.682413\pi$$
0.542211 0.840242i $$-0.317587\pi$$
$$978$$ 0 0
$$979$$ 34.6274 1.10670
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 45.6569 1.45623 0.728114 0.685456i $$-0.240397\pi$$
0.728114 + 0.685456i $$0.240397\pi$$
$$984$$ 0 0
$$985$$ 55.9411 1.78243
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −11.3137 −0.359755
$$990$$ 0 0
$$991$$ 25.7990i 0.819532i 0.912191 + 0.409766i $$0.134390\pi$$
−0.912191 + 0.409766i $$0.865610\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 16.4853i 0.522619i
$$996$$ 0 0
$$997$$ − 61.5980i − 1.95083i −0.220381 0.975414i $$-0.570730\pi$$
0.220381 0.975414i $$-0.429270\pi$$
$$998$$ 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 2304.2.f.a.1151.2 4
3.2 odd 2 2304.2.f.h.1151.4 4
4.3 odd 2 2304.2.f.b.1151.1 4
8.3 odd 2 2304.2.f.h.1151.3 4
8.5 even 2 2304.2.f.g.1151.4 4
12.11 even 2 2304.2.f.g.1151.3 4
16.3 odd 4 1152.2.c.a.1151.4 yes 4
16.5 even 4 1152.2.c.c.1151.1 yes 4
16.11 odd 4 1152.2.c.b.1151.1 yes 4
16.13 even 4 1152.2.c.d.1151.4 yes 4
24.5 odd 2 2304.2.f.b.1151.2 4
24.11 even 2 inner 2304.2.f.a.1151.1 4
48.5 odd 4 1152.2.c.b.1151.4 yes 4
48.11 even 4 1152.2.c.c.1151.4 yes 4
48.29 odd 4 1152.2.c.a.1151.1 4
48.35 even 4 1152.2.c.d.1151.1 yes 4

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