# Properties

 Label 225.4.b.a.199.2 Level $225$ Weight $4$ Character 225.199 Analytic conductor $13.275$ 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] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.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: $$13.2754297513$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 225.199 Dual form 225.4.b.a.199.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000i q^{2} -17.0000 q^{4} +30.0000i q^{7} -45.0000i q^{8} +O(q^{10})$$ $$q+5.00000i q^{2} -17.0000 q^{4} +30.0000i q^{7} -45.0000i q^{8} -50.0000 q^{11} -20.0000i q^{13} -150.000 q^{14} +89.0000 q^{16} -10.0000i q^{17} +44.0000 q^{19} -250.000i q^{22} -120.000i q^{23} +100.000 q^{26} -510.000i q^{28} -50.0000 q^{29} +108.000 q^{31} +85.0000i q^{32} +50.0000 q^{34} +40.0000i q^{37} +220.000i q^{38} -400.000 q^{41} +280.000i q^{43} +850.000 q^{44} +600.000 q^{46} -280.000i q^{47} -557.000 q^{49} +340.000i q^{52} +610.000i q^{53} +1350.00 q^{56} -250.000i q^{58} +50.0000 q^{59} -518.000 q^{61} +540.000i q^{62} +287.000 q^{64} +180.000i q^{67} +170.000i q^{68} -700.000 q^{71} -410.000i q^{73} -200.000 q^{74} -748.000 q^{76} -1500.00i q^{77} +516.000 q^{79} -2000.00i q^{82} -660.000i q^{83} -1400.00 q^{86} +2250.00i q^{88} -1500.00 q^{89} +600.000 q^{91} +2040.00i q^{92} +1400.00 q^{94} +1630.00i q^{97} -2785.00i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 34 q^{4}+O(q^{10})$$ 2 * q - 34 * q^4 $$2 q - 34 q^{4} - 100 q^{11} - 300 q^{14} + 178 q^{16} + 88 q^{19} + 200 q^{26} - 100 q^{29} + 216 q^{31} + 100 q^{34} - 800 q^{41} + 1700 q^{44} + 1200 q^{46} - 1114 q^{49} + 2700 q^{56} + 100 q^{59} - 1036 q^{61} + 574 q^{64} - 1400 q^{71} - 400 q^{74} - 1496 q^{76} + 1032 q^{79} - 2800 q^{86} - 3000 q^{89} + 1200 q^{91} + 2800 q^{94}+O(q^{100})$$ 2 * q - 34 * q^4 - 100 * q^11 - 300 * q^14 + 178 * q^16 + 88 * q^19 + 200 * q^26 - 100 * q^29 + 216 * q^31 + 100 * q^34 - 800 * q^41 + 1700 * q^44 + 1200 * q^46 - 1114 * q^49 + 2700 * q^56 + 100 * q^59 - 1036 * q^61 + 574 * q^64 - 1400 * q^71 - 400 * q^74 - 1496 * q^76 + 1032 * q^79 - 2800 * q^86 - 3000 * q^89 + 1200 * q^91 + 2800 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.00000i 1.76777i 0.467707 + 0.883883i $$0.345080\pi$$
−0.467707 + 0.883883i $$0.654920\pi$$
$$3$$ 0 0
$$4$$ −17.0000 −2.12500
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 30.0000i 1.61985i 0.586535 + 0.809924i $$0.300492\pi$$
−0.586535 + 0.809924i $$0.699508\pi$$
$$8$$ − 45.0000i − 1.98874i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −50.0000 −1.37051 −0.685253 0.728305i $$-0.740308\pi$$
−0.685253 + 0.728305i $$0.740308\pi$$
$$12$$ 0 0
$$13$$ − 20.0000i − 0.426692i −0.976977 0.213346i $$-0.931564\pi$$
0.976977 0.213346i $$-0.0684362\pi$$
$$14$$ −150.000 −2.86351
$$15$$ 0 0
$$16$$ 89.0000 1.39062
$$17$$ − 10.0000i − 0.142668i −0.997452 0.0713340i $$-0.977274\pi$$
0.997452 0.0713340i $$-0.0227256\pi$$
$$18$$ 0 0
$$19$$ 44.0000 0.531279 0.265639 0.964072i $$-0.414417\pi$$
0.265639 + 0.964072i $$0.414417\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 250.000i − 2.42274i
$$23$$ − 120.000i − 1.08790i −0.839117 0.543951i $$-0.816928\pi$$
0.839117 0.543951i $$-0.183072\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 100.000 0.754293
$$27$$ 0 0
$$28$$ − 510.000i − 3.44218i
$$29$$ −50.0000 −0.320164 −0.160082 0.987104i $$-0.551176\pi$$
−0.160082 + 0.987104i $$0.551176\pi$$
$$30$$ 0 0
$$31$$ 108.000 0.625722 0.312861 0.949799i $$-0.398713\pi$$
0.312861 + 0.949799i $$0.398713\pi$$
$$32$$ 85.0000i 0.469563i
$$33$$ 0 0
$$34$$ 50.0000 0.252204
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 40.0000i 0.177729i 0.996044 + 0.0888643i $$0.0283238\pi$$
−0.996044 + 0.0888643i $$0.971676\pi$$
$$38$$ 220.000i 0.939177i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −400.000 −1.52365 −0.761823 0.647785i $$-0.775696\pi$$
−0.761823 + 0.647785i $$0.775696\pi$$
$$42$$ 0 0
$$43$$ 280.000i 0.993014i 0.868033 + 0.496507i $$0.165384\pi$$
−0.868033 + 0.496507i $$0.834616\pi$$
$$44$$ 850.000 2.91233
$$45$$ 0 0
$$46$$ 600.000 1.92316
$$47$$ − 280.000i − 0.868983i −0.900676 0.434491i $$-0.856928\pi$$
0.900676 0.434491i $$-0.143072\pi$$
$$48$$ 0 0
$$49$$ −557.000 −1.62391
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 340.000i 0.906721i
$$53$$ 610.000i 1.58094i 0.612499 + 0.790471i $$0.290164\pi$$
−0.612499 + 0.790471i $$0.709836\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1350.00 3.22145
$$57$$ 0 0
$$58$$ − 250.000i − 0.565976i
$$59$$ 50.0000 0.110330 0.0551648 0.998477i $$-0.482432\pi$$
0.0551648 + 0.998477i $$0.482432\pi$$
$$60$$ 0 0
$$61$$ −518.000 −1.08726 −0.543632 0.839324i $$-0.682951\pi$$
−0.543632 + 0.839324i $$0.682951\pi$$
$$62$$ 540.000i 1.10613i
$$63$$ 0 0
$$64$$ 287.000 0.560547
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 180.000i 0.328216i 0.986442 + 0.164108i $$0.0524746\pi$$
−0.986442 + 0.164108i $$0.947525\pi$$
$$68$$ 170.000i 0.303170i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −700.000 −1.17007 −0.585033 0.811009i $$-0.698919\pi$$
−0.585033 + 0.811009i $$0.698919\pi$$
$$72$$ 0 0
$$73$$ − 410.000i − 0.657354i −0.944442 0.328677i $$-0.893397\pi$$
0.944442 0.328677i $$-0.106603\pi$$
$$74$$ −200.000 −0.314183
$$75$$ 0 0
$$76$$ −748.000 −1.12897
$$77$$ − 1500.00i − 2.22001i
$$78$$ 0 0
$$79$$ 516.000 0.734868 0.367434 0.930050i $$-0.380236\pi$$
0.367434 + 0.930050i $$0.380236\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 2000.00i − 2.69345i
$$83$$ − 660.000i − 0.872824i −0.899747 0.436412i $$-0.856249\pi$$
0.899747 0.436412i $$-0.143751\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1400.00 −1.75542
$$87$$ 0 0
$$88$$ 2250.00i 2.72558i
$$89$$ −1500.00 −1.78651 −0.893257 0.449547i $$-0.851585\pi$$
−0.893257 + 0.449547i $$0.851585\pi$$
$$90$$ 0 0
$$91$$ 600.000 0.691177
$$92$$ 2040.00i 2.31179i
$$93$$ 0 0
$$94$$ 1400.00 1.53616
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1630.00i 1.70620i 0.521747 + 0.853100i $$0.325280\pi$$
−0.521747 + 0.853100i $$0.674720\pi$$
$$98$$ − 2785.00i − 2.87069i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 450.000 0.443333 0.221667 0.975122i $$-0.428850\pi$$
0.221667 + 0.975122i $$0.428850\pi$$
$$102$$ 0 0
$$103$$ 770.000i 0.736605i 0.929706 + 0.368303i $$0.120061\pi$$
−0.929706 + 0.368303i $$0.879939\pi$$
$$104$$ −900.000 −0.848579
$$105$$ 0 0
$$106$$ −3050.00 −2.79474
$$107$$ 660.000i 0.596305i 0.954518 + 0.298152i $$0.0963704\pi$$
−0.954518 + 0.298152i $$0.903630\pi$$
$$108$$ 0 0
$$109$$ −1754.00 −1.54131 −0.770655 0.637253i $$-0.780071\pi$$
−0.770655 + 0.637253i $$0.780071\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2670.00i 2.25260i
$$113$$ 310.000i 0.258074i 0.991640 + 0.129037i $$0.0411886\pi$$
−0.991640 + 0.129037i $$0.958811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 850.000 0.680349
$$117$$ 0 0
$$118$$ 250.000i 0.195037i
$$119$$ 300.000 0.231100
$$120$$ 0 0
$$121$$ 1169.00 0.878287
$$122$$ − 2590.00i − 1.92203i
$$123$$ 0 0
$$124$$ −1836.00 −1.32966
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1070.00i 0.747615i 0.927506 + 0.373808i $$0.121948\pi$$
−0.927506 + 0.373808i $$0.878052\pi$$
$$128$$ 2115.00i 1.46048i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1950.00 −1.30055 −0.650276 0.759698i $$-0.725347\pi$$
−0.650276 + 0.759698i $$0.725347\pi$$
$$132$$ 0 0
$$133$$ 1320.00i 0.860590i
$$134$$ −900.000 −0.580210
$$135$$ 0 0
$$136$$ −450.000 −0.283729
$$137$$ 1050.00i 0.654800i 0.944886 + 0.327400i $$0.106172\pi$$
−0.944886 + 0.327400i $$0.893828\pi$$
$$138$$ 0 0
$$139$$ −1676.00 −1.02271 −0.511354 0.859370i $$-0.670856\pi$$
−0.511354 + 0.859370i $$0.670856\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 3500.00i − 2.06840i
$$143$$ 1000.00i 0.584785i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2050.00 1.16205
$$147$$ 0 0
$$148$$ − 680.000i − 0.377673i
$$149$$ 2050.00 1.12713 0.563566 0.826071i $$-0.309429\pi$$
0.563566 + 0.826071i $$0.309429\pi$$
$$150$$ 0 0
$$151$$ 448.000 0.241442 0.120721 0.992686i $$-0.461479\pi$$
0.120721 + 0.992686i $$0.461479\pi$$
$$152$$ − 1980.00i − 1.05657i
$$153$$ 0 0
$$154$$ 7500.00 3.92446
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 100.000i 0.0508336i 0.999677 + 0.0254168i $$0.00809128\pi$$
−0.999677 + 0.0254168i $$0.991909\pi$$
$$158$$ 2580.00i 1.29907i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 3600.00 1.76223
$$162$$ 0 0
$$163$$ − 1900.00i − 0.913003i −0.889723 0.456501i $$-0.849102\pi$$
0.889723 0.456501i $$-0.150898\pi$$
$$164$$ 6800.00 3.23775
$$165$$ 0 0
$$166$$ 3300.00 1.54295
$$167$$ 1920.00i 0.889665i 0.895614 + 0.444833i $$0.146737\pi$$
−0.895614 + 0.444833i $$0.853263\pi$$
$$168$$ 0 0
$$169$$ 1797.00 0.817934
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 4760.00i − 2.11015i
$$173$$ − 2550.00i − 1.12065i −0.828272 0.560326i $$-0.810676\pi$$
0.828272 0.560326i $$-0.189324\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4450.00 −1.90586
$$177$$ 0 0
$$178$$ − 7500.00i − 3.15814i
$$179$$ 3650.00 1.52410 0.762050 0.647518i $$-0.224193\pi$$
0.762050 + 0.647518i $$0.224193\pi$$
$$180$$ 0 0
$$181$$ −4342.00 −1.78308 −0.891542 0.452937i $$-0.850376\pi$$
−0.891542 + 0.452937i $$0.850376\pi$$
$$182$$ 3000.00i 1.22184i
$$183$$ 0 0
$$184$$ −5400.00 −2.16355
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 500.000i 0.195527i
$$188$$ 4760.00i 1.84659i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3500.00 1.32592 0.662961 0.748654i $$-0.269299\pi$$
0.662961 + 0.748654i $$0.269299\pi$$
$$192$$ 0 0
$$193$$ 3350.00i 1.24942i 0.780856 + 0.624711i $$0.214783\pi$$
−0.780856 + 0.624711i $$0.785217\pi$$
$$194$$ −8150.00 −3.01616
$$195$$ 0 0
$$196$$ 9469.00 3.45080
$$197$$ 90.0000i 0.0325494i 0.999868 + 0.0162747i $$0.00518063\pi$$
−0.999868 + 0.0162747i $$0.994819\pi$$
$$198$$ 0 0
$$199$$ −3664.00 −1.30520 −0.652598 0.757704i $$-0.726321\pi$$
−0.652598 + 0.757704i $$0.726321\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 2250.00i 0.783710i
$$203$$ − 1500.00i − 0.518618i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −3850.00 −1.30215
$$207$$ 0 0
$$208$$ − 1780.00i − 0.593369i
$$209$$ −2200.00 −0.728120
$$210$$ 0 0
$$211$$ −268.000 −0.0874402 −0.0437201 0.999044i $$-0.513921\pi$$
−0.0437201 + 0.999044i $$0.513921\pi$$
$$212$$ − 10370.0i − 3.35950i
$$213$$ 0 0
$$214$$ −3300.00 −1.05413
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3240.00i 1.01357i
$$218$$ − 8770.00i − 2.72468i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −200.000 −0.0608754
$$222$$ 0 0
$$223$$ − 3670.00i − 1.10207i −0.834482 0.551034i $$-0.814233\pi$$
0.834482 0.551034i $$-0.185767\pi$$
$$224$$ −2550.00 −0.760621
$$225$$ 0 0
$$226$$ −1550.00 −0.456214
$$227$$ 3760.00i 1.09938i 0.835368 + 0.549692i $$0.185255\pi$$
−0.835368 + 0.549692i $$0.814745\pi$$
$$228$$ 0 0
$$229$$ 1434.00 0.413805 0.206903 0.978362i $$-0.433662\pi$$
0.206903 + 0.978362i $$0.433662\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2250.00i 0.636723i
$$233$$ 3450.00i 0.970030i 0.874506 + 0.485015i $$0.161186\pi$$
−0.874506 + 0.485015i $$0.838814\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −850.000 −0.234450
$$237$$ 0 0
$$238$$ 1500.00i 0.408532i
$$239$$ −4900.00 −1.32617 −0.663085 0.748544i $$-0.730753\pi$$
−0.663085 + 0.748544i $$0.730753\pi$$
$$240$$ 0 0
$$241$$ 4822.00 1.28885 0.644424 0.764668i $$-0.277097\pi$$
0.644424 + 0.764668i $$0.277097\pi$$
$$242$$ 5845.00i 1.55261i
$$243$$ 0 0
$$244$$ 8806.00 2.31044
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 880.000i − 0.226693i
$$248$$ − 4860.00i − 1.24440i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4650.00 1.16934 0.584672 0.811270i $$-0.301223\pi$$
0.584672 + 0.811270i $$0.301223\pi$$
$$252$$ 0 0
$$253$$ 6000.00i 1.49098i
$$254$$ −5350.00 −1.32161
$$255$$ 0 0
$$256$$ −8279.00 −2.02124
$$257$$ 5130.00i 1.24514i 0.782565 + 0.622569i $$0.213911\pi$$
−0.782565 + 0.622569i $$0.786089\pi$$
$$258$$ 0 0
$$259$$ −1200.00 −0.287893
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 9750.00i − 2.29907i
$$263$$ 1280.00i 0.300107i 0.988678 + 0.150054i $$0.0479446\pi$$
−0.988678 + 0.150054i $$0.952055\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6600.00 −1.52132
$$267$$ 0 0
$$268$$ − 3060.00i − 0.697460i
$$269$$ 3350.00 0.759305 0.379653 0.925129i $$-0.376044\pi$$
0.379653 + 0.925129i $$0.376044\pi$$
$$270$$ 0 0
$$271$$ 5512.00 1.23554 0.617768 0.786361i $$-0.288037\pi$$
0.617768 + 0.786361i $$0.288037\pi$$
$$272$$ − 890.000i − 0.198398i
$$273$$ 0 0
$$274$$ −5250.00 −1.15753
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 4920.00i − 1.06720i −0.845737 0.533600i $$-0.820839\pi$$
0.845737 0.533600i $$-0.179161\pi$$
$$278$$ − 8380.00i − 1.80791i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4500.00 −0.955329 −0.477665 0.878542i $$-0.658517\pi$$
−0.477665 + 0.878542i $$0.658517\pi$$
$$282$$ 0 0
$$283$$ − 6900.00i − 1.44934i −0.689098 0.724669i $$-0.741993\pi$$
0.689098 0.724669i $$-0.258007\pi$$
$$284$$ 11900.0 2.48639
$$285$$ 0 0
$$286$$ −5000.00 −1.03376
$$287$$ − 12000.0i − 2.46808i
$$288$$ 0 0
$$289$$ 4813.00 0.979646
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 6970.00i 1.39688i
$$293$$ − 1530.00i − 0.305063i −0.988299 0.152532i $$-0.951257\pi$$
0.988299 0.152532i $$-0.0487426\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1800.00 0.353456
$$297$$ 0 0
$$298$$ 10250.0i 1.99251i
$$299$$ −2400.00 −0.464199
$$300$$ 0 0
$$301$$ −8400.00 −1.60853
$$302$$ 2240.00i 0.426813i
$$303$$ 0 0
$$304$$ 3916.00 0.738809
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 3040.00i − 0.565153i −0.959245 0.282576i $$-0.908811\pi$$
0.959245 0.282576i $$-0.0911891\pi$$
$$308$$ 25500.0i 4.71752i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5700.00 1.03928 0.519642 0.854384i $$-0.326065\pi$$
0.519642 + 0.854384i $$0.326065\pi$$
$$312$$ 0 0
$$313$$ 3110.00i 0.561622i 0.959763 + 0.280811i $$0.0906034\pi$$
−0.959763 + 0.280811i $$0.909397\pi$$
$$314$$ −500.000 −0.0898619
$$315$$ 0 0
$$316$$ −8772.00 −1.56159
$$317$$ − 950.000i − 0.168320i −0.996452 0.0841598i $$-0.973179\pi$$
0.996452 0.0841598i $$-0.0268206\pi$$
$$318$$ 0 0
$$319$$ 2500.00 0.438787
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 18000.0i 3.11522i
$$323$$ − 440.000i − 0.0757965i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 9500.00 1.61398
$$327$$ 0 0
$$328$$ 18000.0i 3.03013i
$$329$$ 8400.00 1.40762
$$330$$ 0 0
$$331$$ 2292.00 0.380603 0.190302 0.981726i $$-0.439053\pi$$
0.190302 + 0.981726i $$0.439053\pi$$
$$332$$ 11220.0i 1.85475i
$$333$$ 0 0
$$334$$ −9600.00 −1.57272
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7730.00i 1.24950i 0.780827 + 0.624748i $$0.214798\pi$$
−0.780827 + 0.624748i $$0.785202\pi$$
$$338$$ 8985.00i 1.44592i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −5400.00 −0.857555
$$342$$ 0 0
$$343$$ − 6420.00i − 1.01063i
$$344$$ 12600.0 1.97484
$$345$$ 0 0
$$346$$ 12750.0 1.98105
$$347$$ − 1120.00i − 0.173270i −0.996240 0.0866351i $$-0.972389\pi$$
0.996240 0.0866351i $$-0.0276114\pi$$
$$348$$ 0 0
$$349$$ −1186.00 −0.181906 −0.0909529 0.995855i $$-0.528991\pi$$
−0.0909529 + 0.995855i $$0.528991\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 4250.00i − 0.643539i
$$353$$ − 3630.00i − 0.547324i −0.961826 0.273662i $$-0.911765\pi$$
0.961826 0.273662i $$-0.0882350\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 25500.0 3.79634
$$357$$ 0 0
$$358$$ 18250.0i 2.69425i
$$359$$ −1800.00 −0.264625 −0.132312 0.991208i $$-0.542240\pi$$
−0.132312 + 0.991208i $$0.542240\pi$$
$$360$$ 0 0
$$361$$ −4923.00 −0.717743
$$362$$ − 21710.0i − 3.15208i
$$363$$ 0 0
$$364$$ −10200.0 −1.46875
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8490.00i − 1.20756i −0.797151 0.603780i $$-0.793661\pi$$
0.797151 0.603780i $$-0.206339\pi$$
$$368$$ − 10680.0i − 1.51286i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −18300.0 −2.56089
$$372$$ 0 0
$$373$$ 100.000i 0.0138815i 0.999976 + 0.00694076i $$0.00220933\pi$$
−0.999976 + 0.00694076i $$0.997791\pi$$
$$374$$ −2500.00 −0.345647
$$375$$ 0 0
$$376$$ −12600.0 −1.72818
$$377$$ 1000.00i 0.136612i
$$378$$ 0 0
$$379$$ 8084.00 1.09564 0.547820 0.836597i $$-0.315458\pi$$
0.547820 + 0.836597i $$0.315458\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 17500.0i 2.34392i
$$383$$ 9480.00i 1.26477i 0.774656 + 0.632383i $$0.217923\pi$$
−0.774656 + 0.632383i $$0.782077\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −16750.0 −2.20869
$$387$$ 0 0
$$388$$ − 27710.0i − 3.62568i
$$389$$ −10950.0 −1.42722 −0.713608 0.700545i $$-0.752940\pi$$
−0.713608 + 0.700545i $$0.752940\pi$$
$$390$$ 0 0
$$391$$ −1200.00 −0.155209
$$392$$ 25065.0i 3.22952i
$$393$$ 0 0
$$394$$ −450.000 −0.0575398
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 13840.0i − 1.74965i −0.484442 0.874823i $$-0.660977\pi$$
0.484442 0.874823i $$-0.339023\pi$$
$$398$$ − 18320.0i − 2.30728i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −9300.00 −1.15815 −0.579077 0.815273i $$-0.696587\pi$$
−0.579077 + 0.815273i $$0.696587\pi$$
$$402$$ 0 0
$$403$$ − 2160.00i − 0.266991i
$$404$$ −7650.00 −0.942083
$$405$$ 0 0
$$406$$ 7500.00 0.916795
$$407$$ − 2000.00i − 0.243578i
$$408$$ 0 0
$$409$$ 2854.00 0.345040 0.172520 0.985006i $$-0.444809\pi$$
0.172520 + 0.985006i $$0.444809\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 13090.0i − 1.56529i
$$413$$ 1500.00i 0.178717i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1700.00 0.200359
$$417$$ 0 0
$$418$$ − 11000.0i − 1.28715i
$$419$$ 1150.00 0.134084 0.0670420 0.997750i $$-0.478644\pi$$
0.0670420 + 0.997750i $$0.478644\pi$$
$$420$$ 0 0
$$421$$ −11162.0 −1.29217 −0.646084 0.763266i $$-0.723594\pi$$
−0.646084 + 0.763266i $$0.723594\pi$$
$$422$$ − 1340.00i − 0.154574i
$$423$$ 0 0
$$424$$ 27450.0 3.14408
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 15540.0i − 1.76120i
$$428$$ − 11220.0i − 1.26715i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1200.00 0.134111 0.0670556 0.997749i $$-0.478639\pi$$
0.0670556 + 0.997749i $$0.478639\pi$$
$$432$$ 0 0
$$433$$ 1510.00i 0.167589i 0.996483 + 0.0837944i $$0.0267039\pi$$
−0.996483 + 0.0837944i $$0.973296\pi$$
$$434$$ −16200.0 −1.79176
$$435$$ 0 0
$$436$$ 29818.0 3.27528
$$437$$ − 5280.00i − 0.577979i
$$438$$ 0 0
$$439$$ −424.000 −0.0460966 −0.0230483 0.999734i $$-0.507337\pi$$
−0.0230483 + 0.999734i $$0.507337\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 1000.00i − 0.107613i
$$443$$ − 12360.0i − 1.32560i −0.748796 0.662801i $$-0.769368\pi$$
0.748796 0.662801i $$-0.230632\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 18350.0 1.94820
$$447$$ 0 0
$$448$$ 8610.00i 0.908001i
$$449$$ −1300.00 −0.136639 −0.0683194 0.997664i $$-0.521764\pi$$
−0.0683194 + 0.997664i $$0.521764\pi$$
$$450$$ 0 0
$$451$$ 20000.0 2.08817
$$452$$ − 5270.00i − 0.548407i
$$453$$ 0 0
$$454$$ −18800.0 −1.94345
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 7190.00i 0.735961i 0.929834 + 0.367980i $$0.119951\pi$$
−0.929834 + 0.367980i $$0.880049\pi$$
$$458$$ 7170.00i 0.731511i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 150.000 0.0151544 0.00757722 0.999971i $$-0.497588\pi$$
0.00757722 + 0.999971i $$0.497588\pi$$
$$462$$ 0 0
$$463$$ 2670.00i 0.268003i 0.990981 + 0.134002i $$0.0427827\pi$$
−0.990981 + 0.134002i $$0.957217\pi$$
$$464$$ −4450.00 −0.445229
$$465$$ 0 0
$$466$$ −17250.0 −1.71479
$$467$$ 1180.00i 0.116925i 0.998290 + 0.0584624i $$0.0186198\pi$$
−0.998290 + 0.0584624i $$0.981380\pi$$
$$468$$ 0 0
$$469$$ −5400.00 −0.531661
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 2250.00i − 0.219417i
$$473$$ − 14000.0i − 1.36093i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −5100.00 −0.491088
$$477$$ 0 0
$$478$$ − 24500.0i − 2.34436i
$$479$$ −14100.0 −1.34498 −0.672490 0.740106i $$-0.734775\pi$$
−0.672490 + 0.740106i $$0.734775\pi$$
$$480$$ 0 0
$$481$$ 800.000 0.0758355
$$482$$ 24110.0i 2.27838i
$$483$$ 0 0
$$484$$ −19873.0 −1.86636
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 9850.00i 0.916522i 0.888818 + 0.458261i $$0.151527\pi$$
−0.888818 + 0.458261i $$0.848473\pi$$
$$488$$ 23310.0i 2.16228i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2450.00 0.225187 0.112594 0.993641i $$-0.464084\pi$$
0.112594 + 0.993641i $$0.464084\pi$$
$$492$$ 0 0
$$493$$ 500.000i 0.0456772i
$$494$$ 4400.00 0.400740
$$495$$ 0 0
$$496$$ 9612.00 0.870144
$$497$$ − 21000.0i − 1.89533i
$$498$$ 0 0
$$499$$ 17036.0 1.52833 0.764164 0.645021i $$-0.223152\pi$$
0.764164 + 0.645021i $$0.223152\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 23250.0i 2.06713i
$$503$$ 20600.0i 1.82606i 0.407891 + 0.913030i $$0.366264\pi$$
−0.407891 + 0.913030i $$0.633736\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −30000.0 −2.63570
$$507$$ 0 0
$$508$$ − 18190.0i − 1.58868i
$$509$$ 5750.00 0.500716 0.250358 0.968153i $$-0.419452\pi$$
0.250358 + 0.968153i $$0.419452\pi$$
$$510$$ 0 0
$$511$$ 12300.0 1.06481
$$512$$ − 24475.0i − 2.11260i
$$513$$ 0 0
$$514$$ −25650.0 −2.20111
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 14000.0i 1.19095i
$$518$$ − 6000.00i − 0.508928i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 15500.0 1.30339 0.651696 0.758480i $$-0.274058\pi$$
0.651696 + 0.758480i $$0.274058\pi$$
$$522$$ 0 0
$$523$$ − 13940.0i − 1.16549i −0.812653 0.582747i $$-0.801978\pi$$
0.812653 0.582747i $$-0.198022\pi$$
$$524$$ 33150.0 2.76367
$$525$$ 0 0
$$526$$ −6400.00 −0.530520
$$527$$ − 1080.00i − 0.0892705i
$$528$$ 0 0
$$529$$ −2233.00 −0.183529
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 22440.0i − 1.82875i
$$533$$ 8000.00i 0.650128i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 8100.00 0.652736
$$537$$ 0 0
$$538$$ 16750.0i 1.34227i
$$539$$ 27850.0 2.22557
$$540$$ 0 0
$$541$$ −20478.0 −1.62739 −0.813695 0.581292i $$-0.802547\pi$$
−0.813695 + 0.581292i $$0.802547\pi$$
$$542$$ 27560.0i 2.18414i
$$543$$ 0 0
$$544$$ 850.000 0.0669916
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 12040.0i − 0.941121i −0.882368 0.470561i $$-0.844052\pi$$
0.882368 0.470561i $$-0.155948\pi$$
$$548$$ − 17850.0i − 1.39145i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2200.00 −0.170096
$$552$$ 0 0
$$553$$ 15480.0i 1.19037i
$$554$$ 24600.0 1.88656
$$555$$ 0 0
$$556$$ 28492.0 2.17326
$$557$$ 23550.0i 1.79146i 0.444594 + 0.895732i $$0.353348\pi$$
−0.444594 + 0.895732i $$0.646652\pi$$
$$558$$ 0 0
$$559$$ 5600.00 0.423712
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 22500.0i − 1.68880i
$$563$$ 6120.00i 0.458130i 0.973411 + 0.229065i $$0.0735669\pi$$
−0.973411 + 0.229065i $$0.926433\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 34500.0 2.56209
$$567$$ 0 0
$$568$$ 31500.0i 2.32696i
$$569$$ −11700.0 −0.862020 −0.431010 0.902347i $$-0.641843\pi$$
−0.431010 + 0.902347i $$0.641843\pi$$
$$570$$ 0 0
$$571$$ −8188.00 −0.600100 −0.300050 0.953923i $$-0.597003\pi$$
−0.300050 + 0.953923i $$0.597003\pi$$
$$572$$ − 17000.0i − 1.24267i
$$573$$ 0 0
$$574$$ 60000.0 4.36298
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 11690.0i − 0.843433i −0.906728 0.421717i $$-0.861428\pi$$
0.906728 0.421717i $$-0.138572\pi$$
$$578$$ 24065.0i 1.73179i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 19800.0 1.41384
$$582$$ 0 0
$$583$$ − 30500.0i − 2.16669i
$$584$$ −18450.0 −1.30731
$$585$$ 0 0
$$586$$ 7650.00 0.539281
$$587$$ 21060.0i 1.48082i 0.672157 + 0.740408i $$0.265368\pi$$
−0.672157 + 0.740408i $$0.734632\pi$$
$$588$$ 0 0
$$589$$ 4752.00 0.332433
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 3560.00i 0.247154i
$$593$$ 22910.0i 1.58651i 0.608889 + 0.793255i $$0.291615\pi$$
−0.608889 + 0.793255i $$0.708385\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −34850.0 −2.39515
$$597$$ 0 0
$$598$$ − 12000.0i − 0.820596i
$$599$$ −1400.00 −0.0954966 −0.0477483 0.998859i $$-0.515205\pi$$
−0.0477483 + 0.998859i $$0.515205\pi$$
$$600$$ 0 0
$$601$$ −11002.0 −0.746724 −0.373362 0.927686i $$-0.621795\pi$$
−0.373362 + 0.927686i $$0.621795\pi$$
$$602$$ − 42000.0i − 2.84351i
$$603$$ 0 0
$$604$$ −7616.00 −0.513064
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 4630.00i − 0.309598i −0.987946 0.154799i $$-0.950527\pi$$
0.987946 0.154799i $$-0.0494730\pi$$
$$608$$ 3740.00i 0.249469i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5600.00 −0.370788
$$612$$ 0 0
$$613$$ 24040.0i 1.58396i 0.610548 + 0.791979i $$0.290949\pi$$
−0.610548 + 0.791979i $$0.709051\pi$$
$$614$$ 15200.0 0.999059
$$615$$ 0 0
$$616$$ −67500.0 −4.41502
$$617$$ − 1890.00i − 0.123320i −0.998097 0.0616601i $$-0.980361\pi$$
0.998097 0.0616601i $$-0.0196395\pi$$
$$618$$ 0 0
$$619$$ −19244.0 −1.24957 −0.624783 0.780798i $$-0.714813\pi$$
−0.624783 + 0.780798i $$0.714813\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 28500.0i 1.83721i
$$623$$ − 45000.0i − 2.89388i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −15550.0 −0.992816
$$627$$ 0 0
$$628$$ − 1700.00i − 0.108021i
$$629$$ 400.000 0.0253562
$$630$$ 0 0
$$631$$ 15892.0 1.00262 0.501308 0.865269i $$-0.332852\pi$$
0.501308 + 0.865269i $$0.332852\pi$$
$$632$$ − 23220.0i − 1.46146i
$$633$$ 0 0
$$634$$ 4750.00 0.297550
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 11140.0i 0.692909i
$$638$$ 12500.0i 0.775674i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 12600.0 0.776396 0.388198 0.921576i $$-0.373098\pi$$
0.388198 + 0.921576i $$0.373098\pi$$
$$642$$ 0 0
$$643$$ 7260.00i 0.445267i 0.974902 + 0.222633i $$0.0714653\pi$$
−0.974902 + 0.222633i $$0.928535\pi$$
$$644$$ −61200.0 −3.74475
$$645$$ 0 0
$$646$$ 2200.00 0.133990
$$647$$ 7400.00i 0.449651i 0.974399 + 0.224825i $$0.0721812\pi$$
−0.974399 + 0.224825i $$0.927819\pi$$
$$648$$ 0 0
$$649$$ −2500.00 −0.151207
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 32300.0i 1.94013i
$$653$$ 4790.00i 0.287055i 0.989646 + 0.143528i $$0.0458446\pi$$
−0.989646 + 0.143528i $$0.954155\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −35600.0 −2.11882
$$657$$ 0 0
$$658$$ 42000.0i 2.48834i
$$659$$ −1450.00 −0.0857117 −0.0428558 0.999081i $$-0.513646\pi$$
−0.0428558 + 0.999081i $$0.513646\pi$$
$$660$$ 0 0
$$661$$ 11818.0 0.695411 0.347706 0.937604i $$-0.386961\pi$$
0.347706 + 0.937604i $$0.386961\pi$$
$$662$$ 11460.0i 0.672818i
$$663$$ 0 0
$$664$$ −29700.0 −1.73582
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6000.00i 0.348307i
$$668$$ − 32640.0i − 1.89054i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 25900.0 1.49010
$$672$$ 0 0
$$673$$ 5550.00i 0.317885i 0.987288 + 0.158943i $$0.0508085\pi$$
−0.987288 + 0.158943i $$0.949192\pi$$
$$674$$ −38650.0 −2.20882
$$675$$ 0 0
$$676$$ −30549.0 −1.73811
$$677$$ 12930.0i 0.734033i 0.930214 + 0.367016i $$0.119621\pi$$
−0.930214 + 0.367016i $$0.880379\pi$$
$$678$$ 0 0
$$679$$ −48900.0 −2.76378
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 27000.0i − 1.51596i
$$683$$ − 32580.0i − 1.82524i −0.408809 0.912620i $$-0.634056\pi$$
0.408809 0.912620i $$-0.365944\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 32100.0 1.78657
$$687$$ 0 0
$$688$$ 24920.0i 1.38091i
$$689$$ 12200.0 0.674576
$$690$$ 0 0
$$691$$ 10228.0 0.563085 0.281542 0.959549i $$-0.409154\pi$$
0.281542 + 0.959549i $$0.409154\pi$$
$$692$$ 43350.0i 2.38139i
$$693$$ 0 0
$$694$$ 5600.00 0.306301
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4000.00i 0.217376i
$$698$$ − 5930.00i − 0.321567i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −8350.00 −0.449893 −0.224947 0.974371i $$-0.572221\pi$$
−0.224947 + 0.974371i $$0.572221\pi$$
$$702$$ 0 0
$$703$$ 1760.00i 0.0944234i
$$704$$ −14350.0 −0.768233
$$705$$ 0 0
$$706$$ 18150.0 0.967541
$$707$$ 13500.0i 0.718133i
$$708$$ 0 0
$$709$$ 14954.0 0.792115 0.396057 0.918226i $$-0.370378\pi$$
0.396057 + 0.918226i $$0.370378\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 67500.0i 3.55291i
$$713$$ − 12960.0i − 0.680723i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −62050.0 −3.23871
$$717$$ 0 0
$$718$$ − 9000.00i − 0.467795i
$$719$$ 29400.0 1.52494 0.762472 0.647021i $$-0.223985\pi$$
0.762472 + 0.647021i $$0.223985\pi$$
$$720$$ 0 0
$$721$$ −23100.0 −1.19319
$$722$$ − 24615.0i − 1.26880i
$$723$$ 0 0
$$724$$ 73814.0 3.78905
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 16330.0i 0.833076i 0.909118 + 0.416538i $$0.136757\pi$$
−0.909118 + 0.416538i $$0.863243\pi$$
$$728$$ − 27000.0i − 1.37457i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 2800.00 0.141671
$$732$$ 0 0
$$733$$ 30800.0i 1.55201i 0.630726 + 0.776005i $$0.282757\pi$$
−0.630726 + 0.776005i $$0.717243\pi$$
$$734$$ 42450.0 2.13468
$$735$$ 0 0
$$736$$ 10200.0 0.510838
$$737$$ − 9000.00i − 0.449823i
$$738$$ 0 0
$$739$$ 9524.00 0.474081 0.237041 0.971500i $$-0.423823\pi$$
0.237041 + 0.971500i $$0.423823\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 91500.0i − 4.52705i
$$743$$ 28600.0i 1.41216i 0.708134 + 0.706078i $$0.249537\pi$$
−0.708134 + 0.706078i $$0.750463\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −500.000 −0.0245393
$$747$$ 0 0
$$748$$ − 8500.00i − 0.415496i
$$749$$ −19800.0 −0.965923
$$750$$ 0 0
$$751$$ −8252.00 −0.400958 −0.200479 0.979698i $$-0.564250\pi$$
−0.200479 + 0.979698i $$0.564250\pi$$
$$752$$ − 24920.0i − 1.20843i
$$753$$ 0 0
$$754$$ −5000.00 −0.241498
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 24920.0i 1.19648i 0.801318 + 0.598238i $$0.204132\pi$$
−0.801318 + 0.598238i $$0.795868\pi$$
$$758$$ 40420.0i 1.93683i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −27900.0 −1.32901 −0.664503 0.747285i $$-0.731357\pi$$
−0.664503 + 0.747285i $$0.731357\pi$$
$$762$$ 0 0
$$763$$ − 52620.0i − 2.49669i
$$764$$ −59500.0 −2.81758
$$765$$ 0 0
$$766$$ −47400.0 −2.23581
$$767$$ − 1000.00i − 0.0470768i
$$768$$ 0 0
$$769$$ 11506.0 0.539554 0.269777 0.962923i $$-0.413050\pi$$
0.269777 + 0.962923i $$0.413050\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 56950.0i − 2.65502i
$$773$$ 12510.0i 0.582087i 0.956710 + 0.291044i $$0.0940025\pi$$
−0.956710 + 0.291044i $$0.905998\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 73350.0 3.39318
$$777$$ 0 0
$$778$$ − 54750.0i − 2.52299i
$$779$$ −17600.0 −0.809481
$$780$$ 0 0
$$781$$ 35000.0 1.60358
$$782$$ − 6000.00i − 0.274373i
$$783$$ 0 0
$$784$$ −49573.0 −2.25825
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1100.00i 0.0498231i 0.999690 + 0.0249115i $$0.00793041\pi$$
−0.999690 + 0.0249115i $$0.992070\pi$$
$$788$$ − 1530.00i − 0.0691675i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −9300.00 −0.418040
$$792$$ 0 0
$$793$$ 10360.0i 0.463927i
$$794$$ 69200.0 3.09297
$$795$$ 0 0
$$796$$ 62288.0 2.77354
$$797$$ − 4490.00i − 0.199553i −0.995010 0.0997766i $$-0.968187\pi$$
0.995010 0.0997766i $$-0.0318128\pi$$
$$798$$ 0 0
$$799$$ −2800.00 −0.123976
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 46500.0i − 2.04735i
$$803$$ 20500.0i 0.900908i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 10800.0 0.471977
$$807$$ 0 0
$$808$$ − 20250.0i − 0.881674i
$$809$$ 28600.0 1.24292 0.621460 0.783446i $$-0.286540\pi$$
0.621460 + 0.783446i $$0.286540\pi$$
$$810$$ 0 0
$$811$$ 10068.0 0.435925 0.217963 0.975957i $$-0.430059\pi$$
0.217963 + 0.975957i $$0.430059\pi$$
$$812$$ 25500.0i 1.10206i
$$813$$ 0 0
$$814$$ 10000.0 0.430589
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 12320.0i 0.527567i
$$818$$ 14270.0i 0.609950i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 14250.0 0.605759 0.302880 0.953029i $$-0.402052\pi$$
0.302880 + 0.953029i $$0.402052\pi$$
$$822$$ 0 0
$$823$$ 6830.00i 0.289282i 0.989484 + 0.144641i $$0.0462027\pi$$
−0.989484 + 0.144641i $$0.953797\pi$$
$$824$$ 34650.0 1.46491
$$825$$ 0 0
$$826$$ −7500.00 −0.315930
$$827$$ − 8920.00i − 0.375065i −0.982258 0.187533i $$-0.939951\pi$$
0.982258 0.187533i $$-0.0600490\pi$$
$$828$$ 0 0
$$829$$ 3534.00 0.148059 0.0740295 0.997256i $$-0.476414\pi$$
0.0740295 + 0.997256i $$0.476414\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 5740.00i − 0.239181i
$$833$$ 5570.00i 0.231680i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 37400.0 1.54726
$$837$$ 0 0
$$838$$ 5750.00i 0.237029i
$$839$$ −8000.00 −0.329190 −0.164595 0.986361i $$-0.552632\pi$$
−0.164595 + 0.986361i $$0.552632\pi$$
$$840$$ 0 0
$$841$$ −21889.0 −0.897495
$$842$$ − 55810.0i − 2.28425i
$$843$$ 0 0
$$844$$ 4556.00 0.185810
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 35070.0i 1.42269i
$$848$$ 54290.0i 2.19850i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 4800.00 0.193351
$$852$$ 0 0
$$853$$ 5160.00i 0.207122i 0.994623 + 0.103561i $$0.0330237\pi$$
−0.994623 + 0.103561i $$0.966976\pi$$
$$854$$ 77700.0 3.11339
$$855$$ 0 0
$$856$$ 29700.0 1.18589
$$857$$ 7670.00i 0.305720i 0.988248 + 0.152860i $$0.0488484\pi$$
−0.988248 + 0.152860i $$0.951152\pi$$
$$858$$ 0 0
$$859$$ −25804.0 −1.02494 −0.512469 0.858706i $$-0.671269\pi$$
−0.512469 + 0.858706i $$0.671269\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 6000.00i 0.237078i
$$863$$ − 400.000i − 0.0157777i −0.999969 0.00788885i $$-0.997489\pi$$
0.999969 0.00788885i $$-0.00251113\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −7550.00 −0.296258
$$867$$ 0 0
$$868$$ − 55080.0i − 2.15384i
$$869$$ −25800.0 −1.00714
$$870$$ 0 0
$$871$$ 3600.00 0.140047
$$872$$ 78930.0i 3.06526i
$$873$$ 0 0
$$874$$ 26400.0 1.02173
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 35100.0i 1.35147i 0.737143 + 0.675737i $$0.236175\pi$$
−0.737143 + 0.675737i $$0.763825\pi$$
$$878$$ − 2120.00i − 0.0814881i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 18700.0 0.715118 0.357559 0.933891i $$-0.383609\pi$$
0.357559 + 0.933891i $$0.383609\pi$$
$$882$$ 0 0
$$883$$ − 2980.00i − 0.113573i −0.998386 0.0567865i $$-0.981915\pi$$
0.998386 0.0567865i $$-0.0180854\pi$$
$$884$$ 3400.00 0.129360
$$885$$ 0 0
$$886$$ 61800.0 2.34335
$$887$$ − 35880.0i − 1.35821i −0.734041 0.679105i $$-0.762368\pi$$
0.734041 0.679105i $$-0.237632\pi$$
$$888$$ 0 0
$$889$$ −32100.0 −1.21102
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 62390.0i 2.34190i
$$893$$ − 12320.0i − 0.461672i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −63450.0 −2.36575
$$897$$ 0 0
$$898$$ − 6500.00i − 0.241545i
$$899$$ −5400.00 −0.200334
$$900$$ 0 0
$$901$$ 6100.00 0.225550
$$902$$ 100000.i 3.69139i
$$903$$ 0 0
$$904$$ 13950.0 0.513241
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 45240.0i − 1.65620i −0.560584 0.828098i $$-0.689423\pi$$
0.560584 0.828098i $$-0.310577\pi$$
$$908$$ − 63920.0i − 2.33619i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −33200.0 −1.20743 −0.603713 0.797202i $$-0.706313\pi$$
−0.603713 + 0.797202i $$0.706313\pi$$
$$912$$ 0 0
$$913$$ 33000.0i 1.19621i
$$914$$ −35950.0 −1.30101
$$915$$ 0 0
$$916$$ −24378.0 −0.879336
$$917$$ − 58500.0i − 2.10670i
$$918$$ 0 0
$$919$$ −35356.0 −1.26908 −0.634541 0.772889i $$-0.718811\pi$$
−0.634541 + 0.772889i $$0.718811\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 750.000i 0.0267895i
$$923$$ 14000.0i 0.499259i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −13350.0 −0.473767
$$927$$ 0 0
$$928$$ − 4250.00i − 0.150337i
$$929$$ −25700.0 −0.907631 −0.453816 0.891096i $$-0.649938\pi$$
−0.453816 + 0.891096i $$0.649938\pi$$
$$930$$ 0 0
$$931$$ −24508.0 −0.862747
$$932$$ − 58650.0i − 2.06131i
$$933$$ 0 0
$$934$$ −5900.00 −0.206696
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 52890.0i − 1.84401i −0.387173 0.922007i $$-0.626549\pi$$
0.387173 0.922007i $$-0.373451\pi$$
$$938$$ − 27000.0i − 0.939852i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −38050.0 −1.31817 −0.659083 0.752070i $$-0.729055\pi$$
−0.659083 + 0.752070i $$0.729055\pi$$
$$942$$ 0 0
$$943$$ 48000.0i 1.65758i
$$944$$ 4450.00 0.153427
$$945$$ 0 0
$$946$$ 70000.0 2.40581
$$947$$ − 29640.0i − 1.01708i −0.861040 0.508538i $$-0.830186\pi$$
0.861040 0.508538i $$-0.169814\pi$$
$$948$$ 0 0
$$949$$ −8200.00 −0.280488
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 13500.0i − 0.459598i
$$953$$ − 15170.0i − 0.515640i −0.966193 0.257820i $$-0.916996\pi$$
0.966193 0.257820i $$-0.0830041\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 83300.0 2.81811
$$957$$ 0 0
$$958$$ − 70500.0i − 2.37761i
$$959$$ −31500.0 −1.06068
$$960$$ 0 0
$$961$$ −18127.0 −0.608472
$$962$$ 4000.00i 0.134059i
$$963$$ 0 0
$$964$$ −81974.0 −2.73880
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 5470.00i 0.181906i 0.995855 + 0.0909531i $$0.0289913\pi$$
−0.995855 + 0.0909531i $$0.971009\pi$$
$$968$$ − 52605.0i − 1.74668i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −15150.0 −0.500707 −0.250354 0.968154i $$-0.580547\pi$$
−0.250354 + 0.968154i $$0.580547\pi$$
$$972$$ 0 0
$$973$$ − 50280.0i − 1.65663i
$$974$$ −49250.0 −1.62020
$$975$$ 0 0
$$976$$ −46102.0 −1.51198
$$977$$ 31190.0i 1.02135i 0.859775 + 0.510674i $$0.170604\pi$$
−0.859775 + 0.510674i $$0.829396\pi$$
$$978$$ 0 0
$$979$$ 75000.0 2.44843
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12250.0i 0.398079i
$$983$$ 7560.00i 0.245297i 0.992450 + 0.122648i $$0.0391387\pi$$
−0.992450 + 0.122648i $$0.960861\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2500.00 −0.0807467
$$987$$ 0 0
$$988$$ 14960.0i 0.481722i
$$989$$ 33600.0 1.08030
$$990$$ 0 0
$$991$$ 32672.0 1.04729 0.523643 0.851938i $$-0.324573\pi$$
0.523643 + 0.851938i $$0.324573\pi$$
$$992$$ 9180.00i 0.293816i
$$993$$ 0 0
$$994$$ 105000. 3.35050
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 4740.00i 0.150569i 0.997162 + 0.0752845i $$0.0239865\pi$$
−0.997162 + 0.0752845i $$0.976014\pi$$
$$998$$ 85180.0i 2.70173i
$$999$$ 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 225.4.b.a.199.2 2
3.2 odd 2 225.4.b.b.199.1 2
5.2 odd 4 45.4.a.a.1.1 1
5.3 odd 4 225.4.a.h.1.1 1
5.4 even 2 inner 225.4.b.a.199.1 2
15.2 even 4 45.4.a.e.1.1 yes 1
15.8 even 4 225.4.a.a.1.1 1
15.14 odd 2 225.4.b.b.199.2 2
20.7 even 4 720.4.a.bc.1.1 1
35.27 even 4 2205.4.a.a.1.1 1
45.2 even 12 405.4.e.b.271.1 2
45.7 odd 12 405.4.e.n.271.1 2
45.22 odd 12 405.4.e.n.136.1 2
45.32 even 12 405.4.e.b.136.1 2
60.47 odd 4 720.4.a.o.1.1 1
105.62 odd 4 2205.4.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.a.a.1.1 1 5.2 odd 4
45.4.a.e.1.1 yes 1 15.2 even 4
225.4.a.a.1.1 1 15.8 even 4
225.4.a.h.1.1 1 5.3 odd 4
225.4.b.a.199.1 2 5.4 even 2 inner
225.4.b.a.199.2 2 1.1 even 1 trivial
225.4.b.b.199.1 2 3.2 odd 2
225.4.b.b.199.2 2 15.14 odd 2
405.4.e.b.136.1 2 45.32 even 12
405.4.e.b.271.1 2 45.2 even 12
405.4.e.n.136.1 2 45.22 odd 12
405.4.e.n.271.1 2 45.7 odd 12
720.4.a.o.1.1 1 60.47 odd 4
720.4.a.bc.1.1 1 20.7 even 4
2205.4.a.a.1.1 1 35.27 even 4
2205.4.a.t.1.1 1 105.62 odd 4