LMFDB - Embedded newform 60.3.g.a.41.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,3,Mod(41,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("60.41");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	60.g (of order $2$, degree $1$, 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:	$1.63488158616$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.1
Root			$-2.23607i$ of defining polynomial
Character	$\chi$	$=$	60.41
Dual form			60.3.g.a.41.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 2.23607i) q^{3} -2.23607i q^{5} +2.00000 q^{7} +(-1.00000 - 8.94427i) q^{9} +O(q^{10})$	\(q+(2.00000 - 2.23607i) q^{3} -2.23607i q^{5} +2.00000 q^{7} +(-1.00000 - 8.94427i) q^{9} +13.4164i q^{11} +8.00000 q^{13} +(-5.00000 - 4.47214i) q^{15} +13.4164i q^{17} -34.0000 q^{19} +(4.00000 - 4.47214i) q^{21} +40.2492i q^{23} -5.00000 q^{25} +(-22.0000 - 15.6525i) q^{27} -40.2492i q^{29} +14.0000 q^{31} +(30.0000 + 26.8328i) q^{33} -4.47214i q^{35} +56.0000 q^{37} +(16.0000 - 17.8885i) q^{39} -26.8328i q^{41} +8.00000 q^{43} +(-20.0000 + 2.23607i) q^{45} -40.2492i q^{47} -45.0000 q^{49} +(30.0000 + 26.8328i) q^{51} -40.2492i q^{53} +30.0000 q^{55} +(-68.0000 + 76.0263i) q^{57} +13.4164i q^{59} -46.0000 q^{61} +(-2.00000 - 17.8885i) q^{63} -17.8885i q^{65} +32.0000 q^{67} +(90.0000 + 80.4984i) q^{69} +53.6656i q^{71} -106.000 q^{73} +(-10.0000 + 11.1803i) q^{75} +26.8328i q^{77} -22.0000 q^{79} +(-79.0000 + 17.8885i) q^{81} -120.748i q^{83} +30.0000 q^{85} +(-90.0000 - 80.4984i) q^{87} +107.331i q^{89} +16.0000 q^{91} +(28.0000 - 31.3050i) q^{93} +76.0263i q^{95} +122.000 q^{97} +(120.000 - 13.4164i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9	$ 2 q + 4 q^{3} + 4 q^{7} - 2 q^{9} + 16 q^{13} - 10 q^{15} - 68 q^{19} + 8 q^{21} - 10 q^{25} - 44 q^{27} + 28 q^{31} + 60 q^{33} + 112 q^{37} + 32 q^{39} + 16 q^{43} - 40 q^{45} - 90 q^{49} + 60 q^{51} + 60 q^{55} - 136 q^{57} - 92 q^{61} - 4 q^{63} + 64 q^{67} + 180 q^{69} - 212 q^{73} - 20 q^{75} - 44 q^{79} - 158 q^{81} + 60 q^{85} - 180 q^{87} + 32 q^{91} + 56 q^{93} + 244 q^{97} + 240 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9 + 16 * q^13 - 10 * q^15 - 68 * q^19 + 8 * q^21 - 10 * q^25 - 44 * q^27 + 28 * q^31 + 60 * q^33 + 112 * q^37 + 32 * q^39 + 16 * q^43 - 40 * q^45 - 90 * q^49 + 60 * q^51 + 60 * q^55 - 136 * q^57 - 92 * q^61 - 4 * q^63 + 64 * q^67 + 180 * q^69 - 212 * q^73 - 20 * q^75 - 44 * q^79 - 158 * q^81 + 60 * q^85 - 180 * q^87 + 32 * q^91 + 56 * q^93 + 244 * q^97 + 240 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (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 $

$2$		0			0
$2$		0			0
$3$	2.00000	−	2.23607i	0.666667	−	0.745356i
$3$	2.00000	−	2.23607i	0.666667	−	0.745356i
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	2.00000			0.285714			0.142857	−	0.989743i	$-0.454371\pi$
$7$	2.00000			0.285714			0.142857	+	0.989743i	$0.454371\pi$
$8$		0			0
$9$	−1.00000	−	8.94427i	−0.111111	−	0.993808i
$10$		0			0
$11$			13.4164i			1.21967i	0.792527	+	0.609837i	$0.208765\pi$
$11$			13.4164i			1.21967i	−0.792527	+	0.609837i	$0.791235\pi$
$12$		0			0
$13$	8.00000			0.615385			0.307692	−	0.951486i	$-0.400443\pi$
$13$	8.00000			0.615385			0.307692	+	0.951486i	$0.400443\pi$
$14$		0			0
$15$	−5.00000	−	4.47214i	−0.333333	−	0.298142i
$16$		0			0
$17$			13.4164i			0.789200i	0.918853	+	0.394600i	$0.129117\pi$
$17$			13.4164i			0.789200i	−0.918853	+	0.394600i	$0.870883\pi$
$18$		0			0
$19$	−34.0000			−1.78947			−0.894737	−	0.446594i	$-0.852637\pi$
$19$	−34.0000			−1.78947			−0.894737	+	0.446594i	$0.852637\pi$
$20$		0			0
$21$	4.00000	−	4.47214i	0.190476	−	0.212959i
$22$		0			0
$23$			40.2492i			1.74997i	0.484153	+	0.874983i	$0.339128\pi$
$23$			40.2492i			1.74997i	−0.484153	+	0.874983i	$0.660872\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−22.0000	−	15.6525i	−0.814815	−	0.579721i
$28$		0			0
$29$		−	40.2492i		−	1.38790i	−0.720021	−	0.693952i	$-0.755868\pi$
$29$		−	40.2492i		−	1.38790i	0.720021	−	0.693952i	$-0.244132\pi$
$30$		0			0
$31$	14.0000			0.451613			0.225806	−	0.974172i	$-0.427498\pi$
$31$	14.0000			0.451613			0.225806	+	0.974172i	$0.427498\pi$
$32$		0			0
$33$	30.0000	+	26.8328i	0.909091	+	0.813116i
$34$		0			0
$35$		−	4.47214i		−	0.127775i
$36$		0			0
$37$	56.0000			1.51351			0.756757	−	0.653697i	$-0.226783\pi$
$37$	56.0000			1.51351			0.756757	+	0.653697i	$0.226783\pi$
$38$		0			0
$39$	16.0000	−	17.8885i	0.410256	−	0.458681i
$40$		0			0
$41$		−	26.8328i		−	0.654459i	−0.944945	−	0.327229i	$-0.893885\pi$
$41$		−	26.8328i		−	0.654459i	0.944945	−	0.327229i	$-0.106115\pi$
$42$		0			0
$43$	8.00000			0.186047			0.0930233	−	0.995664i	$-0.470347\pi$
$43$	8.00000			0.186047			0.0930233	+	0.995664i	$0.470347\pi$
$44$		0			0
$45$	−20.0000	+	2.23607i	−0.444444	+	0.0496904i
$46$		0			0
$47$		−	40.2492i		−	0.856366i	−0.903692	−	0.428183i	$-0.859154\pi$
$47$		−	40.2492i		−	0.856366i	0.903692	−	0.428183i	$-0.140846\pi$
$48$		0			0
$49$	−45.0000			−0.918367
$50$		0			0
$51$	30.0000	+	26.8328i	0.588235	+	0.526134i
$52$		0			0
$53$		−	40.2492i		−	0.759419i	−0.925106	−	0.379710i	$-0.876024\pi$
$53$		−	40.2492i		−	0.759419i	0.925106	−	0.379710i	$-0.123976\pi$
$54$		0			0
$55$	30.0000			0.545455
$56$		0			0
$57$	−68.0000	+	76.0263i	−1.19298	+	1.33379i
$58$		0			0
$59$			13.4164i			0.227397i	0.993515	+	0.113698i	$0.0362697\pi$
$59$			13.4164i			0.227397i	−0.993515	+	0.113698i	$0.963730\pi$
$60$		0			0
$61$	−46.0000			−0.754098			−0.377049	−	0.926193i	$-0.623061\pi$
$61$	−46.0000			−0.754098			−0.377049	+	0.926193i	$0.623061\pi$
$62$		0			0
$63$	−2.00000	−	17.8885i	−0.0317460	−	0.283945i
$64$		0			0
$65$		−	17.8885i		−	0.275208i
$66$		0			0
$67$	32.0000			0.477612			0.238806	−	0.971067i	$-0.423244\pi$
$67$	32.0000			0.477612			0.238806	+	0.971067i	$0.423244\pi$
$68$		0			0
$69$	90.0000	+	80.4984i	1.30435	+	1.16664i
$70$		0			0
$71$			53.6656i			0.755854i	0.925835	+	0.377927i	$0.123363\pi$
$71$			53.6656i			0.755854i	−0.925835	+	0.377927i	$0.876637\pi$
$72$		0			0
$73$	−106.000			−1.45205			−0.726027	−	0.687666i	$-0.758635\pi$
$73$	−106.000			−1.45205			−0.726027	+	0.687666i	$0.758635\pi$
$74$		0			0
$75$	−10.0000	+	11.1803i	−0.133333	+	0.149071i
$76$		0			0
$77$			26.8328i			0.348478i
$78$		0			0
$79$	−22.0000			−0.278481			−0.139241	−	0.990259i	$-0.544466\pi$
$79$	−22.0000			−0.278481			−0.139241	+	0.990259i	$0.544466\pi$
$80$		0			0
$81$	−79.0000	+	17.8885i	−0.975309	+	0.220846i
$82$		0			0
$83$		−	120.748i		−	1.45479i	−0.686218	−	0.727396i	$-0.740731\pi$
$83$		−	120.748i		−	1.45479i	0.686218	−	0.727396i	$-0.259269\pi$
$84$		0			0
$85$	30.0000			0.352941
$86$		0			0
$87$	−90.0000	−	80.4984i	−1.03448	−	0.925270i
$88$		0			0
$89$			107.331i			1.20597i	0.797753	+	0.602985i	$0.206022\pi$
$89$			107.331i			1.20597i	−0.797753	+	0.602985i	$0.793978\pi$
$90$		0			0
$91$	16.0000			0.175824
$92$		0			0
$93$	28.0000	−	31.3050i	0.301075	−	0.336612i
$94$		0			0
$95$			76.0263i			0.800277i
$96$		0			0
$97$	122.000			1.25773			0.628866	−	0.777514i	$-0.283519\pi$
$97$	122.000			1.25773			0.628866	+	0.777514i	$0.283519\pi$
$98$		0			0
$99$	120.000	−	13.4164i	1.21212	−	0.135519i
$100$		0			0
$101$		−	174.413i		−	1.72686i	−0.504465	−	0.863432i	$-0.668310\pi$
$101$		−	174.413i		−	1.72686i	0.504465	−	0.863432i	$-0.331690\pi$
$102$		0			0
$103$	−46.0000			−0.446602			−0.223301	−	0.974750i	$-0.571683\pi$
$103$	−46.0000			−0.446602			−0.223301	+	0.974750i	$0.571683\pi$
$104$		0			0
$105$	−10.0000	−	8.94427i	−0.0952381	−	0.0851835i
$106$		0			0
$107$			13.4164i			0.125387i	0.998033	+	0.0626935i	$0.0199691\pi$
$107$			13.4164i			0.125387i	−0.998033	+	0.0626935i	$0.980031\pi$
$108$		0			0
$109$	86.0000			0.788991			0.394495	−	0.918898i	$-0.370919\pi$
$109$	86.0000			0.788991			0.394495	+	0.918898i	$0.370919\pi$
$110$		0			0
$111$	112.000	−	125.220i	1.00901	−	1.12811i
$112$		0			0
$113$			93.9149i			0.831105i	0.909569	+	0.415552i	$0.136412\pi$
$113$			93.9149i			0.831105i	−0.909569	+	0.415552i	$0.863588\pi$
$114$		0			0
$115$	90.0000			0.782609
$116$		0			0
$117$	−8.00000	−	71.5542i	−0.0683761	−	0.611574i
$118$		0			0
$119$			26.8328i			0.225486i
$120$		0			0
$121$	−59.0000			−0.487603
$122$		0			0
$123$	−60.0000	−	53.6656i	−0.487805	−	0.436306i
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−34.0000			−0.267717			−0.133858	−	0.991000i	$-0.542737\pi$
$127$	−34.0000			−0.267717			−0.133858	+	0.991000i	$0.542737\pi$
$128$		0			0
$129$	16.0000	−	17.8885i	0.124031	−	0.138671i
$130$		0			0
$131$			147.580i			1.12657i	0.826263	+	0.563284i	$0.190462\pi$
$131$			147.580i			1.12657i	−0.826263	+	0.563284i	$0.809538\pi$
$132$		0			0
$133$	−68.0000			−0.511278
$134$		0			0
$135$	−35.0000	+	49.1935i	−0.259259	+	0.364396i
$136$		0			0
$137$		−	40.2492i		−	0.293790i	−0.989152	−	0.146895i	$-0.953072\pi$
$137$		−	40.2492i		−	0.293790i	0.989152	−	0.146895i	$-0.0469279\pi$
$138$		0			0
$139$	−82.0000			−0.589928			−0.294964	−	0.955508i	$-0.595308\pi$
$139$	−82.0000			−0.589928			−0.294964	+	0.955508i	$0.595308\pi$
$140$		0			0
$141$	−90.0000	−	80.4984i	−0.638298	−	0.570911i
$142$		0			0
$143$			107.331i			0.750568i
$144$		0			0
$145$	−90.0000			−0.620690
$146$		0			0
$147$	−90.0000	+	100.623i	−0.612245	+	0.684511i
$148$		0			0
$149$			147.580i			0.990473i	0.868758	+	0.495237i	$0.164919\pi$
$149$			147.580i			0.990473i	−0.868758	+	0.495237i	$0.835081\pi$
$150$		0			0
$151$	−46.0000			−0.304636			−0.152318	−	0.988332i	$-0.548674\pi$
$151$	−46.0000			−0.304636			−0.152318	+	0.988332i	$0.548674\pi$
$152$		0			0
$153$	120.000	−	13.4164i	0.784314	−	0.0876889i
$154$		0			0
$155$		−	31.3050i		−	0.201967i
$156$		0			0
$157$	92.0000			0.585987			0.292994	−	0.956114i	$-0.405349\pi$
$157$	92.0000			0.585987			0.292994	+	0.956114i	$0.405349\pi$
$158$		0			0
$159$	−90.0000	−	80.4984i	−0.566038	−	0.506280i
$160$		0			0
$161$			80.4984i			0.499990i
$162$		0			0
$163$	68.0000			0.417178			0.208589	−	0.978003i	$-0.433113\pi$
$163$	68.0000			0.417178			0.208589	+	0.978003i	$0.433113\pi$
$164$		0			0
$165$	60.0000	−	67.0820i	0.363636	−	0.406558i
$166$		0			0
$167$			67.0820i			0.401689i	0.979623	+	0.200844i	$0.0643686\pi$
$167$			67.0820i			0.401689i	−0.979623	+	0.200844i	$0.935631\pi$
$168$		0			0
$169$	−105.000			−0.621302
$170$		0			0
$171$	34.0000	+	304.105i	0.198830	+	1.77839i
$172$		0			0
$173$		−	120.748i		−	0.697963i	−0.937130	−	0.348982i	$-0.886528\pi$
$173$		−	120.748i		−	0.697963i	0.937130	−	0.348982i	$-0.113472\pi$
$174$		0			0
$175$	−10.0000			−0.0571429
$176$		0			0
$177$	30.0000	+	26.8328i	0.169492	+	0.151598i
$178$		0			0
$179$		−	281.745i		−	1.57399i	−0.616958	−	0.786996i	$-0.711635\pi$
$179$		−	281.745i		−	1.57399i	0.616958	−	0.786996i	$-0.288365\pi$
$180$		0			0
$181$	194.000			1.07182			0.535912	−	0.844274i	$-0.319968\pi$
$181$	194.000			1.07182			0.535912	+	0.844274i	$0.319968\pi$
$182$		0			0
$183$	−92.0000	+	102.859i	−0.502732	+	0.562072i
$184$		0			0
$185$		−	125.220i		−	0.676864i
$186$		0			0
$187$	−180.000			−0.962567
$188$		0			0
$189$	−44.0000	−	31.3050i	−0.232804	−	0.165635i
$190$		0			0
$191$		−	80.4984i		−	0.421458i	−0.977545	−	0.210729i	$-0.932416\pi$
$191$		−	80.4984i		−	0.421458i	0.977545	−	0.210729i	$-0.0675837\pi$
$192$		0			0
$193$	218.000			1.12953			0.564767	−	0.825251i	$-0.308966\pi$
$193$	218.000			1.12953			0.564767	+	0.825251i	$0.308966\pi$
$194$		0			0
$195$	−40.0000	−	35.7771i	−0.205128	−	0.183472i
$196$		0			0
$197$			93.9149i			0.476725i	0.971176	+	0.238363i	$0.0766107\pi$
$197$			93.9149i			0.476725i	−0.971176	+	0.238363i	$0.923389\pi$
$198$		0			0
$199$	−34.0000			−0.170854			−0.0854271	−	0.996344i	$-0.527225\pi$
$199$	−34.0000			−0.170854			−0.0854271	+	0.996344i	$0.527225\pi$
$200$		0			0
$201$	64.0000	−	71.5542i	0.318408	−	0.355991i
$202$		0			0
$203$		−	80.4984i		−	0.396544i
$204$		0			0
$205$	−60.0000			−0.292683
$206$		0			0
$207$	360.000	−	40.2492i	1.73913	−	0.194441i
$208$		0			0
$209$		−	456.158i		−	2.18257i
$210$		0			0
$211$	−46.0000			−0.218009			−0.109005	−	0.994041i	$-0.534766\pi$
$211$	−46.0000			−0.218009			−0.109005	+	0.994041i	$0.534766\pi$
$212$		0			0
$213$	120.000	+	107.331i	0.563380	+	0.503903i
$214$		0			0
$215$		−	17.8885i		−	0.0832025i
$216$		0			0
$217$	28.0000			0.129032
$218$		0			0
$219$	−212.000	+	237.023i	−0.968037	+	1.08230i
$220$		0			0
$221$			107.331i			0.485662i
$222$		0			0
$223$	398.000			1.78475			0.892377	−	0.451291i	$-0.149036\pi$
$223$	398.000			1.78475			0.892377	+	0.451291i	$0.149036\pi$
$224$		0			0
$225$	5.00000	+	44.7214i	0.0222222	+	0.198762i
$226$		0			0
$227$		−	335.410i		−	1.47758i	−0.673937	−	0.738789i	$-0.735398\pi$
$227$		−	335.410i		−	1.47758i	0.673937	−	0.738789i	$-0.264602\pi$
$228$		0			0
$229$	86.0000			0.375546			0.187773	−	0.982212i	$-0.439873\pi$
$229$	86.0000			0.375546			0.187773	+	0.982212i	$0.439873\pi$
$230$		0			0
$231$	60.0000	+	53.6656i	0.259740	+	0.232319i
$232$		0			0
$233$			308.577i			1.32437i	0.749342	+	0.662183i	$0.230370\pi$
$233$			308.577i			1.32437i	−0.749342	+	0.662183i	$0.769630\pi$
$234$		0			0
$235$	−90.0000			−0.382979
$236$		0			0
$237$	−44.0000	+	49.1935i	−0.185654	+	0.207567i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	134.000			0.556017			0.278008	−	0.960579i	$-0.410326\pi$
$241$	134.000			0.556017			0.278008	+	0.960579i	$0.410326\pi$
$242$		0			0
$243$	−118.000	+	212.426i	−0.485597	+	0.874183i
$244$		0			0
$245$			100.623i			0.410706i
$246$		0			0
$247$	−272.000			−1.10121
$248$		0			0
$249$	−270.000	−	241.495i	−1.08434	−	0.969861i
$250$		0			0
$251$		−	308.577i		−	1.22939i	−0.788764	−	0.614696i	$-0.789279\pi$
$251$		−	308.577i		−	1.22939i	0.788764	−	0.614696i	$-0.210721\pi$
$252$		0			0
$253$	−540.000			−2.13439
$254$		0			0
$255$	60.0000	−	67.0820i	0.235294	−	0.263067i
$256$		0			0
$257$			174.413i			0.678651i	0.940669	+	0.339325i	$0.110199\pi$
$257$			174.413i			0.678651i	−0.940669	+	0.339325i	$0.889801\pi$
$258$		0			0
$259$	112.000			0.432432
$260$		0			0
$261$	−360.000	+	40.2492i	−1.37931	+	0.154212i
$262$		0			0
$263$			254.912i			0.969246i	0.874723	+	0.484623i	$0.161043\pi$
$263$			254.912i			0.969246i	−0.874723	+	0.484623i	$0.838957\pi$
$264$		0			0
$265$	−90.0000			−0.339623
$266$		0			0
$267$	240.000	+	214.663i	0.898876	+	0.803979i
$268$		0			0
$269$			335.410i			1.24688i	0.781872	+	0.623439i	$0.214265\pi$
$269$			335.410i			1.24688i	−0.781872	+	0.623439i	$0.785735\pi$
$270$		0			0
$271$	2.00000			0.00738007			0.00369004	−	0.999993i	$-0.498825\pi$
$271$	2.00000			0.00738007			0.00369004	+	0.999993i	$0.498825\pi$
$272$		0			0
$273$	32.0000	−	35.7771i	0.117216	−	0.131052i
$274$		0			0
$275$		−	67.0820i		−	0.243935i
$276$		0			0
$277$	−448.000			−1.61733			−0.808664	−	0.588270i	$-0.799809\pi$
$277$	−448.000			−1.61733			−0.808664	+	0.588270i	$0.799809\pi$
$278$		0			0
$279$	−14.0000	−	125.220i	−0.0501792	−	0.448817i
$280$		0			0
$281$			187.830i			0.668433i	0.942496	+	0.334217i	$0.108472\pi$
$281$			187.830i			0.668433i	−0.942496	+	0.334217i	$0.891528\pi$
$282$		0			0
$283$	248.000			0.876325			0.438163	−	0.898896i	$-0.355629\pi$
$283$	248.000			0.876325			0.438163	+	0.898896i	$0.355629\pi$
$284$		0			0
$285$	170.000	+	152.053i	0.596491	+	0.533518i
$286$		0			0
$287$		−	53.6656i		−	0.186988i
$288$		0			0
$289$	109.000			0.377163
$290$		0			0
$291$	244.000	−	272.800i	0.838488	−	0.937458i
$292$		0			0
$293$		−	147.580i		−	0.503688i	−0.967768	−	0.251844i	$-0.918963\pi$
$293$		−	147.580i		−	0.503688i	0.967768	−	0.251844i	$-0.0810369\pi$
$294$		0			0
$295$	30.0000			0.101695
$296$		0			0
$297$	210.000	−	295.161i	0.707071	−	0.993808i
$298$		0			0
$299$			321.994i			1.07690i
$300$		0			0
$301$	16.0000			0.0531561
$302$		0			0
$303$	−390.000	−	348.827i	−1.28713	−	1.15124i
$304$		0			0
$305$			102.859i			0.337243i
$306$		0			0
$307$	272.000			0.885993			0.442997	−	0.896523i	$-0.353915\pi$
$307$	272.000			0.885993			0.442997	+	0.896523i	$0.353915\pi$
$308$		0			0
$309$	−92.0000	+	102.859i	−0.297735	+	0.332877i
$310$		0			0
$311$			53.6656i			0.172558i	0.996271	+	0.0862792i	$0.0274977\pi$
$311$			53.6656i			0.172558i	−0.996271	+	0.0862792i	$0.972502\pi$
$312$		0			0
$313$	278.000			0.888179			0.444089	−	0.895982i	$-0.353527\pi$
$313$	278.000			0.888179			0.444089	+	0.895982i	$0.353527\pi$
$314$		0			0
$315$	−40.0000	+	4.47214i	−0.126984	+	0.0141973i
$316$		0			0
$317$			120.748i			0.380907i	0.981696	+	0.190454i	$0.0609959\pi$
$317$			120.748i			0.380907i	−0.981696	+	0.190454i	$0.939004\pi$
$318$		0			0
$319$	540.000			1.69279
$320$		0			0
$321$	30.0000	+	26.8328i	0.0934579	+	0.0835913i
$322$		0			0
$323$		−	456.158i		−	1.41225i
$324$		0			0
$325$	−40.0000			−0.123077
$326$		0			0
$327$	172.000	−	192.302i	0.525994	−	0.588079i
$328$		0			0
$329$		−	80.4984i		−	0.244676i
$330$		0			0
$331$	−598.000			−1.80665			−0.903323	−	0.428960i	$-0.858880\pi$
$331$	−598.000			−1.80665			−0.903323	+	0.428960i	$0.858880\pi$
$332$		0			0
$333$	−56.0000	−	500.879i	−0.168168	−	1.50414i
$334$		0			0
$335$		−	71.5542i		−	0.213595i
$336$		0			0
$337$	−358.000			−1.06231			−0.531157	−	0.847273i	$-0.678243\pi$
$337$	−358.000			−1.06231			−0.531157	+	0.847273i	$0.678243\pi$
$338$		0			0
$339$	210.000	+	187.830i	0.619469	+	0.554070i
$340$		0			0
$341$			187.830i			0.550820i
$342$		0			0
$343$	−188.000			−0.548105
$344$		0			0
$345$	180.000	−	201.246i	0.521739	−	0.583322i
$346$		0			0
$347$			254.912i			0.734616i	0.930099	+	0.367308i	$0.119720\pi$
$347$			254.912i			0.734616i	−0.930099	+	0.367308i	$0.880280\pi$
$348$		0			0
$349$	518.000			1.48424			0.742120	−	0.670267i	$-0.233820\pi$
$349$	518.000			1.48424			0.742120	+	0.670267i	$0.233820\pi$
$350$		0			0
$351$	−176.000	−	125.220i	−0.501425	−	0.356752i
$352$		0			0
$353$		−	281.745i		−	0.798143i	−0.916920	−	0.399072i	$-0.869332\pi$
$353$		−	281.745i		−	0.798143i	0.916920	−	0.399072i	$-0.130668\pi$
$354$		0			0
$355$	120.000			0.338028
$356$		0			0
$357$	60.0000	+	53.6656i	0.168067	+	0.150324i
$358$		0			0
$359$			348.827i			0.971662i	0.874053	+	0.485831i	$0.161483\pi$
$359$			348.827i			0.971662i	−0.874053	+	0.485831i	$0.838517\pi$
$360$		0			0
$361$	795.000			2.20222
$362$		0			0
$363$	−118.000	+	131.928i	−0.325069	+	0.363438i
$364$		0			0
$365$			237.023i			0.649379i
$366$		0			0
$367$	−178.000			−0.485014			−0.242507	−	0.970150i	$-0.577970\pi$
$367$	−178.000			−0.485014			−0.242507	+	0.970150i	$0.577970\pi$
$368$		0			0
$369$	−240.000	+	26.8328i	−0.650407	+	0.0727177i
$370$		0			0
$371$		−	80.4984i		−	0.216977i
$372$		0			0
$373$	−532.000			−1.42627			−0.713137	−	0.701025i	$-0.752726\pi$
$373$	−532.000			−1.42627			−0.713137	+	0.701025i	$0.752726\pi$
$374$		0			0
$375$	25.0000	+	22.3607i	0.0666667	+	0.0596285i
$376$		0			0
$377$		−	321.994i		−	0.854095i
$378$		0			0
$379$	86.0000			0.226913			0.113456	−	0.993543i	$-0.463808\pi$
$379$	86.0000			0.226913			0.113456	+	0.993543i	$0.463808\pi$
$380$		0			0
$381$	−68.0000	+	76.0263i	−0.178478	+	0.199544i
$382$		0			0
$383$		−	120.748i		−	0.315268i	−0.987498	−	0.157634i	$-0.949613\pi$
$383$		−	120.748i		−	0.315268i	0.987498	−	0.157634i	$-0.0503866\pi$
$384$		0			0
$385$	60.0000			0.155844
$386$		0			0
$387$	−8.00000	−	71.5542i	−0.0206718	−	0.184895i
$388$		0			0
$389$			415.909i			1.06917i	0.845113	+	0.534587i	$0.179533\pi$
$389$			415.909i			1.06917i	−0.845113	+	0.534587i	$0.820467\pi$
$390$		0			0
$391$	−540.000			−1.38107
$392$		0			0
$393$	330.000	+	295.161i	0.839695	+	0.751046i
$394$		0			0
$395$			49.1935i			0.124540i
$396$		0			0
$397$	−28.0000			−0.0705290			−0.0352645	−	0.999378i	$-0.511227\pi$
$397$	−28.0000			−0.0705290			−0.0352645	+	0.999378i	$0.511227\pi$
$398$		0			0
$399$	−136.000	+	152.053i	−0.340852	+	0.381084i
$400$		0			0
$401$		−	268.328i		−	0.669148i	−0.942370	−	0.334574i	$-0.891408\pi$
$401$		−	268.328i		−	0.669148i	0.942370	−	0.334574i	$-0.108592\pi$
$402$		0			0
$403$	112.000			0.277916
$404$		0			0
$405$	40.0000	+	176.649i	0.0987654	+	0.436171i
$406$		0			0
$407$			751.319i			1.84599i
$408$		0			0
$409$	−214.000			−0.523227			−0.261614	−	0.965173i	$-0.584255\pi$
$409$	−214.000			−0.523227			−0.261614	+	0.965173i	$0.584255\pi$
$410$		0			0
$411$	−90.0000	−	80.4984i	−0.218978	−	0.195860i
$412$		0			0
$413$			26.8328i			0.0649705i
$414$		0			0
$415$	−270.000			−0.650602
$416$		0			0
$417$	−164.000	+	183.358i	−0.393285	+	0.439706i
$418$		0			0
$419$		−	174.413i		−	0.416261i	−0.978101	−	0.208130i	$-0.933262\pi$
$419$		−	174.413i		−	0.416261i	0.978101	−	0.208130i	$-0.0667379\pi$
$420$		0			0
$421$	−238.000			−0.565321			−0.282660	−	0.959220i	$-0.591217\pi$
$421$	−238.000			−0.565321			−0.282660	+	0.959220i	$0.591217\pi$
$422$		0			0
$423$	−360.000	+	40.2492i	−0.851064	+	0.0951518i
$424$		0			0
$425$		−	67.0820i		−	0.157840i
$426$		0			0
$427$	−92.0000			−0.215457
$428$		0			0
$429$	240.000	+	214.663i	0.559441	+	0.500379i
$430$		0			0
$431$		−	241.495i		−	0.560314i	−0.959954	−	0.280157i	$-0.909613\pi$
$431$		−	241.495i		−	0.560314i	0.959954	−	0.280157i	$-0.0903865\pi$
$432$		0			0
$433$	−382.000			−0.882217			−0.441109	−	0.897454i	$-0.645415\pi$
$433$	−382.000			−0.882217			−0.441109	+	0.897454i	$0.645415\pi$
$434$		0			0
$435$	−180.000	+	201.246i	−0.413793	+	0.462635i
$436$		0			0
$437$		−	1368.47i		−	3.13152i
$438$		0			0
$439$	−274.000			−0.624146			−0.312073	−	0.950058i	$-0.601023\pi$
$439$	−274.000			−0.624146			−0.312073	+	0.950058i	$0.601023\pi$
$440$		0			0
$441$	45.0000	+	402.492i	0.102041	+	0.912681i
$442$		0			0
$443$			764.735i			1.72626i	0.504978	+	0.863132i	$0.331501\pi$
$443$			764.735i			1.72626i	−0.504978	+	0.863132i	$0.668499\pi$
$444$		0			0
$445$	240.000			0.539326
$446$		0			0
$447$	330.000	+	295.161i	0.738255	+	0.660315i
$448$		0			0
$449$		−	778.152i		−	1.73308i	−0.499110	−	0.866539i	$-0.666340\pi$
$449$		−	778.152i		−	1.73308i	0.499110	−	0.866539i	$-0.333660\pi$
$450$		0			0
$451$	360.000			0.798226
$452$		0			0
$453$	−92.0000	+	102.859i	−0.203091	+	0.227062i
$454$		0			0
$455$		−	35.7771i		−	0.0786310i
$456$		0			0
$457$	446.000			0.975930			0.487965	−	0.872863i	$-0.337739\pi$
$457$	446.000			0.975930			0.487965	+	0.872863i	$0.337739\pi$
$458$		0			0
$459$	210.000	−	295.161i	0.457516	−	0.643052i
$460$		0			0
$461$		−	93.9149i		−	0.203720i	−0.994799	−	0.101860i	$-0.967521\pi$
$461$		−	93.9149i		−	0.203720i	0.994799	−	0.101860i	$-0.0324794\pi$
$462$		0			0
$463$	854.000			1.84449			0.922246	−	0.386603i	$-0.126352\pi$
$463$	854.000			1.84449			0.922246	+	0.386603i	$0.126352\pi$
$464$		0			0
$465$	−70.0000	−	62.6099i	−0.150538	−	0.134645i
$466$		0			0
$467$			13.4164i			0.0287289i	0.999897	+	0.0143645i	$0.00457251\pi$
$467$			13.4164i			0.0287289i	−0.999897	+	0.0143645i	$0.995427\pi$
$468$		0			0
$469$	64.0000			0.136461
$470$		0			0
$471$	184.000	−	205.718i	0.390658	−	0.436769i
$472$		0			0
$473$			107.331i			0.226916i
$474$		0			0
$475$	170.000			0.357895
$476$		0			0
$477$	−360.000	+	40.2492i	−0.754717	+	0.0843799i
$478$		0			0
$479$			53.6656i			0.112037i	0.998430	+	0.0560184i	$0.0178406\pi$
$479$			53.6656i			0.112037i	−0.998430	+	0.0560184i	$0.982159\pi$
$480$		0			0
$481$	448.000			0.931393
$482$		0			0
$483$	180.000	+	160.997i	0.372671	+	0.333327i
$484$		0			0
$485$		−	272.800i		−	0.562475i
$486$		0			0
$487$	2.00000			0.00410678			0.00205339	−	0.999998i	$-0.499346\pi$
$487$	2.00000			0.00410678			0.00205339	+	0.999998i	$0.499346\pi$
$488$		0			0
$489$	136.000	−	152.053i	0.278119	−	0.310946i
$490$		0			0
$491$		−	469.574i		−	0.956363i	−0.878261	−	0.478182i	$-0.841296\pi$
$491$		−	469.574i		−	0.956363i	0.878261	−	0.478182i	$-0.158704\pi$
$492$		0			0
$493$	540.000			1.09533
$494$		0			0
$495$	−30.0000	−	268.328i	−0.0606061	−	0.542077i
$496$		0			0
$497$			107.331i			0.215958i
$498$		0			0
$499$	−514.000			−1.03006			−0.515030	−	0.857172i	$-0.672219\pi$
$499$	−514.000			−1.03006			−0.515030	+	0.857172i	$0.672219\pi$
$500$		0			0
$501$	150.000	+	134.164i	0.299401	+	0.267793i
$502$		0			0
$503$		−	657.404i		−	1.30697i	−0.756941	−	0.653483i	$-0.773307\pi$
$503$		−	657.404i		−	1.30697i	0.756941	−	0.653483i	$-0.226693\pi$
$504$		0			0
$505$	−390.000			−0.772277
$506$		0			0
$507$	−210.000	+	234.787i	−0.414201	+	0.463091i
$508$		0			0
$509$		−	308.577i		−	0.606242i	−0.952952	−	0.303121i	$-0.901971\pi$
$509$		−	308.577i		−	0.606242i	0.952952	−	0.303121i	$-0.0980287\pi$
$510$		0			0
$511$	−212.000			−0.414873
$512$		0			0
$513$	748.000	+	532.184i	1.45809	+	1.03740i
$514$		0			0
$515$			102.859i			0.199726i
$516$		0			0
$517$	540.000			1.04449
$518$		0			0
$519$	−270.000	−	241.495i	−0.520231	−	0.465309i
$520$		0			0
$521$			670.820i			1.28756i	0.765209	+	0.643782i	$0.222635\pi$
$521$			670.820i			1.28756i	−0.765209	+	0.643782i	$0.777365\pi$
$522$		0			0
$523$	−832.000			−1.59082			−0.795411	−	0.606070i	$-0.792745\pi$
$523$	−832.000			−1.59082			−0.795411	+	0.606070i	$0.792745\pi$
$524$		0			0
$525$	−20.0000	+	22.3607i	−0.0380952	+	0.0425918i
$526$		0			0
$527$			187.830i			0.356413i
$528$		0			0
$529$	−1091.00			−2.06238
$530$		0			0
$531$	120.000	−	13.4164i	0.225989	−	0.0252663i
$532$		0			0
$533$		−	214.663i		−	0.402744i
$534$		0			0
$535$	30.0000			0.0560748
$536$		0			0
$537$	−630.000	−	563.489i	−1.17318	−	1.04933i
$538$		0			0
$539$		−	603.738i		−	1.12011i
$540$		0			0
$541$	314.000			0.580407			0.290203	−	0.956965i	$-0.406277\pi$
$541$	314.000			0.580407			0.290203	+	0.956965i	$0.406277\pi$
$542$		0			0
$543$	388.000	−	433.797i	0.714549	−	0.798890i
$544$		0			0
$545$		−	192.302i		−	0.352847i
$546$		0			0
$547$	536.000			0.979890			0.489945	−	0.871753i	$-0.337017\pi$
$547$	536.000			0.979890			0.489945	+	0.871753i	$0.337017\pi$
$548$		0			0
$549$	46.0000	+	411.437i	0.0837887	+	0.749429i
$550$		0			0
$551$			1368.47i			2.48362i
$552$		0			0
$553$	−44.0000			−0.0795660
$554$		0			0
$555$	−280.000	−	250.440i	−0.504505	−	0.451243i
$556$		0			0
$557$			898.899i			1.61382i	0.590672	+	0.806911i	$0.298863\pi$
$557$			898.899i			1.61382i	−0.590672	+	0.806911i	$0.701137\pi$
$558$		0			0
$559$	64.0000			0.114490
$560$		0			0
$561$	−360.000	+	402.492i	−0.641711	+	0.717455i
$562$		0			0
$563$			818.401i			1.45364i	0.686827	+	0.726821i	$0.259003\pi$
$563$			818.401i			1.45364i	−0.686827	+	0.726821i	$0.740997\pi$
$564$		0			0
$565$	210.000			0.371681
$566$		0			0
$567$	−158.000	+	35.7771i	−0.278660	+	0.0630989i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−526.000			−0.921191			−0.460595	−	0.887610i	$-0.652364\pi$
$571$	−526.000			−0.921191			−0.460595	+	0.887610i	$0.652364\pi$
$572$		0			0
$573$	−180.000	−	160.997i	−0.314136	−	0.280972i
$574$		0			0
$575$		−	201.246i		−	0.349993i
$576$		0			0
$577$	122.000			0.211438			0.105719	−	0.994396i	$-0.466286\pi$
$577$	122.000			0.211438			0.105719	+	0.994396i	$0.466286\pi$
$578$		0			0
$579$	436.000	−	487.463i	0.753022	−	0.841905i
$580$		0			0
$581$		−	241.495i		−	0.415655i
$582$		0			0
$583$	540.000			0.926244
$584$		0			0
$585$	−160.000	+	17.8885i	−0.273504	+	0.0305787i
$586$		0			0
$587$			228.079i			0.388550i	0.980947	+	0.194275i	$0.0622354\pi$
$587$			228.079i			0.388550i	−0.980947	+	0.194275i	$0.937765\pi$
$588$		0			0
$589$	−476.000			−0.808149
$590$		0			0
$591$	210.000	+	187.830i	0.355330	+	0.317817i
$592$		0			0
$593$			737.902i			1.24435i	0.782876	+	0.622177i	$0.213752\pi$
$593$			737.902i			1.24435i	−0.782876	+	0.622177i	$0.786248\pi$
$594$		0			0
$595$	60.0000			0.100840
$596$		0			0
$597$	−68.0000	+	76.0263i	−0.113903	+	0.127347i
$598$		0			0
$599$		−	80.4984i		−	0.134388i	−0.997740	−	0.0671940i	$-0.978595\pi$
$599$		−	80.4984i		−	0.134388i	0.997740	−	0.0671940i	$-0.0214047\pi$
$600$		0			0
$601$	−766.000			−1.27454			−0.637271	−	0.770640i	$-0.719937\pi$
$601$	−766.000			−1.27454			−0.637271	+	0.770640i	$0.719937\pi$
$602$		0			0
$603$	−32.0000	−	286.217i	−0.0530680	−	0.474655i
$604$		0			0
$605$			131.928i			0.218063i
$606$		0			0
$607$	206.000			0.339374			0.169687	−	0.985498i	$-0.445724\pi$
$607$	206.000			0.339374			0.169687	+	0.985498i	$0.445724\pi$
$608$		0			0
$609$	−180.000	−	160.997i	−0.295567	−	0.264363i
$610$		0			0
$611$		−	321.994i		−	0.526995i
$612$		0			0
$613$	−556.000			−0.907015			−0.453507	−	0.891253i	$-0.649827\pi$
$613$	−556.000			−0.907015			−0.453507	+	0.891253i	$0.649827\pi$
$614$		0			0
$615$	−120.000	+	134.164i	−0.195122	+	0.218153i
$616$		0			0
$617$			389.076i			0.630593i	0.948993	+	0.315296i	$0.102104\pi$
$617$			389.076i			0.630593i	−0.948993	+	0.315296i	$0.897896\pi$
$618$		0			0
$619$	−514.000			−0.830372			−0.415186	−	0.909737i	$-0.636283\pi$
$619$	−514.000			−0.830372			−0.415186	+	0.909737i	$0.636283\pi$
$620$		0			0
$621$	630.000	−	885.483i	1.01449	−	1.42590i
$622$		0			0
$623$			214.663i			0.344563i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−1020.00	−	912.316i	−1.62679	−	1.45505i
$628$		0			0
$629$			751.319i			1.19447i
$630$		0			0
$631$	1094.00			1.73376			0.866878	−	0.498520i	$-0.166123\pi$
$631$	1094.00			1.73376			0.866878	+	0.498520i	$0.166123\pi$
$632$		0			0
$633$	−92.0000	+	102.859i	−0.145340	+	0.162495i
$634$		0			0
$635$			76.0263i			0.119726i
$636$		0			0
$637$	−360.000			−0.565149
$638$		0			0
$639$	480.000	−	53.6656i	0.751174	−	0.0839838i
$640$		0			0
$641$			160.997i			0.251165i	0.992083	+	0.125583i	$0.0400800\pi$
$641$			160.997i			0.251165i	−0.992083	+	0.125583i	$0.959920\pi$
$642$		0			0
$643$	404.000			0.628305			0.314152	−	0.949373i	$-0.398280\pi$
$643$	404.000			0.628305			0.314152	+	0.949373i	$0.398280\pi$
$644$		0			0
$645$	−40.0000	−	35.7771i	−0.0620155	−	0.0554684i
$646$		0			0
$647$		−	1167.23i		−	1.80406i	−0.431672	−	0.902031i	$-0.642076\pi$
$647$		−	1167.23i		−	1.80406i	0.431672	−	0.902031i	$-0.357924\pi$
$648$		0			0
$649$	−180.000			−0.277350
$650$		0			0
$651$	56.0000	−	62.6099i	0.0860215	−	0.0961750i
$652$		0			0
$653$		−	764.735i		−	1.17111i	−0.810632	−	0.585555i	$-0.800876\pi$
$653$		−	764.735i		−	1.17111i	0.810632	−	0.585555i	$-0.199124\pi$
$654$		0			0
$655$	330.000			0.503817
$656$		0			0
$657$	106.000	+	948.093i	0.161339	+	1.44306i
$658$		0			0
$659$			791.568i			1.20117i	0.799563	+	0.600583i	$0.205065\pi$
$659$			791.568i			1.20117i	−0.799563	+	0.600583i	$0.794935\pi$
$660$		0			0
$661$	−118.000			−0.178517			−0.0892587	−	0.996008i	$-0.528450\pi$
$661$	−118.000			−0.178517			−0.0892587	+	0.996008i	$0.528450\pi$
$662$		0			0
$663$	240.000	+	214.663i	0.361991	+	0.323775i
$664$		0			0
$665$			152.053i			0.228651i
$666$		0			0
$667$	1620.00			2.42879
$668$		0			0
$669$	796.000	−	889.955i	1.18984	−	1.33028i
$670$		0			0
$671$		−	617.155i		−	0.919754i
$672$		0			0
$673$	194.000			0.288262			0.144131	−	0.989559i	$-0.453961\pi$
$673$	194.000			0.288262			0.144131	+	0.989559i	$0.453961\pi$
$674$		0			0
$675$	110.000	+	78.2624i	0.162963	+	0.115944i
$676$		0			0
$677$		−	415.909i		−	0.614341i	−0.951655	−	0.307170i	$-0.900618\pi$
$677$		−	415.909i		−	0.614341i	0.951655	−	0.307170i	$-0.0993821\pi$
$678$		0			0
$679$	244.000			0.359352
$680$		0			0
$681$	−750.000	−	670.820i	−1.10132	−	0.985052i
$682$		0			0
$683$			93.9149i			0.137503i	0.997634	+	0.0687517i	$0.0219016\pi$
$683$			93.9149i			0.137503i	−0.997634	+	0.0687517i	$0.978098\pi$
$684$		0			0
$685$	−90.0000			−0.131387
$686$		0			0
$687$	172.000	−	192.302i	0.250364	−	0.279915i
$688$		0			0
$689$		−	321.994i		−	0.467335i
$690$		0			0
$691$	122.000			0.176556			0.0882779	−	0.996096i	$-0.471864\pi$
$691$	122.000			0.176556			0.0882779	+	0.996096i	$0.471864\pi$
$692$		0			0
$693$	240.000	−	26.8328i	0.346320	−	0.0387198i
$694$		0			0
$695$			183.358i			0.263824i
$696$		0			0
$697$	360.000			0.516499
$698$		0			0
$699$	690.000	+	617.155i	0.987124	+	0.882911i
$700$		0			0
$701$		−	576.906i		−	0.822975i	−0.911415	−	0.411488i	$-0.865009\pi$
$701$		−	576.906i		−	0.822975i	0.911415	−	0.411488i	$-0.134991\pi$
$702$		0			0
$703$	−1904.00			−2.70839
$704$		0			0
$705$	−180.000	+	201.246i	−0.255319	+	0.285455i
$706$		0			0
$707$		−	348.827i		−	0.493390i
$708$		0			0
$709$	1166.00			1.64457			0.822285	−	0.569076i	$-0.192699\pi$
$709$	1166.00			1.64457			0.822285	+	0.569076i	$0.192699\pi$
$710$		0			0
$711$	22.0000	+	196.774i	0.0309423	+	0.276757i
$712$		0			0
$713$			563.489i			0.790307i
$714$		0			0
$715$	240.000			0.335664
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	214.663i		−	0.298557i	−0.988795	−	0.149279i	$-0.952305\pi$
$719$		−	214.663i		−	0.298557i	0.988795	−	0.149279i	$-0.0476951\pi$
$720$		0			0
$721$	−92.0000			−0.127601
$722$		0			0
$723$	268.000	−	299.633i	0.370678	−	0.414430i
$724$		0			0
$725$			201.246i			0.277581i
$726$		0			0
$727$	−1318.00			−1.81293			−0.906465	−	0.422281i	$-0.861230\pi$
$727$	−1318.00			−1.81293			−0.906465	+	0.422281i	$0.861230\pi$
$728$		0			0
$729$	239.000	+	688.709i	0.327846	+	0.944731i
$730$		0			0
$731$			107.331i			0.146828i
$732$		0			0
$733$	−136.000			−0.185539			−0.0927694	−	0.995688i	$-0.529572\pi$
$733$	−136.000			−0.185539			−0.0927694	+	0.995688i	$0.529572\pi$
$734$		0			0
$735$	225.000	+	201.246i	0.306122	+	0.273804i
$736$		0			0
$737$			429.325i			0.582531i
$738$		0			0
$739$	−682.000			−0.922869			−0.461434	−	0.887174i	$-0.652665\pi$
$739$	−682.000			−0.922869			−0.461434	+	0.887174i	$0.652665\pi$
$740$		0			0
$741$	−544.000	+	608.210i	−0.734143	+	0.820797i
$742$		0			0
$743$		−	872.067i		−	1.17371i	−0.809692	−	0.586855i	$-0.800366\pi$
$743$		−	872.067i		−	1.17371i	0.809692	−	0.586855i	$-0.199634\pi$
$744$		0			0
$745$	330.000			0.442953
$746$		0			0
$747$	−1080.00	+	120.748i	−1.44578	+	0.161643i
$748$		0			0
$749$			26.8328i			0.0358249i
$750$		0			0
$751$	−658.000			−0.876165			−0.438083	−	0.898935i	$-0.644342\pi$
$751$	−658.000			−0.876165			−0.438083	+	0.898935i	$0.644342\pi$
$752$		0			0
$753$	−690.000	−	617.155i	−0.916335	−	0.819595i
$754$		0			0
$755$			102.859i			0.136237i
$756$		0			0
$757$	1112.00			1.46896			0.734478	−	0.678632i	$-0.237427\pi$
$757$	1112.00			1.46896			0.734478	+	0.678632i	$0.237427\pi$
$758$		0			0
$759$	−1080.00	+	1207.48i	−1.42292	+	1.59088i
$760$		0			0
$761$		−	80.4984i		−	0.105780i	−0.998600	−	0.0528899i	$-0.983157\pi$
$761$		−	80.4984i		−	0.105780i	0.998600	−	0.0528899i	$-0.0168432\pi$
$762$		0			0
$763$	172.000			0.225426
$764$		0			0
$765$	−30.0000	−	268.328i	−0.0392157	−	0.350756i
$766$		0			0
$767$			107.331i			0.139936i
$768$		0			0
$769$	206.000			0.267880			0.133940	−	0.990989i	$-0.457237\pi$
$769$	206.000			0.267880			0.133940	+	0.990989i	$0.457237\pi$
$770$		0			0
$771$	390.000	+	348.827i	0.505837	+	0.452434i
$772$		0			0
$773$		−	1086.73i		−	1.40586i	−0.711260	−	0.702930i	$-0.751875\pi$
$773$		−	1086.73i		−	1.40586i	0.711260	−	0.702930i	$-0.248125\pi$
$774$		0			0
$775$	−70.0000			−0.0903226
$776$		0			0
$777$	224.000	−	250.440i	0.288288	−	0.322316i
$778$		0			0
$779$			912.316i			1.17114i
$780$		0			0
$781$	−720.000			−0.921895
$782$		0			0
$783$	−630.000	+	885.483i	−0.804598	+	1.13088i
$784$		0			0
$785$		−	205.718i		−	0.262061i
$786$		0			0
$787$	−1168.00			−1.48412			−0.742058	−	0.670335i	$-0.766150\pi$
$787$	−1168.00			−1.48412			−0.742058	+	0.670335i	$0.766150\pi$
$788$		0			0
$789$	570.000	+	509.823i	0.722433	+	0.646164i
$790$		0			0
$791$			187.830i			0.237459i
$792$		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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	60.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.23607i) q^{3} -2.23607i q^{5} +2.00000 q^{7} +(-1.00000 - 8.94427i) q^{9} +O(q^{10})\)	\(q+(2.00000 - 2.23607i) q^{3} -2.23607i q^{5} +2.00000 q^{7} +(-1.00000 - 8.94427i) q^{9} +13.4164i q^{11} +8.00000 q^{13} +(-5.00000 - 4.47214i) q^{15} +13.4164i q^{17} -34.0000 q^{19} +(4.00000 - 4.47214i) q^{21} +40.2492i q^{23} -5.00000 q^{25} +(-22.0000 - 15.6525i) q^{27} -40.2492i q^{29} +14.0000 q^{31} +(30.0000 + 26.8328i) q^{33} -4.47214i q^{35} +56.0000 q^{37} +(16.0000 - 17.8885i) q^{39} -26.8328i q^{41} +8.00000 q^{43} +(-20.0000 + 2.23607i) q^{45} -40.2492i q^{47} -45.0000 q^{49} +(30.0000 + 26.8328i) q^{51} -40.2492i q^{53} +30.0000 q^{55} +(-68.0000 + 76.0263i) q^{57} +13.4164i q^{59} -46.0000 q^{61} +(-2.00000 - 17.8885i) q^{63} -17.8885i q^{65} +32.0000 q^{67} +(90.0000 + 80.4984i) q^{69} +53.6656i q^{71} -106.000 q^{73} +(-10.0000 + 11.1803i) q^{75} +26.8328i q^{77} -22.0000 q^{79} +(-79.0000 + 17.8885i) q^{81} -120.748i q^{83} +30.0000 q^{85} +(-90.0000 - 80.4984i) q^{87} +107.331i q^{89} +16.0000 q^{91} +(28.0000 - 31.3050i) q^{93} +76.0263i q^{95} +122.000 q^{97} +(120.000 - 13.4164i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9	\( 2 q + 4 q^{3} + 4 q^{7} - 2 q^{9} + 16 q^{13} - 10 q^{15} - 68 q^{19} + 8 q^{21} - 10 q^{25} - 44 q^{27} + 28 q^{31} + 60 q^{33} + 112 q^{37} + 32 q^{39} + 16 q^{43} - 40 q^{45} - 90 q^{49} + 60 q^{51} + 60 q^{55} - 136 q^{57} - 92 q^{61} - 4 q^{63} + 64 q^{67} + 180 q^{69} - 212 q^{73} - 20 q^{75} - 44 q^{79} - 158 q^{81} + 60 q^{85} - 180 q^{87} + 32 q^{91} + 56 q^{93} + 244 q^{97} + 240 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 + 4 * q^7 - 2 * q^9 + 16 * q^13 - 10 * q^15 - 68 * q^19 + 8 * q^21 - 10 * q^25 - 44 * q^27 + 28 * q^31 + 60 * q^33 + 112 * q^37 + 32 * q^39 + 16 * q^43 - 40 * q^45 - 90 * q^49 + 60 * q^51 + 60 * q^55 - 136 * q^57 - 92 * q^61 - 4 * q^63 + 64 * q^67 + 180 * q^69 - 212 * q^73 - 20 * q^75 - 44 * q^79 - 158 * q^81 + 60 * q^85 - 180 * q^87 + 32 * q^91 + 56 * q^93 + 244 * q^97 + 240 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more