LMFDB - Embedded newform 45.5.g.a.28.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,5,Mod(28,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("45.28");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	45.g (of order $4$, degree $2$, 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:	$4.65164833877$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			28.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	45.28
Dual form			45.5.g.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.00000 + 5.00000i) q^{2} -34.0000i q^{4} +25.0000 q^{5} +(40.0000 - 40.0000i) q^{7} +(90.0000 + 90.0000i) q^{8} +O(q^{10})$	\(q+(-5.00000 + 5.00000i) q^{2} -34.0000i q^{4} +25.0000 q^{5} +(40.0000 - 40.0000i) q^{7} +(90.0000 + 90.0000i) q^{8} +(-125.000 + 125.000i) q^{10} -100.000 q^{11} +(205.000 + 205.000i) q^{13} +400.000i q^{14} -356.000 q^{16} +(235.000 - 235.000i) q^{17} -72.0000i q^{19} -850.000i q^{20} +(500.000 - 500.000i) q^{22} +(340.000 + 340.000i) q^{23} +625.000 q^{25} -2050.00 q^{26} +(-1360.00 - 1360.00i) q^{28} -450.000i q^{29} +428.000 q^{31} +(340.000 - 340.000i) q^{32} +2350.00i q^{34} +(1000.00 - 1000.00i) q^{35} +(-755.000 + 755.000i) q^{37} +(360.000 + 360.000i) q^{38} +(2250.00 + 2250.00i) q^{40} +950.000 q^{41} +(-1220.00 - 1220.00i) q^{43} +3400.00i q^{44} -3400.00 q^{46} +(-320.000 + 320.000i) q^{47} -799.000i q^{49} +(-3125.00 + 3125.00i) q^{50} +(6970.00 - 6970.00i) q^{52} +(505.000 + 505.000i) q^{53} -2500.00 q^{55} +7200.00 q^{56} +(2250.00 + 2250.00i) q^{58} -6300.00i q^{59} -3808.00 q^{61} +(-2140.00 + 2140.00i) q^{62} -2296.00i q^{64} +(5125.00 + 5125.00i) q^{65} +(340.000 - 340.000i) q^{67} +(-7990.00 - 7990.00i) q^{68} +10000.0i q^{70} -3400.00 q^{71} +(415.000 + 415.000i) q^{73} -7550.00i q^{74} -2448.00 q^{76} +(-4000.00 + 4000.00i) q^{77} +6732.00i q^{79} -8900.00 q^{80} +(-4750.00 + 4750.00i) q^{82} +(-680.000 - 680.000i) q^{83} +(5875.00 - 5875.00i) q^{85} +12200.0 q^{86} +(-9000.00 - 9000.00i) q^{88} -2250.00i q^{89} +16400.0 q^{91} +(11560.0 - 11560.0i) q^{92} -3200.00i q^{94} -1800.00i q^{95} +(1615.00 - 1615.00i) q^{97} +(3995.00 + 3995.00i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{2} + 50 q^{5} + 80 q^{7} + 180 q^{8}+O(q^{10}) $ 2 * q - 10 * q^2 + 50 * q^5 + 80 * q^7 + 180 * q^8	$ 2 q - 10 q^{2} + 50 q^{5} + 80 q^{7} + 180 q^{8} - 250 q^{10} - 200 q^{11} + 410 q^{13} - 712 q^{16} + 470 q^{17} + 1000 q^{22} + 680 q^{23} + 1250 q^{25} - 4100 q^{26} - 2720 q^{28} + 856 q^{31} + 680 q^{32} + 2000 q^{35} - 1510 q^{37} + 720 q^{38} + 4500 q^{40} + 1900 q^{41} - 2440 q^{43} - 6800 q^{46} - 640 q^{47} - 6250 q^{50} + 13940 q^{52} + 1010 q^{53} - 5000 q^{55} + 14400 q^{56} + 4500 q^{58} - 7616 q^{61} - 4280 q^{62} + 10250 q^{65} + 680 q^{67} - 15980 q^{68} - 6800 q^{71} + 830 q^{73} - 4896 q^{76} - 8000 q^{77} - 17800 q^{80} - 9500 q^{82} - 1360 q^{83} + 11750 q^{85} + 24400 q^{86} - 18000 q^{88} + 32800 q^{91} + 23120 q^{92} + 3230 q^{97} + 7990 q^{98}+O(q^{100}) $ 2 * q - 10 * q^2 + 50 * q^5 + 80 * q^7 + 180 * q^8 - 250 * q^10 - 200 * q^11 + 410 * q^13 - 712 * q^16 + 470 * q^17 + 1000 * q^22 + 680 * q^23 + 1250 * q^25 - 4100 * q^26 - 2720 * q^28 + 856 * q^31 + 680 * q^32 + 2000 * q^35 - 1510 * q^37 + 720 * q^38 + 4500 * q^40 + 1900 * q^41 - 2440 * q^43 - 6800 * q^46 - 640 * q^47 - 6250 * q^50 + 13940 * q^52 + 1010 * q^53 - 5000 * q^55 + 14400 * q^56 + 4500 * q^58 - 7616 * q^61 - 4280 * q^62 + 10250 * q^65 + 680 * q^67 - 15980 * q^68 - 6800 * q^71 + 830 * q^73 - 4896 * q^76 - 8000 * q^77 - 17800 * q^80 - 9500 * q^82 - 1360 * q^83 + 11750 * q^85 + 24400 * q^86 - 18000 * q^88 + 32800 * q^91 + 23120 * q^92 + 3230 * q^97 + 7990 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−5.00000	+	5.00000i	−1.25000	+	1.25000i	−0.294281	+	0.955719i	$0.595080\pi$
$2$	−5.00000	+	5.00000i	−1.25000	+	1.25000i	−0.955719	+	0.294281i	$0.904920\pi$
$3$		0			0
$3$		0			0
$4$		−	34.0000i		−	2.12500i
$5$	25.0000			1.00000
$5$	25.0000			1.00000
$6$		0			0
$7$	40.0000	−	40.0000i	0.816327	−	0.816327i	−0.169247	−	0.985574i	$-0.554134\pi$
$7$	40.0000	−	40.0000i	0.816327	−	0.816327i	0.985574	+	0.169247i	$0.0541336\pi$
$8$	90.0000	+	90.0000i	1.40625	+	1.40625i
$9$		0			0
$10$	−125.000	+	125.000i	−1.25000	+	1.25000i
$11$	−100.000			−0.826446			−0.413223	−	0.910630i	$-0.635597\pi$
$11$	−100.000			−0.826446			−0.413223	+	0.910630i	$0.635597\pi$
$12$		0			0
$13$	205.000	+	205.000i	1.21302	+	1.21302i	0.970029	+	0.242989i	$0.0781278\pi$
$13$	205.000	+	205.000i	1.21302	+	1.21302i	0.242989	+	0.970029i	$0.421872\pi$
$14$			400.000i			2.04082i
$15$		0			0
$16$	−356.000			−1.39062
$17$	235.000	−	235.000i	0.813149	−	0.813149i	−0.171956	−	0.985105i	$-0.555009\pi$
$17$	235.000	−	235.000i	0.813149	−	0.813149i	0.985105	+	0.171956i	$0.0550086\pi$
$18$		0			0
$19$		−	72.0000i		−	0.199446i	−0.995015	−	0.0997230i	$-0.968204\pi$
$19$		−	72.0000i		−	0.199446i	0.995015	−	0.0997230i	$-0.0317957\pi$
$20$		−	850.000i		−	2.12500i
$21$		0			0
$22$	500.000	−	500.000i	1.03306	−	1.03306i
$23$	340.000	+	340.000i	0.642722	+	0.642722i	0.951224	−	0.308502i	$-0.0998275\pi$
$23$	340.000	+	340.000i	0.642722	+	0.642722i	−0.308502	+	0.951224i	$0.599828\pi$
$24$		0			0
$25$	625.000			1.00000
$26$	−2050.00			−3.03254
$27$		0			0
$28$	−1360.00	−	1360.00i	−1.73469	−	1.73469i
$29$		−	450.000i		−	0.535077i	−0.963547	−	0.267539i	$-0.913790\pi$
$29$		−	450.000i		−	0.535077i	0.963547	−	0.267539i	$-0.0862103\pi$
$30$		0			0
$31$	428.000			0.445369			0.222685	−	0.974891i	$-0.428518\pi$
$31$	428.000			0.445369			0.222685	+	0.974891i	$0.428518\pi$
$32$	340.000	−	340.000i	0.332031	−	0.332031i
$33$		0			0
$34$			2350.00i			2.03287i
$35$	1000.00	−	1000.00i	0.816327	−	0.816327i
$36$		0			0
$37$	−755.000	+	755.000i	−0.551497	+	0.551497i	−0.926873	−	0.375375i	$-0.877514\pi$
$37$	−755.000	+	755.000i	−0.551497	+	0.551497i	0.375375	+	0.926873i	$0.377514\pi$
$38$	360.000	+	360.000i	0.249307	+	0.249307i
$39$		0			0
$40$	2250.00	+	2250.00i	1.40625	+	1.40625i
$41$	950.000			0.565140			0.282570	−	0.959247i	$-0.408813\pi$
$41$	950.000			0.565140			0.282570	+	0.959247i	$0.408813\pi$
$42$		0			0
$43$	−1220.00	−	1220.00i	−0.659816	−	0.659816i	0.295520	−	0.955336i	$-0.404507\pi$
$43$	−1220.00	−	1220.00i	−0.659816	−	0.659816i	−0.955336	+	0.295520i	$0.904507\pi$
$44$			3400.00i			1.75620i
$45$		0			0
$46$	−3400.00			−1.60681
$47$	−320.000	+	320.000i	−0.144862	+	0.144862i	−0.775818	−	0.630956i	$-0.782663\pi$
$47$	−320.000	+	320.000i	−0.144862	+	0.144862i	0.630956	+	0.775818i	$0.282663\pi$
$48$		0			0
$49$		−	799.000i		−	0.332778i
$50$	−3125.00	+	3125.00i	−1.25000	+	1.25000i
$51$		0			0
$52$	6970.00	−	6970.00i	2.57766	−	2.57766i
$53$	505.000	+	505.000i	0.179779	+	0.179779i	0.791260	−	0.611480i	$-0.209426\pi$
$53$	505.000	+	505.000i	0.179779	+	0.179779i	−0.611480	+	0.791260i	$0.709426\pi$
$54$		0			0
$55$	−2500.00			−0.826446
$56$	7200.00			2.29592
$57$		0			0
$58$	2250.00	+	2250.00i	0.668847	+	0.668847i
$59$		−	6300.00i		−	1.80982i	−0.425598	−	0.904912i	$-0.639936\pi$
$59$		−	6300.00i		−	1.80982i	0.425598	−	0.904912i	$-0.360064\pi$
$60$		0			0
$61$	−3808.00			−1.02338			−0.511690	−	0.859170i	$-0.670981\pi$
$61$	−3808.00			−1.02338			−0.511690	+	0.859170i	$0.670981\pi$
$62$	−2140.00	+	2140.00i	−0.556712	+	0.556712i
$63$		0			0
$64$		−	2296.00i		−	0.560547i
$65$	5125.00	+	5125.00i	1.21302	+	1.21302i
$66$		0			0
$67$	340.000	−	340.000i	0.0757407	−	0.0757407i	−0.668222	−	0.743962i	$-0.732944\pi$
$67$	340.000	−	340.000i	0.0757407	−	0.0757407i	0.743962	+	0.668222i	$0.232944\pi$
$68$	−7990.00	−	7990.00i	−1.72794	−	1.72794i
$69$		0			0
$70$			10000.0i			2.04082i
$71$	−3400.00			−0.674469			−0.337235	−	0.941421i	$-0.609492\pi$
$71$	−3400.00			−0.674469			−0.337235	+	0.941421i	$0.609492\pi$
$72$		0			0
$73$	415.000	+	415.000i	0.0778758	+	0.0778758i	0.744972	−	0.667096i	$-0.232463\pi$
$73$	415.000	+	415.000i	0.0778758	+	0.0778758i	−0.667096	+	0.744972i	$0.732463\pi$
$74$		−	7550.00i		−	1.37874i
$75$		0			0
$76$	−2448.00			−0.423823
$77$	−4000.00	+	4000.00i	−0.674650	+	0.674650i
$78$		0			0
$79$			6732.00i			1.07867i	0.842090	+	0.539337i	$0.181325\pi$
$79$			6732.00i			1.07867i	−0.842090	+	0.539337i	$0.818675\pi$
$80$	−8900.00			−1.39062
$81$		0			0
$82$	−4750.00	+	4750.00i	−0.706425	+	0.706425i
$83$	−680.000	−	680.000i	−0.0987081	−	0.0987081i	0.656028	−	0.754736i	$-0.272235\pi$
$83$	−680.000	−	680.000i	−0.0987081	−	0.0987081i	−0.754736	+	0.656028i	$0.772235\pi$
$84$		0			0
$85$	5875.00	−	5875.00i	0.813149	−	0.813149i
$86$	12200.0			1.64954
$87$		0			0
$88$	−9000.00	−	9000.00i	−1.16219	−	1.16219i
$89$		−	2250.00i		−	0.284055i	−0.989863	−	0.142028i	$-0.954638\pi$
$89$		−	2250.00i		−	0.284055i	0.989863	−	0.142028i	$-0.0453622\pi$
$90$		0			0
$91$	16400.0			1.98044
$92$	11560.0	−	11560.0i	1.36578	−	1.36578i
$93$		0			0
$94$		−	3200.00i		−	0.362155i
$95$		−	1800.00i		−	0.199446i
$96$		0			0
$97$	1615.00	−	1615.00i	0.171644	−	0.171644i	−0.616057	−	0.787701i	$-0.711271\pi$
$97$	1615.00	−	1615.00i	0.171644	−	0.171644i	0.787701	+	0.616057i	$0.211271\pi$
$98$	3995.00	+	3995.00i	0.415973	+	0.415973i
$99$		0			0
$100$		−	21250.0i		−	2.12500i
$101$	−13600.0			−1.33320			−0.666601	−	0.745414i	$-0.732252\pi$
$101$	−13600.0			−1.33320			−0.666601	+	0.745414i	$0.732252\pi$
$102$		0			0
$103$	−5780.00	−	5780.00i	−0.544820	−	0.544820i	0.380118	−	0.924938i	$-0.375883\pi$
$103$	−5780.00	−	5780.00i	−0.544820	−	0.544820i	−0.924938	+	0.380118i	$0.875883\pi$
$104$			36900.0i			3.41161i
$105$		0			0
$106$	−5050.00			−0.449448
$107$	−12860.0	+	12860.0i	−1.12324	+	1.12324i	−0.131991	+	0.991251i	$0.542137\pi$
$107$	−12860.0	+	12860.0i	−1.12324	+	1.12324i	−0.991251	+	0.131991i	$0.957863\pi$
$108$		0			0
$109$			8262.00i			0.695396i	0.937607	+	0.347698i	$0.113037\pi$
$109$			8262.00i			0.695396i	−0.937607	+	0.347698i	$0.886963\pi$
$110$	12500.0	−	12500.0i	1.03306	−	1.03306i
$111$		0			0
$112$	−14240.0	+	14240.0i	−1.13520	+	1.13520i
$113$	16405.0	+	16405.0i	1.28475	+	1.28475i	0.937931	+	0.346821i	$0.112739\pi$
$113$	16405.0	+	16405.0i	1.28475	+	1.28475i	0.346821	+	0.937931i	$0.387261\pi$
$114$		0			0
$115$	8500.00	+	8500.00i	0.642722	+	0.642722i
$116$	−15300.0			−1.13704
$117$		0			0
$118$	31500.0	+	31500.0i	2.26228	+	2.26228i
$119$		−	18800.0i		−	1.32759i
$120$		0			0
$121$	−4641.00			−0.316987
$122$	19040.0	−	19040.0i	1.27923	−	1.27923i
$123$		0			0
$124$		−	14552.0i		−	0.946410i
$125$	15625.0			1.00000
$126$		0			0
$127$	−13940.0	+	13940.0i	−0.864282	+	0.864282i	−0.991832	−	0.127550i	$-0.959289\pi$
$127$	−13940.0	+	13940.0i	−0.864282	+	0.864282i	0.127550	+	0.991832i	$0.459289\pi$
$128$	16920.0	+	16920.0i	1.03271	+	1.03271i
$129$		0			0
$130$	−51250.0			−3.03254
$131$	−22900.0			−1.33442			−0.667211	−	0.744869i	$-0.732512\pi$
$131$	−22900.0			−1.33442			−0.667211	+	0.744869i	$0.732512\pi$
$132$		0			0
$133$	−2880.00	−	2880.00i	−0.162813	−	0.162813i
$134$			3400.00i			0.189352i
$135$		0			0
$136$	42300.0			2.28698
$137$	−7925.00	+	7925.00i	−0.422239	+	0.422239i	−0.885974	−	0.463735i	$-0.846509\pi$
$137$	−7925.00	+	7925.00i	−0.422239	+	0.422239i	0.463735	+	0.885974i	$0.346509\pi$
$138$		0			0
$139$		−	27792.0i		−	1.43843i	−0.694785	−	0.719217i	$-0.744501\pi$
$139$		−	27792.0i		−	1.43843i	0.694785	−	0.719217i	$-0.255499\pi$
$140$	−34000.0	−	34000.0i	−1.73469	−	1.73469i
$141$		0			0
$142$	17000.0	−	17000.0i	0.843087	−	0.843087i
$143$	−20500.0	−	20500.0i	−1.00249	−	1.00249i
$144$		0			0
$145$		−	11250.0i		−	0.535077i
$146$	−4150.00			−0.194689
$147$		0			0
$148$	25670.0	+	25670.0i	1.17193	+	1.17193i
$149$			25200.0i			1.13508i	0.823344	+	0.567542i	$0.192106\pi$
$149$			25200.0i			1.13508i	−0.823344	+	0.567542i	$0.807894\pi$
$150$		0			0
$151$	−22852.0			−1.00224			−0.501118	−	0.865379i	$-0.667078\pi$
$151$	−22852.0			−1.00224			−0.501118	+	0.865379i	$0.667078\pi$
$152$	6480.00	−	6480.00i	0.280471	−	0.280471i
$153$		0			0
$154$		−	40000.0i		−	1.68663i
$155$	10700.0			0.445369
$156$		0			0
$157$	−1325.00	+	1325.00i	−0.0537547	+	0.0537547i	−0.733473	−	0.679718i	$-0.762102\pi$
$157$	−1325.00	+	1325.00i	−0.0537547	+	0.0537547i	0.679718	+	0.733473i	$0.262102\pi$
$158$	−33660.0	−	33660.0i	−1.34834	−	1.34834i
$159$		0			0
$160$	8500.00	−	8500.00i	0.332031	−	0.332031i
$161$	27200.0			1.04934
$162$		0			0
$163$	−22400.0	−	22400.0i	−0.843088	−	0.843088i	0.146171	−	0.989259i	$-0.453305\pi$
$163$	−22400.0	−	22400.0i	−0.843088	−	0.843088i	−0.989259	+	0.146171i	$0.953305\pi$
$164$		−	32300.0i		−	1.20092i
$165$		0			0
$166$	6800.00			0.246770
$167$	27880.0	−	27880.0i	0.999677	−	0.999677i	−0.000322656	−	1.00000i	$-0.500103\pi$
$167$	27880.0	−	27880.0i	0.999677	−	0.999677i	1.00000		0.000322656i	$0.000102705\pi$
$168$		0			0
$169$			55489.0i			1.94282i
$170$			58750.0i			2.03287i
$171$		0			0
$172$	−41480.0	+	41480.0i	−1.40211	+	1.40211i
$173$	19975.0	+	19975.0i	0.667413	+	0.667413i	0.957116	−	0.289704i	$-0.0935567\pi$
$173$	19975.0	+	19975.0i	0.667413	+	0.667413i	−0.289704	+	0.957116i	$0.593557\pi$
$174$		0			0
$175$	25000.0	−	25000.0i	0.816327	−	0.816327i
$176$	35600.0			1.14928
$177$		0			0
$178$	11250.0	+	11250.0i	0.355069	+	0.355069i
$179$		−	45900.0i		−	1.43254i	−0.697823	−	0.716270i	$-0.745848\pi$
$179$		−	45900.0i		−	1.43254i	0.697823	−	0.716270i	$-0.254152\pi$
$180$		0			0
$181$	15878.0			0.484662			0.242331	−	0.970194i	$-0.422088\pi$
$181$	15878.0			0.484662			0.242331	+	0.970194i	$0.422088\pi$
$182$	−82000.0	+	82000.0i	−2.47555	+	2.47555i
$183$		0			0
$184$			61200.0i			1.80766i
$185$	−18875.0	+	18875.0i	−0.551497	+	0.551497i
$186$		0			0
$187$	−23500.0	+	23500.0i	−0.672024	+	0.672024i
$188$	10880.0	+	10880.0i	0.307832	+	0.307832i
$189$		0			0
$190$	9000.00	+	9000.00i	0.249307	+	0.249307i
$191$	17000.0			0.465996			0.232998	−	0.972477i	$-0.425146\pi$
$191$	17000.0			0.465996			0.232998	+	0.972477i	$0.425146\pi$
$192$		0			0
$193$	−31025.0	−	31025.0i	−0.832908	−	0.832908i	0.155005	−	0.987914i	$-0.450461\pi$
$193$	−31025.0	−	31025.0i	−0.832908	−	0.832908i	−0.987914	+	0.155005i	$0.950461\pi$
$194$			16150.0i			0.429110i
$195$		0			0
$196$	−27166.0			−0.707153
$197$	−665.000	+	665.000i	−0.0171352	+	0.0171352i	−0.715622	−	0.698487i	$-0.753857\pi$
$197$	−665.000	+	665.000i	−0.0171352	+	0.0171352i	0.698487	+	0.715622i	$0.253857\pi$
$198$		0			0
$199$			30852.0i			0.779071i	0.921011	+	0.389536i	$0.127364\pi$
$199$			30852.0i			0.779071i	−0.921011	+	0.389536i	$0.872636\pi$
$200$	56250.0	+	56250.0i	1.40625	+	1.40625i
$201$		0			0
$202$	68000.0	−	68000.0i	1.66650	−	1.66650i
$203$	−18000.0	−	18000.0i	−0.436798	−	0.436798i
$204$		0			0
$205$	23750.0			0.565140
$206$	57800.0			1.36205
$207$		0			0
$208$	−72980.0	−	72980.0i	−1.68685	−	1.68685i
$209$			7200.00i			0.164831i
$210$		0			0
$211$	77792.0			1.74731			0.873655	−	0.486546i	$-0.161743\pi$
$211$	77792.0			1.74731			0.873655	+	0.486546i	$0.161743\pi$
$212$	17170.0	−	17170.0i	0.382031	−	0.382031i
$213$		0			0
$214$		−	128600.i		−	2.80811i
$215$	−30500.0	−	30500.0i	−0.659816	−	0.659816i
$216$		0			0
$217$	17120.0	−	17120.0i	0.363567	−	0.363567i
$218$	−41310.0	−	41310.0i	−0.869245	−	0.869245i
$219$		0			0
$220$			85000.0i			1.75620i
$221$	96350.0			1.97273
$222$		0			0
$223$	32980.0	+	32980.0i	0.663195	+	0.663195i	0.956132	−	0.292937i	$-0.0946327\pi$
$223$	32980.0	+	32980.0i	0.663195	+	0.663195i	−0.292937	+	0.956132i	$0.594633\pi$
$224$		−	27200.0i		−	0.542092i
$225$		0			0
$226$	−164050.			−3.21188
$227$	20740.0	−	20740.0i	0.402492	−	0.402492i	−0.476618	−	0.879110i	$-0.658138\pi$
$227$	20740.0	−	20740.0i	0.402492	−	0.402492i	0.879110	+	0.476618i	$0.158138\pi$
$228$		0			0
$229$		−	25632.0i		−	0.488778i	−0.969677	−	0.244389i	$-0.921413\pi$
$229$		−	25632.0i		−	0.488778i	0.969677	−	0.244389i	$-0.0785874\pi$
$230$	−85000.0			−1.60681
$231$		0			0
$232$	40500.0	−	40500.0i	0.752452	−	0.752452i
$233$	−5525.00	−	5525.00i	−0.101770	−	0.101770i	0.654388	−	0.756159i	$-0.272926\pi$
$233$	−5525.00	−	5525.00i	−0.101770	−	0.101770i	−0.756159	+	0.654388i	$0.772926\pi$
$234$		0			0
$235$	−8000.00	+	8000.00i	−0.144862	+	0.144862i
$236$	−214200.			−3.84588
$237$		0			0
$238$	94000.0	+	94000.0i	1.65949	+	1.65949i
$239$			86400.0i			1.51258i	0.654237	+	0.756289i	$0.272990\pi$
$239$			86400.0i			1.51258i	−0.654237	+	0.756289i	$0.727010\pi$
$240$		0			0
$241$	32912.0			0.566657			0.283328	−	0.959023i	$-0.408561\pi$
$241$	32912.0			0.566657			0.283328	+	0.959023i	$0.408561\pi$
$242$	23205.0	−	23205.0i	0.396233	−	0.396233i
$243$		0			0
$244$			129472.i			2.17468i
$245$		−	19975.0i		−	0.332778i
$246$		0			0
$247$	14760.0	−	14760.0i	0.241932	−	0.241932i
$248$	38520.0	+	38520.0i	0.626301	+	0.626301i
$249$		0			0
$250$	−78125.0	+	78125.0i	−1.25000	+	1.25000i
$251$	−54700.0			−0.868240			−0.434120	−	0.900855i	$-0.642941\pi$
$251$	−54700.0			−0.868240			−0.434120	+	0.900855i	$0.642941\pi$
$252$		0			0
$253$	−34000.0	−	34000.0i	−0.531175	−	0.531175i
$254$		−	139400.i		−	2.16070i
$255$		0			0
$256$	−132464.			−2.02124
$257$	28645.0	−	28645.0i	0.433693	−	0.433693i	−0.456189	−	0.889883i	$-0.650786\pi$
$257$	28645.0	−	28645.0i	0.433693	−	0.433693i	0.889883	+	0.456189i	$0.150786\pi$
$258$		0			0
$259$			60400.0i			0.900404i
$260$	174250.	−	174250.i	2.57766	−	2.57766i
$261$		0			0
$262$	114500.	−	114500.i	1.66803	−	1.66803i
$263$	−5360.00	−	5360.00i	−0.0774914	−	0.0774914i	0.667299	−	0.744790i	$-0.267450\pi$
$263$	−5360.00	−	5360.00i	−0.0774914	−	0.0774914i	−0.744790	+	0.667299i	$0.767450\pi$
$264$		0			0
$265$	12625.0	+	12625.0i	0.179779	+	0.179779i
$266$	28800.0			0.407033
$267$		0			0
$268$	−11560.0	−	11560.0i	−0.160949	−	0.160949i
$269$			68400.0i			0.945261i	0.881261	+	0.472630i	$0.156695\pi$
$269$			68400.0i			0.945261i	−0.881261	+	0.472630i	$0.843305\pi$
$270$		0			0
$271$	−57868.0			−0.787952			−0.393976	−	0.919121i	$-0.628901\pi$
$271$	−57868.0			−0.787952			−0.393976	+	0.919121i	$0.628901\pi$
$272$	−83660.0	+	83660.0i	−1.13079	+	1.13079i
$273$		0			0
$274$		−	79250.0i		−	1.05560i
$275$	−62500.0			−0.826446
$276$		0			0
$277$	−67235.0	+	67235.0i	−0.876266	+	0.876266i	−0.993146	−	0.116880i	$-0.962711\pi$
$277$	−67235.0	+	67235.0i	−0.876266	+	0.876266i	0.116880	+	0.993146i	$0.462711\pi$
$278$	138960.	+	138960.i	1.79804	+	1.79804i
$279$		0			0
$280$	180000.			2.29592
$281$	−97750.0			−1.23795			−0.618976	−	0.785410i	$-0.712452\pi$
$281$	−97750.0			−1.23795			−0.618976	+	0.785410i	$0.712452\pi$
$282$		0			0
$283$	−47900.0	−	47900.0i	−0.598085	−	0.598085i	0.341718	−	0.939803i	$-0.388991\pi$
$283$	−47900.0	−	47900.0i	−0.598085	−	0.598085i	−0.939803	+	0.341718i	$0.888991\pi$
$284$			115600.i			1.43325i
$285$		0			0
$286$	205000.			2.50624
$287$	38000.0	−	38000.0i	0.461339	−	0.461339i
$288$		0			0
$289$		−	26929.0i		−	0.322422i
$290$	56250.0	+	56250.0i	0.668847	+	0.668847i
$291$		0			0
$292$	14110.0	−	14110.0i	0.165486	−	0.165486i
$293$	−48215.0	−	48215.0i	−0.561626	−	0.561626i	0.368143	−	0.929769i	$-0.379994\pi$
$293$	−48215.0	−	48215.0i	−0.561626	−	0.561626i	−0.929769	+	0.368143i	$0.879994\pi$
$294$		0			0
$295$		−	157500.i		−	1.80982i
$296$	−135900.			−1.55109
$297$		0			0
$298$	−126000.	−	126000.i	−1.41886	−	1.41886i
$299$			139400.i			1.55927i
$300$		0			0
$301$	−97600.0			−1.07725
$302$	114260.	−	114260.i	1.25280	−	1.25280i
$303$		0			0
$304$			25632.0i			0.277355i
$305$	−95200.0			−1.02338
$306$		0			0
$307$	−53720.0	+	53720.0i	−0.569980	+	0.569980i	−0.932123	−	0.362143i	$-0.882045\pi$
$307$	−53720.0	+	53720.0i	−0.569980	+	0.569980i	0.362143	+	0.932123i	$0.382045\pi$
$308$	136000.	+	136000.i	1.43363	+	1.43363i
$309$		0			0
$310$	−53500.0	+	53500.0i	−0.556712	+	0.556712i
$311$	136400.			1.41024			0.705121	−	0.709087i	$-0.250893\pi$
$311$	136400.			1.41024			0.705121	+	0.709087i	$0.250893\pi$
$312$		0			0
$313$	−48065.0	−	48065.0i	−0.490614	−	0.490614i	0.417885	−	0.908500i	$-0.362771\pi$
$313$	−48065.0	−	48065.0i	−0.490614	−	0.490614i	−0.908500	+	0.417885i	$0.862771\pi$
$314$		−	13250.0i		−	0.134387i
$315$		0			0
$316$	228888.			2.29218
$317$	−12425.0	+	12425.0i	−0.123645	+	0.123645i	−0.766222	−	0.642576i	$-0.777866\pi$
$317$	−12425.0	+	12425.0i	−0.123645	+	0.123645i	0.642576	+	0.766222i	$0.277866\pi$
$318$		0			0
$319$			45000.0i			0.442213i
$320$		−	57400.0i		−	0.560547i
$321$		0			0
$322$	−136000.	+	136000.i	−1.31168	+	1.31168i
$323$	−16920.0	−	16920.0i	−0.162179	−	0.162179i
$324$		0			0
$325$	128125.	+	128125.i	1.21302	+	1.21302i
$326$	224000.			2.10772
$327$		0			0
$328$	85500.0	+	85500.0i	0.794728	+	0.794728i
$329$			25600.0i			0.236509i
$330$		0			0
$331$	−13528.0			−0.123475			−0.0617373	−	0.998092i	$-0.519664\pi$
$331$	−13528.0			−0.123475			−0.0617373	+	0.998092i	$0.519664\pi$
$332$	−23120.0	+	23120.0i	−0.209755	+	0.209755i
$333$		0			0
$334$			278800.i			2.49919i
$335$	8500.00	−	8500.00i	0.0757407	−	0.0757407i
$336$		0			0
$337$	20815.0	−	20815.0i	0.183281	−	0.183281i	−0.609503	−	0.792784i	$-0.708631\pi$
$337$	20815.0	−	20815.0i	0.183281	−	0.183281i	0.792784	+	0.609503i	$0.208631\pi$
$338$	−277445.	−	277445.i	−2.42853	−	2.42853i
$339$		0			0
$340$	−199750.	−	199750.i	−1.72794	−	1.72794i
$341$	−42800.0			−0.368074
$342$		0			0
$343$	64080.0	+	64080.0i	0.544671	+	0.544671i
$344$		−	219600.i		−	1.85573i
$345$		0			0
$346$	−199750.			−1.66853
$347$	−158780.	+	158780.i	−1.31867	+	1.31867i	−0.403845	+	0.914828i	$0.632326\pi$
$347$	−158780.	+	158780.i	−1.31867	+	1.31867i	−0.914828	+	0.403845i	$0.867674\pi$
$348$		0			0
$349$		−	116352.i		−	0.955263i	−0.878560	−	0.477632i	$-0.841495\pi$
$349$		−	116352.i		−	0.955263i	0.878560	−	0.477632i	$-0.158505\pi$
$350$			250000.i			2.04082i
$351$		0			0
$352$	−34000.0	+	34000.0i	−0.274406	+	0.274406i
$353$	18085.0	+	18085.0i	0.145134	+	0.145134i	0.775940	−	0.630806i	$-0.217276\pi$
$353$	18085.0	+	18085.0i	0.145134	+	0.145134i	−0.630806	+	0.775940i	$0.717276\pi$
$354$		0			0
$355$	−85000.0			−0.674469
$356$	−76500.0			−0.603617
$357$		0			0
$358$	229500.	+	229500.i	1.79067	+	1.79067i
$359$		−	223200.i		−	1.73183i	−0.500191	−	0.865915i	$-0.666737\pi$
$359$		−	223200.i		−	1.73183i	0.500191	−	0.865915i	$-0.333263\pi$
$360$		0			0
$361$	125137.			0.960221
$362$	−79390.0	+	79390.0i	−0.605827	+	0.605827i
$363$		0			0
$364$		−	557600.i		−	4.20843i
$365$	10375.0	+	10375.0i	0.0778758	+	0.0778758i
$366$		0			0
$367$	78880.0	−	78880.0i	0.585645	−	0.585645i	−0.350804	−	0.936449i	$-0.614092\pi$
$367$	78880.0	−	78880.0i	0.585645	−	0.585645i	0.936449	+	0.350804i	$0.114092\pi$
$368$	−121040.	−	121040.i	−0.893785	−	0.893785i
$369$		0			0
$370$		−	188750.i		−	1.37874i
$371$	40400.0			0.293517
$372$		0			0
$373$	83725.0	+	83725.0i	0.601780	+	0.601780i	0.940785	−	0.339005i	$-0.110090\pi$
$373$	83725.0	+	83725.0i	0.601780	+	0.601780i	−0.339005	+	0.940785i	$0.610090\pi$
$374$		−	235000.i		−	1.68006i
$375$		0			0
$376$	−57600.0			−0.407424
$377$	92250.0	−	92250.0i	0.649058	−	0.649058i
$378$		0			0
$379$		−	101232.i		−	0.704757i	−0.935858	−	0.352378i	$-0.885373\pi$
$379$		−	101232.i		−	0.704757i	0.935858	−	0.352378i	$-0.114627\pi$
$380$	−61200.0			−0.423823
$381$		0			0
$382$	−85000.0	+	85000.0i	−0.582495	+	0.582495i
$383$	142540.	+	142540.i	0.971716	+	0.971716i	0.999611	−	0.0278952i	$-0.00888046\pi$
$383$	142540.	+	142540.i	0.971716	+	0.971716i	−0.0278952	+	0.999611i	$0.508880\pi$
$384$		0			0
$385$	−100000.	+	100000.i	−0.674650	+	0.674650i
$386$	310250.			2.08227
$387$		0			0
$388$	−54910.0	−	54910.0i	−0.364744	−	0.364744i
$389$		−	100800.i		−	0.666134i	−0.942903	−	0.333067i	$-0.891917\pi$
$389$		−	100800.i		−	0.666134i	0.942903	−	0.333067i	$-0.108083\pi$
$390$		0			0
$391$	159800.			1.04526
$392$	71910.0	−	71910.0i	0.467969	−	0.467969i
$393$		0			0
$394$		−	6650.00i		−	0.0428380i
$395$			168300.i			1.07867i
$396$		0			0
$397$	123805.	−	123805.i	0.785520	−	0.785520i	−0.195236	−	0.980756i	$-0.562547\pi$
$397$	123805.	−	123805.i	0.785520	−	0.785520i	0.980756	+	0.195236i	$0.0625474\pi$
$398$	−154260.	−	154260.i	−0.973839	−	0.973839i
$399$		0			0
$400$	−222500.			−1.39062
$401$	−25600.0			−0.159203			−0.0796015	−	0.996827i	$-0.525365\pi$
$401$	−25600.0			−0.159203			−0.0796015	+	0.996827i	$0.525365\pi$
$402$		0			0
$403$	87740.0	+	87740.0i	0.540241	+	0.540241i
$404$			462400.i			2.83306i
$405$		0			0
$406$	180000.			1.09199
$407$	75500.0	−	75500.0i	0.455783	−	0.455783i
$408$		0			0
$409$			313938.i			1.87671i	0.345672	+	0.938355i	$0.387651\pi$
$409$			313938.i			1.87671i	−0.345672	+	0.938355i	$0.612349\pi$
$410$	−118750.	+	118750.i	−0.706425	+	0.706425i
$411$		0			0
$412$	−196520.	+	196520.i	−1.15774	+	1.15774i
$413$	−252000.	−	252000.i	−1.47741	−	1.47741i
$414$		0			0
$415$	−17000.0	−	17000.0i	−0.0987081	−	0.0987081i
$416$	139400.			0.805520
$417$		0			0
$418$	−36000.0	−	36000.0i	−0.206039	−	0.206039i
$419$			197100.i			1.12269i	0.827583	+	0.561343i	$0.189715\pi$
$419$			197100.i			1.12269i	−0.827583	+	0.561343i	$0.810285\pi$
$420$		0			0
$421$	−111232.			−0.627575			−0.313787	−	0.949493i	$-0.601598\pi$
$421$	−111232.			−0.627575			−0.313787	+	0.949493i	$0.601598\pi$
$422$	−388960.	+	388960.i	−2.18414	+	2.18414i
$423$		0			0
$424$			90900.0i			0.505629i
$425$	146875.	−	146875.i	0.813149	−	0.813149i
$426$		0			0
$427$	−152320.	+	152320.i	−0.835413	+	0.835413i
$428$	437240.	+	437240.i	2.38689	+	2.38689i
$429$		0			0
$430$	305000.			1.64954
$431$	151400.			0.815026			0.407513	−	0.913199i	$-0.366396\pi$
$431$	151400.			0.815026			0.407513	+	0.913199i	$0.366396\pi$
$432$		0			0
$433$	−117215.	−	117215.i	−0.625183	−	0.625183i	0.321669	−	0.946852i	$-0.395756\pi$
$433$	−117215.	−	117215.i	−0.625183	−	0.625183i	−0.946852	+	0.321669i	$0.895756\pi$
$434$			171200.i			0.908917i
$435$		0			0
$436$	280908.			1.47772
$437$	24480.0	−	24480.0i	0.128188	−	0.128188i
$438$		0			0
$439$		−	158508.i		−	0.822474i	−0.911528	−	0.411237i	$-0.865097\pi$
$439$		−	158508.i		−	0.822474i	0.911528	−	0.411237i	$-0.134903\pi$
$440$	−225000.	−	225000.i	−1.16219	−	1.16219i
$441$		0			0
$442$	−481750.	+	481750.i	−2.46591	+	2.46591i
$443$	142420.	+	142420.i	0.725711	+	0.725711i	0.969762	−	0.244052i	$-0.0784766\pi$
$443$	142420.	+	142420.i	0.725711	+	0.725711i	−0.244052	+	0.969762i	$0.578477\pi$
$444$		0			0
$445$		−	56250.0i		−	0.284055i
$446$	−329800.			−1.65799
$447$		0			0
$448$	−91840.0	−	91840.0i	−0.457589	−	0.457589i
$449$		−	367200.i		−	1.82142i	−0.413047	−	0.910710i	$-0.635535\pi$
$449$		−	367200.i		−	1.82142i	0.413047	−	0.910710i	$-0.364465\pi$
$450$		0			0
$451$	−95000.0			−0.467058
$452$	557770.	−	557770.i	2.73010	−	2.73010i
$453$		0			0
$454$			207400.i			1.00623i
$455$	410000.			1.98044
$456$		0			0
$457$	278545.	−	278545.i	1.33371	−	1.33371i	0.431695	−	0.902019i	$-0.357916\pi$
$457$	278545.	−	278545.i	1.33371	−	1.33371i	0.902019	−	0.431695i	$-0.142084\pi$
$458$	128160.	+	128160.i	0.610972	+	0.610972i
$459$		0			0
$460$	289000.	−	289000.i	1.36578	−	1.36578i
$461$	−197200.			−0.927908			−0.463954	−	0.885859i	$-0.653570\pi$
$461$	−197200.			−0.927908			−0.463954	+	0.885859i	$0.653570\pi$
$462$		0			0
$463$	101320.	+	101320.i	0.472643	+	0.472643i	0.902769	−	0.430126i	$-0.141531\pi$
$463$	101320.	+	101320.i	0.472643	+	0.472643i	−0.430126	+	0.902769i	$0.641531\pi$
$464$			160200.i			0.744092i
$465$		0			0
$466$	55250.0			0.254425
$467$	−122480.	+	122480.i	−0.561606	+	0.561606i	−0.929763	−	0.368158i	$-0.879989\pi$
$467$	−122480.	+	122480.i	−0.561606	+	0.561606i	0.368158	+	0.929763i	$0.379989\pi$
$468$		0			0
$469$		−	27200.0i		−	0.123658i
$470$		−	80000.0i		−	0.362155i
$471$		0			0
$472$	567000.	−	567000.i	2.54507	−	2.54507i
$473$	122000.	+	122000.i	0.545303	+	0.545303i
$474$		0			0
$475$		−	45000.0i		−	0.199446i
$476$	−639200.			−2.82113
$477$		0			0
$478$	−432000.	−	432000.i	−1.89072	−	1.89072i
$479$		−	14400.0i		−	0.0627612i	−0.999508	−	0.0313806i	$-0.990010\pi$
$479$		−	14400.0i		−	0.0627612i	0.999508	−	0.0313806i	$-0.00999040\pi$
$480$		0			0
$481$	−309550.			−1.33795
$482$	−164560.	+	164560.i	−0.708321	+	0.708321i
$483$		0			0
$484$			157794.i			0.673596i
$485$	40375.0	−	40375.0i	0.171644	−	0.171644i
$486$		0			0
$487$	−226700.	+	226700.i	−0.955858	+	0.955858i	−0.999066	−	0.0432076i	$-0.986242\pi$
$487$	−226700.	+	226700.i	−0.955858	+	0.955858i	0.0432076	+	0.999066i	$0.486242\pi$
$488$	−342720.	−	342720.i	−1.43913	−	1.43913i
$489$		0			0
$490$	99875.0	+	99875.0i	0.415973	+	0.415973i
$491$	−354100.			−1.46880			−0.734400	−	0.678716i	$-0.762537\pi$
$491$	−354100.			−1.46880			−0.734400	+	0.678716i	$0.762537\pi$
$492$		0			0
$493$	−105750.	−	105750.i	−0.435097	−	0.435097i
$494$			147600.i			0.604829i
$495$		0			0
$496$	−152368.			−0.619342
$497$	−136000.	+	136000.i	−0.550587	+	0.550587i
$498$		0			0
$499$		−	227448.i		−	0.913442i	−0.889610	−	0.456721i	$-0.849024\pi$
$499$		−	227448.i		−	0.913442i	0.889610	−	0.456721i	$-0.150976\pi$
$500$		−	531250.i		−	2.12500i
$501$		0			0
$502$	273500.	−	273500.i	1.08530	−	1.08530i
$503$	164800.	+	164800.i	0.651360	+	0.651360i	0.953321	−	0.301960i	$-0.0976410\pi$
$503$	164800.	+	164800.i	0.651360	+	0.651360i	−0.301960	+	0.953321i	$0.597641\pi$
$504$		0			0
$505$	−340000.			−1.33320
$506$	340000.			1.32794
$507$		0			0
$508$	473960.	+	473960.i	1.83660	+	1.83660i
$509$		−	420750.i		−	1.62401i	−0.583651	−	0.812005i	$-0.698376\pi$
$509$		−	420750.i		−	1.62401i	0.583651	−	0.812005i	$-0.301624\pi$
$510$		0			0
$511$	33200.0			0.127144
$512$	391600.	−	391600.i	1.49384	−	1.49384i
$513$		0			0
$514$			286450.i			1.08423i
$515$	−144500.	−	144500.i	−0.544820	−	0.544820i
$516$		0			0
$517$	32000.0	−	32000.0i	0.119721	−	0.119721i
$518$	−302000.	−	302000.i	−1.12550	−	1.12550i
$519$		0			0
$520$			922500.i			3.41161i
$521$	56000.0			0.206306			0.103153	−	0.994665i	$-0.467107\pi$
$521$	56000.0			0.206306			0.103153	+	0.994665i	$0.467107\pi$
$522$		0			0
$523$	262360.	+	262360.i	0.959167	+	0.959167i	0.999198	−	0.0400314i	$-0.0127458\pi$
$523$	262360.	+	262360.i	0.959167	+	0.959167i	−0.0400314	+	0.999198i	$0.512746\pi$
$524$			778600.i			2.83564i
$525$		0			0
$526$	53600.0			0.193728
$527$	100580.	−	100580.i	0.362152	−	0.362152i
$528$		0			0
$529$		−	48641.0i		−	0.173817i
$530$	−126250.			−0.449448
$531$		0			0
$532$	−97920.0	+	97920.0i	−0.345978	+	0.345978i
$533$	194750.	+	194750.i	0.685525	+	0.685525i
$534$		0			0
$535$	−321500.	+	321500.i	−1.12324	+	1.12324i
$536$	61200.0			0.213021
$537$		0			0
$538$	−342000.	−	342000.i	−1.18158	−	1.18158i
$539$			79900.0i			0.275023i
$540$		0			0
$541$	−298438.			−1.01967			−0.509835	−	0.860272i	$-0.670294\pi$
$541$	−298438.			−1.01967			−0.509835	+	0.860272i	$0.670294\pi$
$542$	289340.	−	289340.i	0.984940	−	0.984940i
$543$		0			0
$544$		−	159800.i		−	0.539982i
$545$			206550.i			0.695396i
$546$		0			0
$547$	72580.0	−	72580.0i	0.242573	−	0.242573i	−0.575341	−	0.817914i	$-0.695131\pi$
$547$	72580.0	−	72580.0i	0.242573	−	0.242573i	0.817914	+	0.575341i	$0.195131\pi$
$548$	269450.	+	269450.i	0.897257	+	0.897257i
$549$		0			0
$550$	312500.	−	312500.i	1.03306	−	1.03306i
$551$	−32400.0			−0.106719
$552$		0			0
$553$	269280.	+	269280.i	0.880550	+	0.880550i
$554$		−	672350.i		−	2.19066i
$555$		0			0
$556$	−944928.			−3.05667
$557$	−319175.	+	319175.i	−1.02877	+	1.02877i	−0.0291968	+	0.999574i	$0.509295\pi$
$557$	−319175.	+	319175.i	−1.02877	+	1.02877i	−0.999574	+	0.0291968i	$0.990705\pi$
$558$		0			0
$559$		−	500200.i		−	1.60074i
$560$	−356000.	+	356000.i	−1.13520	+	1.13520i
$561$		0			0
$562$	488750.	−	488750.i	1.54744	−	1.54744i
$563$	−187940.	−	187940.i	−0.592929	−	0.592929i	0.345493	−	0.938421i	$-0.387712\pi$
$563$	−187940.	−	187940.i	−0.592929	−	0.592929i	−0.938421	+	0.345493i	$0.887712\pi$
$564$		0			0
$565$	410125.	+	410125.i	1.28475	+	1.28475i
$566$	479000.			1.49521
$567$		0			0
$568$	−306000.	−	306000.i	−0.948473	−	0.948473i
$569$			223200.i			0.689397i	0.938713	+	0.344699i	$0.112019\pi$
$569$			223200.i			0.689397i	−0.938713	+	0.344699i	$0.887981\pi$
$570$		0			0
$571$	468032.			1.43550			0.717750	−	0.696301i	$-0.245172\pi$
$571$	468032.			1.43550			0.717750	+	0.696301i	$0.245172\pi$
$572$	−697000.	+	697000.i	−2.13030	+	2.13030i
$573$		0			0
$574$			380000.i			1.15335i
$575$	212500.	+	212500.i	0.642722	+	0.642722i
$576$		0			0
$577$	325855.	−	325855.i	0.978752	−	0.978752i	−0.0210267	−	0.999779i	$-0.506693\pi$
$577$	325855.	−	325855.i	0.978752	−	0.978752i	0.999779	+	0.0210267i	$0.00669350\pi$
$578$	134645.	+	134645.i	0.403027	+	0.403027i
$579$		0			0
$580$	−382500.			−1.13704
$581$	−54400.0			−0.161156
$582$		0			0
$583$	−50500.0	−	50500.0i	−0.148578	−	0.148578i
$584$			74700.0i			0.219026i
$585$		0			0
$586$	482150.			1.40406
$587$	−112460.	+	112460.i	−0.326379	+	0.326379i	−0.851208	−	0.524829i	$-0.824129\pi$
$587$	−112460.	+	112460.i	−0.326379	+	0.326379i	0.524829	+	0.851208i	$0.324129\pi$
$588$		0			0
$589$		−	30816.0i		−	0.0888271i
$590$	787500.	+	787500.i	2.26228	+	2.26228i
$591$		0			0
$592$	268780.	−	268780.i	0.766926	−	0.766926i
$593$	−398645.	−	398645.i	−1.13364	−	1.13364i	−0.989566	−	0.144078i	$-0.953978\pi$
$593$	−398645.	−	398645.i	−1.13364	−	1.13364i	−0.144078	−	0.989566i	$-0.546022\pi$
$594$		0			0
$595$		−	470000.i		−	1.32759i
$596$	856800.			2.41205
$597$		0			0
$598$	−697000.	−	697000.i	−1.94908	−	1.94908i
$599$		−	336600.i		−	0.938124i	−0.883165	−	0.469062i	$-0.844592\pi$
$599$		−	336600.i		−	0.938124i	0.883165	−	0.469062i	$-0.155408\pi$
$600$		0			0
$601$	−352.000			−0.000974527			−0.000487263	−	1.00000i	$-0.500155\pi$
$601$	−352.000			−0.000974527			−0.000487263		1.00000i	$0.500155\pi$
$602$	488000.	−	488000.i	1.34656	−	1.34656i
$603$		0			0
$604$			776968.i			2.12975i
$605$	−116025.			−0.316987
$606$		0			0
$607$	−323840.	+	323840.i	−0.878928	+	0.878928i	−0.993424	−	0.114496i	$-0.963475\pi$
$607$	−323840.	+	323840.i	−0.878928	+	0.878928i	0.114496	+	0.993424i	$0.463475\pi$
$608$	−24480.0	−	24480.0i	−0.0662223	−	0.0662223i
$609$		0			0
$610$	476000.	−	476000.i	1.27923	−	1.27923i
$611$	−131200.			−0.351440
$612$		0			0
$613$	267715.	+	267715.i	0.712446	+	0.712446i	0.967046	−	0.254601i	$-0.0819440\pi$
$613$	267715.	+	267715.i	0.712446	+	0.712446i	−0.254601	+	0.967046i	$0.581944\pi$
$614$		−	537200.i		−	1.42495i
$615$		0			0
$616$	−720000.			−1.89745
$617$	−34085.0	+	34085.0i	−0.0895350	+	0.0895350i	−0.750456	−	0.660921i	$-0.770166\pi$
$617$	−34085.0	+	34085.0i	−0.0895350	+	0.0895350i	0.660921	+	0.750456i	$0.270166\pi$
$618$		0			0
$619$			126072.i			0.329031i	0.986374	+	0.164516i	$0.0526061\pi$
$619$			126072.i			0.329031i	−0.986374	+	0.164516i	$0.947394\pi$
$620$		−	363800.i		−	0.946410i
$621$		0			0
$622$	−682000.	+	682000.i	−1.76280	+	1.76280i
$623$	−90000.0	−	90000.0i	−0.231882	−	0.231882i
$624$		0			0
$625$	390625.			1.00000
$626$	480650.			1.22654
$627$		0			0
$628$	45050.0	+	45050.0i	0.114229	+	0.114229i
$629$			354850.i			0.896899i
$630$		0			0
$631$	440372.			1.10601			0.553007	−	0.833176i	$-0.313480\pi$
$631$	440372.			1.10601			0.553007	+	0.833176i	$0.313480\pi$
$632$	−605880.	+	605880.i	−1.51688	+	1.51688i
$633$		0			0
$634$		−	124250.i		−	0.309113i
$635$	−348500.	+	348500.i	−0.864282	+	0.864282i
$636$		0			0
$637$	163795.	−	163795.i	0.403666	−	0.403666i
$638$	−225000.	−	225000.i	−0.552766	−	0.552766i
$639$		0			0
$640$	423000.	+	423000.i	1.03271	+	1.03271i
$641$	−90550.0			−0.220380			−0.110190	−	0.993911i	$-0.535146\pi$
$641$	−90550.0			−0.220380			−0.110190	+	0.993911i	$0.535146\pi$
$642$		0			0
$643$	30760.0	+	30760.0i	0.0743985	+	0.0743985i	0.743327	−	0.668928i	$-0.233247\pi$
$643$	30760.0	+	30760.0i	0.0743985	+	0.0743985i	−0.668928	+	0.743327i	$0.733247\pi$
$644$		−	924800.i		−	2.22985i
$645$		0			0
$646$	169200.			0.405448
$647$	166600.	−	166600.i	0.397985	−	0.397985i	−0.479537	−	0.877522i	$-0.659195\pi$
$647$	166600.	−	166600.i	0.397985	−	0.397985i	0.877522	+	0.479537i	$0.159195\pi$
$648$		0			0
$649$			630000.i			1.49572i
$650$	−1.28125e6			−3.03254
$651$		0			0
$652$	−761600.	+	761600.i	−1.79156	+	1.79156i
$653$	349945.	+	349945.i	0.820679	+	0.820679i	0.986205	−	0.165526i	$-0.0529322\pi$
$653$	349945.	+	349945.i	0.820679	+	0.820679i	−0.165526	+	0.986205i	$0.552932\pi$
$654$		0			0
$655$	−572500.			−1.33442
$656$	−338200.			−0.785898
$657$		0			0
$658$	−128000.	−	128000.i	−0.295637	−	0.295637i
$659$			407700.i			0.938793i	0.882987	+	0.469397i	$0.155529\pi$
$659$			407700.i			0.938793i	−0.882987	+	0.469397i	$0.844471\pi$
$660$		0			0
$661$	−740992.			−1.69594			−0.847970	−	0.530044i	$-0.822175\pi$
$661$	−740992.			−1.69594			−0.847970	+	0.530044i	$0.822175\pi$
$662$	67640.0	−	67640.0i	0.154343	−	0.154343i
$663$		0			0
$664$		−	122400.i		−	0.277616i
$665$	−72000.0	−	72000.0i	−0.162813	−	0.162813i
$666$		0			0
$667$	153000.	−	153000.i	0.343906	−	0.343906i
$668$	−947920.	−	947920.i	−2.12431	−	2.12431i
$669$		0			0
$670$			85000.0i			0.189352i
$671$	380800.			0.845769
$672$		0			0
$673$	−258575.	−	258575.i	−0.570895	−	0.570895i	0.361483	−	0.932379i	$-0.382270\pi$
$673$	−258575.	−	258575.i	−0.570895	−	0.570895i	−0.932379	+	0.361483i	$0.882270\pi$
$674$			208150.i			0.458202i
$675$		0			0
$676$	1.88663e6			4.12850
$677$	499945.	−	499945.i	1.09080	−	1.09080i	0.0953562	−	0.995443i	$-0.469601\pi$
$677$	499945.	−	499945.i	1.09080	−	1.09080i	0.995443	−	0.0953562i	$-0.0303990\pi$
$678$		0			0
$679$		−	129200.i		−	0.280235i
$680$	1.05750e6			2.28698
$681$		0			0
$682$	214000.	−	214000.i	0.460092	−	0.460092i
$683$	−266840.	−	266840.i	−0.572018	−	0.572018i	0.360674	−	0.932692i	$-0.382547\pi$
$683$	−266840.	−	266840.i	−0.572018	−	0.572018i	−0.932692	+	0.360674i	$0.882547\pi$
$684$		0			0
$685$	−198125.	+	198125.i	−0.422239	+	0.422239i
$686$	−640800.			−1.36168
$687$		0			0
$688$	434320.	+	434320.i	0.917557	+	0.917557i
$689$			207050.i			0.436151i
$690$		0			0
$691$	−141112.			−0.295534			−0.147767	−	0.989022i	$-0.547209\pi$
$691$	−141112.			−0.295534			−0.147767	+	0.989022i	$0.547209\pi$
$692$	679150.	−	679150.i	1.41825	−	1.41825i
$693$		0			0
$694$		−	1.58780e6i		−	3.29668i
$695$		−	694800.i		−	1.43843i
$696$		0			0
$697$	223250.	−	223250.i	0.459543	−	0.459543i
$698$	581760.	+	581760.i	1.19408	+	1.19408i
$699$		0			0
$700$	−850000.	−	850000.i	−1.73469	−	1.73469i
$701$	708050.			1.44088			0.720440	−	0.693517i	$-0.243940\pi$
$701$	708050.			1.44088			0.720440	+	0.693517i	$0.243940\pi$
$702$		0			0
$703$	54360.0	+	54360.0i	0.109994	+	0.109994i
$704$			229600.i			0.463262i
$705$		0			0
$706$	−180850.			−0.362835
$707$	−544000.	+	544000.i	−1.08833	+	1.08833i
$708$		0			0
$709$		−	474912.i		−	0.944758i	−0.881395	−	0.472379i	$-0.843395\pi$
$709$		−	474912.i		−	0.944758i	0.881395	−	0.472379i	$-0.156605\pi$
$710$	425000.	−	425000.i	0.843087	−	0.843087i
$711$		0			0
$712$	202500.	−	202500.i	0.399452	−	0.399452i
$713$	145520.	+	145520.i	0.286249	+	0.286249i
$714$		0			0
$715$	−512500.	−	512500.i	−1.00249	−	1.00249i
$716$	−1.56060e6			−3.04415
$717$		0			0
$718$	1.11600e6	+	1.11600e6i	2.16479	+	2.16479i
$719$		−	527400.i		−	1.02019i	−0.860117	−	0.510097i	$-0.829610\pi$
$719$		−	527400.i		−	1.02019i	0.860117	−	0.510097i	$-0.170390\pi$
$720$		0			0
$721$	−462400.			−0.889503
$722$	−625685.	+	625685.i	−1.20028	+	1.20028i
$723$		0			0
$724$		−	539852.i		−	1.02991i
$725$		−	281250.i		−	0.535077i
$726$		0			0
$727$	−315860.	+	315860.i	−0.597621	+	0.597621i	−0.939679	−	0.342058i	$-0.888876\pi$
$727$	−315860.	+	315860.i	−0.597621	+	0.597621i	0.342058	+	0.939679i	$0.388876\pi$
$728$	1.47600e6	+	1.47600e6i	2.78499	+	2.78499i
$729$		0			0
$730$	−103750.			−0.194689
$731$	−573400.			−1.07306
$732$		0			0
$733$	−325325.	−	325325.i	−0.605494	−	0.605494i	0.336272	−	0.941765i	$-0.390834\pi$
$733$	−325325.	−	325325.i	−0.605494	−	0.605494i	−0.941765	+	0.336272i	$0.890834\pi$
$734$			788800.i			1.46411i
$735$		0			0
$736$	231200.			0.426808
$737$	−34000.0	+	34000.0i	−0.0625956	+	0.0625956i
$738$		0			0
$739$			388008.i			0.710480i	0.934775	+	0.355240i	$0.115601\pi$
$739$			388008.i			0.710480i	−0.934775	+	0.355240i	$0.884399\pi$
$740$	641750.	+	641750.i	1.17193	+	1.17193i
$741$		0			0
$742$	−202000.	+	202000.i	−0.366896	+	0.366896i
$743$	−256700.	−	256700.i	−0.464995	−	0.464995i	0.435294	−	0.900289i	$-0.356645\pi$
$743$	−256700.	−	256700.i	−0.464995	−	0.464995i	−0.900289	+	0.435294i	$0.856645\pi$
$744$		0			0
$745$			630000.i			1.13508i
$746$	−837250.			−1.50445
$747$		0			0
$748$	799000.	+	799000.i	1.42805	+	1.42805i
$749$			1.02880e6i			1.83386i
$750$		0			0
$751$	532148.			0.943523			0.471762	−	0.881726i	$-0.343618\pi$
$751$	532148.			0.943523			0.471762	+	0.881726i	$0.343618\pi$
$752$	113920.	−	113920.i	0.201449	−	0.201449i
$753$		0			0
$754$			922500.i			1.62265i
$755$	−571300.			−1.00224
$756$		0			0
$757$	54685.0	−	54685.0i	0.0954281	−	0.0954281i	−0.657781	−	0.753209i	$-0.728505\pi$
$757$	54685.0	−	54685.0i	0.0954281	−	0.0954281i	0.753209	+	0.657781i	$0.228505\pi$
$758$	506160.	+	506160.i	0.880946	+	0.880946i
$759$		0			0
$760$	162000.	−	162000.i	0.280471	−	0.280471i
$761$	399200.			0.689321			0.344660	−	0.938727i	$-0.387994\pi$
$761$	399200.			0.689321			0.344660	+	0.938727i	$0.387994\pi$
$762$		0			0
$763$	330480.	+	330480.i	0.567670	+	0.567670i
$764$		−	578000.i		−	0.990241i
$765$		0			0
$766$	−1.42540e6			−2.42929
$767$	1.29150e6	−	1.29150e6i	2.19535	−	2.19535i
$768$		0			0
$769$		−	714528.i		−	1.20828i	−0.796879	−	0.604139i	$-0.793517\pi$
$769$		−	714528.i		−	1.20828i	0.796879	−	0.604139i	$-0.206483\pi$
$770$		−	1.00000e6i		−	1.68663i
$771$		0			0
$772$	−1.05485e6	+	1.05485e6i	−1.76993	+	1.76993i
$773$	360055.	+	360055.i	0.602573	+	0.602573i	0.940995	−	0.338421i	$-0.109893\pi$
$773$	360055.	+	360055.i	0.602573	+	0.602573i	−0.338421	+	0.940995i	$0.609893\pi$
$774$		0			0
$775$	267500.			0.445369
$776$	290700.			0.482749
$777$		0			0
$778$	504000.	+	504000.i	0.832667	+	0.832667i
$779$

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	45.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.00000 + 5.00000i) q^{2} -34.0000i q^{4} +25.0000 q^{5} +(40.0000 - 40.0000i) q^{7} +(90.0000 + 90.0000i) q^{8} +O(q^{10})\)	\(q+(-5.00000 + 5.00000i) q^{2} -34.0000i q^{4} +25.0000 q^{5} +(40.0000 - 40.0000i) q^{7} +(90.0000 + 90.0000i) q^{8} +(-125.000 + 125.000i) q^{10} -100.000 q^{11} +(205.000 + 205.000i) q^{13} +400.000i q^{14} -356.000 q^{16} +(235.000 - 235.000i) q^{17} -72.0000i q^{19} -850.000i q^{20} +(500.000 - 500.000i) q^{22} +(340.000 + 340.000i) q^{23} +625.000 q^{25} -2050.00 q^{26} +(-1360.00 - 1360.00i) q^{28} -450.000i q^{29} +428.000 q^{31} +(340.000 - 340.000i) q^{32} +2350.00i q^{34} +(1000.00 - 1000.00i) q^{35} +(-755.000 + 755.000i) q^{37} +(360.000 + 360.000i) q^{38} +(2250.00 + 2250.00i) q^{40} +950.000 q^{41} +(-1220.00 - 1220.00i) q^{43} +3400.00i q^{44} -3400.00 q^{46} +(-320.000 + 320.000i) q^{47} -799.000i q^{49} +(-3125.00 + 3125.00i) q^{50} +(6970.00 - 6970.00i) q^{52} +(505.000 + 505.000i) q^{53} -2500.00 q^{55} +7200.00 q^{56} +(2250.00 + 2250.00i) q^{58} -6300.00i q^{59} -3808.00 q^{61} +(-2140.00 + 2140.00i) q^{62} -2296.00i q^{64} +(5125.00 + 5125.00i) q^{65} +(340.000 - 340.000i) q^{67} +(-7990.00 - 7990.00i) q^{68} +10000.0i q^{70} -3400.00 q^{71} +(415.000 + 415.000i) q^{73} -7550.00i q^{74} -2448.00 q^{76} +(-4000.00 + 4000.00i) q^{77} +6732.00i q^{79} -8900.00 q^{80} +(-4750.00 + 4750.00i) q^{82} +(-680.000 - 680.000i) q^{83} +(5875.00 - 5875.00i) q^{85} +12200.0 q^{86} +(-9000.00 - 9000.00i) q^{88} -2250.00i q^{89} +16400.0 q^{91} +(11560.0 - 11560.0i) q^{92} -3200.00i q^{94} -1800.00i q^{95} +(1615.00 - 1615.00i) q^{97} +(3995.00 + 3995.00i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{2} + 50 q^{5} + 80 q^{7} + 180 q^{8}+O(q^{10}) \) 2 * q - 10 * q^2 + 50 * q^5 + 80 * q^7 + 180 * q^8	\( 2 q - 10 q^{2} + 50 q^{5} + 80 q^{7} + 180 q^{8} - 250 q^{10} - 200 q^{11} + 410 q^{13} - 712 q^{16} + 470 q^{17} + 1000 q^{22} + 680 q^{23} + 1250 q^{25} - 4100 q^{26} - 2720 q^{28} + 856 q^{31} + 680 q^{32} + 2000 q^{35} - 1510 q^{37} + 720 q^{38} + 4500 q^{40} + 1900 q^{41} - 2440 q^{43} - 6800 q^{46} - 640 q^{47} - 6250 q^{50} + 13940 q^{52} + 1010 q^{53} - 5000 q^{55} + 14400 q^{56} + 4500 q^{58} - 7616 q^{61} - 4280 q^{62} + 10250 q^{65} + 680 q^{67} - 15980 q^{68} - 6800 q^{71} + 830 q^{73} - 4896 q^{76} - 8000 q^{77} - 17800 q^{80} - 9500 q^{82} - 1360 q^{83} + 11750 q^{85} + 24400 q^{86} - 18000 q^{88} + 32800 q^{91} + 23120 q^{92} + 3230 q^{97} + 7990 q^{98}+O(q^{100}) \) 2 * q - 10 * q^2 + 50 * q^5 + 80 * q^7 + 180 * q^8 - 250 * q^10 - 200 * q^11 + 410 * q^13 - 712 * q^16 + 470 * q^17 + 1000 * q^22 + 680 * q^23 + 1250 * q^25 - 4100 * q^26 - 2720 * q^28 + 856 * q^31 + 680 * q^32 + 2000 * q^35 - 1510 * q^37 + 720 * q^38 + 4500 * q^40 + 1900 * q^41 - 2440 * q^43 - 6800 * q^46 - 640 * q^47 - 6250 * q^50 + 13940 * q^52 + 1010 * q^53 - 5000 * q^55 + 14400 * q^56 + 4500 * q^58 - 7616 * q^61 - 4280 * q^62 + 10250 * q^65 + 680 * q^67 - 15980 * q^68 - 6800 * q^71 + 830 * q^73 - 4896 * q^76 - 8000 * q^77 - 17800 * q^80 - 9500 * q^82 - 1360 * q^83 + 11750 * q^85 + 24400 * q^86 - 18000 * q^88 + 32800 * q^91 + 23120 * q^92 + 3230 * q^97 + 7990 * q^98

Introduction

L-functions

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Database

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