LMFDB - Embedded newform 225.4.b.b.199.1

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \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	$-0.654920\pi$
$2$	− 5.00000i	− 1.76777i	0.467707	−	0.883883i	$-0.345080\pi$
$3$	0	0
$3$	0	0
$4$	−17.0000	−2.12500
$5$	0	0
$5$	0	0
$6$	0	0
$7$	30.0000i	1.61985i	0.586535	+	0.809924i	$0.300492\pi$
$7$	30.0000i	1.61985i	−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	$-0.259692\pi$
$11$	50.0000	1.37051	0.685253	+	0.728305i	$0.259692\pi$
$12$	0	0
$13$	− 20.0000i	− 0.426692i	−0.976977	−	0.213346i	$-0.931564\pi$
$13$	− 20.0000i	− 0.426692i	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	$0.0227256\pi$
$17$	10.0000i	0.142668i	−0.997452	+	0.0713340i	$0.977274\pi$
$18$	0	0
$19$	44.0000	0.531279	0.265639	−	0.964072i	$-0.414417\pi$
$19$	44.0000	0.531279	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	$0.183072\pi$
$23$	120.000i	1.08790i	−0.839117	+	0.543951i	$0.816928\pi$
$24$	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	$-0.448824\pi$
$29$	50.0000	0.320164	0.160082	+	0.987104i	$0.448824\pi$
$30$	0	0
$31$	108.000	0.625722	0.312861	−	0.949799i	$-0.398713\pi$
$31$	108.000	0.625722	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	$0.0283238\pi$
$37$	40.0000i	0.177729i	−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	$-0.224304\pi$
$41$	400.000	1.52365	0.761823	+	0.647785i	$0.224304\pi$
$42$	0	0
$43$	280.000i	0.993014i	0.868033	+	0.496507i	$0.165384\pi$
$43$	280.000i	0.993014i	−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	$0.143072\pi$
$47$	280.000i	0.868983i	−0.900676	+	0.434491i	$0.856928\pi$
$48$	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	$-0.709836\pi$
$53$	− 610.000i	− 1.58094i	0.612499	−	0.790471i	$-0.290164\pi$
$54$	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	$-0.517568\pi$
$59$	−50.0000	−0.110330	−0.0551648	+	0.998477i	$0.517568\pi$
$60$	0	0
$61$	−518.000	−1.08726	−0.543632	−	0.839324i	$-0.682951\pi$
$61$	−518.000	−1.08726	−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	$0.0524746\pi$
$67$	180.000i	0.328216i	−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	$-0.301081\pi$
$71$	700.000	1.17007	0.585033	+	0.811009i	$0.301081\pi$
$72$	0	0
$73$	− 410.000i	− 0.657354i	−0.944442	−	0.328677i	$-0.893397\pi$
$73$	− 410.000i	− 0.657354i	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	$-0.380236\pi$
$79$	516.000	0.734868	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	$0.143751\pi$
$83$	660.000i	0.872824i	−0.899747	+	0.436412i	$0.856249\pi$
$84$	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	$-0.148415\pi$
$89$	1500.00	1.78651	0.893257	+	0.449547i	$0.148415\pi$
$90$	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	$0.325280\pi$
$97$	1630.00i	1.70620i	−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	$-0.571150\pi$
$101$	−450.000	−0.443333	−0.221667	+	0.975122i	$0.571150\pi$
$102$	0	0
$103$	770.000i	0.736605i	0.929706	+	0.368303i	$0.120061\pi$
$103$	770.000i	0.736605i	−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	$-0.903630\pi$
$107$	− 660.000i	− 0.596305i	0.954518	−	0.298152i	$-0.0963704\pi$
$108$	0	0
$109$	−1754.00	−1.54131	−0.770655	−	0.637253i	$-0.780071\pi$
$109$	−1754.00	−1.54131	−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	$-0.958811\pi$
$113$	− 310.000i	− 0.258074i	0.991640	−	0.129037i	$-0.0411886\pi$
$114$	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	$0.121948\pi$
$127$	1070.00i	0.747615i	−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	$-0.274653\pi$
$131$	1950.00	1.30055	0.650276	+	0.759698i	$0.274653\pi$
$132$	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	$-0.893828\pi$
$137$	− 1050.00i	− 0.654800i	0.944886	−	0.327400i	$-0.106172\pi$
$138$	0	0
$139$	−1676.00	−1.02271	−0.511354	−	0.859370i	$-0.670856\pi$
$139$	−1676.00	−1.02271	−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	$-0.690571\pi$
$149$	−2050.00	−1.12713	−0.563566	+	0.826071i	$0.690571\pi$
$150$	0	0
$151$	448.000	0.241442	0.120721	−	0.992686i	$-0.461479\pi$
$151$	448.000	0.241442	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	$0.00809128\pi$
$157$	100.000i	0.0508336i	−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	$-0.849102\pi$
$163$	− 1900.00i	− 0.913003i	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	$-0.853263\pi$
$167$	− 1920.00i	− 0.889665i	0.895614	−	0.444833i	$-0.146737\pi$
$168$	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	$0.189324\pi$
$173$	2550.00i	1.12065i	−0.828272	+	0.560326i	$0.810676\pi$
$174$	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	$-0.775807\pi$
$179$	−3650.00	−1.52410	−0.762050	+	0.647518i	$0.775807\pi$
$180$	0	0
$181$	−4342.00	−1.78308	−0.891542	−	0.452937i	$-0.850376\pi$
$181$	−4342.00	−1.78308	−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	$-0.730701\pi$
$191$	−3500.00	−1.32592	−0.662961	+	0.748654i	$0.730701\pi$
$192$	0	0
$193$	3350.00i	1.24942i	0.780856	+	0.624711i	$0.214783\pi$
$193$	3350.00i	1.24942i	−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	$-0.994819\pi$
$197$	− 90.0000i	− 0.0325494i	0.999868	−	0.0162747i	$-0.00518063\pi$
$198$	0	0
$199$	−3664.00	−1.30520	−0.652598	−	0.757704i	$-0.726321\pi$
$199$	−3664.00	−1.30520	−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	$-0.513921\pi$
$211$	−268.000	−0.0874402	−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	$-0.814233\pi$
$223$	− 3670.00i	− 1.10207i	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	$-0.814745\pi$
$227$	− 3760.00i	− 1.09938i	0.835368	−	0.549692i	$-0.185255\pi$
$228$	0	0
$229$	1434.00	0.413805	0.206903	−	0.978362i	$-0.433662\pi$
$229$	1434.00	0.413805	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	$-0.838814\pi$
$233$	− 3450.00i	− 0.970030i	0.874506	−	0.485015i	$-0.161186\pi$
$234$	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	$-0.269247\pi$
$239$	4900.00	1.32617	0.663085	+	0.748544i	$0.269247\pi$
$240$	0	0
$241$	4822.00	1.28885	0.644424	−	0.764668i	$-0.277097\pi$
$241$	4822.00	1.28885	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	$-0.698777\pi$
$251$	−4650.00	−1.16934	−0.584672	+	0.811270i	$0.698777\pi$
$252$	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	$-0.786089\pi$
$257$	− 5130.00i	− 1.24514i	0.782565	−	0.622569i	$-0.213911\pi$
$258$	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	$-0.952055\pi$
$263$	− 1280.00i	− 0.300107i	0.988678	−	0.150054i	$-0.0479446\pi$
$264$	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	$-0.623956\pi$
$269$	−3350.00	−0.759305	−0.379653	+	0.925129i	$0.623956\pi$
$270$	0	0
$271$	5512.00	1.23554	0.617768	−	0.786361i	$-0.288037\pi$
$271$	5512.00	1.23554	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	$-0.820839\pi$
$277$	− 4920.00i	− 1.06720i	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	$-0.341483\pi$
$281$	4500.00	0.955329	0.477665	+	0.878542i	$0.341483\pi$
$282$	0	0
$283$	− 6900.00i	− 1.44934i	−0.689098	−	0.724669i	$-0.741993\pi$
$283$	− 6900.00i	− 1.44934i	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	$0.0487426\pi$
$293$	1530.00i	0.305063i	−0.988299	+	0.152532i	$0.951257\pi$
$294$	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	$-0.908811\pi$
$307$	− 3040.00i	− 0.565153i	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	$-0.673935\pi$
$311$	−5700.00	−1.03928	−0.519642	+	0.854384i	$0.673935\pi$
$312$	0	0
$313$	3110.00i	0.561622i	0.959763	+	0.280811i	$0.0906034\pi$
$313$	3110.00i	0.561622i	−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	$0.0268206\pi$
$317$	950.000i	0.168320i	−0.996452	+	0.0841598i	$0.973179\pi$
$318$	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	$-0.439053\pi$
$331$	2292.00	0.380603	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	$0.214798\pi$
$337$	7730.00i	1.24950i	−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	$0.0276114\pi$
$347$	1120.00i	0.173270i	−0.996240	+	0.0866351i	$0.972389\pi$
$348$	0	0
$349$	−1186.00	−0.181906	−0.0909529	−	0.995855i	$-0.528991\pi$
$349$	−1186.00	−0.181906	−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	$0.0882350\pi$
$353$	3630.00i	0.547324i	−0.961826	+	0.273662i	$0.911765\pi$
$354$	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	$-0.457760\pi$
$359$	1800.00	0.264625	0.132312	+	0.991208i	$0.457760\pi$
$360$	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	$-0.793661\pi$
$367$	− 8490.00i	− 1.20756i	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	$0.00220933\pi$
$373$	100.000i	0.0138815i	−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	$-0.315458\pi$
$379$	8084.00	1.09564	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	$-0.782077\pi$
$383$	− 9480.00i	− 1.26477i	0.774656	−	0.632383i	$-0.217923\pi$
$384$	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	$-0.247060\pi$
$389$	10950.0	1.42722	0.713608	+	0.700545i	$0.247060\pi$
$390$	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	$-0.660977\pi$
$397$	− 13840.0i	− 1.74965i	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	$-0.303413\pi$
$401$	9300.00	1.15815	0.579077	+	0.815273i	$0.303413\pi$
$402$	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	$-0.444809\pi$
$409$	2854.00	0.345040	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	$-0.521356\pi$
$419$	−1150.00	−0.134084	−0.0670420	+	0.997750i	$0.521356\pi$
$420$	0	0
$421$	−11162.0	−1.29217	−0.646084	−	0.763266i	$-0.723594\pi$
$421$	−11162.0	−1.29217	−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	$-0.521361\pi$
$431$	−1200.00	−0.134111	−0.0670556	+	0.997749i	$0.521361\pi$
$432$	0	0
$433$	1510.00i	0.167589i	0.996483	+	0.0837944i	$0.0267039\pi$
$433$	1510.00i	0.167589i	−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	$-0.507337\pi$
$439$	−424.000	−0.0460966	−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	$0.230632\pi$
$443$	12360.0i	1.32560i	−0.748796	+	0.662801i	$0.769368\pi$
$444$	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	$-0.478236\pi$
$449$	1300.00	0.136639	0.0683194	+	0.997664i	$0.478236\pi$
$450$	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	$0.119951\pi$
$457$	7190.00i	0.735961i	−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	$-0.502412\pi$
$461$	−150.000	−0.0151544	−0.00757722	+	0.999971i	$0.502412\pi$
$462$	0	0
$463$	2670.00i	0.268003i	0.990981	+	0.134002i	$0.0427827\pi$
$463$	2670.00i	0.268003i	−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	$-0.981380\pi$
$467$	− 1180.00i	− 0.116925i	0.998290	−	0.0584624i	$-0.0186198\pi$
$468$	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	$-0.265225\pi$
$479$	14100.0	1.34498	0.672490	+	0.740106i	$0.265225\pi$
$480$	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	$0.151527\pi$
$487$	9850.00i	0.916522i	−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	$-0.535916\pi$
$491$	−2450.00	−0.225187	−0.112594	+	0.993641i	$0.535916\pi$
$492$	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	$-0.223152\pi$
$499$	17036.0	1.52833	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	$-0.633736\pi$
$503$	− 20600.0i	− 1.82606i	0.407891	−	0.913030i	$-0.366264\pi$
$504$	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	$-0.580548\pi$
$509$	−5750.00	−0.500716	−0.250358	+	0.968153i	$0.580548\pi$
$510$	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	$-0.725942\pi$
$521$	−15500.0	−1.30339	−0.651696	+	0.758480i	$0.725942\pi$
$522$	0	0
$523$	− 13940.0i	− 1.16549i	−0.812653	−	0.582747i	$-0.801978\pi$
$523$	− 13940.0i	− 1.16549i	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	$-0.802547\pi$
$541$	−20478.0	−1.62739	−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	$-0.844052\pi$
$547$	− 12040.0i	− 0.941121i	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	$-0.646652\pi$
$557$	− 23550.0i	− 1.79146i	0.444594	−	0.895732i	$-0.353348\pi$
$558$	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	$-0.926433\pi$
$563$	− 6120.00i	− 0.458130i	0.973411	−	0.229065i	$-0.0735669\pi$
$564$	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	$-0.358157\pi$
$569$	11700.0	0.862020	0.431010	+	0.902347i	$0.358157\pi$
$570$	0	0
$571$	−8188.00	−0.600100	−0.300050	−	0.953923i	$-0.597003\pi$
$571$	−8188.00	−0.600100	−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	$-0.861428\pi$
$577$	− 11690.0i	− 0.843433i	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	$-0.734632\pi$
$587$	− 21060.0i	− 1.48082i	0.672157	−	0.740408i	$-0.265368\pi$
$588$	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	$-0.708385\pi$
$593$	− 22910.0i	− 1.58651i	0.608889	−	0.793255i	$-0.291615\pi$
$594$	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	$-0.484795\pi$
$599$	1400.00	0.0954966	0.0477483	+	0.998859i	$0.484795\pi$
$600$	0	0
$601$	−11002.0	−0.746724	−0.373362	−	0.927686i	$-0.621795\pi$
$601$	−11002.0	−0.746724	−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	$-0.950527\pi$
$607$	− 4630.00i	− 0.309598i	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	$0.290949\pi$
$613$	24040.0i	1.58396i	−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	$0.0196395\pi$
$617$	1890.00i	0.123320i	−0.998097	+	0.0616601i	$0.980361\pi$
$618$	0	0
$619$	−19244.0	−1.24957	−0.624783	−	0.780798i	$-0.714813\pi$
$619$	−19244.0	−1.24957	−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	$-0.332852\pi$
$631$	15892.0	1.00262	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	$-0.626902\pi$
$641$	−12600.0	−0.776396	−0.388198	+	0.921576i	$0.626902\pi$
$642$	0	0
$643$	7260.00i	0.445267i	0.974902	+	0.222633i	$0.0714653\pi$
$643$	7260.00i	0.445267i	−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	$-0.927819\pi$
$647$	− 7400.00i	− 0.449651i	0.974399	−	0.224825i	$-0.0721812\pi$
$648$	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	$-0.954155\pi$
$653$	− 4790.00i	− 0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$654$	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	$-0.486354\pi$
$659$	1450.00	0.0857117	0.0428558	+	0.999081i	$0.486354\pi$
$660$	0	0
$661$	11818.0	0.695411	0.347706	−	0.937604i	$-0.386961\pi$
$661$	11818.0	0.695411	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	$0.0508085\pi$
$673$	5550.00i	0.317885i	−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	$-0.880379\pi$
$677$	− 12930.0i	− 0.734033i	0.930214	−	0.367016i	$-0.119621\pi$
$678$	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	$0.365944\pi$
$683$	32580.0i	1.82524i	−0.408809	+	0.912620i	$0.634056\pi$
$684$	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	$-0.409154\pi$
$691$	10228.0	0.563085	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	$-0.427779\pi$
$701$	8350.00	0.449893	0.224947	+	0.974371i	$0.427779\pi$
$702$	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	$-0.370378\pi$
$709$	14954.0	0.792115	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	$-0.776015\pi$
$719$	−29400.0	−1.52494	−0.762472	+	0.647021i	$0.776015\pi$
$720$	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	$0.136757\pi$
$727$	16330.0i	0.833076i	−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	$0.282757\pi$
$733$	30800.0i	1.55201i	−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	$-0.423823\pi$
$739$	9524.00	0.474081	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	$-0.750463\pi$
$743$	− 28600.0i	− 1.41216i	0.708134	−	0.706078i	$-0.249537\pi$
$744$	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	$-0.564250\pi$
$751$	−8252.00	−0.400958	−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	$0.204132\pi$
$757$	24920.0i	1.19648i	−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	$-0.268643\pi$
$761$	27900.0	1.32901	0.664503	+	0.747285i	$0.268643\pi$
$762$	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	$-0.413050\pi$
$769$	11506.0	0.539554	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	$-0.905998\pi$
$773$	− 12510.0i	− 0.582087i	0.956710	−	0.291044i	$-0.0940025\pi$
$774$	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	$0.00793041\pi$
$787$	1100.00i	0.0498231i	−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	$0.0318128\pi$
$797$	4490.00i	0.199553i	−0.995010	+	0.0997766i	$0.968187\pi$
$798$	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	$-0.713460\pi$
$809$	−28600.0	−1.24292	−0.621460	+	0.783446i	$0.713460\pi$
$810$	0	0
$811$	10068.0	0.435925	0.217963	−	0.975957i	$-0.430059\pi$
$811$	10068.0	0.435925	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	$-0.597948\pi$
$821$	−14250.0	−0.605759	−0.302880	+	0.953029i	$0.597948\pi$
$822$	0	0
$823$	6830.00i	0.289282i	0.989484	+	0.144641i	$0.0462027\pi$
$823$	6830.00i	0.289282i	−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	$0.0600490\pi$
$827$	8920.00i	0.375065i	−0.982258	+	0.187533i	$0.939951\pi$
$828$	0	0
$829$	3534.00	0.148059	0.0740295	−	0.997256i	$-0.476414\pi$
$829$	3534.00	0.148059	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	$-0.447368\pi$
$839$	8000.00	0.329190	0.164595	+	0.986361i	$0.447368\pi$
$840$	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	$0.0330237\pi$
$853$	5160.00i	0.207122i	−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	$-0.951152\pi$
$857$	− 7670.00i	− 0.305720i	0.988248	−	0.152860i	$-0.0488484\pi$
$858$	0	0
$859$	−25804.0	−1.02494	−0.512469	−	0.858706i	$-0.671269\pi$
$859$	−25804.0	−1.02494	−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	$0.00251113\pi$
$863$	400.000i	0.0157777i	−0.999969	+	0.00788885i	$0.997489\pi$
$864$	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	$0.236175\pi$
$877$	35100.0i	1.35147i	−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	$-0.616391\pi$
$881$	−18700.0	−0.715118	−0.357559	+	0.933891i	$0.616391\pi$
$882$	0	0
$883$	− 2980.00i	− 0.113573i	−0.998386	−	0.0567865i	$-0.981915\pi$
$883$	− 2980.00i	− 0.113573i	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	$0.237632\pi$
$887$	35880.0i	1.35821i	−0.734041	+	0.679105i	$0.762368\pi$
$888$	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	$-0.689423\pi$
$907$	− 45240.0i	− 1.65620i	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	$-0.293687\pi$
$911$	33200.0	1.20743	0.603713	+	0.797202i	$0.293687\pi$
$912$	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	$-0.718811\pi$
$919$	−35356.0	−1.26908	−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	$-0.350062\pi$
$929$	25700.0	0.907631	0.453816	+	0.891096i	$0.350062\pi$
$930$	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	$-0.626549\pi$
$937$	− 52890.0i	− 1.84401i	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	$-0.270945\pi$
$941$	38050.0	1.31817	0.659083	+	0.752070i	$0.270945\pi$
$942$	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	$0.169814\pi$
$947$	29640.0i	1.01708i	−0.861040	+	0.508538i	$0.830186\pi$
$948$	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	$0.0830041\pi$
$953$	15170.0i	0.515640i	−0.966193	+	0.257820i	$0.916996\pi$
$954$	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	$0.0289913\pi$
$967$	5470.00i	0.181906i	−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	$-0.419453\pi$
$971$	15150.0	0.500707	0.250354	+	0.968154i	$0.419453\pi$
$972$	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	$-0.829396\pi$
$977$	− 31190.0i	− 1.02135i	0.859775	−	0.510674i	$-0.170604\pi$
$978$	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	$-0.960861\pi$
$983$	− 7560.00i	− 0.245297i	0.992450	−	0.122648i	$-0.0391387\pi$
$984$	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	$-0.324573\pi$
$991$	32672.0	1.04729	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	$0.0239865\pi$
$997$	4740.00i	0.150569i	−0.997162	+	0.0752845i	$0.976014\pi$
$998$	− 85180.0i	− 2.70173i
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more