LMFDB - Embedded newform 45.5.g.a.37.1

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			37.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	45.37
Dual form			45.5.g.a.28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-5.00000 - 5.00000i) q^{2} +34.0000i q^{4} +25.0000 q^{5} +(40.0000 + 40.0000i) q^{7} +(90.0000 - 90.0000i) q^{8} +(-125.000 - 125.000i) q^{10} -100.000 q^{11} +(205.000 - 205.000i) q^{13} -400.000i q^{14} -356.000 q^{16} +(235.000 + 235.000i) q^{17} +72.0000i q^{19} +850.000i q^{20} +(500.000 + 500.000i) q^{22} +(340.000 - 340.000i) q^{23} +625.000 q^{25} -2050.00 q^{26} +(-1360.00 + 1360.00i) q^{28} +450.000i q^{29} +428.000 q^{31} +(340.000 + 340.000i) q^{32} -2350.00i q^{34} +(1000.00 + 1000.00i) q^{35} +(-755.000 - 755.000i) q^{37} +(360.000 - 360.000i) q^{38} +(2250.00 - 2250.00i) q^{40} +950.000 q^{41} +(-1220.00 + 1220.00i) q^{43} -3400.00i q^{44} -3400.00 q^{46} +(-320.000 - 320.000i) q^{47} +799.000i q^{49} +(-3125.00 - 3125.00i) q^{50} +(6970.00 + 6970.00i) q^{52} +(505.000 - 505.000i) q^{53} -2500.00 q^{55} +7200.00 q^{56} +(2250.00 - 2250.00i) q^{58} +6300.00i q^{59} -3808.00 q^{61} +(-2140.00 - 2140.00i) q^{62} +2296.00i q^{64} +(5125.00 - 5125.00i) q^{65} +(340.000 + 340.000i) q^{67} +(-7990.00 + 7990.00i) q^{68} -10000.0i q^{70} -3400.00 q^{71} +(415.000 - 415.000i) q^{73} +7550.00i q^{74} -2448.00 q^{76} +(-4000.00 - 4000.00i) q^{77} -6732.00i q^{79} -8900.00 q^{80} +(-4750.00 - 4750.00i) q^{82} +(-680.000 + 680.000i) q^{83} +(5875.00 + 5875.00i) q^{85} +12200.0 q^{86} +(-9000.00 + 9000.00i) q^{88} +2250.00i q^{89} +16400.0 q^{91} +(11560.0 + 11560.0i) q^{92} +3200.00i q^{94} +1800.00i q^{95} +(1615.00 + 1615.00i) q^{97} +(3995.00 - 3995.00i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{2} + 50 q^{5} + 80 q^{7} + 180 q^{8} - 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}+ \cdots + 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

Character values

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

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

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	−0.955719	−	0.294281i	$-0.904920\pi$
$2$	−5.00000	−	5.00000i	−1.25000	−	1.25000i	−0.294281	−	0.955719i	$-0.595080\pi$
$3$		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	0.985574	−	0.169247i	$-0.0541336\pi$
$7$	40.0000	+	40.0000i	0.816327	+	0.816327i	−0.169247	+	0.985574i	$0.554134\pi$
$8$	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	0.242989	−	0.970029i	$-0.421872\pi$
$13$	205.000	−	205.000i	1.21302	−	1.21302i	0.970029	−	0.242989i	$-0.0781278\pi$
$14$		−	400.000i		−	2.04082i
$15$		0			0
$16$	−356.000			−1.39062
$17$	235.000	+	235.000i	0.813149	+	0.813149i	0.985105	−	0.171956i	$-0.0550086\pi$
$17$	235.000	+	235.000i	0.813149	+	0.813149i	−0.171956	+	0.985105i	$0.555009\pi$
$18$		0			0
$19$			72.0000i			0.199446i	0.995015	+	0.0997230i	$0.0317957\pi$
$19$			72.0000i			0.199446i	−0.995015	+	0.0997230i	$0.968204\pi$
$20$			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	−0.308502	−	0.951224i	$-0.599828\pi$
$23$	340.000	−	340.000i	0.642722	−	0.642722i	0.951224	+	0.308502i	$0.0998275\pi$
$24$		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	$0.0862103\pi$
$29$			450.000i			0.535077i	−0.963547	+	0.267539i	$0.913790\pi$
$30$		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	0.375375	−	0.926873i	$-0.377514\pi$
$37$	−755.000	−	755.000i	−0.551497	−	0.551497i	−0.926873	+	0.375375i	$0.877514\pi$
$38$	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	−0.955336	−	0.295520i	$-0.904507\pi$
$43$	−1220.00	+	1220.00i	−0.659816	+	0.659816i	0.295520	+	0.955336i	$0.404507\pi$
$44$		−	3400.00i		−	1.75620i
$45$		0			0
$46$	−3400.00			−1.60681
$47$	−320.000	−	320.000i	−0.144862	−	0.144862i	0.630956	−	0.775818i	$-0.282663\pi$
$47$	−320.000	−	320.000i	−0.144862	−	0.144862i	−0.775818	+	0.630956i	$0.782663\pi$
$48$		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	−0.611480	−	0.791260i	$-0.709426\pi$
$53$	505.000	−	505.000i	0.179779	−	0.179779i	0.791260	+	0.611480i	$0.209426\pi$
$54$		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	$0.360064\pi$
$59$			6300.00i			1.80982i	−0.425598	+	0.904912i	$0.639936\pi$
$60$		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	0.743962	−	0.668222i	$-0.232944\pi$
$67$	340.000	+	340.000i	0.0757407	+	0.0757407i	−0.668222	+	0.743962i	$0.732944\pi$
$68$	−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	−0.667096	−	0.744972i	$-0.732463\pi$
$73$	415.000	−	415.000i	0.0778758	−	0.0778758i	0.744972	+	0.667096i	$0.232463\pi$
$74$			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	$-0.818675\pi$
$79$		−	6732.00i		−	1.07867i	0.842090	−	0.539337i	$-0.181325\pi$
$80$	−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	−0.754736	−	0.656028i	$-0.772235\pi$
$83$	−680.000	+	680.000i	−0.0987081	+	0.0987081i	0.656028	+	0.754736i	$0.272235\pi$
$84$		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	$0.0453622\pi$
$89$			2250.00i			0.284055i	−0.989863	+	0.142028i	$0.954638\pi$
$90$		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	0.787701	−	0.616057i	$-0.211271\pi$
$97$	1615.00	+	1615.00i	0.171644	+	0.171644i	−0.616057	+	0.787701i	$0.711271\pi$
$98$	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	−0.924938	−	0.380118i	$-0.875883\pi$
$103$	−5780.00	+	5780.00i	−0.544820	+	0.544820i	0.380118	+	0.924938i	$0.375883\pi$
$104$		−	36900.0i		−	3.41161i
$105$		0			0
$106$	−5050.00			−0.449448
$107$	−12860.0	−	12860.0i	−1.12324	−	1.12324i	−0.991251	−	0.131991i	$-0.957863\pi$
$107$	−12860.0	−	12860.0i	−1.12324	−	1.12324i	−0.131991	−	0.991251i	$-0.542137\pi$
$108$		0			0
$109$		−	8262.00i		−	0.695396i	−0.937607	−	0.347698i	$-0.886963\pi$
$109$		−	8262.00i		−	0.695396i	0.937607	−	0.347698i	$-0.113037\pi$
$110$	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	0.346821	−	0.937931i	$-0.387261\pi$
$113$	16405.0	−	16405.0i	1.28475	−	1.28475i	0.937931	−	0.346821i	$-0.112739\pi$
$114$		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	0.127550	−	0.991832i	$-0.459289\pi$
$127$	−13940.0	−	13940.0i	−0.864282	−	0.864282i	−0.991832	+	0.127550i	$0.959289\pi$
$128$	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	0.463735	−	0.885974i	$-0.346509\pi$
$137$	−7925.00	−	7925.00i	−0.422239	−	0.422239i	−0.885974	+	0.463735i	$0.846509\pi$
$138$		0			0
$139$			27792.0i			1.43843i	0.694785	+	0.719217i	$0.255499\pi$
$139$			27792.0i			1.43843i	−0.694785	+	0.719217i	$0.744501\pi$
$140$	−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	$-0.807894\pi$
$149$		−	25200.0i		−	1.13508i	0.823344	−	0.567542i	$-0.192106\pi$
$150$		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	0.679718	−	0.733473i	$-0.262102\pi$
$157$	−1325.00	−	1325.00i	−0.0537547	−	0.0537547i	−0.733473	+	0.679718i	$0.762102\pi$
$158$	−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	−0.989259	−	0.146171i	$-0.953305\pi$
$163$	−22400.0	+	22400.0i	−0.843088	+	0.843088i	0.146171	+	0.989259i	$0.453305\pi$
$164$			32300.0i			1.20092i
$165$		0			0
$166$	6800.00			0.246770
$167$	27880.0	+	27880.0i	0.999677	+	0.999677i	1.00000		0.000322656i	$-0.000102705\pi$
$167$	27880.0	+	27880.0i	0.999677	+	0.999677i	−0.000322656		1.00000i	$0.500103\pi$
$168$		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	−0.289704	−	0.957116i	$-0.593557\pi$
$173$	19975.0	−	19975.0i	0.667413	−	0.667413i	0.957116	+	0.289704i	$0.0935567\pi$
$174$		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	$0.254152\pi$
$179$			45900.0i			1.43254i	−0.697823	+	0.716270i	$0.745848\pi$
$180$		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	−0.987914	−	0.155005i	$-0.950461\pi$
$193$	−31025.0	+	31025.0i	−0.832908	+	0.832908i	0.155005	+	0.987914i	$0.450461\pi$
$194$		−	16150.0i		−	0.429110i
$195$		0			0
$196$	−27166.0			−0.707153
$197$	−665.000	−	665.000i	−0.0171352	−	0.0171352i	0.698487	−	0.715622i	$-0.253857\pi$
$197$	−665.000	−	665.000i	−0.0171352	−	0.0171352i	−0.715622	+	0.698487i	$0.753857\pi$
$198$		0			0
$199$		−	30852.0i		−	0.779071i	−0.921011	−	0.389536i	$-0.872636\pi$
$199$		−	30852.0i		−	0.779071i	0.921011	−	0.389536i	$-0.127364\pi$
$200$	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	−0.292937	−	0.956132i	$-0.594633\pi$
$223$	32980.0	−	32980.0i	0.663195	−	0.663195i	0.956132	+	0.292937i	$0.0946327\pi$
$224$			27200.0i			0.542092i
$225$		0			0
$226$	−164050.			−3.21188
$227$	20740.0	+	20740.0i	0.402492	+	0.402492i	0.879110	−	0.476618i	$-0.158138\pi$
$227$	20740.0	+	20740.0i	0.402492	+	0.402492i	−0.476618	+	0.879110i	$0.658138\pi$
$228$		0			0
$229$			25632.0i			0.488778i	0.969677	+	0.244389i	$0.0785874\pi$
$229$			25632.0i			0.488778i	−0.969677	+	0.244389i	$0.921413\pi$
$230$	−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	−0.756159	−	0.654388i	$-0.772926\pi$
$233$	−5525.00	+	5525.00i	−0.101770	+	0.101770i	0.654388	+	0.756159i	$0.272926\pi$
$234$		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	$-0.727010\pi$
$239$		−	86400.0i		−	1.51258i	0.654237	−	0.756289i	$-0.272990\pi$
$240$		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	0.889883	−	0.456189i	$-0.150786\pi$
$257$	28645.0	+	28645.0i	0.433693	+	0.433693i	−0.456189	+	0.889883i	$0.650786\pi$
$258$		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	−0.744790	−	0.667299i	$-0.767450\pi$
$263$	−5360.00	+	5360.00i	−0.0774914	+	0.0774914i	0.667299	+	0.744790i	$0.267450\pi$
$264$		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	$-0.843305\pi$
$269$		−	68400.0i		−	0.945261i	0.881261	−	0.472630i	$-0.156695\pi$
$270$		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	0.116880	−	0.993146i	$-0.462711\pi$
$277$	−67235.0	−	67235.0i	−0.876266	−	0.876266i	−0.993146	+	0.116880i	$0.962711\pi$
$278$	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	−0.939803	−	0.341718i	$-0.888991\pi$
$283$	−47900.0	+	47900.0i	−0.598085	+	0.598085i	0.341718	+	0.939803i	$0.388991\pi$
$284$		−	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	−0.929769	−	0.368143i	$-0.879994\pi$
$293$	−48215.0	+	48215.0i	−0.561626	+	0.561626i	0.368143	+	0.929769i	$0.379994\pi$
$294$		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	0.362143	−	0.932123i	$-0.382045\pi$
$307$	−53720.0	−	53720.0i	−0.569980	−	0.569980i	−0.932123	+	0.362143i	$0.882045\pi$
$308$	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	−0.908500	−	0.417885i	$-0.862771\pi$
$313$	−48065.0	+	48065.0i	−0.490614	+	0.490614i	0.417885	+	0.908500i	$0.362771\pi$
$314$			13250.0i			0.134387i
$315$		0			0
$316$	228888.			2.29218
$317$	−12425.0	−	12425.0i	−0.123645	−	0.123645i	0.642576	−	0.766222i	$-0.277866\pi$
$317$	−12425.0	−	12425.0i	−0.123645	−	0.123645i	−0.766222	+	0.642576i	$0.777866\pi$
$318$		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	0.792784	−	0.609503i	$-0.208631\pi$
$337$	20815.0	+	20815.0i	0.183281	+	0.183281i	−0.609503	+	0.792784i	$0.708631\pi$
$338$	−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	−0.914828	−	0.403845i	$-0.867674\pi$
$347$	−158780.	−	158780.i	−1.31867	−	1.31867i	−0.403845	−	0.914828i	$-0.632326\pi$
$348$		0			0
$349$			116352.i			0.955263i	0.878560	+	0.477632i	$0.158505\pi$
$349$			116352.i			0.955263i	−0.878560	+	0.477632i	$0.841495\pi$
$350$		−	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	−0.630806	−	0.775940i	$-0.717276\pi$
$353$	18085.0	−	18085.0i	0.145134	−	0.145134i	0.775940	+	0.630806i	$0.217276\pi$
$354$		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	$0.333263\pi$
$359$			223200.i			1.73183i	−0.500191	+	0.865915i	$0.666737\pi$
$360$		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	0.936449	−	0.350804i	$-0.114092\pi$
$367$	78880.0	+	78880.0i	0.585645	+	0.585645i	−0.350804	+	0.936449i	$0.614092\pi$
$368$	−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	−0.339005	−	0.940785i	$-0.610090\pi$
$373$	83725.0	−	83725.0i	0.601780	−	0.601780i	0.940785	+	0.339005i	$0.110090\pi$
$374$			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	$0.114627\pi$
$379$			101232.i			0.704757i	−0.935858	+	0.352378i	$0.885373\pi$
$380$	−61200.0			−0.423823
$381$		0			0
$382$	−85000.0	−	85000.0i	−0.582495	−	0.582495i
$383$	142540.	−	142540.i	0.971716	−	0.971716i	−0.0278952	−	0.999611i	$-0.508880\pi$
$383$	142540.	−	142540.i	0.971716	−	0.971716i	0.999611	+	0.0278952i	$0.00888046\pi$
$384$		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	$0.108083\pi$
$389$			100800.i			0.666134i	−0.942903	+	0.333067i	$0.891917\pi$
$390$		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	0.980756	−	0.195236i	$-0.0625474\pi$
$397$	123805.	+	123805.i	0.785520	+	0.785520i	−0.195236	+	0.980756i	$0.562547\pi$
$398$	−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	$-0.612349\pi$
$409$		−	313938.i		−	1.87671i	0.345672	−	0.938355i	$-0.387651\pi$
$410$	−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	$-0.810285\pi$
$419$		−	197100.i		−	1.12269i	0.827583	−	0.561343i	$-0.189715\pi$
$420$		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	−0.946852	−	0.321669i	$-0.895756\pi$
$433$	−117215.	+	117215.i	−0.625183	+	0.625183i	0.321669	+	0.946852i	$0.395756\pi$
$434$		−	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	$0.134903\pi$
$439$			158508.i			0.822474i	−0.911528	+	0.411237i	$0.865097\pi$
$440$	−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	−0.244052	−	0.969762i	$-0.578477\pi$
$443$	142420.	−	142420.i	0.725711	−	0.725711i	0.969762	+	0.244052i	$0.0784766\pi$
$444$		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	$0.364465\pi$
$449$			367200.i			1.82142i	−0.413047	+	0.910710i	$0.635535\pi$
$450$		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	0.902019	+	0.431695i	$0.142084\pi$
$457$	278545.	+	278545.i	1.33371	+	1.33371i	0.431695	+	0.902019i	$0.357916\pi$
$458$	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	−0.430126	−	0.902769i	$-0.641531\pi$
$463$	101320.	−	101320.i	0.472643	−	0.472643i	0.902769	+	0.430126i	$0.141531\pi$
$464$		−	160200.i		−	0.744092i
$465$		0			0
$466$	55250.0			0.254425
$467$	−122480.	−	122480.i	−0.561606	−	0.561606i	0.368158	−	0.929763i	$-0.379989\pi$
$467$	−122480.	−	122480.i	−0.561606	−	0.561606i	−0.929763	+	0.368158i	$0.879989\pi$
$468$		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	$0.00999040\pi$
$479$			14400.0i			0.0627612i	−0.999508	+	0.0313806i	$0.990010\pi$
$480$		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	0.0432076	−	0.999066i	$-0.486242\pi$
$487$	−226700.	−	226700.i	−0.955858	−	0.955858i	−0.999066	+	0.0432076i	$0.986242\pi$
$488$	−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	$0.150976\pi$
$499$			227448.i			0.913442i	−0.889610	+	0.456721i	$0.849024\pi$
$500$			531250.i			2.12500i
$501$		0			0
$502$	273500.	+	273500.i	1.08530	+	1.08530i
$503$	164800.	−	164800.i	0.651360	−	0.651360i	−0.301960	−	0.953321i	$-0.597641\pi$
$503$	164800.	−	164800.i	0.651360	−	0.651360i	0.953321	+	0.301960i	$0.0976410\pi$
$504$		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	$0.301624\pi$
$509$			420750.i			1.62401i	−0.583651	+	0.812005i	$0.698376\pi$
$510$		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	−0.0400314	−	0.999198i	$-0.512746\pi$
$523$	262360.	−	262360.i	0.959167	−	0.959167i	0.999198	+	0.0400314i	$0.0127458\pi$
$524$		−	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	0.817914	−	0.575341i	$-0.195131\pi$
$547$	72580.0	+	72580.0i	0.242573	+	0.242573i	−0.575341	+	0.817914i	$0.695131\pi$
$548$	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	−0.999574	−	0.0291968i	$-0.990705\pi$
$557$	−319175.	−	319175.i	−1.02877	−	1.02877i	−0.0291968	−	0.999574i	$-0.509295\pi$
$558$		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	−0.938421	−	0.345493i	$-0.887712\pi$
$563$	−187940.	+	187940.i	−0.592929	+	0.592929i	0.345493	+	0.938421i	$0.387712\pi$
$564$		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	$-0.887981\pi$
$569$		−	223200.i		−	0.689397i	0.938713	−	0.344699i	$-0.112019\pi$
$570$		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	0.999779	−	0.0210267i	$-0.00669350\pi$
$577$	325855.	+	325855.i	0.978752	+	0.978752i	−0.0210267	+	0.999779i	$0.506693\pi$
$578$	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	0.524829	−	0.851208i	$-0.324129\pi$
$587$	−112460.	−	112460.i	−0.326379	−	0.326379i	−0.851208	+	0.524829i	$0.824129\pi$
$588$		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	−0.144078	+	0.989566i	$0.546022\pi$
$593$	−398645.	+	398645.i	−1.13364	+	1.13364i	−0.989566	+	0.144078i	$0.953978\pi$
$594$		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	$0.155408\pi$
$599$			336600.i			0.938124i	−0.883165	+	0.469062i	$0.844592\pi$
$600$		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	0.114496	−	0.993424i	$-0.463475\pi$
$607$	−323840.	−	323840.i	−0.878928	−	0.878928i	−0.993424	+	0.114496i	$0.963475\pi$
$608$	−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	−0.254601	−	0.967046i	$-0.581944\pi$
$613$	267715.	−	267715.i	0.712446	−	0.712446i	0.967046	+	0.254601i	$0.0819440\pi$
$614$			537200.i			1.42495i
$615$		0			0
$616$	−720000.			−1.89745
$617$	−34085.0	−	34085.0i	−0.0895350	−	0.0895350i	0.660921	−	0.750456i	$-0.270166\pi$
$617$	−34085.0	−	34085.0i	−0.0895350	−	0.0895350i	−0.750456	+	0.660921i	$0.770166\pi$
$618$		0			0
$619$		−	126072.i		−	0.329031i	−0.986374	−	0.164516i	$-0.947394\pi$
$619$		−	126072.i		−	0.329031i	0.986374	−	0.164516i	$-0.0526061\pi$
$620$			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	−0.668928	−	0.743327i	$-0.733247\pi$
$643$	30760.0	−	30760.0i	0.0743985	−	0.0743985i	0.743327	+	0.668928i	$0.233247\pi$
$644$			924800.i			2.22985i
$645$		0			0
$646$	169200.			0.405448
$647$	166600.	+	166600.i	0.397985	+	0.397985i	0.877522	−	0.479537i	$-0.159195\pi$
$647$	166600.	+	166600.i	0.397985	+	0.397985i	−0.479537	+	0.877522i	$0.659195\pi$
$648$		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	−0.165526	−	0.986205i	$-0.552932\pi$
$653$	349945.	−	349945.i	0.820679	−	0.820679i	0.986205	+	0.165526i	$0.0529322\pi$
$654$		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	$-0.844471\pi$
$659$		−	407700.i		−	0.938793i	0.882987	−	0.469397i	$-0.155529\pi$
$660$		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	−0.932379	−	0.361483i	$-0.882270\pi$
$673$	−258575.	+	258575.i	−0.570895	+	0.570895i	0.361483	+	0.932379i	$0.382270\pi$
$674$		−	208150.i		−	0.458202i
$675$		0			0
$676$	1.88663e6			4.12850
$677$	499945.	+	499945.i	1.09080	+	1.09080i	0.995443	+	0.0953562i	$0.0303990\pi$
$677$	499945.	+	499945.i	1.09080	+	1.09080i	0.0953562	+	0.995443i	$0.469601\pi$
$678$		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	−0.932692	−	0.360674i	$-0.882547\pi$
$683$	−266840.	+	266840.i	−0.572018	+	0.572018i	0.360674	+	0.932692i	$0.382547\pi$
$684$		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	$0.156605\pi$
$709$			474912.i			0.944758i	−0.881395	+	0.472379i	$0.843395\pi$
$710$	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	$0.170390\pi$
$719$			527400.i			1.02019i	−0.860117	+	0.510097i	$0.829610\pi$
$720$		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	0.342058	−	0.939679i	$-0.388876\pi$
$727$	−315860.	−	315860.i	−0.597621	−	0.597621i	−0.939679	+	0.342058i	$0.888876\pi$
$728$	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	−0.941765	−	0.336272i	$-0.890834\pi$
$733$	−325325.	+	325325.i	−0.605494	+	0.605494i	0.336272	+	0.941765i	$0.390834\pi$
$734$		−	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	$-0.884399\pi$
$739$		−	388008.i		−	0.710480i	0.934775	−	0.355240i	$-0.115601\pi$
$740$	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	−0.900289	−	0.435294i	$-0.856645\pi$
$743$	−256700.	+	256700.i	−0.464995	+	0.464995i	0.435294	+	0.900289i	$0.356645\pi$
$744$		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	0.753209	−	0.657781i	$-0.228505\pi$
$757$	54685.0	+	54685.0i	0.0954281	+	0.0954281i	−0.657781	+	0.753209i	$0.728505\pi$
$758$	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	$0.206483\pi$
$769$			714528.i			1.20828i	−0.796879	+	0.604139i	$0.793517\pi$
$770$			1.00000e6i			1.68663i
$771$		0			0
$772$	−1.05485e6	−	1.05485e6i	−1.76993	−	1.76993i
$773$	360055.	−	360055.i	0.602573	−	0.602573i	−0.338421	−	0.940995i	$-0.609893\pi$
$773$	360055.	−	360055.i	0.602573	−	0.602573i	0.940995	+	0.338421i	$0.109893\pi$
$774$		0			0
$775$	267500.			0.445369
$776$	290700.			0.482749
$777$		0			0
$778$	504000.	−	504000.i	0.832667	−	0.832667i
$779$			68400.0i			0.112715i
$780$		0			0

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)

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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more