LMFDB - Embedded newform 208.4.w.a.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:	$\Z[a_1, \ldots, a_{5}]$
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.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.50000 - 6.06218i) q^{3} -13.8564i q^{5} +(-19.5000 - 11.2583i) q^{7} +(-11.0000 + 19.0526i) q^{9} +O(q^{10})$	\(q+(-3.50000 - 6.06218i) q^{3} -13.8564i q^{5} +(-19.5000 - 11.2583i) q^{7} +(-11.0000 + 19.0526i) q^{9} +(19.5000 - 11.2583i) q^{11} +(-13.0000 - 45.0333i) q^{13} +(-84.0000 + 48.4974i) q^{15} +(-13.5000 + 23.3827i) q^{17} +(76.5000 + 44.1673i) q^{19} +157.617i q^{21} +(28.5000 + 49.3634i) q^{23} -67.0000 q^{25} -35.0000 q^{27} +(34.5000 + 59.7558i) q^{29} -72.7461i q^{31} +(-136.500 - 78.8083i) q^{33} +(-156.000 + 270.200i) q^{35} +(-34.5000 + 19.9186i) q^{37} +(-227.500 + 236.425i) q^{39} +(-340.500 + 196.588i) q^{41} +(-42.5000 + 73.6122i) q^{43} +(264.000 + 152.420i) q^{45} -342.946i q^{47} +(82.0000 + 142.028i) q^{49} +189.000 q^{51} +426.000 q^{53} +(-156.000 - 270.200i) q^{55} -618.342i q^{57} +(16.5000 + 9.52628i) q^{59} +(8.50000 - 14.7224i) q^{61} +(429.000 - 247.683i) q^{63} +(-624.000 + 180.133i) q^{65} +(-142.500 + 82.2724i) q^{67} +(199.500 - 345.544i) q^{69} +(-505.500 - 291.851i) q^{71} -1004.59i q^{73} +(234.500 + 406.166i) q^{75} -507.000 q^{77} +1244.00 q^{79} +(419.500 + 726.595i) q^{81} -426.084i q^{83} +(324.000 + 187.061i) q^{85} +(241.500 - 418.290i) q^{87} +(265.500 - 153.286i) q^{89} +(-253.500 + 1024.51i) q^{91} +(-441.000 + 254.611i) q^{93} +(612.000 - 1060.02i) q^{95} +(1069.50 + 617.476i) q^{97} +495.367i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 7 q^{3} - 39 q^{7} - 22 q^{9}+O(q^{10}) $ 2 * q - 7 * q^3 - 39 * q^7 - 22 * q^9	$ 2 q - 7 q^{3} - 39 q^{7} - 22 q^{9} + 39 q^{11} - 26 q^{13} - 168 q^{15} - 27 q^{17} + 153 q^{19} + 57 q^{23} - 134 q^{25} - 70 q^{27} + 69 q^{29} - 273 q^{33} - 312 q^{35} - 69 q^{37} - 455 q^{39} - 681 q^{41} - 85 q^{43} + 528 q^{45} + 164 q^{49} + 378 q^{51} + 852 q^{53} - 312 q^{55} + 33 q^{59} + 17 q^{61} + 858 q^{63} - 1248 q^{65} - 285 q^{67} + 399 q^{69} - 1011 q^{71} + 469 q^{75} - 1014 q^{77} + 2488 q^{79} + 839 q^{81} + 648 q^{85} + 483 q^{87} + 531 q^{89} - 507 q^{91} - 882 q^{93} + 1224 q^{95} + 2139 q^{97}+O(q^{100}) $ 2 * q - 7 * q^3 - 39 * q^7 - 22 * q^9 + 39 * q^11 - 26 * q^13 - 168 * q^15 - 27 * q^17 + 153 * q^19 + 57 * q^23 - 134 * q^25 - 70 * q^27 + 69 * q^29 - 273 * q^33 - 312 * q^35 - 69 * q^37 - 455 * q^39 - 681 * q^41 - 85 * q^43 + 528 * q^45 + 164 * q^49 + 378 * q^51 + 852 * q^53 - 312 * q^55 + 33 * q^59 + 17 * q^61 + 858 * q^63 - 1248 * q^65 - 285 * q^67 + 399 * q^69 - 1011 * q^71 + 469 * q^75 - 1014 * q^77 + 2488 * q^79 + 839 * q^81 + 648 * q^85 + 483 * q^87 + 531 * q^89 - 507 * q^91 - 882 * q^93 + 1224 * q^95 + 2139 * 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
$3$	−3.50000	−	6.06218i	−0.673575	−	1.16667i	−0.976883	−	0.213774i	$-0.931424\pi$
$3$	−3.50000	−	6.06218i	−0.673575	−	1.16667i	0.303308	−	0.952893i	$-0.401909\pi$
$4$		0			0
$5$		−	13.8564i		−	1.23935i	−0.784857	−	0.619677i	$-0.787263\pi$
$5$		−	13.8564i		−	1.23935i	0.784857	−	0.619677i	$-0.212737\pi$
$6$		0			0
$7$	−19.5000	−	11.2583i	−1.05290	−	0.607893i	−0.129441	−	0.991587i	$-0.541318\pi$
$7$	−19.5000	−	11.2583i	−1.05290	−	0.607893i	−0.923460	+	0.383694i	$0.874652\pi$
$8$		0			0
$9$	−11.0000	+	19.0526i	−0.407407	+	0.705650i
$10$		0			0
$11$	19.5000	−	11.2583i	0.534497	−	0.308592i	−0.208349	−	0.978055i	$-0.566809\pi$
$11$	19.5000	−	11.2583i	0.534497	−	0.308592i	0.742846	+	0.669462i	$0.233475\pi$
$12$		0			0
$13$	−13.0000	−	45.0333i	−0.277350	−	0.960769i
$13$	−13.0000	−	45.0333i	−0.277350	−	0.960769i
$14$		0			0
$15$	−84.0000	+	48.4974i	−1.44591	+	0.834799i
$16$		0			0
$17$	−13.5000	+	23.3827i	−0.192602	+	0.333596i	−0.946112	−	0.323840i	$-0.895026\pi$
$17$	−13.5000	+	23.3827i	−0.192602	+	0.333596i	0.753510	+	0.657437i	$0.228359\pi$
$18$		0			0
$19$	76.5000	+	44.1673i	0.923700	+	0.533299i	0.884814	−	0.465945i	$-0.154286\pi$
$19$	76.5000	+	44.1673i	0.923700	+	0.533299i	0.0388865	+	0.999244i	$0.487619\pi$
$20$		0			0
$21$			157.617i			1.63785i
$22$		0			0
$23$	28.5000	+	49.3634i	0.258377	+	0.447521i	0.965807	−	0.259261i	$-0.0834791\pi$
$23$	28.5000	+	49.3634i	0.258377	+	0.447521i	−0.707431	+	0.706783i	$0.750146\pi$
$24$		0			0
$25$	−67.0000			−0.536000
$26$		0			0
$27$	−35.0000			−0.249472
$28$		0			0
$29$	34.5000	+	59.7558i	0.220913	+	0.382633i	0.955086	−	0.296330i	$-0.0957628\pi$
$29$	34.5000	+	59.7558i	0.220913	+	0.382633i	−0.734172	+	0.678963i	$0.762430\pi$
$30$		0			0
$31$		−	72.7461i		−	0.421471i	−0.977543	−	0.210735i	$-0.932414\pi$
$31$		−	72.7461i		−	0.421471i	0.977543	−	0.210735i	$-0.0675858\pi$
$32$		0			0
$33$	−136.500	−	78.8083i	−0.720048	−	0.415720i
$34$		0			0
$35$	−156.000	+	270.200i	−0.753395	+	1.30492i
$36$		0			0
$37$	−34.5000	+	19.9186i	−0.153291	+	0.0885026i	−0.574683	−	0.818376i	$-0.694875\pi$
$37$	−34.5000	+	19.9186i	−0.153291	+	0.0885026i	0.421393	+	0.906878i	$0.361541\pi$
$38$		0			0
$39$	−227.500	+	236.425i	−0.934081	+	0.970725i
$40$		0			0
$41$	−340.500	+	196.588i	−1.29700	+	0.748826i	−0.979886	−	0.199560i	$-0.936049\pi$
$41$	−340.500	+	196.588i	−1.29700	+	0.748826i	−0.317118	+	0.948386i	$0.602715\pi$
$42$		0			0
$43$	−42.5000	+	73.6122i	−0.150725	+	0.261064i	−0.931494	−	0.363756i	$-0.881494\pi$
$43$	−42.5000	+	73.6122i	−0.150725	+	0.261064i	0.780769	+	0.624820i	$0.214828\pi$
$44$		0			0
$45$	264.000	+	152.420i	0.874551	+	0.504922i
$46$		0			0
$47$		−	342.946i		−	1.06434i	−0.846639	−	0.532168i	$-0.821377\pi$
$47$		−	342.946i		−	1.06434i	0.846639	−	0.532168i	$-0.178623\pi$
$48$		0			0
$49$	82.0000	+	142.028i	0.239067	+	0.414076i
$50$		0			0
$51$	189.000			0.518927
$52$		0			0
$53$	426.000			1.10407			0.552034	−	0.833822i	$-0.313852\pi$
$53$	426.000			1.10407			0.552034	+	0.833822i	$0.313852\pi$
$54$		0			0
$55$	−156.000	−	270.200i	−0.382455	−	0.662432i
$56$		0			0
$57$		−	618.342i		−	1.43687i
$58$		0			0
$59$	16.5000	+	9.52628i	0.0364088	+	0.0210206i	0.518094	−	0.855324i	$-0.326642\pi$
$59$	16.5000	+	9.52628i	0.0364088	+	0.0210206i	−0.481685	+	0.876344i	$0.659975\pi$
$60$		0			0
$61$	8.50000	−	14.7224i	0.0178412	−	0.0309019i	−0.856967	−	0.515371i	$-0.827654\pi$
$61$	8.50000	−	14.7224i	0.0178412	−	0.0309019i	0.874808	+	0.484469i	$0.160987\pi$
$62$		0			0
$63$	429.000	−	247.683i	0.857919	−	0.495320i
$64$		0			0
$65$	−624.000	+	180.133i	−1.19073	+	0.343735i
$66$		0			0
$67$	−142.500	+	82.2724i	−0.259838	+	0.150018i	−0.624261	−	0.781216i	$-0.714600\pi$
$67$	−142.500	+	82.2724i	−0.259838	+	0.150018i	0.364423	+	0.931234i	$0.381266\pi$
$68$		0			0
$69$	199.500	−	345.544i	0.348072	−	0.602879i
$70$		0			0
$71$	−505.500	−	291.851i	−0.844955	−	0.487835i	0.0139904	−	0.999902i	$-0.495547\pi$
$71$	−505.500	−	291.851i	−0.844955	−	0.487835i	−0.858945	+	0.512067i	$0.828880\pi$
$72$		0			0
$73$		−	1004.59i		−	1.61066i	−0.592826	−	0.805331i	$-0.701988\pi$
$73$		−	1004.59i		−	1.61066i	0.592826	−	0.805331i	$-0.298012\pi$
$74$		0			0
$75$	234.500	+	406.166i	0.361036	+	0.625333i
$76$		0			0
$77$	−507.000			−0.750364
$78$		0			0
$79$	1244.00			1.77166			0.885829	−	0.464012i	$-0.153591\pi$
$79$	1244.00			1.77166			0.885829	+	0.464012i	$0.153591\pi$
$80$		0			0
$81$	419.500	+	726.595i	0.575446	+	0.996701i
$82$		0			0
$83$		−	426.084i		−	0.563480i	−0.959491	−	0.281740i	$-0.909088\pi$
$83$		−	426.084i		−	0.563480i	0.959491	−	0.281740i	$-0.0909116\pi$
$84$		0			0
$85$	324.000	+	187.061i	0.413444	+	0.238702i
$86$		0			0
$87$	241.500	−	418.290i	0.297604	−	0.515465i
$88$		0			0
$89$	265.500	−	153.286i	0.316213	−	0.182566i	−0.333490	−	0.942753i	$-0.608226\pi$
$89$	265.500	−	153.286i	0.316213	−	0.182566i	0.649703	+	0.760188i	$0.274893\pi$
$90$		0			0
$91$	−253.500	+	1024.51i	−0.292022	+	1.18019i
$92$		0			0
$93$	−441.000	+	254.611i	−0.491716	+	0.283892i
$94$		0			0
$95$	612.000	−	1060.02i	0.660946	−	1.14479i
$96$		0			0
$97$	1069.50	+	617.476i	1.11950	+	0.646342i	0.941273	−	0.337647i	$-0.109631\pi$
$97$	1069.50	+	617.476i	1.11950	+	0.646342i	0.178225	+	0.983990i	$0.442965\pi$
$98$		0			0
$99$			495.367i			0.502891i
$100$		0			0
$101$	−979.500	−	1696.54i	−0.964989	−	1.67141i	−0.709645	−	0.704560i	$-0.751144\pi$
$101$	−979.500	−	1696.54i	−0.964989	−	1.67141i	−0.255345	−	0.966850i	$-0.582189\pi$
$102$		0			0
$103$	−1856.00			−1.77551			−0.887753	−	0.460320i	$-0.847735\pi$
$103$	−1856.00			−1.77551			−0.887753	+	0.460320i	$0.847735\pi$
$104$		0			0
$105$	2184.00			2.02987
$106$		0			0
$107$	−127.500	−	220.836i	−0.115195	−	0.199524i	0.802663	−	0.596433i	$-0.203416\pi$
$107$	−127.500	−	220.836i	−0.115195	−	0.199524i	−0.917858	+	0.396909i	$0.870083\pi$
$108$		0			0
$109$			609.682i			0.535752i	0.963453	+	0.267876i	$0.0863217\pi$
$109$			609.682i			0.535752i	−0.963453	+	0.267876i	$0.913678\pi$
$110$		0			0
$111$	241.500	+	139.430i	0.206506	+	0.119226i
$112$		0			0
$113$	−205.500	+	355.936i	−0.171078	+	0.296316i	−0.938797	−	0.344471i	$-0.888058\pi$
$113$	−205.500	+	355.936i	−0.171078	+	0.296316i	0.767719	+	0.640787i	$0.221392\pi$
$114$		0			0
$115$	684.000	−	394.908i	0.554638	−	0.320220i
$116$		0			0
$117$	1001.00	+	247.683i	0.790961	+	0.195712i
$118$		0			0
$119$	526.500	−	303.975i	0.405581	−	0.234162i
$120$		0			0
$121$	−412.000	+	713.605i	−0.309542	+	0.536142i
$122$		0			0
$123$	2383.50	+	1376.11i	1.74726	+	1.00878i
$124$		0			0
$125$		−	803.672i		−	0.575061i
$126$		0			0
$127$	−1121.50	−	1942.49i	−0.783599	−	1.35723i	−0.929833	−	0.367983i	$-0.880049\pi$
$127$	−1121.50	−	1942.49i	−0.783599	−	1.35723i	0.146234	−	0.989250i	$-0.453285\pi$
$128$		0			0
$129$	595.000			0.406099
$130$		0			0
$131$	372.000			0.248105			0.124053	−	0.992276i	$-0.460411\pi$
$131$	372.000			0.248105			0.124053	+	0.992276i	$0.460411\pi$
$132$		0			0
$133$	−994.500	−	1722.52i	−0.648377	−	1.12302i
$134$		0			0
$135$			484.974i			0.309185i
$136$		0			0
$137$	−1030.50	−	594.959i	−0.642639	−	0.371028i	0.142991	−	0.989724i	$-0.454328\pi$
$137$	−1030.50	−	594.959i	−0.642639	−	0.371028i	−0.785630	+	0.618696i	$0.787661\pi$
$138$		0			0
$139$	−1272.50	+	2204.03i	−0.776490	+	1.34492i	0.157464	+	0.987525i	$0.449668\pi$
$139$	−1272.50	+	2204.03i	−0.776490	+	1.34492i	−0.933953	+	0.357395i	$0.883665\pi$
$140$		0			0
$141$	−2079.00	+	1200.31i	−1.24173	+	0.716911i
$142$		0			0
$143$	−760.500	−	731.791i	−0.444729	−	0.427940i
$144$		0			0
$145$	828.000	−	478.046i	0.474218	−	0.273790i
$146$		0			0
$147$	574.000	−	994.197i	0.322059	−	0.557823i
$148$		0			0
$149$	1129.50	+	652.117i	0.621022	+	0.358547i	0.777267	−	0.629171i	$-0.216606\pi$
$149$	1129.50	+	652.117i	0.621022	+	0.358547i	−0.156245	+	0.987718i	$0.549939\pi$
$150$		0			0
$151$		−	86.6025i		−	0.0466729i	−0.999728	−	0.0233365i	$-0.992571\pi$
$151$		−	86.6025i		−	0.0466729i	0.999728	−	0.0233365i	$-0.00742890\pi$
$152$		0			0
$153$	−297.000	−	514.419i	−0.156935	−	0.271819i
$154$		0			0
$155$	−1008.00			−0.522352
$156$		0			0
$157$	−1534.00			−0.779787			−0.389893	−	0.920860i	$-0.627488\pi$
$157$	−1534.00			−0.779787			−0.389893	+	0.920860i	$0.627488\pi$
$158$		0			0
$159$	−1491.00	−	2582.49i	−0.743673	−	1.28808i
$160$		0			0
$161$		−	1283.45i		−	0.628261i
$162$		0			0
$163$	1414.50	+	816.662i	0.679707	+	0.392429i	0.799745	−	0.600340i	$-0.204968\pi$
$163$	1414.50	+	816.662i	0.679707	+	0.392429i	−0.120038	+	0.992769i	$0.538302\pi$
$164$		0			0
$165$	−1092.00	+	1891.40i	−0.515225	+	0.892395i
$166$		0			0
$167$	−1408.50	+	813.198i	−0.652653	+	0.376809i	−0.789472	−	0.613787i	$-0.789645\pi$
$167$	−1408.50	+	813.198i	−0.652653	+	0.376809i	0.136819	+	0.990596i	$0.456312\pi$
$168$		0			0
$169$	−1859.00	+	1170.87i	−0.846154	+	0.532939i
$170$		0			0
$171$	−1683.00	+	971.681i	−0.752645	+	0.434540i
$172$		0			0
$173$	436.500	−	756.040i	0.191829	−	0.332258i	−0.754027	−	0.656843i	$-0.771891\pi$
$173$	436.500	−	756.040i	0.191829	−	0.332258i	0.945857	+	0.324585i	$0.105225\pi$
$174$		0			0
$175$	1306.50	+	754.308i	0.564355	+	0.325830i
$176$		0			0
$177$		−	133.368i		−	0.0566359i
$178$		0			0
$179$	−643.500	−	1114.57i	−0.268701	−	0.465403i	0.699826	−	0.714314i	$-0.253261\pi$
$179$	−643.500	−	1114.57i	−0.268701	−	0.465403i	−0.968527	+	0.248910i	$0.919928\pi$
$180$		0			0
$181$	2.00000			0.000821319			0.000410660	−	1.00000i	$-0.499869\pi$
$181$	2.00000			0.000821319			0.000410660		1.00000i	$0.499869\pi$
$182$		0			0
$183$	−119.000			−0.0480696
$184$		0			0
$185$	276.000	+	478.046i	0.109686	+	0.189982i
$186$		0			0
$187$			607.950i			0.237742i
$188$		0			0
$189$	682.500	+	394.042i	0.262670	+	0.151652i
$190$		0			0
$191$	−1420.50	+	2460.38i	−0.538135	+	0.932077i	0.460870	+	0.887468i	$0.347538\pi$
$191$	−1420.50	+	2460.38i	−0.538135	+	0.932077i	−0.999005	+	0.0446092i	$0.985796\pi$
$192$		0			0
$193$	−3676.50	+	2122.63i	−1.37119	+	0.791659i	−0.991078	−	0.133281i	$-0.957449\pi$
$193$	−3676.50	+	2122.63i	−1.37119	+	0.791659i	−0.380115	+	0.924939i	$0.624115\pi$
$194$		0			0
$195$	3276.00	+	3152.33i	1.20307	+	1.15766i
$196$		0			0
$197$	2383.50	−	1376.11i	0.862017	−	0.497686i	−0.00267023	−	0.999996i	$-0.500850\pi$
$197$	2383.50	−	1376.11i	0.862017	−	0.497686i	0.864687	+	0.502311i	$0.167517\pi$
$198$		0			0
$199$	−842.500	+	1459.25i	−0.300117	+	0.519818i	−0.976162	−	0.217042i	$-0.930359\pi$
$199$	−842.500	+	1459.25i	−0.300117	+	0.519818i	0.676045	+	0.736860i	$0.263692\pi$
$200$		0			0
$201$	997.500	+	575.907i	0.350041	+	0.202096i
$202$		0			0
$203$		−	1553.65i		−	0.537167i
$204$		0			0
$205$	2724.00	+	4718.11i	0.928061	+	1.60745i
$206$		0			0
$207$	−1254.00			−0.421058
$208$		0			0
$209$	1989.00			0.658287
$210$		0			0
$211$	840.500	+	1455.79i	0.274229	+	0.474979i	0.969940	−	0.243343i	$-0.0782439\pi$
$211$	840.500	+	1455.79i	0.274229	+	0.474979i	−0.695711	+	0.718322i	$0.744911\pi$
$212$		0			0
$213$			4085.91i			1.31437i
$214$		0			0
$215$	1020.00	+	588.897i	0.323551	+	0.186802i
$216$		0			0
$217$	−819.000	+	1418.55i	−0.256209	+	0.443767i
$218$		0			0
$219$	−6090.00	+	3516.06i	−1.87911	+	1.08490i
$220$		0			0
$221$	1228.50	+	303.975i	0.373927	+	0.0925229i
$222$		0			0
$223$	3547.50	−	2048.15i	1.06528	−	0.615042i	0.138394	−	0.990377i	$-0.455806\pi$
$223$	3547.50	−	2048.15i	1.06528	−	0.615042i	0.926889	+	0.375336i	$0.122473\pi$
$224$		0			0
$225$	737.000	−	1276.52i	0.218370	−	0.378229i
$226$		0			0
$227$	−379.500	−	219.104i	−0.110962	−	0.0640638i	0.443492	−	0.896278i	$-0.353739\pi$
$227$	−379.500	−	219.104i	−0.110962	−	0.0640638i	−0.554454	+	0.832215i	$0.687073\pi$
$228$		0			0
$229$		−	180.133i		−	0.0519805i	−0.999662	−	0.0259903i	$-0.991726\pi$
$229$		−	180.133i		−	0.0519805i	0.999662	−	0.0259903i	$-0.00827389\pi$
$230$		0			0
$231$	1774.50	+	3073.52i	0.505427	+	0.875424i
$232$		0			0
$233$	−5778.00			−1.62459			−0.812295	−	0.583247i	$-0.801782\pi$
$233$	−5778.00			−1.62459			−0.812295	+	0.583247i	$0.801782\pi$
$234$		0			0
$235$	−4752.00			−1.31909
$236$		0			0
$237$	−4354.00	−	7541.35i	−1.19334	−	2.06693i
$238$		0			0
$239$		−	1860.22i		−	0.503464i	−0.967797	−	0.251732i	$-0.919000\pi$
$239$		−	1860.22i		−	0.503464i	0.967797	−	0.251732i	$-0.0810001\pi$
$240$		0			0
$241$	1783.50	+	1029.70i	0.476703	+	0.275224i	0.719041	−	0.694967i	$-0.244581\pi$
$241$	1783.50	+	1029.70i	0.476703	+	0.275224i	−0.242339	+	0.970192i	$0.577915\pi$
$242$		0			0
$243$	2464.00	−	4267.77i	0.650476	−	1.12666i
$244$		0			0
$245$	1968.00	−	1136.23i	0.513187	−	0.296289i
$246$		0			0
$247$	994.500	−	4019.22i	0.256188	−	1.03537i
$248$		0			0
$249$	−2583.00	+	1491.30i	−0.657393	+	0.379546i
$250$		0			0
$251$	2245.50	−	3889.32i	0.564680	−	0.978055i	−0.432399	−	0.901682i	$-0.642333\pi$
$251$	2245.50	−	3889.32i	0.564680	−	0.978055i	0.997079	−	0.0763724i	$-0.0243338\pi$
$252$		0			0
$253$	1111.50	+	641.725i	0.276203	+	0.159466i
$254$		0			0
$255$		−	2618.86i		−	0.643135i
$256$		0			0
$257$	−2725.50	−	4720.70i	−0.661525	−	1.14580i	−0.980215	−	0.197936i	$-0.936576\pi$
$257$	−2725.50	−	4720.70i	−0.661525	−	1.14580i	0.318690	−	0.947859i	$-0.396757\pi$
$258$		0			0
$259$	897.000			0.215200
$260$		0			0
$261$	−1518.00			−0.360007
$262$		0			0
$263$	−391.500	−	678.098i	−0.0917906	−	0.158986i	0.816474	−	0.577382i	$-0.195926\pi$
$263$	−391.500	−	678.098i	−0.0917906	−	0.158986i	−0.908265	+	0.418396i	$0.862592\pi$
$264$		0			0
$265$		−	5902.83i		−	1.36833i
$266$		0			0
$267$	−1858.50	−	1073.01i	−0.425986	−	0.245943i
$268$		0			0
$269$	2542.50	−	4403.74i	0.576279	−	0.998144i	−0.419623	−	0.907699i	$-0.637838\pi$
$269$	2542.50	−	4403.74i	0.576279	−	0.998144i	0.995901	−	0.0904453i	$-0.0288290\pi$
$270$		0			0
$271$	1147.50	−	662.509i	0.257216	−	0.148504i	−0.365848	−	0.930675i	$-0.619221\pi$
$271$	1147.50	−	662.509i	0.257216	−	0.148504i	0.623064	+	0.782171i	$0.285888\pi$
$272$		0			0
$273$	7098.00	−	2049.02i	1.57359	−	0.454257i
$274$		0			0
$275$	−1306.50	+	754.308i	−0.286491	+	0.165405i
$276$		0			0
$277$	1710.50	−	2962.67i	0.371025	−	0.642635i	−0.618698	−	0.785629i	$-0.712340\pi$
$277$	1710.50	−	2962.67i	0.371025	−	0.642635i	0.989724	+	0.142994i	$0.0456730\pi$
$278$		0			0
$279$	1386.00	+	800.207i	0.297411	+	0.171710i
$280$		0			0
$281$		−	810.600i		−	0.172087i	−0.996291	−	0.0860433i	$-0.972578\pi$
$281$		−	810.600i		−	0.172087i	0.996291	−	0.0860433i	$-0.0274223\pi$
$282$		0			0
$283$	3588.50	+	6215.46i	0.753760	+	1.30555i	0.945988	+	0.324201i	$0.105095\pi$
$283$	3588.50	+	6215.46i	0.753760	+	1.30555i	−0.192228	+	0.981350i	$0.561571\pi$
$284$		0			0
$285$	−8568.00			−1.78079
$286$		0			0
$287$	8853.00			1.82082
$288$		0			0
$289$	2092.00	+	3623.45i	0.425809	+	0.737523i
$290$		0			0
$291$		−	8644.67i		−	1.74144i
$292$		0			0
$293$	8065.50	+	4656.62i	1.60816	+	0.928473i	0.989781	+	0.142595i	$0.0455445\pi$
$293$	8065.50	+	4656.62i	1.60816	+	0.928473i	0.618381	+	0.785878i	$0.287789\pi$
$294$		0			0
$295$	132.000	−	228.631i	0.0260520	−	0.0451234i
$296$		0			0
$297$	−682.500	+	394.042i	−0.133342	+	0.0769852i
$298$		0			0
$299$	1852.50	−	1925.17i	0.358304	−	0.372360i
$300$		0			0
$301$	1657.50	−	956.958i	0.317398	−	0.183250i
$302$		0			0
$303$	−6856.50	+	11875.8i	−1.29999	+	2.25164i
$304$		0			0
$305$	−204.000	−	117.779i	−0.0382984	−	0.0221116i
$306$		0			0
$307$		−	4777.00i		−	0.888070i	−0.896009	−	0.444035i	$-0.853547\pi$
$307$		−	4777.00i		−	0.888070i	0.896009	−	0.444035i	$-0.146453\pi$
$308$		0			0
$309$	6496.00	+	11251.4i	1.19594	+	2.07142i
$310$		0			0
$311$	−6192.00			−1.12899			−0.564495	−	0.825436i	$-0.690929\pi$
$311$	−6192.00			−1.12899			−0.564495	+	0.825436i	$0.690929\pi$
$312$		0			0
$313$	−770.000			−0.139051			−0.0695255	−	0.997580i	$-0.522149\pi$
$313$	−770.000			−0.139051			−0.0695255	+	0.997580i	$0.522149\pi$
$314$		0			0
$315$	−3432.00	−	5944.40i	−0.613877	−	1.06327i
$316$		0			0
$317$		−	8057.50i		−	1.42762i	−0.700341	−	0.713808i	$-0.746969\pi$
$317$		−	8057.50i		−	1.42762i	0.700341	−	0.713808i	$-0.253031\pi$
$318$		0			0
$319$	1345.50	+	776.825i	0.236155	+	0.136344i
$320$		0			0
$321$	−892.500	+	1545.86i	−0.155185	+	0.268789i
$322$		0			0
$323$	−2065.50	+	1192.52i	−0.355813	+	0.205429i
$324$		0			0
$325$	871.000	+	3017.23i	0.148660	+	0.514972i
$326$		0			0
$327$	3696.00	−	2133.89i	0.625044	−	0.360869i
$328$		0			0
$329$	−3861.00	+	6687.45i	−0.647002	+	1.12064i
$330$		0			0
$331$	4570.50	+	2638.78i	0.758965	+	0.438189i	0.828924	−	0.559361i	$-0.188954\pi$
$331$	4570.50	+	2638.78i	0.758965	+	0.438189i	−0.0699590	+	0.997550i	$0.522287\pi$
$332$		0			0
$333$		−	876.418i		−	0.144226i
$334$		0			0
$335$	1140.00	+	1974.54i	0.185925	+	0.322031i
$336$		0			0
$337$	8278.00			1.33808			0.669038	−	0.743228i	$-0.266706\pi$
$337$	8278.00			1.33808			0.669038	+	0.743228i	$0.266706\pi$
$338$		0			0
$339$	2877.00			0.460936
$340$		0			0
$341$	−819.000	−	1418.55i	−0.130063	−	0.225275i
$342$		0			0
$343$			4030.48i			0.634477i
$344$		0			0
$345$	−4788.00	−	2764.35i	−0.747180	−	0.431385i
$346$		0			0
$347$	3433.50	−	5947.00i	0.531181	−	0.920033i	−0.468156	−	0.883646i	$-0.655082\pi$
$347$	3433.50	−	5947.00i	0.531181	−	0.920033i	0.999338	−	0.0363875i	$-0.0115851\pi$
$348$		0			0
$349$	10525.5	−	6076.90i	1.61438	−	0.932060i	0.626036	−	0.779794i	$-0.284676\pi$
$349$	10525.5	−	6076.90i	1.61438	−	0.932060i	0.988340	−	0.152266i	$-0.0486571\pi$
$350$		0			0
$351$	455.000	+	1576.17i	0.0691912	+	0.239685i
$352$		0			0
$353$	5029.50	−	2903.78i	0.758338	−	0.437827i	−0.0703608	−	0.997522i	$-0.522415\pi$
$353$	5029.50	−	2903.78i	0.758338	−	0.437827i	0.828699	+	0.559695i	$0.189082\pi$
$354$		0			0
$355$	−4044.00	+	7004.41i	−0.604601	+	1.04720i
$356$		0			0
$357$	−3685.50	−	2127.82i	−0.546379	−	0.315452i
$358$		0			0
$359$			1340.61i			0.197088i	0.995133	+	0.0985439i	$0.0314185\pi$
$359$			1340.61i			0.197088i	−0.995133	+	0.0985439i	$0.968581\pi$
$360$		0			0
$361$	472.000	+	817.528i	0.0688147	+	0.119191i
$362$		0			0
$363$	5768.00			0.833999
$364$		0			0
$365$	−13920.0			−1.99618
$366$		0			0
$367$	1832.50	+	3173.98i	0.260642	+	0.451446i	0.966413	−	0.256995i	$-0.0827324\pi$
$367$	1832.50	+	3173.98i	0.260642	+	0.451446i	−0.705770	+	0.708441i	$0.749399\pi$
$368$		0			0
$369$		−	8649.86i		−	1.22031i
$370$		0			0
$371$	−8307.00	−	4796.05i	−1.16247	−	0.671155i
$372$		0			0
$373$	−2685.50	+	4651.42i	−0.372788	+	0.645688i	−0.989993	−	0.141114i	$-0.954931\pi$
$373$	−2685.50	+	4651.42i	−0.372788	+	0.645688i	0.617205	+	0.786802i	$0.288265\pi$
$374$		0			0
$375$	−4872.00	+	2812.85i	−0.670904	+	0.387347i
$376$		0			0
$377$	2242.50	−	2330.47i	0.306352	−	0.318370i
$378$		0			0
$379$	9967.50	−	5754.74i	1.35091	−	0.779950i	0.362536	−	0.931970i	$-0.381911\pi$
$379$	9967.50	−	5754.74i	1.35091	−	0.779950i	0.988377	+	0.152020i	$0.0485778\pi$
$380$		0			0
$381$	−7850.50	+	13597.5i	−1.05563	+	1.82840i
$382$		0			0
$383$	−2095.50	−	1209.84i	−0.279569	−	0.161409i	0.353659	−	0.935374i	$-0.384937\pi$
$383$	−2095.50	−	1209.84i	−0.279569	−	0.161409i	−0.633228	+	0.773965i	$0.718271\pi$
$384$		0			0
$385$			7025.20i			0.929967i
$386$		0			0
$387$	−935.000	−	1619.47i	−0.122813	−	0.212719i
$388$		0			0
$389$	−9858.00			−1.28489			−0.642443	−	0.766334i	$-0.722079\pi$
$389$	−9858.00			−1.28489			−0.642443	+	0.766334i	$0.722079\pi$
$390$		0			0
$391$	−1539.00			−0.199055
$392$		0			0
$393$	−1302.00	−	2255.13i	−0.167118	−	0.289456i
$394$		0			0
$395$		−	17237.4i		−	2.19571i
$396$		0			0
$397$	−7552.50	−	4360.44i	−0.954784	−	0.551245i	−0.0602200	−	0.998185i	$-0.519180\pi$
$397$	−7552.50	−	4360.44i	−0.954784	−	0.551245i	−0.894564	+	0.446941i	$0.852514\pi$
$398$		0			0
$399$	−6961.50	+	12057.7i	−0.873461	+	1.51288i
$400$		0			0
$401$	−6568.50	+	3792.33i	−0.817993	+	0.472269i	−0.849724	−	0.527228i	$-0.823231\pi$
$401$	−6568.50	+	3792.33i	−0.817993	+	0.472269i	0.0317308	+	0.999496i	$0.489898\pi$
$402$		0			0
$403$	−3276.00	+	945.700i	−0.404936	+	0.116895i
$404$		0			0
$405$	10068.0	−	5812.76i	1.23527	−	0.713181i
$406$		0			0
$407$	−448.500	+	776.825i	−0.0546224	+	0.0946088i
$408$		0			0
$409$	3727.50	+	2152.07i	0.450643	+	0.260179i	0.708102	−	0.706110i	$-0.249552\pi$
$409$	3727.50	+	2152.07i	0.450643	+	0.260179i	−0.257459	+	0.966289i	$0.582885\pi$
$410$		0			0
$411$			8329.43i			0.999661i
$412$		0			0
$413$	−214.500	−	371.525i	−0.0255565	−	0.0442652i
$414$		0			0
$415$	−5904.00			−0.698352
$416$		0			0
$417$	17815.0			2.09210
$418$		0			0
$419$	2698.50	+	4673.94i	0.314631	+	0.544957i	0.979359	−	0.202129i	$-0.0647859\pi$
$419$	2698.50	+	4673.94i	0.314631	+	0.544957i	−0.664728	+	0.747085i	$0.731453\pi$
$420$		0			0
$421$		−	7260.76i		−	0.840541i	−0.907399	−	0.420270i	$-0.861935\pi$
$421$		−	7260.76i		−	0.840541i	0.907399	−	0.420270i	$-0.138065\pi$
$422$		0			0
$423$	6534.00	+	3772.41i	0.751050	+	0.433619i
$424$		0			0
$425$	904.500	−	1566.64i	0.103235	−	0.178808i
$426$		0			0
$427$	−331.500	+	191.392i	−0.0375700	+	0.0216911i
$428$		0			0
$429$	−1774.50	+	7171.56i	−0.199706	+	0.807100i
$430$		0			0
$431$	421.500	−	243.353i	0.0471066	−	0.0271970i	−0.476262	−	0.879304i	$-0.658009\pi$
$431$	421.500	−	243.353i	0.0471066	−	0.0271970i	0.523368	+	0.852107i	$0.324675\pi$
$432$		0			0
$433$	−6069.50	+	10512.7i	−0.673629	+	1.16676i	0.303238	+	0.952915i	$0.401932\pi$
$433$	−6069.50	+	10512.7i	−0.673629	+	1.16676i	−0.976867	+	0.213846i	$0.931401\pi$
$434$		0			0
$435$	−5796.00	−	3346.32i	−0.638844	−	0.368836i
$436$		0			0
$437$			5035.07i			0.551167i
$438$		0			0
$439$	230.500	+	399.238i	0.0250596	+	0.0434045i	0.878283	−	0.478141i	$-0.158689\pi$
$439$	230.500	+	399.238i	0.0250596	+	0.0434045i	−0.853224	+	0.521545i	$0.825356\pi$
$440$		0			0
$441$	−3608.00			−0.389591
$442$		0			0
$443$	−12156.0			−1.30372			−0.651861	−	0.758338i	$-0.726012\pi$
$443$	−12156.0			−1.30372			−0.651861	+	0.758338i	$0.726012\pi$
$444$		0			0
$445$	−2124.00	−	3678.88i	−0.226263	−	0.391900i
$446$		0			0
$447$		−	9129.64i		−	0.966034i
$448$		0			0
$449$	−256.500	−	148.090i	−0.0269599	−	0.0155653i	0.486459	−	0.873703i	$-0.338288\pi$
$449$	−256.500	−	148.090i	−0.0269599	−	0.0155653i	−0.513419	+	0.858138i	$0.671621\pi$
$450$		0			0
$451$	−4426.50	+	7666.92i	−0.462164	+	0.800491i
$452$		0			0
$453$	−525.000	+	303.109i	−0.0544518	+	0.0314377i
$454$		0			0
$455$	14196.0	+	3512.60i	1.46268	+	0.361919i
$456$		0			0
$457$	529.500	−	305.707i	0.0541990	−	0.0312918i	−0.472656	−	0.881247i	$-0.656705\pi$
$457$	529.500	−	305.707i	0.0541990	−	0.0312918i	0.526855	+	0.849955i	$0.323371\pi$
$458$		0			0
$459$	472.500	−	818.394i	0.0480488	−	0.0832230i
$460$		0			0
$461$	−11368.5	−	6563.61i	−1.14855	−	0.663119i	−0.200020	−	0.979792i	$-0.564101\pi$
$461$	−11368.5	−	6563.61i	−1.14855	−	0.663119i	−0.948535	+	0.316673i	$0.897434\pi$
$462$		0			0
$463$		−	834.848i		−	0.0837985i	−0.999122	−	0.0418992i	$-0.986659\pi$
$463$		−	834.848i		−	0.0837985i	0.999122	−	0.0418992i	$-0.0133408\pi$
$464$		0			0
$465$	3528.00	+	6110.68i	0.351843	+	0.609410i
$466$		0			0
$467$	−14496.0			−1.43639			−0.718196	−	0.695841i	$-0.755032\pi$
$467$	−14496.0			−1.43639			−0.718196	+	0.695841i	$0.755032\pi$
$468$		0			0
$469$	3705.00			0.364778
$470$		0			0
$471$	5369.00	+	9299.38i	0.525245	+	0.909751i
$472$		0			0
$473$			1913.92i			0.186051i
$474$		0			0
$475$	−5125.50	−	2959.21i	−0.495103	−	0.285848i
$476$		0			0
$477$	−4686.00	+	8116.39i	−0.449805	+	0.779086i
$478$		0			0
$479$	7705.50	−	4448.77i	0.735017	−	0.424362i	−0.0852376	−	0.996361i	$-0.527165\pi$
$479$	7705.50	−	4448.77i	0.735017	−	0.424362i	0.820255	+	0.571998i	$0.193832\pi$
$480$		0			0
$481$	1345.50	+	1294.71i	0.127546	+	0.122731i
$482$		0			0
$483$	−7780.50	+	4492.07i	−0.732971	+	0.423181i
$484$		0			0
$485$	8556.00	−	14819.4i	0.801047	−	1.38745i
$486$		0			0
$487$	−4117.50	−	2377.24i	−0.383125	−	0.221197i	0.296052	−	0.955172i	$-0.404330\pi$
$487$	−4117.50	−	2377.24i	−0.383125	−	0.221197i	−0.679177	+	0.733975i	$0.737663\pi$
$488$		0			0
$489$		−	11433.3i		−	1.05732i
$490$		0			0
$491$	−817.500	−	1415.95i	−0.0751390	−	0.130145i	0.826008	−	0.563659i	$-0.190607\pi$
$491$	−817.500	−	1415.95i	−0.0751390	−	0.130145i	−0.901147	+	0.433514i	$0.857273\pi$
$492$		0			0
$493$	−1863.00			−0.170193
$494$		0			0
$495$	6864.00			0.623260
$496$		0			0
$497$	6571.50	+	11382.2i	0.593103	+	1.02728i
$498$		0			0
$499$		−	14434.9i		−	1.29498i	−0.762074	−	0.647490i	$-0.775819\pi$
$499$		−	14434.9i		−	1.29498i	0.762074	−	0.647490i	$-0.224181\pi$
$500$		0			0
$501$	9859.50	+	5692.38i	0.879222	+	0.507619i
$502$		0			0
$503$	6343.50	−	10987.3i	0.562312	−	0.973952i	−0.434983	−	0.900439i	$-0.643246\pi$
$503$	6343.50	−	10987.3i	0.562312	−	0.973952i	0.997294	−	0.0735133i	$-0.0234211\pi$
$504$		0			0
$505$	−23508.0	+	13572.4i	−2.07147	+	1.19596i
$506$		0			0
$507$	13604.5	+	7171.56i	1.19171	+	0.628205i
$508$		0			0
$509$	−4978.50	+	2874.34i	−0.433533	+	0.250300i	−0.700850	−	0.713308i	$-0.747196\pi$
$509$	−4978.50	+	2874.34i	−0.433533	+	0.250300i	0.267318	+	0.963608i	$0.413863\pi$
$510$		0			0
$511$	−11310.0	+	19589.5i	−0.979109	+	1.69587i
$512$		0			0
$513$	−2677.50	−	1545.86i	−0.230438	−	0.133043i
$514$		0			0
$515$			25717.5i			2.20048i
$516$		0			0
$517$	−3861.00	−	6687.45i	−0.328446	−	0.568885i
$518$		0			0
$519$	−6111.00			−0.516846
$520$		0			0
$521$	6054.00			0.509080			0.254540	−	0.967062i	$-0.418076\pi$
$521$	6054.00			0.509080			0.254540	+	0.967062i	$0.418076\pi$
$522$		0			0
$523$	−7401.50	−	12819.8i	−0.618824	−	1.07183i	−0.989701	−	0.143153i	$-0.954276\pi$
$523$	−7401.50	−	12819.8i	−0.618824	−	1.07183i	0.370877	−	0.928682i	$-0.379057\pi$
$524$		0			0
$525$		−	10560.3i		−	0.877885i
$526$		0			0
$527$	1701.00	+	982.073i	0.140601	+	0.0811760i
$528$		0			0
$529$	4459.00	−	7723.21i	0.366483	−	0.634767i
$530$		0			0
$531$	−363.000	+	209.578i	−0.0296664	+	0.0171279i
$532$		0			0
$533$	13279.5	+	12778.2i	1.07917	+	1.03843i
$534$		0			0
$535$	−3060.00	+	1766.69i	−0.247281	+	0.142768i
$536$		0			0
$537$	−4504.50	+	7802.02i	−0.361980	+	0.626969i
$538$		0			0
$539$	3198.00	+	1846.37i	0.255561	+	0.147548i
$540$		0			0
$541$			21470.5i			1.70626i	0.521695	+	0.853132i	$0.325300\pi$
$541$			21470.5i			1.70626i	−0.521695	+	0.853132i	$0.674700\pi$
$542$		0			0
$543$	−7.00000	−	12.1244i	−0.000553221	−	0.000958206i
$544$		0			0
$545$	8448.00			0.663986
$546$		0			0
$547$	13516.0			1.05649			0.528247	−	0.849091i	$-0.322849\pi$
$547$	13516.0			1.05649			0.528247	+	0.849091i	$0.322849\pi$
$548$		0			0
$549$	187.000	+	323.894i	0.0145373	+	0.0251793i
$550$		0			0
$551$			6095.09i			0.471251i
$552$		0			0
$553$	−24258.0	−	14005.4i	−1.86538	−	1.07698i
$554$		0			0
$555$	1932.00	−	3346.32i	0.147764	−	0.255934i
$556$		0			0
$557$	2503.50	−	1445.40i	0.190443	−	0.109952i	−0.401747	−	0.915751i	$-0.631597\pi$
$557$	2503.50	−	1445.40i	0.190443	−	0.109952i	0.592190	+	0.805798i	$0.298264\pi$
$558$		0			0
$559$	3867.50	+	956.958i	0.292626	+	0.0724061i
$560$		0			0
$561$	3685.50	−	2127.82i	0.277365	−	0.160137i
$562$		0			0
$563$	5791.50	−	10031.2i	0.433539	−	0.750912i	−0.563636	−	0.826023i	$-0.690598\pi$
$563$	5791.50	−	10031.2i	0.433539	−	0.750912i	0.997175	+	0.0751113i	$0.0239312\pi$
$564$		0			0
$565$	4932.00	+	2847.49i	0.367240	+	0.212026i
$566$		0			0
$567$		−	18891.5i		−	1.39924i
$568$		0			0
$569$	−6439.50	−	11153.5i	−0.474443	−	0.821759i	0.525129	−	0.851023i	$-0.324017\pi$
$569$	−6439.50	−	11153.5i	−0.474443	−	0.821759i	−0.999572	+	0.0292638i	$0.990684\pi$
$570$		0			0
$571$	11636.0			0.852805			0.426402	−	0.904534i	$-0.359781\pi$
$571$	11636.0			0.852805			0.426402	+	0.904534i	$0.359781\pi$
$572$		0			0
$573$	19887.0			1.44990
$574$		0			0
$575$	−1909.50	−	3307.35i	−0.138490	−	0.239871i
$576$		0			0
$577$		−	12311.4i		−	0.888269i	−0.895960	−	0.444134i	$-0.853511\pi$
$577$		−	12311.4i		−	0.888269i	0.895960	−	0.444134i	$-0.146489\pi$
$578$		0			0
$579$	25735.5	+	14858.4i	1.84720	+	1.06648i
$580$		0			0
$581$	−4797.00	+	8308.65i	−0.342535	+	0.593289i
$582$		0			0
$583$	8307.00	−	4796.05i	0.590121	−	0.340707i
$584$		0			0
$585$	3432.00	−	13870.3i	0.242557	−	0.980282i
$586$		0			0
$587$	13549.5	−	7822.81i	0.952722	−	0.550054i	0.0587964	−	0.998270i	$-0.481274\pi$
$587$	13549.5	−	7822.81i	0.952722	−	0.550054i	0.893925	+	0.448216i	$0.147940\pi$
$588$		0			0
$589$	3213.00	−	5565.08i	0.224770	−	0.389313i
$590$		0			0
$591$	−16684.5	−	9632.80i	−1.16127	−	0.670458i
$592$		0			0
$593$		−	25821.4i		−	1.78813i	−0.447942	−	0.894063i	$-0.647843\pi$
$593$		−	25821.4i		−	1.78813i	0.447942	−	0.894063i	$-0.352157\pi$
$594$		0			0
$595$	−4212.00	−	7295.40i	−0.290210	−	0.502659i
$596$		0			0
$597$	11795.0			0.808605
$598$		0			0
$599$	−1668.00			−0.113777			−0.0568887	−	0.998381i	$-0.518118\pi$
$599$	−1668.00			−0.113777			−0.0568887	+	0.998381i	$0.518118\pi$
$600$		0			0
$601$	−6849.50	−	11863.7i	−0.464887	−	0.805207i	0.534310	−	0.845289i	$-0.320572\pi$
$601$	−6849.50	−	11863.7i	−0.464887	−	0.805207i	−0.999196	+	0.0400813i	$0.987238\pi$
$602$		0			0
$603$		−	3619.99i		−	0.244473i
$604$		0			0
$605$	9888.00	+	5708.84i	0.664470	+	0.383632i
$606$		0			0
$607$	−11586.5	+	20068.4i	−0.774764	+	1.34193i	0.160164	+	0.987090i	$0.448798\pi$
$607$	−11586.5	+	20068.4i	−0.774764	+	1.34193i	−0.934927	+	0.354839i	$0.884536\pi$
$608$		0			0
$609$	−9418.50	+	5437.77i	−0.626694	+	0.361822i
$610$		0			0
$611$	−15444.0	+	4458.30i	−1.02258	+	0.295194i
$612$		0			0
$613$	14389.5	−	8307.78i	0.948102	−	0.547387i	0.0556111	−	0.998453i	$-0.482289\pi$
$613$	14389.5	−	8307.78i	0.948102	−	0.547387i	0.892491	+	0.451066i	$0.148956\pi$
$614$		0			0
$615$	19068.0	−	33026.7i	1.25024	−	2.16547i
$616$		0			0
$617$	24589.5	+	14196.8i	1.60443	+	0.926321i	0.990585	+	0.136897i	$0.0437130\pi$
$617$	24589.5	+	14196.8i	1.60443	+	0.926321i	0.613849	+	0.789423i	$0.289620\pi$
$618$		0			0
$619$		−	6245.78i		−	0.405556i	−0.979225	−	0.202778i	$-0.935003\pi$
$619$		−	6245.78i		−	0.405556i	0.979225	−	0.202778i	$-0.0649969\pi$
$620$		0			0
$621$	−997.500	−	1727.72i	−0.0644578	−	0.111644i
$622$		0			0
$623$	−6903.00			−0.443921
$624$		0			0
$625$	−19511.0			−1.24870
$626$		0			0
$627$	−6961.50	−	12057.7i	−0.443406	−	0.768002i
$628$		0			0
$629$		−	1075.60i		−	0.0681830i
$630$		0			0
$631$	−19381.5	−	11189.9i	−1.22277	−	0.705964i	−0.257259	−	0.966342i	$-0.582819\pi$
$631$	−19381.5	−	11189.9i	−1.22277	−	0.705964i	−0.965507	+	0.260378i	$0.916153\pi$
$632$		0			0
$633$	5883.50	−	10190.5i	0.369428	−	0.639869i
$634$		0			0
$635$	−26916.0	+	15540.0i	−1.68209	+	0.971157i
$636$		0			0
$637$	5330.00	−	5539.10i	0.331526	−	0.344532i
$638$		0			0
$639$	11121.0	−	6420.71i	0.688482	−	0.397495i
$640$		0			0
$641$	−9913.50	+	17170.7i	−0.610858	+	1.05804i	0.380239	+	0.924888i	$0.375842\pi$
$641$	−9913.50	+	17170.7i	−0.610858	+	1.05804i	−0.991096	+	0.133148i	$0.957491\pi$
$642$		0			0
$643$	7318.50	+	4225.34i	0.448855	+	0.259146i	0.707346	−	0.706867i	$-0.249892\pi$
$643$	7318.50	+	4225.34i	0.448855	+	0.259146i	−0.258492	+	0.966013i	$0.583225\pi$
$644$		0			0
$645$		−	8244.56i		−	0.503301i
$646$		0			0
$647$	1474.50	+	2553.91i	0.0895959	+	0.155185i	0.907340	−	0.420397i	$-0.138109\pi$
$647$	1474.50	+	2553.91i	0.0895959	+	0.155185i	−0.817744	+	0.575581i	$0.804776\pi$
$648$		0			0
$649$	429.000			0.0259472
$650$		0			0
$651$	11466.0			0.690304
$652$		0			0
$653$	−6019.50	−	10426.1i	−0.360737	−	0.624815i	0.627345	−	0.778741i	$-0.284141\pi$
$653$	−6019.50	−	10426.1i	−0.360737	−	0.624815i	−0.988082	+	0.153926i	$0.950808\pi$
$654$		0			0
$655$		−	5154.58i		−	0.307490i
$656$		0			0
$657$	19140.0	+	11050.5i	1.13656	+	0.656196i
$658$		0			0
$659$	1681.50	−	2912.44i	0.0993960	−	0.172159i	−0.812039	−	0.583603i	$-0.801642\pi$
$659$	1681.50	−	2912.44i	0.0993960	−	0.172159i	0.911435	+	0.411445i	$0.134976\pi$
$660$		0			0
$661$	8797.50	−	5079.24i	0.517675	−	0.298880i	−0.218308	−	0.975880i	$-0.570054\pi$
$661$	8797.50	−	5079.24i	0.517675	−	0.298880i	0.735983	+	0.677000i	$0.236720\pi$
$662$		0			0
$663$	−2457.00	−	8511.30i	−0.143925	−	0.498569i
$664$		0			0
$665$	−23868.0	+	13780.2i	−1.39182	+	0.803569i
$666$		0			0
$667$	−1966.50	+	3406.08i	−0.114158	+	0.197727i
$668$		0			0
$669$	−24832.5	−	14337.1i	−1.43510	−	0.828554i
$670$		0			0
$671$		−	382.783i		−	0.0220226i
$672$		0			0
$673$	9084.50	+	15734.8i	0.520329	+	0.901237i	0.999721	+	0.0236358i	$0.00752419\pi$
$673$	9084.50	+	15734.8i	0.520329	+	0.901237i	−0.479391	+	0.877601i	$0.659142\pi$
$674$		0			0
$675$	2345.00			0.133717
$676$		0			0
$677$	9042.00			0.513312			0.256656	−	0.966503i	$-0.417379\pi$
$677$	9042.00			0.513312			0.256656	+	0.966503i	$0.417379\pi$
$678$		0			0
$679$	−13903.5	−	24081.6i	−0.785813	−	1.36107i
$680$		0			0
$681$			3067.46i			0.172607i
$682$		0			0
$683$	10792.5	+	6231.05i	0.604632	+	0.349084i	0.770862	−	0.637003i	$-0.219826\pi$
$683$	10792.5	+	6231.05i	0.604632	+	0.349084i	−0.166230	+	0.986087i	$0.553159\pi$
$684$		0			0
$685$	−8244.00	+	14279.0i	−0.459835	+	0.796458i
$686$		0			0
$687$	−1092.00	+	630.466i	−0.0606440	+	0.0350128i
$688$		0			0
$689$	−5538.00	−	19184.2i	−0.306213	−	1.06075i
$690$		0			0
$691$	3739.50	−	2159.00i	0.205872	−	0.118860i	−0.393520	−	0.919316i	$-0.628743\pi$
$691$	3739.50	−	2159.00i	0.205872	−	0.118860i	0.599391	+	0.800456i	$0.295409\pi$
$692$		0			0
$693$	5577.00	−	9659.65i	0.305704	−	0.529494i
$694$		0			0
$695$	30540.0	+	17632.3i	1.66683	+	0.962346i
$696$		0			0
$697$		−	10615.7i		−	0.576901i
$698$		0			0
$699$	20223.0	+	35027.3i	1.09428	+	1.89535i
$700$		0			0
$701$	−18270.0			−0.984377			−0.492189	−	0.870489i	$-0.663803\pi$
$701$	−18270.0			−0.984377			−0.492189	+	0.870489i	$0.663803\pi$
$702$		0			0
$703$	−3519.00			−0.188793
$704$		0			0
$705$	16632.0	+	28807.5i	0.888507	+	1.53894i
$706$		0			0
$707$			44110.1i			2.34644i
$708$		0			0
$709$	1411.50	+	814.930i	0.0747673	+	0.0431669i	0.536918	−	0.843635i	$-0.319589\pi$
$709$	1411.50	+	814.930i	0.0747673	+	0.0431669i	−0.462150	+	0.886802i	$0.652922\pi$
$710$		0			0
$711$	−13684.0	+	23701.4i	−0.721786	+	1.25017i
$712$		0			0
$713$	3591.00	−	2073.26i	0.188617	−	0.108898i
$714$		0			0
$715$	−10140.0	+	10537.8i	−0.530370	+	0.551177i
$716$		0			0
$717$	−11277.0	+	6510.78i	−0.587374	+	0.339121i
$718$		0			0
$719$	4915.50	−	8513.90i	0.254961	−	0.441606i	−0.709924	−	0.704279i	$-0.751271\pi$
$719$	4915.50	−	8513.90i	0.254961	−	0.441606i	0.964885	+	0.262673i	$0.0846039\pi$
$720$		0			0
$721$	36192.0	+	20895.5i	1.86943	+	1.07932i
$722$		0			0
$723$		−	14415.9i		−	0.741537i
$724$		0			0
$725$	−2311.50	−	4003.64i	−0.118410	−	0.205091i
$726$		0			0
$727$	15464.0			0.788897			0.394448	−	0.918918i	$-0.370936\pi$
$727$	15464.0			0.788897			0.394448	+	0.918918i	$0.370936\pi$
$728$		0			0
$729$	−11843.0			−0.601687
$730$		0			0
$731$	−1147.50	−	1987.53i	−0.0580599	−	0.100563i
$732$		0			0
$733$			12616.3i			0.635733i	0.948136	+	0.317866i	$0.102966\pi$
$733$			12616.3i			0.635733i	−0.948136	+	0.317866i	$0.897034\pi$
$734$		0			0
$735$	−13776.0	−	7953.58i	−0.691341	−	0.399146i
$736$		0			0
$737$	−1852.50	+	3208.62i	−0.0925885	+	0.160368i
$738$		0			0
$739$	14101.5	−	8141.50i	0.701938	−	0.405264i	−0.106131	−	0.994352i	$-0.533846\pi$
$739$	14101.5	−	8141.50i	0.701938	−	0.405264i	0.808069	+	0.589088i	$0.200513\pi$
$740$		0			0
$741$	−27846.0	+	8038.45i	−1.38050	+	0.398515i
$742$		0			0
$743$	−9358.50	+	5403.13i	−0.462086	+	0.266786i	−0.712921	−	0.701244i	$-0.752628\pi$
$743$	−9358.50	+	5403.13i	−0.462086	+	0.266786i	0.250835	+	0.968030i	$0.419295\pi$
$744$		0			0
$745$	9036.00	−	15650.8i	0.444367	−	0.769666i
$746$		0			0
$747$	8118.00	+	4686.93i	0.397620	+	0.229566i
$748$		0			0
$749$			5741.75i			0.280105i
$750$		0			0
$751$	−6807.50	−	11790.9i	−0.330771	−	0.572913i	0.651892	−	0.758312i	$-0.273976\pi$
$751$	−6807.50	−	11790.9i	−0.330771	−	0.572913i	−0.982663	+	0.185399i	$0.940642\pi$
$752$		0			0
$753$	−31437.0			−1.52142
$754$		0			0
$755$	−1200.00			−0.0578443
$756$		0			0
$757$	−2775.50	−	4807.31i	−0.133259	−	0.230812i	0.791672	−	0.610947i	$-0.209211\pi$
$757$	−2775.50	−	4807.31i	−0.133259	−	0.230812i	−0.924931	+	0.380135i	$0.875878\pi$
$758$		0			0
$759$		−	8984.15i		−	0.429649i
$760$		0			0
$761$	8731.50	+	5041.13i	0.415922	+	0.240133i	0.693331	−	0.720619i	$-0.256142\pi$
$761$	8731.50	+	5041.13i	0.415922	+	0.240133i	−0.277409	+	0.960752i	$0.589476\pi$
$762$		0			0
$763$	6864.00	−	11888.8i	0.325680	−	0.564093i
$764$		0			0
$765$	−7128.00	+	4115.35i	−0.336880	+	0.194498i
$766$		0			0
$767$	214.500	−	866.891i	0.0100980	−	0.0408105i
$768$		0			0
$769$	25771.5	−	14879.2i	1.20851	−	0.697733i	0.246076	−	0.969250i	$-0.420859\pi$
$769$	25771.5	−	14879.2i	1.20851	−	0.697733i	0.962434	+	0.271517i	$0.0875253\pi$
$770$		0			0
$771$	−19078.5	+	33044.9i	−0.891174	+	1.54356i
$772$		0			0
$773$	24019.5	+	13867.7i	1.11762	+	0.645259i	0.940793	−	0.338983i	$-0.110083\pi$
$773$	24019.5	+	13867.7i	1.11762	+	0.645259i	0.176829	+	0.984242i	$0.443416\pi$
$774$		0			0
$775$			4873.99i			0.225908i
$776$		0			0
$777$	−3139.50	−	5437.77i	−0.144954	−	0.251067i
$778$		0			0
$779$	−34731.0			−1.59739
$780$		0			0
$781$	−13143.0			−0.602168
$782$		0			0
$783$	−1207.50	−	2091.45i	−0.0551118	−	0.0954564i
$784$		0			0
$785$			21255.7i			0.966432i

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)

\(f(q)\)	\(=\)	\(q+(-3.50000 - 6.06218i) q^{3} -13.8564i q^{5} +(-19.5000 - 11.2583i) q^{7} +(-11.0000 + 19.0526i) q^{9} +O(q^{10})\)	\(q+(-3.50000 - 6.06218i) q^{3} -13.8564i q^{5} +(-19.5000 - 11.2583i) q^{7} +(-11.0000 + 19.0526i) q^{9} +(19.5000 - 11.2583i) q^{11} +(-13.0000 - 45.0333i) q^{13} +(-84.0000 + 48.4974i) q^{15} +(-13.5000 + 23.3827i) q^{17} +(76.5000 + 44.1673i) q^{19} +157.617i q^{21} +(28.5000 + 49.3634i) q^{23} -67.0000 q^{25} -35.0000 q^{27} +(34.5000 + 59.7558i) q^{29} -72.7461i q^{31} +(-136.500 - 78.8083i) q^{33} +(-156.000 + 270.200i) q^{35} +(-34.5000 + 19.9186i) q^{37} +(-227.500 + 236.425i) q^{39} +(-340.500 + 196.588i) q^{41} +(-42.5000 + 73.6122i) q^{43} +(264.000 + 152.420i) q^{45} -342.946i q^{47} +(82.0000 + 142.028i) q^{49} +189.000 q^{51} +426.000 q^{53} +(-156.000 - 270.200i) q^{55} -618.342i q^{57} +(16.5000 + 9.52628i) q^{59} +(8.50000 - 14.7224i) q^{61} +(429.000 - 247.683i) q^{63} +(-624.000 + 180.133i) q^{65} +(-142.500 + 82.2724i) q^{67} +(199.500 - 345.544i) q^{69} +(-505.500 - 291.851i) q^{71} -1004.59i q^{73} +(234.500 + 406.166i) q^{75} -507.000 q^{77} +1244.00 q^{79} +(419.500 + 726.595i) q^{81} -426.084i q^{83} +(324.000 + 187.061i) q^{85} +(241.500 - 418.290i) q^{87} +(265.500 - 153.286i) q^{89} +(-253.500 + 1024.51i) q^{91} +(-441.000 + 254.611i) q^{93} +(612.000 - 1060.02i) q^{95} +(1069.50 + 617.476i) q^{97} +495.367i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 7 q^{3} - 39 q^{7} - 22 q^{9}+O(q^{10}) \) 2 * q - 7 * q^3 - 39 * q^7 - 22 * q^9	\( 2 q - 7 q^{3} - 39 q^{7} - 22 q^{9} + 39 q^{11} - 26 q^{13} - 168 q^{15} - 27 q^{17} + 153 q^{19} + 57 q^{23} - 134 q^{25} - 70 q^{27} + 69 q^{29} - 273 q^{33} - 312 q^{35} - 69 q^{37} - 455 q^{39} - 681 q^{41} - 85 q^{43} + 528 q^{45} + 164 q^{49} + 378 q^{51} + 852 q^{53} - 312 q^{55} + 33 q^{59} + 17 q^{61} + 858 q^{63} - 1248 q^{65} - 285 q^{67} + 399 q^{69} - 1011 q^{71} + 469 q^{75} - 1014 q^{77} + 2488 q^{79} + 839 q^{81} + 648 q^{85} + 483 q^{87} + 531 q^{89} - 507 q^{91} - 882 q^{93} + 1224 q^{95} + 2139 q^{97}+O(q^{100}) \) 2 * q - 7 * q^3 - 39 * q^7 - 22 * q^9 + 39 * q^11 - 26 * q^13 - 168 * q^15 - 27 * q^17 + 153 * q^19 + 57 * q^23 - 134 * q^25 - 70 * q^27 + 69 * q^29 - 273 * q^33 - 312 * q^35 - 69 * q^37 - 455 * q^39 - 681 * q^41 - 85 * q^43 + 528 * q^45 + 164 * q^49 + 378 * q^51 + 852 * q^53 - 312 * q^55 + 33 * q^59 + 17 * q^61 + 858 * q^63 - 1248 * q^65 - 285 * q^67 + 399 * q^69 - 1011 * q^71 + 469 * q^75 - 1014 * q^77 + 2488 * q^79 + 839 * q^81 + 648 * q^85 + 483 * q^87 + 531 * q^89 - 507 * q^91 - 882 * q^93 + 1224 * q^95 + 2139 * q^97

Introduction

L-functions

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