LMFDB - Embedded newform 208.4.w.b.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(17,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("208.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.w (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	208.49
Dual form			208.4.w.b.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{3} +1.73205i q^{5} +(12.0000 + 6.92820i) q^{7} +(11.5000 - 19.9186i) q^{9} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{3} +1.73205i q^{5} +(12.0000 + 6.92820i) q^{7} +(11.5000 - 19.9186i) q^{9} +(-12.0000 + 6.92820i) q^{11} +(45.5000 - 11.2583i) q^{13} +(-3.00000 + 1.73205i) q^{15} +(-58.5000 + 101.325i) q^{17} +(99.0000 + 57.1577i) q^{19} +27.7128i q^{21} +(-39.0000 - 67.5500i) q^{23} +122.000 q^{25} +100.000 q^{27} +(70.5000 + 122.110i) q^{29} +155.885i q^{31} +(-24.0000 - 13.8564i) q^{33} +(-12.0000 + 20.7846i) q^{35} +(-124.500 + 71.8801i) q^{37} +(65.0000 + 67.5500i) q^{39} +(235.500 - 135.966i) q^{41} +(52.0000 - 90.0666i) q^{43} +(34.5000 + 19.9186i) q^{45} +301.377i q^{47} +(-75.5000 - 130.770i) q^{49} -234.000 q^{51} +93.0000 q^{53} +(-12.0000 - 20.7846i) q^{55} +228.631i q^{57} +(246.000 + 142.028i) q^{59} +(-72.5000 + 125.574i) q^{61} +(276.000 - 159.349i) q^{63} +(19.5000 + 78.8083i) q^{65} +(681.000 - 393.176i) q^{67} +(78.0000 - 135.100i) q^{69} +(-915.000 - 528.275i) q^{71} -458.993i q^{73} +(122.000 + 211.310i) q^{75} -192.000 q^{77} -1276.00 q^{79} +(-210.500 - 364.597i) q^{81} -789.815i q^{83} +(-175.500 - 101.325i) q^{85} +(-141.000 + 244.219i) q^{87} +(-846.000 + 488.438i) q^{89} +(624.000 + 180.133i) q^{91} +(-270.000 + 155.885i) q^{93} +(-99.0000 + 171.473i) q^{95} +(174.000 + 100.459i) q^{97} +318.697i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 24 q^{7} + 23 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 24 * q^7 + 23 * q^9	$ 2 q + 2 q^{3} + 24 q^{7} + 23 q^{9} - 24 q^{11} + 91 q^{13} - 6 q^{15} - 117 q^{17} + 198 q^{19} - 78 q^{23} + 244 q^{25} + 200 q^{27} + 141 q^{29} - 48 q^{33} - 24 q^{35} - 249 q^{37} + 130 q^{39} + 471 q^{41} + 104 q^{43} + 69 q^{45} - 151 q^{49} - 468 q^{51} + 186 q^{53} - 24 q^{55} + 492 q^{59} - 145 q^{61} + 552 q^{63} + 39 q^{65} + 1362 q^{67} + 156 q^{69} - 1830 q^{71} + 244 q^{75} - 384 q^{77} - 2552 q^{79} - 421 q^{81} - 351 q^{85} - 282 q^{87} - 1692 q^{89} + 1248 q^{91} - 540 q^{93} - 198 q^{95} + 348 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 24 * q^7 + 23 * q^9 - 24 * q^11 + 91 * q^13 - 6 * q^15 - 117 * q^17 + 198 * q^19 - 78 * q^23 + 244 * q^25 + 200 * q^27 + 141 * q^29 - 48 * q^33 - 24 * q^35 - 249 * q^37 + 130 * q^39 + 471 * q^41 + 104 * q^43 + 69 * q^45 - 151 * q^49 - 468 * q^51 + 186 * q^53 - 24 * q^55 + 492 * q^59 - 145 * q^61 + 552 * q^63 + 39 * q^65 + 1362 * q^67 + 156 * q^69 - 1830 * q^71 + 244 * q^75 - 384 * q^77 - 2552 * q^79 - 421 * q^81 - 351 * q^85 - 282 * q^87 - 1692 * q^89 + 1248 * q^91 - 540 * q^93 - 198 * q^95 + 348 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	1.00000	+	1.73205i	0.192450	+	0.333333i	0.946062	−	0.323987i	$-0.105023\pi$
$3$	1.00000	+	1.73205i	0.192450	+	0.333333i	−0.753612	+	0.657320i	$0.771690\pi$
$4$		0			0
$5$			1.73205i			0.154919i	0.996995	+	0.0774597i	$0.0246809\pi$
$5$			1.73205i			0.154919i	−0.996995	+	0.0774597i	$0.975319\pi$
$6$		0			0
$7$	12.0000	+	6.92820i	0.647939	+	0.374088i	0.787666	−	0.616102i	$-0.211289\pi$
$7$	12.0000	+	6.92820i	0.647939	+	0.374088i	−0.139727	+	0.990190i	$0.544623\pi$
$8$		0			0
$9$	11.5000	−	19.9186i	0.425926	−	0.737725i
$10$		0			0
$11$	−12.0000	+	6.92820i	−0.328921	+	0.189903i	−0.655362	−	0.755315i	$-0.727484\pi$
$11$	−12.0000	+	6.92820i	−0.328921	+	0.189903i	0.326441	+	0.945218i	$0.394151\pi$
$12$		0			0
$13$	45.5000	−	11.2583i	0.970725	−	0.240192i
$13$	45.5000	−	11.2583i	0.970725	−	0.240192i
$14$		0			0
$15$	−3.00000	+	1.73205i	−0.0516398	+	0.0298142i
$16$		0			0
$17$	−58.5000	+	101.325i	−0.834608	+	1.44558i	0.0597414	+	0.998214i	$0.480972\pi$
$17$	−58.5000	+	101.325i	−0.834608	+	1.44558i	−0.894349	+	0.447369i	$0.852361\pi$
$18$		0			0
$19$	99.0000	+	57.1577i	1.19538	+	0.690151i	0.959521	−	0.281637i	$-0.0908774\pi$
$19$	99.0000	+	57.1577i	1.19538	+	0.690151i	0.235856	+	0.971788i	$0.424211\pi$
$20$		0			0
$21$			27.7128i			0.287973i
$22$		0			0
$23$	−39.0000	−	67.5500i	−0.353568	−	0.612398i	0.633304	−	0.773903i	$-0.281698\pi$
$23$	−39.0000	−	67.5500i	−0.353568	−	0.612398i	−0.986872	+	0.161506i	$0.948365\pi$
$24$		0			0
$25$	122.000			0.976000
$26$		0			0
$27$	100.000			0.712778
$28$		0			0
$29$	70.5000	+	122.110i	0.451432	+	0.781903i	0.998475	−	0.0552014i	$-0.0175801\pi$
$29$	70.5000	+	122.110i	0.451432	+	0.781903i	−0.547043	+	0.837104i	$0.684247\pi$
$30$		0			0
$31$			155.885i			0.903151i	0.892233	+	0.451576i	$0.149138\pi$
$31$			155.885i			0.903151i	−0.892233	+	0.451576i	$0.850862\pi$
$32$		0			0
$33$	−24.0000	−	13.8564i	−0.126602	−	0.0730937i
$34$		0			0
$35$	−12.0000	+	20.7846i	−0.0579534	+	0.100378i
$36$		0			0
$37$	−124.500	+	71.8801i	−0.553180	+	0.319379i	−0.750404	−	0.660980i	$-0.770141\pi$
$37$	−124.500	+	71.8801i	−0.553180	+	0.319379i	0.197223	+	0.980359i	$0.436808\pi$
$38$		0			0
$39$	65.0000	+	67.5500i	0.266880	+	0.277350i
$40$		0			0
$41$	235.500	−	135.966i	0.897047	−	0.517910i	0.0208059	−	0.999784i	$-0.493377\pi$
$41$	235.500	−	135.966i	0.897047	−	0.517910i	0.876241	+	0.481873i	$0.160043\pi$
$42$		0			0
$43$	52.0000	−	90.0666i	0.184417	−	0.319419i	−0.758963	−	0.651134i	$-0.774294\pi$
$43$	52.0000	−	90.0666i	0.184417	−	0.319419i	0.943380	+	0.331714i	$0.107627\pi$
$44$		0			0
$45$	34.5000	+	19.9186i	0.114288	+	0.0659842i
$46$		0			0
$47$			301.377i			0.935326i	0.883907	+	0.467663i	$0.154904\pi$
$47$			301.377i			0.935326i	−0.883907	+	0.467663i	$0.845096\pi$
$48$		0			0
$49$	−75.5000	−	130.770i	−0.220117	−	0.381253i
$50$		0			0
$51$	−234.000			−0.642481
$52$		0			0
$53$	93.0000			0.241029			0.120514	−	0.992712i	$-0.461546\pi$
$53$	93.0000			0.241029			0.120514	+	0.992712i	$0.461546\pi$
$54$		0			0
$55$	−12.0000	−	20.7846i	−0.0294196	−	0.0509563i
$56$		0			0
$57$			228.631i			0.531279i
$58$		0			0
$59$	246.000	+	142.028i	0.542822	+	0.313398i	0.746222	−	0.665698i	$-0.231866\pi$
$59$	246.000	+	142.028i	0.542822	+	0.313398i	−0.203400	+	0.979096i	$0.565199\pi$
$60$		0			0
$61$	−72.5000	+	125.574i	−0.152175	+	0.263575i	−0.932027	−	0.362389i	$-0.881961\pi$
$61$	−72.5000	+	125.574i	−0.152175	+	0.263575i	0.779852	+	0.625964i	$0.215294\pi$
$62$		0			0
$63$	276.000	−	159.349i	0.551948	−	0.318667i
$64$		0			0
$65$	19.5000	+	78.8083i	0.0372104	+	0.150384i
$66$		0			0
$67$	681.000	−	393.176i	1.24175	−	0.716926i	0.272301	−	0.962212i	$-0.412215\pi$
$67$	681.000	−	393.176i	1.24175	−	0.716926i	0.969451	+	0.245286i	$0.0788819\pi$
$68$		0			0
$69$	78.0000	−	135.100i	0.136088	−	0.235712i
$70$		0			0
$71$	−915.000	−	528.275i	−1.52944	−	0.883025i	−0.999385	−	0.0350641i	$-0.988836\pi$
$71$	−915.000	−	528.275i	−1.52944	−	0.883025i	−0.530059	−	0.847961i	$-0.677830\pi$
$72$		0			0
$73$		−	458.993i		−	0.735906i	−0.929844	−	0.367953i	$-0.880059\pi$
$73$		−	458.993i		−	0.735906i	0.929844	−	0.367953i	$-0.119941\pi$
$74$		0			0
$75$	122.000	+	211.310i	0.187831	+	0.325333i
$76$		0			0
$77$	−192.000			−0.284161
$78$		0			0
$79$	−1276.00			−1.81723			−0.908615	−	0.417634i	$-0.862859\pi$
$79$	−1276.00			−1.81723			−0.908615	+	0.417634i	$0.862859\pi$
$80$		0			0
$81$	−210.500	−	364.597i	−0.288752	−	0.500133i
$82$		0			0
$83$		−	789.815i		−	1.04450i	−0.852793	−	0.522250i	$-0.825093\pi$
$83$		−	789.815i		−	1.04450i	0.852793	−	0.522250i	$-0.174907\pi$
$84$		0			0
$85$	−175.500	−	101.325i	−0.223949	−	0.129297i
$86$		0			0
$87$	−141.000	+	244.219i	−0.173756	+	0.300955i
$88$		0			0
$89$	−846.000	+	488.438i	−1.00759	+	0.581734i	−0.910486	−	0.413540i	$-0.864292\pi$
$89$	−846.000	+	488.438i	−1.00759	+	0.581734i	−0.0971073	+	0.995274i	$0.530959\pi$
$90$		0			0
$91$	624.000	+	180.133i	0.718824	+	0.207507i
$92$		0			0
$93$	−270.000	+	155.885i	−0.301050	+	0.173812i
$94$		0			0
$95$	−99.0000	+	171.473i	−0.106918	+	0.185187i
$96$		0			0
$97$	174.000	+	100.459i	0.182134	+	0.105155i	0.588295	−	0.808646i	$-0.299799\pi$
$97$	174.000	+	100.459i	0.182134	+	0.105155i	−0.406161	+	0.913802i	$0.633133\pi$
$98$		0			0
$99$			318.697i			0.323538i
$100$		0			0
$101$	−214.500	−	371.525i	−0.211322	−	0.366021i	0.740806	−	0.671719i	$-0.234444\pi$
$101$	−214.500	−	371.525i	−0.211322	−	0.366021i	−0.952129	+	0.305698i	$0.901110\pi$
$102$		0			0
$103$	−182.000			−0.174107			−0.0870534	−	0.996204i	$-0.527745\pi$
$103$	−182.000			−0.174107			−0.0870534	+	0.996204i	$0.527745\pi$
$104$		0			0
$105$	−48.0000			−0.0446126
$106$		0			0
$107$	−753.000	−	1304.23i	−0.680330	−	1.17837i	−0.974880	−	0.222729i	$-0.928503\pi$
$107$	−753.000	−	1304.23i	−0.680330	−	1.17837i	0.294551	−	0.955636i	$-0.404830\pi$
$108$		0			0
$109$		−	1551.92i		−	1.36373i	−0.731477	−	0.681866i	$-0.761169\pi$
$109$		−	1551.92i		−	1.36373i	0.731477	−	0.681866i	$-0.238831\pi$
$110$		0			0
$111$	−249.000	−	143.760i	−0.212919	−	0.122929i
$112$		0			0
$113$	343.500	−	594.959i	0.285962	−	0.495302i	−0.686880	−	0.726771i	$-0.741020\pi$
$113$	343.500	−	594.959i	0.285962	−	0.495302i	0.972842	+	0.231470i	$0.0743534\pi$
$114$		0			0
$115$	117.000	−	67.5500i	0.0948722	−	0.0547745i
$116$		0			0
$117$	299.000	−	1035.77i	0.236261	−	0.818433i
$118$		0			0
$119$	−1404.00	+	810.600i	−1.08155	+	0.624433i
$120$		0			0
$121$	−569.500	+	986.403i	−0.427874	+	0.741099i
$122$		0			0
$123$	471.000	+	271.932i	0.345273	+	0.199344i
$124$		0			0
$125$			427.817i			0.306121i
$126$		0			0
$127$	143.000	+	247.683i	0.0999149	+	0.173058i	0.911649	−	0.410969i	$-0.134810\pi$
$127$	143.000	+	247.683i	0.0999149	+	0.173058i	−0.811734	+	0.584027i	$0.801476\pi$
$128$		0			0
$129$	208.000			0.141964
$130$		0			0
$131$	1974.00			1.31656			0.658279	−	0.752774i	$-0.271285\pi$
$131$	1974.00			1.31656			0.658279	+	0.752774i	$0.271285\pi$
$132$		0			0
$133$	792.000	+	1371.78i	0.516354	+	0.894352i
$134$		0			0
$135$			173.205i			0.110423i
$136$		0			0
$137$	−733.500	−	423.486i	−0.457424	−	0.264094i	0.253536	−	0.967326i	$-0.418406\pi$
$137$	−733.500	−	423.486i	−0.457424	−	0.264094i	−0.710961	+	0.703232i	$0.751740\pi$
$138$		0			0
$139$	118.000	−	204.382i	0.0720045	−	0.124716i	−0.827775	−	0.561060i	$-0.810394\pi$
$139$	118.000	−	204.382i	0.0720045	−	0.124716i	0.899780	+	0.436344i	$0.143727\pi$
$140$		0			0
$141$	−522.000	+	301.377i	−0.311775	+	0.180004i
$142$		0			0
$143$	−468.000	+	450.333i	−0.273679	+	0.263348i
$144$		0			0
$145$	−211.500	+	122.110i	−0.121132	+	0.0699355i
$146$		0			0
$147$	151.000	−	261.540i	0.0847229	−	0.146744i
$148$		0			0
$149$	−40.5000	−	23.3827i	−0.0222677	−	0.0128563i	0.488825	−	0.872382i	$-0.337426\pi$
$149$	−40.5000	−	23.3827i	−0.0222677	−	0.0128563i	−0.511093	+	0.859526i	$0.670759\pi$
$150$		0			0
$151$		−	1770.16i		−	0.953995i	−0.878905	−	0.476998i	$-0.841725\pi$
$151$		−	1770.16i		−	0.953995i	0.878905	−	0.476998i	$-0.158275\pi$
$152$		0			0
$153$	1345.50	+	2330.47i	0.710962	+	1.23142i
$154$		0			0
$155$	−270.000			−0.139916
$156$		0			0
$157$	1211.00			0.615594			0.307797	−	0.951452i	$-0.400408\pi$
$157$	1211.00			0.615594			0.307797	+	0.951452i	$0.400408\pi$
$158$		0			0
$159$	93.0000	+	161.081i	0.0463860	+	0.0803430i
$160$		0			0
$161$		−	1080.80i		−	0.529062i
$162$		0			0
$163$	870.000	+	502.295i	0.418059	+	0.241367i	0.694247	−	0.719737i	$-0.255738\pi$
$163$	870.000	+	502.295i	0.418059	+	0.241367i	−0.276187	+	0.961104i	$0.589071\pi$
$164$		0			0
$165$	24.0000	−	41.5692i	0.0113236	−	0.0196131i
$166$		0			0
$167$	−792.000	+	457.261i	−0.366987	+	0.211880i	−0.672141	−	0.740423i	$-0.734625\pi$
$167$	−792.000	+	457.261i	−0.366987	+	0.211880i	0.305154	+	0.952303i	$0.401292\pi$
$168$		0			0
$169$	1943.50	−	1024.51i	0.884615	−	0.466321i
$170$		0			0
$171$	2277.00	−	1314.63i	1.01828	−	0.587906i
$172$		0			0
$173$	−1287.00	+	2229.15i	−0.565600	+	0.979648i	0.431394	+	0.902164i	$0.358022\pi$
$173$	−1287.00	+	2229.15i	−0.565600	+	0.979648i	−0.996994	+	0.0774841i	$0.975311\pi$
$174$		0			0
$175$	1464.00	+	845.241i	0.632389	+	0.365110i
$176$		0			0
$177$			568.113i			0.241254i
$178$		0			0
$179$	1872.00	+	3242.40i	0.781675	+	1.35390i	0.930965	+	0.365108i	$0.118968\pi$
$179$	1872.00	+	3242.40i	0.781675	+	1.35390i	−0.149290	+	0.988793i	$0.547699\pi$
$180$		0			0
$181$	−637.000			−0.261590			−0.130795	−	0.991409i	$-0.541753\pi$
$181$	−637.000			−0.261590			−0.130795	+	0.991409i	$0.541753\pi$
$182$		0			0
$183$	−290.000			−0.117144
$184$		0			0
$185$	−124.500	−	215.640i	−0.0494780	−	0.0856983i
$186$		0			0
$187$		−	1621.20i		−	0.633978i
$188$		0			0
$189$	1200.00	+	692.820i	0.461837	+	0.266642i
$190$		0			0
$191$	−1299.00	+	2249.93i	−0.492106	+	0.852353i	−0.999959	−	0.00909077i	$-0.997106\pi$
$191$	−1299.00	+	2249.93i	−0.492106	+	0.852353i	0.507852	+	0.861444i	$0.330440\pi$
$192$		0			0
$193$	967.500	−	558.586i	0.360840	−	0.208331i	−0.308609	−	0.951189i	$-0.599863\pi$
$193$	967.500	−	558.586i	0.360840	−	0.208331i	0.669449	+	0.742858i	$0.266530\pi$
$194$		0			0
$195$	−117.000	+	112.583i	−0.0429669	+	0.0413449i
$196$		0			0
$197$	1776.00	−	1025.37i	0.642308	−	0.370837i	−0.143195	−	0.989695i	$-0.545738\pi$
$197$	1776.00	−	1025.37i	0.642308	−	0.370837i	0.785503	+	0.618858i	$0.212404\pi$
$198$		0			0
$199$	−1261.00	+	2184.12i	−0.449196	+	0.778030i	−0.998334	−	0.0577019i	$-0.981623\pi$
$199$	−1261.00	+	2184.12i	−0.449196	+	0.778030i	0.549138	+	0.835732i	$0.314956\pi$
$200$		0			0
$201$	1362.00	+	786.351i	0.477951	+	0.275945i
$202$		0			0
$203$			1953.75i			0.675500i
$204$		0			0
$205$	235.500	+	407.898i	0.0802343	+	0.138970i
$206$		0			0
$207$	−1794.00			−0.602375
$208$		0			0
$209$	−1584.00			−0.524247
$210$		0			0
$211$	521.000	+	902.398i	0.169986	+	0.294425i	0.938415	−	0.345511i	$-0.112294\pi$
$211$	521.000	+	902.398i	0.169986	+	0.294425i	−0.768428	+	0.639936i	$0.778961\pi$
$212$		0			0
$213$		−	2113.10i		−	0.679753i
$214$		0			0
$215$	156.000	+	90.0666i	0.0494842	+	0.0285697i
$216$		0			0
$217$	−1080.00	+	1870.61i	−0.337858	+	0.585187i
$218$		0			0
$219$	795.000	−	458.993i	0.245302	−	0.141625i
$220$		0			0
$221$	−1521.00	+	5268.90i	−0.462957	+	1.60373i
$222$		0			0
$223$	2085.00	−	1203.78i	0.626107	−	0.361483i	−0.153136	−	0.988205i	$-0.548937\pi$
$223$	2085.00	−	1203.78i	0.626107	−	0.361483i	0.779243	+	0.626722i	$0.215604\pi$
$224$		0			0
$225$	1403.00	−	2430.07i	0.415704	−	0.720020i
$226$		0			0
$227$	−2085.00	−	1203.78i	−0.609631	−	0.351971i	0.163190	−	0.986595i	$-0.447822\pi$
$227$	−2085.00	−	1203.78i	−0.609631	−	0.351971i	−0.772821	+	0.634624i	$0.781155\pi$
$228$		0			0
$229$		−	2508.01i		−	0.723729i	−0.932231	−	0.361864i	$-0.882140\pi$
$229$		−	2508.01i		−	0.723729i	0.932231	−	0.361864i	$-0.117860\pi$
$230$		0			0
$231$	−192.000	−	332.554i	−0.0546869	−	0.0947205i
$232$		0			0
$233$	−5850.00			−1.64483			−0.822417	−	0.568885i	$-0.807375\pi$
$233$	−5850.00			−1.64483			−0.822417	+	0.568885i	$0.807375\pi$
$234$		0			0
$235$	−522.000			−0.144900
$236$		0			0
$237$	−1276.00	−	2210.10i	−0.349726	−	0.605744i
$238$		0			0
$239$		−	5383.21i		−	1.45695i	−0.685072	−	0.728475i	$-0.740229\pi$
$239$		−	5383.21i		−	1.45695i	0.685072	−	0.728475i	$-0.259771\pi$
$240$		0			0
$241$	4258.50	+	2458.65i	1.13823	+	0.657159i	0.945992	−	0.324189i	$-0.105091\pi$
$241$	4258.50	+	2458.65i	1.13823	+	0.657159i	0.192240	+	0.981348i	$0.438425\pi$
$242$		0			0
$243$	1771.00	−	3067.46i	0.467530	−	0.809785i
$244$		0			0
$245$	226.500	−	130.770i	0.0590635	−	0.0341003i
$246$		0			0
$247$	5148.00	+	1486.10i	1.32615	+	0.382827i
$248$		0			0
$249$	1368.00	−	789.815i	0.348167	−	0.201014i
$250$		0			0
$251$	−1989.00	+	3445.05i	−0.500178	+	0.866333i	0.499822	+	0.866128i	$0.333399\pi$
$251$	−1989.00	+	3445.05i	−0.500178	+	0.866333i	−1.00000		0.000205037i	$0.999935\pi$
$252$		0			0
$253$	936.000	+	540.400i	0.232592	+	0.134287i
$254$		0			0
$255$		−	405.300i		−	0.0995328i
$256$		0			0
$257$	−1033.50	−	1790.07i	−0.250848	−	0.434482i	0.712911	−	0.701254i	$-0.247376\pi$
$257$	−1033.50	−	1790.07i	−0.250848	−	0.434482i	−0.963760	+	0.266772i	$0.914043\pi$
$258$		0			0
$259$	−1992.00			−0.477903
$260$		0			0
$261$	3243.00			0.769106
$262$		0			0
$263$	−1026.00	−	1777.08i	−0.240555	−	0.416653i	0.720318	−	0.693644i	$-0.243996\pi$
$263$	−1026.00	−	1777.08i	−0.240555	−	0.416653i	−0.960872	+	0.276991i	$0.910663\pi$
$264$		0			0
$265$			161.081i			0.0373400i
$266$		0			0
$267$	−1692.00	−	976.877i	−0.387823	−	0.223910i
$268$		0			0
$269$	−1665.00	+	2883.86i	−0.377386	+	0.653652i	−0.990681	−	0.136202i	$-0.956510\pi$
$269$	−1665.00	+	2883.86i	−0.377386	+	0.653652i	0.613295	+	0.789854i	$0.289844\pi$
$270$		0			0
$271$	−2430.00	+	1402.96i	−0.544694	+	0.314479i	−0.746979	−	0.664848i	$-0.768496\pi$
$271$	−2430.00	+	1402.96i	−0.544694	+	0.314479i	0.202285	+	0.979327i	$0.435163\pi$
$272$		0			0
$273$	312.000	+	1260.93i	0.0691689	+	0.279543i
$274$		0			0
$275$	−1464.00	+	845.241i	−0.321027	+	0.185345i
$276$		0			0
$277$	−188.500	+	326.492i	−0.0408876	+	0.0708194i	−0.885745	−	0.464172i	$-0.846352\pi$
$277$	−188.500	+	326.492i	−0.0408876	+	0.0708194i	0.844857	+	0.534992i	$0.179685\pi$
$278$		0			0
$279$	3105.00	+	1792.67i	0.666278	+	0.384676i
$280$		0			0
$281$			36.3731i			0.00772183i	0.999993	+	0.00386092i	$0.00122897\pi$
$281$			36.3731i			0.00772183i	−0.999993	+	0.00386092i	$0.998771\pi$
$282$		0			0
$283$	−3562.00	−	6169.56i	−0.748194	−	1.29591i	−0.948688	−	0.316215i	$-0.897588\pi$
$283$	−3562.00	−	6169.56i	−0.748194	−	1.29591i	0.200493	−	0.979695i	$-0.435745\pi$
$284$		0			0
$285$	−396.000			−0.0823053
$286$		0			0
$287$	3768.00			0.774976
$288$		0			0
$289$	−4388.00	−	7600.24i	−0.893141	−	1.54696i
$290$		0			0
$291$			401.836i			0.0809486i
$292$		0			0
$293$	−7207.50	−	4161.25i	−1.43709	−	0.829703i	−0.439441	−	0.898271i	$-0.644823\pi$
$293$	−7207.50	−	4161.25i	−1.43709	−	0.829703i	−0.997646	+	0.0685685i	$0.978157\pi$
$294$		0			0
$295$	−246.000	+	426.084i	−0.0485514	+	0.0840936i
$296$		0			0
$297$	−1200.00	+	692.820i	−0.234448	+	0.135359i
$298$		0			0
$299$	−2535.00	−	2634.45i	−0.490310	−	0.509546i
$300$		0			0
$301$	1248.00	−	720.533i	0.238982	−	0.137976i
$302$		0			0
$303$	429.000	−	743.050i	0.0813380	−	0.140882i
$304$		0			0
$305$	−217.500	−	125.574i	−0.0408328	−	0.0235748i
$306$		0			0
$307$		−	2220.49i		−	0.412801i	−0.978468	−	0.206401i	$-0.933825\pi$
$307$		−	2220.49i		−	0.412801i	0.978468	−	0.206401i	$-0.0661750\pi$
$308$		0			0
$309$	−182.000	−	315.233i	−0.0335069	−	0.0580356i
$310$		0			0
$311$	−4914.00			−0.895972			−0.447986	−	0.894041i	$-0.647859\pi$
$311$	−4914.00			−0.895972			−0.447986	+	0.894041i	$0.647859\pi$
$312$		0			0
$313$	−518.000			−0.0935434			−0.0467717	−	0.998906i	$-0.514893\pi$
$313$	−518.000			−0.0935434			−0.0467717	+	0.998906i	$0.514893\pi$
$314$		0			0
$315$	276.000	+	478.046i	0.0493677	+	0.0855074i
$316$		0			0
$317$		−	3916.17i		−	0.693861i	−0.937891	−	0.346930i	$-0.887224\pi$
$317$		−	3916.17i		−	0.693861i	0.937891	−	0.346930i	$-0.112776\pi$
$318$		0			0
$319$	−1692.00	−	976.877i	−0.296971	−	0.171456i
$320$		0			0
$321$	1506.00	−	2608.47i	0.261859	−	0.453553i
$322$		0			0
$323$	−11583.0	+	6687.45i	−1.99534	+	1.15201i
$324$		0			0
$325$	5551.00	−	1373.52i	0.947428	−	0.234428i
$326$		0			0
$327$	2688.00	−	1551.92i	0.454577	−	0.262450i
$328$		0			0
$329$	−2088.00	+	3616.52i	−0.349894	+	0.606034i
$330$		0			0
$331$	6456.00	+	3727.37i	1.07207	+	0.618958i	0.928745	−	0.370719i	$-0.120889\pi$
$331$	6456.00	+	3727.37i	1.07207	+	0.618958i	0.143321	+	0.989676i	$0.454222\pi$
$332$		0			0
$333$			3306.48i			0.544127i
$334$		0			0
$335$	681.000	+	1179.53i	0.111066	+	0.192371i
$336$		0			0
$337$	−3575.00			−0.577871			−0.288936	−	0.957349i	$-0.593301\pi$
$337$	−3575.00			−0.577871			−0.288936	+	0.957349i	$0.593301\pi$
$338$		0			0
$339$	1374.00			0.220134
$340$		0			0
$341$	−1080.00	−	1870.61i	−0.171511	−	0.297066i
$342$		0			0
$343$		−	6845.06i		−	1.07755i
$344$		0			0
$345$	234.000	+	135.100i	0.0365163	+	0.0210827i
$346$		0			0
$347$	−3483.00	+	6032.73i	−0.538839	+	0.933297i	0.460128	+	0.887853i	$0.347804\pi$
$347$	−3483.00	+	6032.73i	−0.538839	+	0.933297i	−0.998967	+	0.0454442i	$0.985530\pi$
$348$		0			0
$349$	−5760.00	+	3325.54i	−0.883455	+	0.510063i	−0.871796	−	0.489869i	$-0.837045\pi$
$349$	−5760.00	+	3325.54i	−0.883455	+	0.510063i	−0.0116588	+	0.999932i	$0.503711\pi$
$350$		0			0
$351$	4550.00	−	1125.83i	0.691912	−	0.171204i
$352$		0			0
$353$	4876.50	−	2815.45i	0.735269	−	0.424508i	−0.0850777	−	0.996374i	$-0.527114\pi$
$353$	4876.50	−	2815.45i	0.735269	−	0.424508i	0.820347	+	0.571867i	$0.193781\pi$
$354$		0			0
$355$	915.000	−	1584.83i	0.136798	−	0.236940i
$356$		0			0
$357$	−2808.00	−	1621.20i	−0.416289	−	0.240344i
$358$		0			0
$359$		−	7129.12i		−	1.04808i	−0.851694	−	0.524040i	$-0.824424\pi$
$359$		−	7129.12i		−	1.04808i	0.851694	−	0.524040i	$-0.175576\pi$
$360$		0			0
$361$	3104.50	+	5377.15i	0.452617	+	0.783956i
$362$		0			0
$363$	−2278.00			−0.329377
$364$		0			0
$365$	795.000			0.114006
$366$		0			0
$367$	1.00000	+	1.73205i	0.000142233	+	0.000246355i	0.866097	−	0.499877i	$-0.166621\pi$
$367$	1.00000	+	1.73205i	0.000142233	+	0.000246355i	−0.865954	+	0.500123i	$0.833288\pi$
$368$		0			0
$369$		−	6254.44i		−	0.882366i
$370$		0			0
$371$	1116.00	+	644.323i	0.156172	+	0.0901660i
$372$		0			0
$373$	−1749.50	+	3030.22i	−0.242857	+	0.420641i	−0.961527	−	0.274711i	$-0.911418\pi$
$373$	−1749.50	+	3030.22i	−0.242857	+	0.420641i	0.718670	+	0.695351i	$0.244751\pi$
$374$		0			0
$375$	−741.000	+	427.817i	−0.102040	+	0.0589129i
$376$		0			0
$377$	4582.50	+	4762.27i	0.626023	+	0.650582i
$378$		0			0
$379$	−4779.00	+	2759.16i	−0.647706	+	0.373953i	−0.787577	−	0.616216i	$-0.788665\pi$
$379$	−4779.00	+	2759.16i	−0.647706	+	0.373953i	0.139871	+	0.990170i	$0.455331\pi$
$380$		0			0
$381$	−286.000	+	495.367i	−0.0384573	+	0.0666100i
$382$		0			0
$383$	6378.00	+	3682.34i	0.850915	+	0.491276i	0.860960	−	0.508673i	$-0.169864\pi$
$383$	6378.00	+	3682.34i	0.850915	+	0.491276i	−0.0100443	+	0.999950i	$0.503197\pi$
$384$		0			0
$385$		−	332.554i		−	0.0440221i
$386$		0			0
$387$	−1196.00	−	2071.53i	−0.157096	−	0.272098i
$388$		0			0
$389$	−1209.00			−0.157580			−0.0787901	−	0.996891i	$-0.525106\pi$
$389$	−1209.00			−0.157580			−0.0787901	+	0.996891i	$0.525106\pi$
$390$		0			0
$391$	9126.00			1.18036
$392$		0			0
$393$	1974.00	+	3419.07i	0.253372	+	0.438853i
$394$		0			0
$395$		−	2210.10i		−	0.281524i
$396$		0			0
$397$	10128.0	+	5847.40i	1.28038	+	0.739226i	0.976917	−	0.213618i	$-0.0685248\pi$
$397$	10128.0	+	5847.40i	1.28038	+	0.739226i	0.303460	+	0.952844i	$0.401858\pi$
$398$		0			0
$399$	−1584.00	+	2743.57i	−0.198745	+	0.344236i
$400$		0			0
$401$	−2581.50	+	1490.43i	−0.321481	+	0.185607i	−0.652053	−	0.758174i	$-0.726092\pi$
$401$	−2581.50	+	1490.43i	−0.321481	+	0.185607i	0.330571	+	0.943781i	$0.392759\pi$
$402$		0			0
$403$	1755.00	+	7092.75i	0.216930	+	0.876712i
$404$		0			0
$405$	631.500	−	364.597i	0.0774802	−	0.0447332i
$406$		0			0
$407$	996.000	−	1725.12i	0.121302	−	0.210101i
$408$		0			0
$409$	37.5000	+	21.6506i	0.00453363	+	0.00261749i	0.502265	−	0.864714i	$-0.332500\pi$
$409$	37.5000	+	21.6506i	0.00453363	+	0.00261749i	−0.497731	+	0.867331i	$0.665833\pi$
$410$		0			0
$411$		−	1693.95i		−	0.203300i
$412$		0			0
$413$	1968.00	+	3408.68i	0.234477	+	0.406126i
$414$		0			0
$415$	1368.00			0.161813
$416$		0			0
$417$	472.000			0.0554291
$418$		0			0
$419$	−4731.00	−	8194.33i	−0.551610	−	0.955416i	−0.998159	−	0.0606569i	$-0.980680\pi$
$419$	−4731.00	−	8194.33i	−0.551610	−	0.955416i	0.446549	−	0.894759i	$-0.352653\pi$
$420$		0			0
$421$		−	7068.50i		−	0.818284i	−0.912471	−	0.409142i	$-0.865828\pi$
$421$		−	7068.50i		−	0.818284i	0.912471	−	0.409142i	$-0.134172\pi$
$422$		0			0
$423$	6003.00	+	3465.83i	0.690014	+	0.398380i
$424$		0			0
$425$	−7137.00	+	12361.6i	−0.814577	+	1.41089i
$426$		0			0
$427$	−1740.00	+	1004.59i	−0.197200	+	0.113854i
$428$		0			0
$429$	−1248.00	−	360.267i	−0.140452	−	0.0405451i
$430$		0			0
$431$	8598.00	−	4964.06i	0.960907	−	0.554780i	0.0644552	−	0.997921i	$-0.479469\pi$
$431$	8598.00	−	4964.06i	0.960907	−	0.554780i	0.896452	+	0.443140i	$0.146136\pi$
$432$		0			0
$433$	3308.50	−	5730.49i	0.367197	−	0.636004i	−0.621929	−	0.783074i	$-0.713651\pi$
$433$	3308.50	−	5730.49i	0.367197	−	0.636004i	0.989126	+	0.147070i	$0.0469841\pi$
$434$		0			0
$435$	−423.000	−	244.219i	−0.0466237	−	0.0269182i
$436$		0			0
$437$		−	8916.60i		−	0.976061i
$438$		0			0
$439$	6994.00	+	12114.0i	0.760377	+	1.31701i	0.942656	+	0.333765i	$0.108319\pi$
$439$	6994.00	+	12114.0i	0.760377	+	1.31701i	−0.182280	+	0.983247i	$0.558348\pi$
$440$		0			0
$441$	−3473.00			−0.375013
$442$		0			0
$443$	−2004.00			−0.214928			−0.107464	−	0.994209i	$-0.534273\pi$
$443$	−2004.00			−0.214928			−0.107464	+	0.994209i	$0.534273\pi$
$444$		0			0
$445$	−846.000	−	1465.31i	−0.0901219	−	0.156096i
$446$		0			0
$447$		−	93.5307i		−	0.00989676i
$448$		0			0
$449$	7866.00	+	4541.44i	0.826769	+	0.477336i	0.852745	−	0.522327i	$-0.174936\pi$
$449$	7866.00	+	4541.44i	0.826769	+	0.477336i	−0.0259758	+	0.999663i	$0.508269\pi$
$450$		0			0
$451$	−1884.00	+	3263.18i	−0.196705	+	0.340704i
$452$		0			0
$453$	3066.00	−	1770.16i	0.317998	−	0.183596i
$454$		0			0
$455$	−312.000	+	1080.80i	−0.0321468	+	0.111360i
$456$		0			0
$457$	2185.50	−	1261.80i	0.223705	−	0.129156i	−0.383959	−	0.923350i	$-0.625440\pi$
$457$	2185.50	−	1261.80i	0.223705	−	0.129156i	0.607665	+	0.794194i	$0.292106\pi$
$458$		0			0
$459$	−5850.00	+	10132.5i	−0.594890	+	1.03038i
$460$		0			0
$461$	16963.5	+	9793.88i	1.71382	+	0.989472i	0.929270	+	0.369400i	$0.120437\pi$
$461$	16963.5	+	9793.88i	1.71382	+	0.989472i	0.784545	+	0.620072i	$0.212897\pi$
$462$		0			0
$463$			8632.54i			0.866497i	0.901274	+	0.433249i	$0.142633\pi$
$463$			8632.54i			0.866497i	−0.901274	+	0.433249i	$0.857367\pi$
$464$		0			0
$465$	−270.000	−	467.654i	−0.0269268	−	0.0466385i
$466$		0			0
$467$	−5460.00			−0.541025			−0.270512	−	0.962716i	$-0.587193\pi$
$467$	−5460.00			−0.541025			−0.270512	+	0.962716i	$0.587193\pi$
$468$		0			0
$469$	10896.0			1.07277
$470$		0			0
$471$	1211.00	+	2097.51i	0.118471	+	0.205198i
$472$		0			0
$473$			1441.07i			0.140085i
$474$		0			0
$475$	12078.0	+	6973.24i	1.16669	+	0.673587i
$476$		0			0
$477$	1069.50	−	1852.43i	0.102660	−	0.177813i
$478$		0			0
$479$	2211.00	−	1276.52i	0.210904	−	0.121766i	−0.390827	−	0.920464i	$-0.627811\pi$
$479$	2211.00	−	1276.52i	0.210904	−	0.121766i	0.601732	+	0.798698i	$0.294478\pi$
$480$		0			0
$481$	−4855.50	+	4672.21i	−0.460274	+	0.442899i
$482$		0			0
$483$	1872.00	−	1080.80i	0.176354	−	0.101818i
$484$		0			0
$485$	−174.000	+	301.377i	−0.0162906	+	0.0282161i
$486$		0			0
$487$	−9378.00	−	5414.39i	−0.872603	−	0.503798i	−0.00439074	−	0.999990i	$-0.501398\pi$
$487$	−9378.00	−	5414.39i	−0.872603	−	0.503798i	−0.868212	+	0.496193i	$0.834731\pi$
$488$		0			0
$489$			2009.18i			0.185804i
$490$		0			0
$491$	5694.00	+	9862.30i	0.523354	+	0.906475i	0.999631	+	0.0271797i	$0.00865264\pi$
$491$	5694.00	+	9862.30i	0.523354	+	0.906475i	−0.476277	+	0.879295i	$0.658014\pi$
$492$		0			0
$493$	−16497.0			−1.50707
$494$		0			0
$495$	−552.000			−0.0501223
$496$		0			0
$497$	−7320.00	−	12678.6i	−0.660658	−	1.14429i
$498$		0			0
$499$		−	17677.3i		−	1.58586i	−0.609311	−	0.792931i	$-0.708554\pi$
$499$		−	17677.3i		−	1.58586i	0.609311	−	0.792931i	$-0.291446\pi$
$500$		0			0
$501$	−1584.00	−	914.523i	−0.141253	−	0.0815526i
$502$		0			0
$503$	1938.00	−	3356.71i	0.171792	−	0.297552i	−0.767255	−	0.641343i	$-0.778378\pi$
$503$	1938.00	−	3356.71i	0.171792	−	0.297552i	0.939046	+	0.343791i	$0.111711\pi$
$504$		0			0
$505$	643.500	−	371.525i	0.0567037	−	0.0327379i
$506$		0			0
$507$	3718.00	+	2341.73i	0.325685	+	0.205128i
$508$		0			0
$509$	−14779.5	+	8532.95i	−1.28701	+	0.743058i	−0.978120	−	0.208039i	$-0.933292\pi$
$509$	−14779.5	+	8532.95i	−1.28701	+	0.743058i	−0.308893	+	0.951097i	$0.599958\pi$
$510$		0			0
$511$	3180.00	−	5507.92i	0.275293	−	0.476822i
$512$		0			0
$513$	9900.00	+	5715.77i	0.852038	+	0.491925i
$514$		0			0
$515$		−	315.233i		−	0.0269725i
$516$		0			0
$517$	−2088.00	−	3616.52i	−0.177621	−	0.307649i
$518$		0			0
$519$	−5148.00			−0.435399
$520$		0			0
$521$	2121.00			0.178355			0.0891773	−	0.996016i	$-0.471576\pi$
$521$	2121.00			0.178355			0.0891773	+	0.996016i	$0.471576\pi$
$522$		0			0
$523$	−5732.00	−	9928.12i	−0.479241	−	0.830069i	0.520476	−	0.853876i	$-0.325755\pi$
$523$	−5732.00	−	9928.12i	−0.479241	−	0.830069i	−0.999717	+	0.0238072i	$0.992421\pi$
$524$		0			0
$525$			3380.96i			0.281062i
$526$		0			0
$527$	−15795.0	−	9119.25i	−1.30558	−	0.753777i
$528$		0			0
$529$	3041.50	−	5268.03i	0.249979	−	0.432977i
$530$		0			0
$531$	5658.00	−	3266.65i	0.462404	−	0.266969i
$532$		0			0
$533$	9184.50	−	8837.79i	0.746388	−	0.718212i
$534$		0			0
$535$	2259.00	−	1304.23i	0.182552	−	0.105396i
$536$		0			0
$537$	−3744.00	+	6484.80i	−0.300867	+	0.521117i
$538$		0			0
$539$	1812.00	+	1046.16i	0.144802	+	0.0836016i
$540$		0			0
$541$			4764.87i			0.378665i	0.981913	+	0.189333i	$0.0606324\pi$
$541$			4764.87i			0.378665i	−0.981913	+	0.189333i	$0.939368\pi$
$542$		0			0
$543$	−637.000	−	1103.32i	−0.0503431	−	0.0871968i
$544$		0			0
$545$	2688.00			0.211268
$546$		0			0
$547$	−6554.00			−0.512301			−0.256151	−	0.966637i	$-0.582454\pi$
$547$	−6554.00			−0.512301			−0.256151	+	0.966637i	$0.582454\pi$
$548$		0			0
$549$	1667.50	+	2888.19i	0.129631	+	0.224527i
$550$		0			0
$551$			16118.5i			1.24622i
$552$		0			0
$553$	−15312.0	−	8840.39i	−1.17745	−	0.679804i
$554$		0			0
$555$	249.000	−	431.281i	0.0190441	−	0.0329853i
$556$		0			0
$557$	−15685.5	+	9056.03i	−1.19321	+	0.688898i	−0.959032	−	0.283297i	$-0.908572\pi$
$557$	−15685.5	+	9056.03i	−1.19321	+	0.688898i	−0.234174	+	0.972195i	$0.575239\pi$
$558$		0			0
$559$	1352.00	−	4683.47i	0.102296	−	0.354364i
$560$		0			0
$561$	2808.00	−	1621.20i	0.211326	−	0.122009i
$562$		0			0
$563$	−6084.00	+	10537.8i	−0.455435	+	0.788837i	−0.998713	−	0.0507160i	$-0.983850\pi$
$563$	−6084.00	+	10537.8i	−0.455435	+	0.788837i	0.543278	+	0.839553i	$0.317183\pi$
$564$		0			0
$565$	1030.50	+	594.959i	0.0767318	+	0.0443011i
$566$		0			0
$567$		−	5833.55i		−	0.432074i
$568$		0			0
$569$	3861.00	+	6687.45i	0.284467	+	0.492711i	0.972480	−	0.232988i	$-0.0748502\pi$
$569$	3861.00	+	6687.45i	0.284467	+	0.492711i	−0.688013	+	0.725698i	$0.741517\pi$
$570$		0			0
$571$	−11440.0			−0.838440			−0.419220	−	0.907885i	$-0.637696\pi$
$571$	−11440.0			−0.838440			−0.419220	+	0.907885i	$0.637696\pi$
$572$		0			0
$573$	−5196.00			−0.378824
$574$		0			0
$575$	−4758.00	−	8241.10i	−0.345082	−	0.597700i
$576$		0			0
$577$		−	15444.7i		−	1.11433i	−0.830400	−	0.557167i	$-0.811888\pi$
$577$		−	15444.7i		−	1.11433i	0.830400	−	0.557167i	$-0.188112\pi$
$578$		0			0
$579$	1935.00	+	1117.17i	0.138887	+	0.0801867i
$580$		0			0
$581$	5472.00	−	9477.78i	0.390735	−	0.676772i
$582$		0			0
$583$	−1116.00	+	644.323i	−0.0792796	+	0.0457721i
$584$		0			0
$585$	1794.00	+	517.883i	0.126791	+	0.0366014i
$586$		0			0
$587$	12186.0	−	7035.59i	0.856848	−	0.494702i	−0.00610719	−	0.999981i	$-0.501944\pi$
$587$	12186.0	−	7035.59i	0.856848	−	0.494702i	0.862956	+	0.505280i	$0.168611\pi$
$588$		0			0
$589$	−8910.00	+	15432.6i	−0.623311	+	1.07961i
$590$		0			0
$591$	3552.00	+	2050.75i	0.247225	+	0.142735i
$592$		0			0
$593$		−	26938.6i		−	1.86549i	−0.360538	−	0.932745i	$-0.617407\pi$
$593$		−	26938.6i		−	1.86549i	0.360538	−	0.932745i	$-0.382593\pi$
$594$		0			0
$595$	−1404.00	−	2431.80i	−0.0967368	−	0.167553i
$596$		0			0
$597$	−5044.00			−0.345791
$598$		0			0
$599$	10554.0			0.719908			0.359954	−	0.932970i	$-0.382792\pi$
$599$	10554.0			0.719908			0.359954	+	0.932970i	$0.382792\pi$
$600$		0			0
$601$	7415.50	+	12844.0i	0.503302	+	0.871745i	0.999993	+	0.00381713i	$0.00121503\pi$
$601$	7415.50	+	12844.0i	0.503302	+	0.871745i	−0.496691	+	0.867928i	$0.665452\pi$
$602$		0			0
$603$		−	18086.1i		−	1.22143i
$604$		0			0
$605$	−1708.50	−	986.403i	−0.114811	−	0.0662859i
$606$		0			0
$607$	−3977.00	+	6888.37i	−0.265933	+	0.460610i	−0.967808	−	0.251691i	$-0.919013\pi$
$607$	−3977.00	+	6888.37i	−0.265933	+	0.460610i	0.701874	+	0.712301i	$0.252347\pi$
$608$		0			0
$609$	−3384.00	+	1953.75i	−0.225167	+	0.130000i
$610$		0			0
$611$	3393.00	+	13712.6i	0.224658	+	0.907945i
$612$		0			0
$613$	21841.5	−	12610.2i	1.43910	−	0.830866i	0.441315	−	0.897352i	$-0.354512\pi$
$613$	21841.5	−	12610.2i	1.43910	−	0.830866i	0.997787	+	0.0664859i	$0.0211787\pi$
$614$		0			0
$615$	−471.000	+	815.796i	−0.0308822	+	0.0534895i
$616$		0			0
$617$	−15055.5	−	8692.30i	−0.982353	−	0.567162i	−0.0793731	−	0.996845i	$-0.525292\pi$
$617$	−15055.5	−	8692.30i	−0.982353	−	0.567162i	−0.902980	+	0.429683i	$0.858625\pi$
$618$		0			0
$619$		−	8209.92i		−	0.533093i	−0.963822	−	0.266547i	$-0.914117\pi$
$619$		−	8209.92i		−	0.533093i	0.963822	−	0.266547i	$-0.0858826\pi$
$620$		0			0
$621$	−3900.00	−	6755.00i	−0.252015	−	0.436504i
$622$		0			0
$623$	−13536.0			−0.870479
$624$		0			0
$625$	14509.0			0.928576
$626$		0			0
$627$	−1584.00	−	2743.57i	−0.100891	−	0.174749i
$628$		0			0
$629$		−	16819.9i		−	1.06622i
$630$		0			0
$631$	−11142.0	−	6432.84i	−0.702941	−	0.405843i	0.105501	−	0.994419i	$-0.466355\pi$
$631$	−11142.0	−	6432.84i	−0.702941	−	0.405843i	−0.808442	+	0.588576i	$0.799689\pi$
$632$		0			0
$633$	−1042.00	+	1804.80i	−0.0654278	+	0.113324i
$634$		0			0
$635$	−429.000	+	247.683i	−0.0268100	+	0.0154788i
$636$		0			0
$637$	−4907.50	−	5100.02i	−0.305247	−	0.317222i
$638$		0			0
$639$	−21045.0	+	12150.3i	−1.30286	+	0.752206i
$640$		0			0
$641$	−3100.50	+	5370.22i	−0.191049	+	0.330907i	−0.945598	−	0.325337i	$-0.894522\pi$
$641$	−3100.50	+	5370.22i	−0.191049	+	0.330907i	0.754549	+	0.656244i	$0.227856\pi$
$642$		0			0
$643$	14568.0	+	8410.84i	0.893477	+	0.515849i	0.875078	−	0.483981i	$-0.160810\pi$
$643$	14568.0	+	8410.84i	0.893477	+	0.515849i	0.0183989	+	0.999831i	$0.494143\pi$
$644$		0			0
$645$			360.267i			0.0219930i
$646$		0			0
$647$	−6747.00	−	11686.1i	−0.409972	−	0.710092i	0.584914	−	0.811095i	$-0.301128\pi$
$647$	−6747.00	−	11686.1i	−0.409972	−	0.710092i	−0.994886	+	0.101003i	$0.967795\pi$
$648$		0			0
$649$	−3936.00			−0.238061
$650$		0			0
$651$	−4320.00			−0.260083
$652$		0			0
$653$	5667.00	+	9815.53i	0.339612	+	0.588226i	0.984360	−	0.176170i	$-0.0563708\pi$
$653$	5667.00	+	9815.53i	0.339612	+	0.588226i	−0.644747	+	0.764396i	$0.723037\pi$
$654$		0			0
$655$			3419.07i			0.203960i
$656$		0			0
$657$	−9142.50	−	5278.42i	−0.542896	−	0.313441i
$658$		0			0
$659$	6618.00	−	11462.7i	0.391200	−	0.677578i	−0.601408	−	0.798942i	$-0.705393\pi$
$659$	6618.00	−	11462.7i	0.391200	−	0.677578i	0.992608	+	0.121364i	$0.0387268\pi$
$660$		0			0
$661$	−10264.5	+	5926.21i	−0.603998	+	0.348718i	−0.770613	−	0.637304i	$-0.780050\pi$
$661$	−10264.5	+	5926.21i	−0.603998	+	0.348718i	0.166615	+	0.986022i	$0.446716\pi$
$662$		0			0
$663$	−10647.0	+	2634.45i	−0.623673	+	0.154319i
$664$		0			0
$665$	−2376.00	+	1371.78i	−0.138552	+	0.0799932i
$666$		0			0
$667$	5499.00	−	9524.55i	0.319224	−	0.552911i
$668$		0			0
$669$	4170.00	+	2407.55i	0.240989	+	0.139135i
$670$		0			0
$671$		−	2009.18i		−	0.115594i
$672$		0			0
$673$	−4010.50	−	6946.39i	−0.229708	−	0.397866i	0.728014	−	0.685563i	$-0.240444\pi$
$673$	−4010.50	−	6946.39i	−0.229708	−	0.397866i	−0.957722	+	0.287697i	$0.907110\pi$
$674$		0			0
$675$	12200.0			0.695671
$676$		0			0
$677$	−21630.0			−1.22793			−0.613965	−	0.789333i	$-0.710426\pi$
$677$	−21630.0			−1.22793			−0.613965	+	0.789333i	$0.710426\pi$
$678$		0			0
$679$	1392.00	+	2411.01i	0.0786746	+	0.136268i
$680$		0			0
$681$		−	4815.10i		−	0.270947i
$682$		0			0
$683$	22983.0	+	13269.2i	1.28758	+	0.743387i	0.978223	−	0.207557i	$-0.0665514\pi$
$683$	22983.0	+	13269.2i	1.28758	+	0.743387i	0.309361	+	0.950945i	$0.399885\pi$
$684$		0			0
$685$	733.500	−	1270.46i	0.0409133	−	0.0708639i
$686$		0			0
$687$	4344.00	−	2508.01i	0.241243	−	0.139282i
$688$		0			0
$689$	4231.50	−	1047.02i	0.233973	−	0.0578933i
$690$		0			0
$691$	720.000	−	415.692i	0.0396383	−	0.0228852i	−0.480050	−	0.877241i	$-0.659381\pi$
$691$	720.000	−	415.692i	0.0396383	−	0.0228852i	0.519688	+	0.854356i	$0.326048\pi$
$692$		0			0
$693$	−2208.00	+	3824.37i	−0.121032	+	0.209633i
$694$		0			0
$695$	354.000	+	204.382i	0.0193208	+	0.0111549i
$696$		0			0
$697$			31816.0i			1.72901i
$698$		0			0
$699$	−5850.00	−	10132.5i	−0.316548	−	0.548278i
$700$		0			0
$701$	30186.0			1.62640			0.813202	−	0.581981i	$-0.197722\pi$
$701$	30186.0			1.62640			0.813202	+	0.581981i	$0.197722\pi$
$702$		0			0
$703$	−16434.0			−0.881679
$704$		0			0
$705$	−522.000	−	904.131i	−0.0278860	−	0.0483000i
$706$		0			0
$707$		−	5944.40i		−	0.316212i
$708$		0			0
$709$	−10288.5	−	5940.07i	−0.544983	−	0.314646i	0.202113	−	0.979362i	$-0.435219\pi$
$709$	−10288.5	−	5940.07i	−0.544983	−	0.314646i	−0.747096	+	0.664716i	$0.768552\pi$
$710$		0			0
$711$	−14674.0	+	25416.1i	−0.774006	+	1.34062i
$712$		0			0
$713$	10530.0	−	6079.50i	0.553088	−	0.319325i
$714$		0			0
$715$	−780.000	−	810.600i	−0.0407977	−	0.0423982i
$716$		0			0
$717$	9324.00	−	5383.21i	0.485650	−	0.280390i
$718$		0			0
$719$	9204.00	−	15941.8i	0.477401	−	0.826883i	−0.522264	−	0.852784i	$-0.674912\pi$
$719$	9204.00	−	15941.8i	0.477401	−	0.826883i	0.999665	+	0.0259014i	$0.00824561\pi$
$720$		0			0
$721$	−2184.00	−	1260.93i	−0.112811	−	0.0651312i
$722$		0			0
$723$			9834.58i			0.505881i
$724$		0			0
$725$	8601.00	+	14897.4i	0.440597	+	0.763137i
$726$		0			0
$727$	−21112.0			−1.07703			−0.538515	−	0.842616i	$-0.681014\pi$
$727$	−21112.0			−1.07703			−0.538515	+	0.842616i	$0.681014\pi$
$728$		0			0
$729$	−4283.00			−0.217599
$730$		0			0
$731$	6084.00	+	10537.8i	0.307832	+	0.533180i
$732$		0			0
$733$			23959.5i			1.20732i	0.797243	+	0.603658i	$0.206291\pi$
$733$			23959.5i			1.20732i	−0.797243	+	0.603658i	$0.793709\pi$
$734$		0			0
$735$	453.000	+	261.540i	0.0227335	+	0.0131252i
$736$		0			0
$737$	−5448.00	+	9436.21i	−0.272293	+	0.471625i
$738$		0			0
$739$	−2742.00	+	1583.09i	−0.136490	+	0.0788025i	−0.566690	−	0.823931i	$-0.691776\pi$
$739$	−2742.00	+	1583.09i	−0.136490	+	0.0788025i	0.430200	+	0.902734i	$0.358443\pi$
$740$		0			0
$741$	2574.00	+	10402.7i	0.127609	+	0.515726i
$742$		0			0
$743$	26070.0	−	15051.5i	1.28723	−	0.743185i	0.309075	−	0.951038i	$-0.399981\pi$
$743$	26070.0	−	15051.5i	1.28723	−	0.743185i	0.978160	+	0.207852i	$0.0666474\pi$
$744$		0			0
$745$	40.5000	−	70.1481i	0.00199168	−	0.00344970i
$746$		0			0
$747$	−15732.0	−	9082.87i	−0.770554	−	0.444880i
$748$		0			0
$749$		−	20867.7i		−	1.01801i
$750$		0			0
$751$	14248.0	+	24678.3i	0.692299	+	1.19910i	0.971083	+	0.238744i	$0.0767356\pi$
$751$	14248.0	+	24678.3i	0.692299	+	1.19910i	−0.278783	+	0.960354i	$0.589931\pi$
$752$		0			0
$753$	−7956.00			−0.385037
$754$		0			0
$755$	3066.00			0.147792
$756$		0			0
$757$	−8711.00	−	15087.9i	−0.418239	−	0.724411i	0.577524	−	0.816374i	$-0.304019\pi$
$757$	−8711.00	−	15087.9i	−0.418239	−	0.724411i	−0.995762	+	0.0919633i	$0.970686\pi$
$758$		0			0
$759$			2161.60i			0.103374i
$760$		0			0
$761$	35790.0	+	20663.4i	1.70484	+	0.984292i	0.940695	+	0.339252i	$0.110174\pi$
$761$	35790.0	+	20663.4i	1.70484	+	0.984292i	0.764149	+	0.645040i	$0.223159\pi$
$762$		0			0
$763$	10752.0	−	18623.0i	0.510155	−	0.883615i
$764$		0			0
$765$	−4036.50	+	2330.47i	−0.190771	+	0.110142i
$766$		0			0
$767$	12792.0	+	3692.73i	0.602206	+	0.173842i
$768$		0			0
$769$	−12186.0	+	7035.59i	−0.571441	+	0.329922i	−0.757725	−	0.652574i	$-0.773689\pi$
$769$	−12186.0	+	7035.59i	−0.571441	+	0.329922i	0.186283	+	0.982496i	$0.440356\pi$
$770$		0			0
$771$	2067.00	−	3580.15i	0.0965515	−	0.167232i
$772$		0			0
$773$	174.000	+	100.459i	0.00809618	+	0.00467433i	0.504043	−	0.863679i	$-0.331845\pi$
$773$	174.000	+	100.459i	0.00809618	+	0.00467433i	−0.495946	+	0.868353i	$0.665179\pi$
$774$		0			0
$775$			19017.9i			0.881476i
$776$		0			0
$777$	−1992.00	−	3450.25i	−0.0919725	−	0.159301i
$778$		0			0
$779$	31086.0			1.42975
$780$		0			0
$781$	14640.0			0.670756
$782$		0			0
$783$	7050.00	+	12211.0i	0.321771	+	0.557323i
$784$		0			0
$785$			2097.51i			0.0953675i

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{3} +1.73205i q^{5} +(12.0000 + 6.92820i) q^{7} +(11.5000 - 19.9186i) q^{9} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{3} +1.73205i q^{5} +(12.0000 + 6.92820i) q^{7} +(11.5000 - 19.9186i) q^{9} +(-12.0000 + 6.92820i) q^{11} +(45.5000 - 11.2583i) q^{13} +(-3.00000 + 1.73205i) q^{15} +(-58.5000 + 101.325i) q^{17} +(99.0000 + 57.1577i) q^{19} +27.7128i q^{21} +(-39.0000 - 67.5500i) q^{23} +122.000 q^{25} +100.000 q^{27} +(70.5000 + 122.110i) q^{29} +155.885i q^{31} +(-24.0000 - 13.8564i) q^{33} +(-12.0000 + 20.7846i) q^{35} +(-124.500 + 71.8801i) q^{37} +(65.0000 + 67.5500i) q^{39} +(235.500 - 135.966i) q^{41} +(52.0000 - 90.0666i) q^{43} +(34.5000 + 19.9186i) q^{45} +301.377i q^{47} +(-75.5000 - 130.770i) q^{49} -234.000 q^{51} +93.0000 q^{53} +(-12.0000 - 20.7846i) q^{55} +228.631i q^{57} +(246.000 + 142.028i) q^{59} +(-72.5000 + 125.574i) q^{61} +(276.000 - 159.349i) q^{63} +(19.5000 + 78.8083i) q^{65} +(681.000 - 393.176i) q^{67} +(78.0000 - 135.100i) q^{69} +(-915.000 - 528.275i) q^{71} -458.993i q^{73} +(122.000 + 211.310i) q^{75} -192.000 q^{77} -1276.00 q^{79} +(-210.500 - 364.597i) q^{81} -789.815i q^{83} +(-175.500 - 101.325i) q^{85} +(-141.000 + 244.219i) q^{87} +(-846.000 + 488.438i) q^{89} +(624.000 + 180.133i) q^{91} +(-270.000 + 155.885i) q^{93} +(-99.0000 + 171.473i) q^{95} +(174.000 + 100.459i) q^{97} +318.697i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 24 q^{7} + 23 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 24 * q^7 + 23 * q^9	\( 2 q + 2 q^{3} + 24 q^{7} + 23 q^{9} - 24 q^{11} + 91 q^{13} - 6 q^{15} - 117 q^{17} + 198 q^{19} - 78 q^{23} + 244 q^{25} + 200 q^{27} + 141 q^{29} - 48 q^{33} - 24 q^{35} - 249 q^{37} + 130 q^{39} + 471 q^{41} + 104 q^{43} + 69 q^{45} - 151 q^{49} - 468 q^{51} + 186 q^{53} - 24 q^{55} + 492 q^{59} - 145 q^{61} + 552 q^{63} + 39 q^{65} + 1362 q^{67} + 156 q^{69} - 1830 q^{71} + 244 q^{75} - 384 q^{77} - 2552 q^{79} - 421 q^{81} - 351 q^{85} - 282 q^{87} - 1692 q^{89} + 1248 q^{91} - 540 q^{93} - 198 q^{95} + 348 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 24 * q^7 + 23 * q^9 - 24 * q^11 + 91 * q^13 - 6 * q^15 - 117 * q^17 + 198 * q^19 - 78 * q^23 + 244 * q^25 + 200 * q^27 + 141 * q^29 - 48 * q^33 - 24 * q^35 - 249 * q^37 + 130 * q^39 + 471 * q^41 + 104 * q^43 + 69 * q^45 - 151 * q^49 - 468 * q^51 + 186 * q^53 - 24 * q^55 + 492 * q^59 - 145 * q^61 + 552 * q^63 + 39 * q^65 + 1362 * q^67 + 156 * q^69 - 1830 * q^71 + 244 * q^75 - 384 * q^77 - 2552 * q^79 - 421 * q^81 - 351 * q^85 - 282 * q^87 - 1692 * q^89 + 1248 * q^91 - 540 * q^93 - 198 * q^95 + 348 * q^97

Introduction

L-functions

Modular forms

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