LMFDB - Embedded newform 208.4.f.e.129.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(129,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

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("208.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, 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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 170x^{8} + 8945x^{6} + 145432x^{4} + 614160x^{2} + 20736 $ x^10 + 170x^8 + 8945x^6 + 145432x^4 + 614160x^2 + 20736
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-8.27625i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.4.f.e.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.27625 q^{3} -2.27999i q^{5} +28.5449i q^{7} +59.0489 q^{9} +O(q^{10})$	$q-9.27625 q^{3} -2.27999i q^{5} +28.5449i q^{7} +59.0489 q^{9} +0.604833i q^{11} +(-19.1535 + 42.7802i) q^{13} +21.1498i q^{15} -41.0109 q^{17} -129.485i q^{19} -264.790i q^{21} -73.8974 q^{23} +119.802 q^{25} -297.293 q^{27} -214.532 q^{29} -126.175i q^{31} -5.61058i q^{33} +65.0822 q^{35} -309.470i q^{37} +(177.673 - 396.840i) q^{39} +88.1488i q^{41} +245.304 q^{43} -134.631i q^{45} -68.5441i q^{47} -471.814 q^{49} +380.428 q^{51} -613.275 q^{53} +1.37901 q^{55} +1201.14i q^{57} -840.276i q^{59} +587.808 q^{61} +1685.55i q^{63} +(97.5384 + 43.6699i) q^{65} +606.554i q^{67} +685.491 q^{69} +507.627i q^{71} -177.127i q^{73} -1111.31 q^{75} -17.2649 q^{77} -143.122 q^{79} +1163.45 q^{81} -773.287i q^{83} +93.5046i q^{85} +1990.06 q^{87} -1402.90i q^{89} +(-1221.16 - 546.736i) q^{91} +1170.43i q^{93} -295.226 q^{95} -468.225i q^{97} +35.7147i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{3} + 72 q^{9}+O(q^{10}) $ 10 * q - 6 * q^3 + 72 * q^9	$ 10 q - 6 q^{3} + 72 q^{9} - 24 q^{13} + 58 q^{17} - 180 q^{23} + 28 q^{25} - 354 q^{27} - 392 q^{29} - 154 q^{35} + 532 q^{39} + 234 q^{43} - 128 q^{49} + 510 q^{51} - 1244 q^{53} + 576 q^{55} - 56 q^{61} + 566 q^{65} + 1748 q^{69} - 472 q^{75} - 304 q^{77} - 1908 q^{79} + 2282 q^{81} + 64 q^{87} - 582 q^{91} + 2340 q^{95}+O(q^{100}) $ 10 * q - 6 * q^3 + 72 * q^9 - 24 * q^13 + 58 * q^17 - 180 * q^23 + 28 * q^25 - 354 * q^27 - 392 * q^29 - 154 * q^35 + 532 * q^39 + 234 * q^43 - 128 * q^49 + 510 * q^51 - 1244 * q^53 + 576 * q^55 - 56 * q^61 + 566 * q^65 + 1748 * q^69 - 472 * q^75 - 304 * q^77 - 1908 * q^79 + 2282 * q^81 + 64 * q^87 - 582 * q^91 + 2340 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−9.27625			−1.78522			−0.892608	−	0.450834i	$-0.851127\pi$
$3$	−9.27625			−1.78522			−0.892608	+	0.450834i	$0.851127\pi$
$4$		0			0
$5$		−	2.27999i		−	0.203929i	−0.994788	−	0.101964i	$-0.967487\pi$
$5$		−	2.27999i		−	0.203929i	0.994788	−	0.101964i	$-0.0325128\pi$
$6$		0			0
$7$			28.5449i			1.54128i	0.637269	+	0.770641i	$0.280064\pi$
$7$			28.5449i			1.54128i	−0.637269	+	0.770641i	$0.719936\pi$
$8$		0			0
$9$	59.0489			2.18699
$10$		0			0
$11$			0.604833i			0.0165785i	0.999966	+	0.00828927i	$0.00263859\pi$
$11$			0.604833i			0.0165785i	−0.999966	+	0.00828927i	$0.997361\pi$
$12$		0			0
$13$	−19.1535	+	42.7802i	−0.408633	+	0.912699i
$13$	−19.1535	+	42.7802i	−0.408633	+	0.912699i
$14$		0			0
$15$			21.1498i			0.364057i
$16$		0			0
$17$	−41.0109			−0.585095			−0.292548	−	0.956251i	$-0.594503\pi$
$17$	−41.0109			−0.585095			−0.292548	+	0.956251i	$0.594503\pi$
$18$		0			0
$19$		−	129.485i		−	1.56347i	−0.623609	−	0.781737i	$-0.714334\pi$
$19$		−	129.485i		−	1.56347i	0.623609	−	0.781737i	$-0.285666\pi$
$20$		0			0
$21$		−	264.790i		−	2.75152i
$22$		0			0
$23$	−73.8974			−0.669943			−0.334971	−	0.942228i	$-0.608727\pi$
$23$	−73.8974			−0.669943			−0.334971	+	0.942228i	$0.608727\pi$
$24$		0			0
$25$	119.802			0.958413
$26$		0			0
$27$	−297.293			−2.11904
$28$		0			0
$29$	−214.532			−1.37371			−0.686856	−	0.726793i	$-0.741010\pi$
$29$	−214.532			−1.37371			−0.686856	+	0.726793i	$0.741010\pi$
$30$		0			0
$31$		−	126.175i		−	0.731023i	−0.930807	−	0.365512i	$-0.880894\pi$
$31$		−	126.175i		−	0.731023i	0.930807	−	0.365512i	$-0.119106\pi$
$32$		0			0
$33$		−	5.61058i		−	0.0295963i
$34$		0			0
$35$	65.0822			0.314312
$36$		0			0
$37$		−	309.470i		−	1.37504i	−0.726164	−	0.687522i	$-0.758699\pi$
$37$		−	309.470i		−	1.37504i	0.726164	−	0.687522i	$-0.241301\pi$
$38$		0			0
$39$	177.673	−	396.840i	0.729498	−	1.62936i
$40$		0			0
$41$			88.1488i			0.335769i	0.985807	+	0.167885i	$0.0536936\pi$
$41$			88.1488i			0.335769i	−0.985807	+	0.167885i	$0.946306\pi$
$42$		0			0
$43$	245.304			0.869967			0.434983	−	0.900438i	$-0.356754\pi$
$43$	245.304			0.869967			0.434983	+	0.900438i	$0.356754\pi$
$44$		0			0
$45$		−	134.631i		−	0.445991i
$46$		0			0
$47$		−	68.5441i		−	0.212727i	−0.994327	−	0.106364i	$-0.966079\pi$
$47$		−	68.5441i		−	0.212727i	0.994327	−	0.106364i	$-0.0339208\pi$
$48$		0			0
$49$	−471.814			−1.37555
$50$		0			0
$51$	380.428			1.04452
$52$		0			0
$53$	−613.275			−1.58943			−0.794716	−	0.606982i	$-0.792380\pi$
$53$	−613.275			−1.58943			−0.794716	+	0.606982i	$0.792380\pi$
$54$		0			0
$55$	1.37901			0.00338084
$56$		0			0
$57$			1201.14i			2.79114i
$58$		0			0
$59$		−	840.276i		−	1.85415i	−0.374882	−	0.927073i	$-0.622317\pi$
$59$		−	840.276i		−	1.85415i	0.374882	−	0.927073i	$-0.377683\pi$
$60$		0			0
$61$	587.808			1.23379			0.616894	−	0.787046i	$-0.288391\pi$
$61$	587.808			1.23379			0.616894	+	0.787046i	$0.288391\pi$
$62$		0			0
$63$			1685.55i			3.37078i
$64$		0			0
$65$	97.5384	+	43.6699i	0.186125	+	0.0833320i
$66$		0			0
$67$			606.554i			1.10601i	0.833180	+	0.553003i	$0.186518\pi$
$67$			606.554i			1.10601i	−0.833180	+	0.553003i	$0.813482\pi$
$68$		0			0
$69$	685.491			1.19599
$70$		0			0
$71$			507.627i			0.848510i	0.905543	+	0.424255i	$0.139464\pi$
$71$			507.627i			0.848510i	−0.905543	+	0.424255i	$0.860536\pi$
$72$		0			0
$73$		−	177.127i		−	0.283989i	−0.989867	−	0.141994i	$-0.954649\pi$
$73$		−	177.127i		−	0.283989i	0.989867	−	0.141994i	$-0.0453515\pi$
$74$		0			0
$75$	−1111.31			−1.71097
$76$		0			0
$77$	−17.2649			−0.0255522
$78$		0			0
$79$	−143.122			−0.203829			−0.101914	−	0.994793i	$-0.532497\pi$
$79$	−143.122			−0.203829			−0.101914	+	0.994793i	$0.532497\pi$
$80$		0			0
$81$	1163.45			1.59595
$82$		0			0
$83$		−	773.287i		−	1.02264i	−0.859390	−	0.511321i	$-0.829156\pi$
$83$		−	773.287i		−	1.02264i	0.859390	−	0.511321i	$-0.170844\pi$
$84$		0			0
$85$			93.5046i			0.119318i
$86$		0			0
$87$	1990.06			2.45237
$88$		0			0
$89$		−	1402.90i		−	1.67086i	−0.549593	−	0.835432i	$-0.685217\pi$
$89$		−	1402.90i		−	1.67086i	0.549593	−	0.835432i	$-0.314783\pi$
$90$		0			0
$91$	−1221.16	−	546.736i	−1.40673	−	0.629819i
$92$		0			0
$93$			1170.43i			1.30503i
$94$		0			0
$95$	−295.226			−0.318837
$96$		0			0
$97$		−	468.225i		−	0.490114i	−0.969509	−	0.245057i	$-0.921193\pi$
$97$		−	468.225i		−	0.490114i	0.969509	−	0.245057i	$-0.0788066\pi$
$98$		0			0
$99$			35.7147i			0.0362572i
$100$		0			0
$101$	−399.959			−0.394033			−0.197017	−	0.980400i	$-0.563125\pi$
$101$	−399.959			−0.394033			−0.197017	+	0.980400i	$0.563125\pi$
$102$		0			0
$103$	−1404.11			−1.34321			−0.671607	−	0.740907i	$-0.734396\pi$
$103$	−1404.11			−1.34321			−0.671607	+	0.740907i	$0.734396\pi$
$104$		0			0
$105$	−603.719			−0.561114
$106$		0			0
$107$	1596.54			1.44246			0.721232	−	0.692694i	$-0.243576\pi$
$107$	1596.54			1.44246			0.721232	+	0.692694i	$0.243576\pi$
$108$		0			0
$109$		−	247.807i		−	0.217758i	−0.994055	−	0.108879i	$-0.965274\pi$
$109$		−	247.807i		−	0.217758i	0.994055	−	0.108879i	$-0.0347261\pi$
$110$		0			0
$111$			2870.73i			2.45475i
$112$		0			0
$113$	423.360			0.352445			0.176223	−	0.984350i	$-0.443612\pi$
$113$	423.360			0.352445			0.176223	+	0.984350i	$0.443612\pi$
$114$		0			0
$115$			168.486i			0.136621i
$116$		0			0
$117$	−1130.99	+	2526.12i	−0.893678	+	1.99607i
$118$		0			0
$119$		−	1170.66i		−	0.901797i
$120$		0			0
$121$	1330.63			0.999725
$122$		0			0
$123$		−	817.691i		−	0.599420i
$124$		0			0
$125$		−	558.146i		−	0.399377i
$126$		0			0
$127$	640.855			0.447769			0.223884	−	0.974616i	$-0.428126\pi$
$127$	640.855			0.447769			0.223884	+	0.974616i	$0.428126\pi$
$128$		0			0
$129$	−2275.51			−1.55308
$130$		0			0
$131$	−809.348			−0.539794			−0.269897	−	0.962889i	$-0.586990\pi$
$131$	−809.348			−0.539794			−0.269897	+	0.962889i	$0.586990\pi$
$132$		0			0
$133$	3696.15			2.40975
$134$		0			0
$135$			677.826i			0.432133i
$136$		0			0
$137$		−	1666.32i		−	1.03915i	−0.854426	−	0.519573i	$-0.826091\pi$
$137$		−	1666.32i		−	1.03915i	0.854426	−	0.519573i	$-0.173909\pi$
$138$		0			0
$139$	−2473.29			−1.50922			−0.754609	−	0.656174i	$-0.772174\pi$
$139$	−2473.29			−1.50922			−0.754609	+	0.656174i	$0.772174\pi$
$140$		0			0
$141$			635.833i			0.379764i
$142$		0			0
$143$	−25.8748	−	11.5847i	−0.0151312	−	0.00677454i
$144$		0			0
$145$			489.132i			0.280139i
$146$		0			0
$147$	4376.66			2.45565
$148$		0			0
$149$			2014.63i			1.10769i	0.832621	+	0.553843i	$0.186839\pi$
$149$			2014.63i			1.10769i	−0.832621	+	0.553843i	$0.813161\pi$
$150$		0			0
$151$		−	389.182i		−	0.209743i	−0.994486	−	0.104872i	$-0.966557\pi$
$151$		−	389.182i		−	0.209743i	0.994486	−	0.104872i	$-0.0334432\pi$
$152$		0			0
$153$	−2421.65			−1.27960
$154$		0			0
$155$	−287.678			−0.149077
$156$		0			0
$157$	−1485.77			−0.755270			−0.377635	−	0.925954i	$-0.623263\pi$
$157$	−1485.77			−0.755270			−0.377635	+	0.925954i	$0.623263\pi$
$158$		0			0
$159$	5688.90			2.83748
$160$		0			0
$161$		−	2109.40i		−	1.03257i
$162$		0			0
$163$			2345.72i			1.12718i	0.826054	+	0.563591i	$0.190581\pi$
$163$			2345.72i			1.12718i	−0.826054	+	0.563591i	$0.809419\pi$
$164$		0			0
$165$	−12.7921			−0.00603553
$166$		0			0
$167$		−	1978.23i		−	0.916648i	−0.888785	−	0.458324i	$-0.848450\pi$
$167$		−	1978.23i		−	0.916648i	0.888785	−	0.458324i	$-0.151550\pi$
$168$		0			0
$169$	−1463.29	−	1638.78i	−0.666038	−	0.745918i
$170$		0			0
$171$		−	7645.97i		−	3.41931i
$172$		0			0
$173$	−1892.71			−0.831791			−0.415895	−	0.909412i	$-0.636532\pi$
$173$	−1892.71			−0.831791			−0.415895	+	0.909412i	$0.636532\pi$
$174$		0			0
$175$			3419.73i			1.47718i
$176$		0			0
$177$			7794.61i			3.31005i
$178$		0			0
$179$	−4002.27			−1.67119			−0.835597	−	0.549343i	$-0.814878\pi$
$179$	−4002.27			−1.67119			−0.835597	+	0.549343i	$0.814878\pi$
$180$		0			0
$181$	1128.60			0.463469			0.231734	−	0.972779i	$-0.425560\pi$
$181$	1128.60			0.463469			0.231734	+	0.972779i	$0.425560\pi$
$182$		0			0
$183$	−5452.65			−2.20258
$184$		0			0
$185$	−705.590			−0.280411
$186$		0			0
$187$		−	24.8048i		−	0.00970002i
$188$		0			0
$189$		−	8486.22i		−	3.26604i
$190$		0			0
$191$	3133.94			1.18724			0.593622	−	0.804744i	$-0.297697\pi$
$191$	3133.94			1.18724			0.593622	+	0.804744i	$0.297697\pi$
$192$		0			0
$193$		−	2115.69i		−	0.789070i	−0.918881	−	0.394535i	$-0.870906\pi$
$193$		−	2115.69i		−	0.789070i	0.918881	−	0.394535i	$-0.129094\pi$
$194$		0			0
$195$	−904.791	−	405.093i	−0.332274	−	0.148766i
$196$		0			0
$197$			14.4068i			0.00521038i	0.999997	+	0.00260519i	$0.000829259\pi$
$197$			14.4068i			0.00521038i	−0.999997	+	0.00260519i	$0.999171\pi$
$198$		0			0
$199$	−1412.24			−0.503072			−0.251536	−	0.967848i	$-0.580936\pi$
$199$	−1412.24			−0.503072			−0.251536	+	0.967848i	$0.580936\pi$
$200$		0			0
$201$		−	5626.55i		−	1.97446i
$202$		0			0
$203$		−	6123.81i		−	2.11728i
$204$		0			0
$205$	200.979			0.0684729
$206$		0			0
$207$	−4363.56			−1.46516
$208$		0			0
$209$	78.3170			0.0259201
$210$		0			0
$211$	−2908.92			−0.949093			−0.474546	−	0.880231i	$-0.657388\pi$
$211$	−2908.92			−0.949093			−0.474546	+	0.880231i	$0.657388\pi$
$212$		0			0
$213$		−	4708.87i		−	1.51477i
$214$		0			0
$215$		−	559.292i		−	0.177411i
$216$		0			0
$217$	3601.66			1.12671
$218$		0			0
$219$			1643.08i			0.506981i
$220$		0			0
$221$	785.504	−	1754.46i	0.239089	−	0.534016i
$222$		0			0
$223$		−	354.193i		−	0.106361i	−0.998585	−	0.0531805i	$-0.983064\pi$
$223$		−	354.193i		−	0.106361i	0.998585	−	0.0531805i	$-0.0169359\pi$
$224$		0			0
$225$	7074.15			2.09604
$226$		0			0
$227$			4122.96i			1.20551i	0.797927	+	0.602754i	$0.205930\pi$
$227$			4122.96i			1.20551i	−0.797927	+	0.602754i	$0.794070\pi$
$228$		0			0
$229$			4250.04i			1.22642i	0.789920	+	0.613210i	$0.210122\pi$
$229$			4250.04i			1.22642i	−0.789920	+	0.613210i	$0.789878\pi$
$230$		0			0
$231$	160.154			0.0456162
$232$		0			0
$233$	−1590.88			−0.447304			−0.223652	−	0.974669i	$-0.571798\pi$
$233$	−1590.88			−0.447304			−0.223652	+	0.974669i	$0.571798\pi$
$234$		0			0
$235$	−156.280			−0.0433812
$236$		0			0
$237$	1327.64			0.363879
$238$		0			0
$239$			1030.76i			0.278972i	0.990224	+	0.139486i	$0.0445451\pi$
$239$			1030.76i			0.278972i	−0.990224	+	0.139486i	$0.955455\pi$
$240$		0			0
$241$			4582.38i			1.22480i	0.790547	+	0.612401i	$0.209796\pi$
$241$			4582.38i			1.22480i	−0.790547	+	0.612401i	$0.790204\pi$
$242$		0			0
$243$	−2765.52			−0.730076
$244$		0			0
$245$			1075.73i			0.280514i
$246$		0			0
$247$	5539.41	+	2480.10i	1.42698	+	0.638887i
$248$		0			0
$249$			7173.20i			1.82564i
$250$		0			0
$251$	3084.18			0.775584			0.387792	−	0.921747i	$-0.373238\pi$
$251$	3084.18			0.775584			0.387792	+	0.921747i	$0.373238\pi$
$252$		0			0
$253$		−	44.6956i		−	0.0111067i
$254$		0			0
$255$		−	867.372i		−	0.213008i
$256$		0			0
$257$	−2328.69			−0.565212			−0.282606	−	0.959236i	$-0.591199\pi$
$257$	−2328.69			−0.565212			−0.282606	+	0.959236i	$0.591199\pi$
$258$		0			0
$259$	8833.81			2.11933
$260$		0			0
$261$	−12667.9			−3.00430
$262$		0			0
$263$	−3795.43			−0.889873			−0.444936	−	0.895562i	$-0.646774\pi$
$263$	−3795.43			−0.889873			−0.444936	+	0.895562i	$0.646774\pi$
$264$		0			0
$265$			1398.26i			0.324131i
$266$		0			0
$267$			13013.6i			2.98285i
$268$		0			0
$269$	−6178.54			−1.40042			−0.700209	−	0.713938i	$-0.746910\pi$
$269$	−6178.54			−1.40042			−0.700209	+	0.713938i	$0.746910\pi$
$270$		0			0
$271$		−	322.020i		−	0.0721819i	−0.999349	−	0.0360910i	$-0.988509\pi$
$271$		−	322.020i		−	0.0721819i	0.999349	−	0.0360910i	$-0.0114906\pi$
$272$		0			0
$273$	11327.8	+	5071.66i	2.51131	+	1.12436i
$274$		0			0
$275$			72.4600i			0.0158891i
$276$		0			0
$277$	4183.15			0.907369			0.453684	−	0.891162i	$-0.350109\pi$
$277$	4183.15			0.907369			0.453684	+	0.891162i	$0.350109\pi$
$278$		0			0
$279$		−	7450.49i		−	1.59874i
$280$		0			0
$281$			392.013i			0.0832226i	0.999134	+	0.0416113i	$0.0132491\pi$
$281$			392.013i			0.0832226i	−0.999134	+	0.0416113i	$0.986751\pi$
$282$		0			0
$283$	4119.24			0.865241			0.432621	−	0.901576i	$-0.357589\pi$
$283$	4119.24			0.865241			0.432621	+	0.901576i	$0.357589\pi$
$284$		0			0
$285$	2738.59			0.569193
$286$		0			0
$287$	−2516.20			−0.517515
$288$		0			0
$289$	−3231.10			−0.657664
$290$		0			0
$291$			4343.37i			0.874959i
$292$		0			0
$293$		−	4885.80i		−	0.974169i	−0.873355	−	0.487085i	$-0.838060\pi$
$293$		−	4885.80i		−	0.974169i	0.873355	−	0.487085i	$-0.161940\pi$
$294$		0			0
$295$	−1915.82			−0.378113
$296$		0			0
$297$		−	179.813i		−	0.0351306i
$298$		0			0
$299$	1415.40	−	3161.35i	0.273761	−	0.611456i
$300$		0			0
$301$			7002.20i			1.34086i
$302$		0			0
$303$	3710.12			0.703435
$304$		0			0
$305$		−	1340.20i		−	0.251605i
$306$		0			0
$307$			2024.02i			0.376277i	0.982142	+	0.188139i	$0.0602454\pi$
$307$			2024.02i			0.376277i	−0.982142	+	0.188139i	$0.939755\pi$
$308$		0			0
$309$	13024.9			2.39793
$310$		0			0
$311$	1272.97			0.232102			0.116051	−	0.993243i	$-0.462976\pi$
$311$	1272.97			0.232102			0.116051	+	0.993243i	$0.462976\pi$
$312$		0			0
$313$	375.693			0.0678448			0.0339224	−	0.999424i	$-0.489200\pi$
$313$	375.693			0.0678448			0.0339224	+	0.999424i	$0.489200\pi$
$314$		0			0
$315$	3843.03			0.687398
$316$		0			0
$317$			10369.7i			1.83728i	0.395094	+	0.918641i	$0.370712\pi$
$317$			10369.7i			1.83728i	−0.395094	+	0.918641i	$0.629288\pi$
$318$		0			0
$319$		−	129.756i		−	0.0227741i
$320$		0			0
$321$	−14809.9			−2.57511
$322$		0			0
$323$			5310.32i			0.914780i
$324$		0			0
$325$	−2294.62	+	5125.13i	−0.391639	+	0.874742i
$326$		0			0
$327$			2298.72i			0.388744i
$328$		0			0
$329$	1956.59			0.327873
$330$		0			0
$331$		−	2272.03i		−	0.377287i	−0.982046	−	0.188644i	$-0.939591\pi$
$331$		−	2272.03i		−	0.377287i	0.982046	−	0.188644i	$-0.0604091\pi$
$332$		0			0
$333$		−	18273.9i		−	3.00721i
$334$		0			0
$335$	1382.94			0.225546
$336$		0			0
$337$	−2984.47			−0.482417			−0.241209	−	0.970473i	$-0.577544\pi$
$337$	−2984.47			−0.482417			−0.241209	+	0.970473i	$0.577544\pi$
$338$		0			0
$339$	−3927.19			−0.629191
$340$		0			0
$341$	76.3148			0.0121193
$342$		0			0
$343$		−	3676.98i		−	0.578829i
$344$		0			0
$345$		−	1562.91i		−	0.243897i
$346$		0			0
$347$	−4329.42			−0.669786			−0.334893	−	0.942256i	$-0.608700\pi$
$347$	−4329.42			−0.669786			−0.334893	+	0.942256i	$0.608700\pi$
$348$		0			0
$349$		−	4025.97i		−	0.617494i	−0.951144	−	0.308747i	$-0.900090\pi$
$349$		−	4025.97i		−	0.617494i	0.951144	−	0.308747i	$-0.0999097\pi$
$350$		0			0
$351$	5694.21	−	12718.3i	0.865910	−	1.93405i
$352$		0			0
$353$		−	6907.56i		−	1.04151i	−0.853707	−	0.520754i	$-0.825651\pi$
$353$		−	6907.56i		−	1.04151i	0.853707	−	0.520754i	$-0.174349\pi$
$354$		0			0
$355$	1157.38			0.173036
$356$		0			0
$357$			10859.3i			1.60990i
$358$		0			0
$359$		−	1518.19i		−	0.223195i	−0.993754	−	0.111597i	$-0.964403\pi$
$359$		−	1518.19i		−	0.223195i	0.993754	−	0.111597i	$-0.0355967\pi$
$360$		0			0
$361$	−9907.47			−1.44445
$362$		0			0
$363$	−12343.3			−1.78472
$364$		0			0
$365$	−403.848			−0.0579134
$366$		0			0
$367$	−456.984			−0.0649983			−0.0324992	−	0.999472i	$-0.510347\pi$
$367$	−456.984			−0.0649983			−0.0324992	+	0.999472i	$0.510347\pi$
$368$		0			0
$369$			5205.09i			0.734325i
$370$		0			0
$371$		−	17505.9i		−	2.44976i
$372$		0			0
$373$	908.186			0.126070			0.0630350	−	0.998011i	$-0.479922\pi$
$373$	908.186			0.126070			0.0630350	+	0.998011i	$0.479922\pi$
$374$		0			0
$375$			5177.50i			0.712973i
$376$		0			0
$377$	4109.05	−	9177.73i	0.561344	−	1.25379i
$378$		0			0
$379$			9874.70i			1.33834i	0.743111	+	0.669168i	$0.233349\pi$
$379$			9874.70i			1.33834i	−0.743111	+	0.669168i	$0.766651\pi$
$380$		0			0
$381$	−5944.73			−0.799364
$382$		0			0
$383$			7726.79i			1.03086i	0.856931	+	0.515432i	$0.172368\pi$
$383$			7726.79i			1.03086i	−0.856931	+	0.515432i	$0.827632\pi$
$384$		0			0
$385$			39.3639i			0.00521083i
$386$		0			0
$387$	14484.9			1.90261
$388$		0			0
$389$	6461.27			0.842157			0.421079	−	0.907024i	$-0.361652\pi$
$389$	6461.27			0.842157			0.421079	+	0.907024i	$0.361652\pi$
$390$		0			0
$391$	3030.60			0.391980
$392$		0			0
$393$	7507.72			0.963649
$394$		0			0
$395$			326.317i			0.0415666i
$396$		0			0
$397$			9730.13i			1.23008i	0.788497	+	0.615039i	$0.210860\pi$
$397$			9730.13i			1.23008i	−0.788497	+	0.615039i	$0.789140\pi$
$398$		0			0
$399$	−34286.5			−4.30193
$400$		0			0
$401$		−	10566.7i		−	1.31590i	−0.753061	−	0.657950i	$-0.771424\pi$
$401$		−	10566.7i		−	1.31590i	0.753061	−	0.657950i	$-0.228576\pi$
$402$		0			0
$403$	5397.79	+	2416.70i	0.667204	+	0.298720i
$404$		0			0
$405$		−	2652.65i		−	0.325460i
$406$		0			0
$407$	187.178			0.0227962
$408$		0			0
$409$		−	10627.3i		−	1.28481i	−0.766366	−	0.642405i	$-0.777937\pi$
$409$		−	10627.3i		−	1.28481i	0.766366	−	0.642405i	$-0.222063\pi$
$410$		0			0
$411$			15457.2i			1.85510i
$412$		0			0
$413$	23985.6			2.85776
$414$		0			0
$415$	−1763.09			−0.208546
$416$		0			0
$417$	22942.8			2.69428
$418$		0			0
$419$	−1384.56			−0.161433			−0.0807164	−	0.996737i	$-0.525721\pi$
$419$	−1384.56			−0.161433			−0.0807164	+	0.996737i	$0.525721\pi$
$420$		0			0
$421$			4777.19i			0.553031i	0.961010	+	0.276515i	$0.0891796\pi$
$421$			4777.19i			0.553031i	−0.961010	+	0.276515i	$0.910820\pi$
$422$		0			0
$423$		−	4047.45i		−	0.465234i
$424$		0			0
$425$	−4913.18			−0.560763
$426$		0			0
$427$			16778.9i			1.90162i
$428$		0			0
$429$	240.022	+	107.462i	0.0270125	+	0.0120940i
$430$		0			0
$431$		−	8535.98i		−	0.953976i	−0.878910	−	0.476988i	$-0.841729\pi$
$431$		−	8535.98i		−	0.953976i	0.878910	−	0.476988i	$-0.158271\pi$
$432$		0			0
$433$	637.729			0.0707789			0.0353895	−	0.999374i	$-0.488733\pi$
$433$	637.729			0.0707789			0.0353895	+	0.999374i	$0.488733\pi$
$434$		0			0
$435$		−	4537.31i		−	0.500109i
$436$		0			0
$437$			9568.64i			1.04744i
$438$		0			0
$439$	1780.56			0.193580			0.0967901	−	0.995305i	$-0.469142\pi$
$439$	1780.56			0.193580			0.0967901	+	0.995305i	$0.469142\pi$
$440$		0			0
$441$	−27860.1			−3.00832
$442$		0			0
$443$	−5257.36			−0.563848			−0.281924	−	0.959437i	$-0.590973\pi$
$443$	−5257.36			−0.563848			−0.281924	+	0.959437i	$0.590973\pi$
$444$		0			0
$445$	−3198.60			−0.340737
$446$		0			0
$447$		−	18688.2i		−	1.97746i
$448$		0			0
$449$			4175.11i			0.438832i	0.975631	+	0.219416i	$0.0704152\pi$
$449$			4175.11i			0.438832i	−0.975631	+	0.219416i	$0.929585\pi$
$450$		0			0
$451$	−53.3153			−0.00556656
$452$		0			0
$453$			3610.15i			0.374437i
$454$		0			0
$455$	−1246.55	+	2784.23i	−0.128438	+	0.286872i
$456$		0			0
$457$		−	9325.52i		−	0.954550i	−0.878754	−	0.477275i	$-0.841625\pi$
$457$		−	9325.52i		−	0.954550i	0.878754	−	0.477275i	$-0.158375\pi$
$458$		0			0
$459$	12192.3			1.23984
$460$		0			0
$461$		−	17815.8i		−	1.79993i	−0.435966	−	0.899963i	$-0.643593\pi$
$461$		−	17815.8i		−	1.79993i	0.435966	−	0.899963i	$-0.356407\pi$
$462$		0			0
$463$		−	13941.4i		−	1.39938i	−0.714448	−	0.699689i	$-0.753322\pi$
$463$		−	13941.4i		−	1.39938i	0.714448	−	0.699689i	$-0.246678\pi$
$464$		0			0
$465$	2668.57			0.266134
$466$		0			0
$467$	−17855.6			−1.76929			−0.884644	−	0.466268i	$-0.845598\pi$
$467$	−17855.6			−1.76929			−0.884644	+	0.466268i	$0.845598\pi$
$468$		0			0
$469$	−17314.0			−1.70467
$470$		0			0
$471$	13782.4			1.34832
$472$		0			0
$473$			148.368i			0.0144228i
$474$		0			0
$475$		−	15512.6i		−	1.49845i
$476$		0			0
$477$	−36213.2			−3.47608
$478$		0			0
$479$		−	5268.36i		−	0.502542i	−0.967917	−	0.251271i	$-0.919151\pi$
$479$		−	5268.36i		−	0.502542i	0.967917	−	0.251271i	$-0.0808485\pi$
$480$		0			0
$481$	13239.2	+	5927.45i	1.25500	+	0.561888i
$482$		0			0
$483$			19567.3i			1.84336i
$484$		0			0
$485$	−1067.55			−0.0999482
$486$		0			0
$487$		−	5676.35i		−	0.528172i	−0.964499	−	0.264086i	$-0.914930\pi$
$487$		−	5676.35i		−	0.528172i	0.964499	−	0.264086i	$-0.0850704\pi$
$488$		0			0
$489$		−	21759.5i		−	2.01226i
$490$		0			0
$491$	12342.0			1.13439			0.567197	−	0.823582i	$-0.308028\pi$
$491$	12342.0			1.13439			0.567197	+	0.823582i	$0.308028\pi$
$492$		0			0
$493$	8798.18			0.803752
$494$		0			0
$495$	81.4292			0.00739388
$496$		0			0
$497$	−14490.2			−1.30779
$498$		0			0
$499$		−	16955.3i		−	1.52109i	−0.649285	−	0.760545i	$-0.724932\pi$
$499$		−	16955.3i		−	1.52109i	0.649285	−	0.760545i	$-0.275068\pi$
$500$		0			0
$501$			18350.6i			1.63641i
$502$		0			0
$503$	−10583.7			−0.938178			−0.469089	−	0.883151i	$-0.655418\pi$
$503$	−10583.7			−0.938178			−0.469089	+	0.883151i	$0.655418\pi$
$504$		0			0
$505$			911.902i			0.0803547i
$506$		0			0
$507$	13573.8	+	15201.7i	1.18902	+	1.33162i
$508$		0			0
$509$		−	13349.6i		−	1.16249i	−0.813728	−	0.581246i	$-0.802565\pi$
$509$		−	13349.6i		−	1.16249i	0.813728	−	0.581246i	$-0.197435\pi$
$510$		0			0
$511$	5056.09			0.437707
$512$		0			0
$513$			38495.1i			3.31306i
$514$		0			0
$515$			3201.36i			0.273920i
$516$		0			0
$517$	41.4577			0.00352671
$518$		0			0
$519$	17557.2			1.48493
$520$		0			0
$521$	−7321.30			−0.615647			−0.307823	−	0.951443i	$-0.599601\pi$
$521$	−7321.30			−0.615647			−0.307823	+	0.951443i	$0.599601\pi$
$522$		0			0
$523$	−6772.54			−0.566238			−0.283119	−	0.959085i	$-0.591369\pi$
$523$	−6772.54			−0.566238			−0.283119	+	0.959085i	$0.591369\pi$
$524$		0			0
$525$		−	31722.3i		−	2.63709i
$526$		0			0
$527$			5174.56i			0.427718i
$528$		0			0
$529$	−6706.17			−0.551177
$530$		0			0
$531$		−	49617.3i		−	4.05501i
$532$		0			0
$533$	−3771.02	−	1688.36i	−0.306456	−	0.137206i
$534$		0			0
$535$		−	3640.10i		−	0.294160i
$536$		0			0
$537$	37126.1			2.98344
$538$		0			0
$539$		−	285.368i		−	0.0228046i
$540$		0			0
$541$			2816.12i			0.223798i	0.993720	+	0.111899i	$0.0356933\pi$
$541$			2816.12i			0.223798i	−0.993720	+	0.111899i	$0.964307\pi$
$542$		0			0
$543$	−10469.1			−0.827391
$544$		0			0
$545$	−564.998			−0.0444070
$546$		0			0
$547$	946.909			0.0740163			0.0370082	−	0.999315i	$-0.488217\pi$
$547$	946.909			0.0740163			0.0370082	+	0.999315i	$0.488217\pi$
$548$		0			0
$549$	34709.4			2.69829
$550$		0			0
$551$			27778.8i			2.14776i
$552$		0			0
$553$		−	4085.41i		−	0.314158i
$554$		0			0
$555$	6545.23			0.500594
$556$		0			0
$557$			6635.00i			0.504729i	0.967632	+	0.252364i	$0.0812081\pi$
$557$			6635.00i			0.504729i	−0.967632	+	0.252364i	$0.918792\pi$
$558$		0			0
$559$	−4698.44	+	10494.2i	−0.355497	+	0.794018i
$560$		0			0
$561$			230.095i			0.0173166i
$562$		0			0
$563$	−14327.3			−1.07251			−0.536254	−	0.844056i	$-0.680161\pi$
$563$	−14327.3			−1.07251			−0.536254	+	0.844056i	$0.680161\pi$
$564$		0			0
$565$		−	965.256i		−	0.0718737i
$566$		0			0
$567$			33210.6i			2.45981i
$568$		0			0
$569$	−20996.9			−1.54699			−0.773493	−	0.633805i	$-0.781492\pi$
$569$	−20996.9			−1.54699			−0.773493	+	0.633805i	$0.781492\pi$
$570$		0			0
$571$	−3078.19			−0.225601			−0.112800	−	0.993618i	$-0.535982\pi$
$571$	−3078.19			−0.225601			−0.112800	+	0.993618i	$0.535982\pi$
$572$		0			0
$573$	−29071.2			−2.11949
$574$		0			0
$575$	−8853.04			−0.642082
$576$		0			0
$577$			2220.02i			0.160174i	0.996788	+	0.0800871i	$0.0255199\pi$
$577$			2220.02i			0.160174i	−0.996788	+	0.0800871i	$0.974480\pi$
$578$		0			0
$579$			19625.6i			1.40866i
$580$		0			0
$581$	22073.4			1.57618
$582$		0			0
$583$		−	370.929i		−	0.0263504i
$584$		0			0
$585$	5759.53	+	2578.66i	0.407055	+	0.182247i
$586$		0			0
$587$		−	21570.1i		−	1.51669i	−0.651855	−	0.758343i	$-0.726009\pi$
$587$		−	21570.1i		−	1.51669i	0.651855	−	0.758343i	$-0.273991\pi$
$588$		0			0
$589$	−16337.8			−1.14293
$590$		0			0
$591$		−	133.642i		−	0.00930166i
$592$		0			0
$593$			17778.7i			1.23117i	0.788071	+	0.615584i	$0.211080\pi$
$593$			17778.7i			1.23117i	−0.788071	+	0.615584i	$0.788920\pi$
$594$		0			0
$595$	−2669.08			−0.183902
$596$		0			0
$597$	13100.3			0.898092
$598$		0			0
$599$	6024.84			0.410965			0.205483	−	0.978661i	$-0.434124\pi$
$599$	6024.84			0.410965			0.205483	+	0.978661i	$0.434124\pi$
$600$		0			0
$601$	−14058.2			−0.954153			−0.477077	−	0.878862i	$-0.658304\pi$
$601$	−14058.2			−0.954153			−0.477077	+	0.878862i	$0.658304\pi$
$602$		0			0
$603$			35816.3i			2.41883i
$604$		0			0
$605$		−	3033.83i		−	0.203873i
$606$		0			0
$607$	16830.7			1.12543			0.562716	−	0.826650i	$-0.309757\pi$
$607$	16830.7			1.12543			0.562716	+	0.826650i	$0.309757\pi$
$608$		0			0
$609$			56806.1i			3.77980i
$610$		0			0
$611$	2932.33	+	1312.86i	0.194156	+	0.0869274i
$612$		0			0
$613$		−	9954.11i		−	0.655861i	−0.944702	−	0.327931i	$-0.893649\pi$
$613$		−	9954.11i		−	0.655861i	0.944702	−	0.327931i	$-0.106351\pi$
$614$		0			0
$615$	−1864.33			−0.122239
$616$		0			0
$617$			19391.0i			1.26524i	0.774462	+	0.632621i	$0.218021\pi$
$617$			19391.0i			1.26524i	−0.774462	+	0.632621i	$0.781979\pi$
$618$		0			0
$619$			60.1151i			0.00390344i	0.999998	+	0.00195172i	$0.000621253\pi$
$619$			60.1151i			0.00390344i	−0.999998	+	0.00195172i	$0.999379\pi$
$620$		0			0
$621$	21969.2			1.41964
$622$		0			0
$623$	40045.7			2.57527
$624$		0			0
$625$	13702.6			0.876969
$626$		0			0
$627$	−726.488			−0.0462730
$628$		0			0
$629$			12691.7i			0.804531i
$630$		0			0
$631$		−	15910.7i		−	1.00379i	−0.864928	−	0.501896i	$-0.832636\pi$
$631$		−	15910.7i		−	1.00379i	0.864928	−	0.501896i	$-0.167364\pi$
$632$		0			0
$633$	26983.9			1.69434
$634$		0			0
$635$		−	1461.14i		−	0.0913129i
$636$		0			0
$637$	9036.89	−	20184.3i	0.562095	−	1.25546i
$638$		0			0
$639$			29974.8i			1.85569i
$640$		0			0
$641$	10812.8			0.666271			0.333135	−	0.942879i	$-0.391893\pi$
$641$	10812.8			0.666271			0.333135	+	0.942879i	$0.391893\pi$
$642$		0			0
$643$			15062.1i			0.923783i	0.886937	+	0.461891i	$0.152829\pi$
$643$			15062.1i			0.923783i	−0.886937	+	0.461891i	$0.847171\pi$
$644$		0			0
$645$			5188.13i			0.316717i
$646$		0			0
$647$	−21953.9			−1.33400			−0.667001	−	0.745057i	$-0.732422\pi$
$647$	−21953.9			−1.33400			−0.667001	+	0.745057i	$0.732422\pi$
$648$		0			0
$649$	508.226			0.0307390
$650$		0			0
$651$	−33409.9			−2.01143
$652$		0			0
$653$	−9906.28			−0.593664			−0.296832	−	0.954930i	$-0.595930\pi$
$653$	−9906.28			−0.593664			−0.296832	+	0.954930i	$0.595930\pi$
$654$		0			0
$655$			1845.31i			0.110080i
$656$		0			0
$657$		−	10459.2i		−	0.621082i
$658$		0			0
$659$	17156.6			1.01415			0.507077	−	0.861901i	$-0.330726\pi$
$659$	17156.6			1.01415			0.507077	+	0.861901i	$0.330726\pi$
$660$		0			0
$661$		−	1654.70i		−	0.0973681i	−0.998814	−	0.0486840i	$-0.984497\pi$
$661$		−	1654.70i		−	0.0973681i	0.998814	−	0.0486840i	$-0.0155027\pi$
$662$		0			0
$663$	−7286.53	+	16274.8i	−0.426826	+	0.953333i
$664$		0			0
$665$		−	8427.20i		−	0.491418i
$666$		0			0
$667$	15853.4			0.920309
$668$		0			0
$669$			3285.58i			0.189877i
$670$		0			0
$671$			355.525i			0.0204544i
$672$		0			0
$673$	−1227.71			−0.0703190			−0.0351595	−	0.999382i	$-0.511194\pi$
$673$	−1227.71			−0.0703190			−0.0351595	+	0.999382i	$0.511194\pi$
$674$		0			0
$675$	−35616.2			−2.03092
$676$		0			0
$677$	−16939.3			−0.961638			−0.480819	−	0.876820i	$-0.659660\pi$
$677$	−16939.3			−0.961638			−0.480819	+	0.876820i	$0.659660\pi$
$678$		0			0
$679$	13365.5			0.755404
$680$		0			0
$681$		−	38245.6i		−	2.15209i
$682$		0			0
$683$		−	16572.3i		−	0.928434i	−0.885721	−	0.464217i	$-0.846336\pi$
$683$		−	16572.3i		−	0.928434i	0.885721	−	0.464217i	$-0.153664\pi$
$684$		0			0
$685$	−3799.19			−0.211912
$686$		0			0
$687$		−	39424.4i		−	2.18942i
$688$		0			0
$689$	11746.4	−	26236.0i	0.649494	−	1.45067i
$690$		0			0
$691$		−	3836.25i		−	0.211198i	−0.994409	−	0.105599i	$-0.966324\pi$
$691$		−	3836.25i		−	0.211198i	0.994409	−	0.105599i	$-0.0336760\pi$
$692$		0			0
$693$	−1019.47			−0.0558825
$694$		0			0
$695$			5639.07i			0.307773i
$696$		0			0
$697$		−	3615.07i		−	0.196457i
$698$		0			0
$699$	14757.4			0.798533
$700$		0			0
$701$	16178.6			0.871692			0.435846	−	0.900021i	$-0.356449\pi$
$701$	16178.6			0.871692			0.435846	+	0.900021i	$0.356449\pi$
$702$		0			0
$703$	−40071.9			−2.14984
$704$		0			0
$705$	1449.69			0.0774448
$706$		0			0
$707$		−	11416.8i		−	0.607317i
$708$		0			0
$709$			26689.0i			1.41372i	0.707354	+	0.706859i	$0.249888\pi$
$709$			26689.0i			1.41372i	−0.707354	+	0.706859i	$0.750112\pi$
$710$		0			0
$711$	−8451.19			−0.445773
$712$		0			0
$713$			9324.02i			0.489744i
$714$		0			0
$715$	−26.4130	+	58.9944i	−0.00138152	+	0.00308569i
$716$		0			0
$717$		−	9561.60i		−	0.498026i
$718$		0			0
$719$	25906.0			1.34371			0.671857	−	0.740681i	$-0.265497\pi$
$719$	25906.0			1.34371			0.671857	+	0.740681i	$0.265497\pi$
$720$		0			0
$721$		−	40080.3i		−	2.07027i
$722$		0			0
$723$		−	42507.3i		−	2.18654i
$724$		0			0
$725$	−25701.3			−1.31658
$726$		0			0
$727$	−8386.29			−0.427827			−0.213914	−	0.976853i	$-0.568621\pi$
$727$	−8386.29			−0.427827			−0.213914	+	0.976853i	$0.568621\pi$
$728$		0			0
$729$	−5759.42			−0.292609
$730$		0			0
$731$	−10060.2			−0.509013
$732$		0			0
$733$		−	289.490i		−	0.0145874i	−0.999973	−	0.00729371i	$-0.997678\pi$
$733$		−	289.490i		−	0.0145874i	0.999973	−	0.00729371i	$-0.00232168\pi$
$734$		0			0
$735$		−	9978.76i		−	0.500778i
$736$		0			0
$737$	−366.864			−0.0183359
$738$		0			0
$739$			17933.9i			0.892705i	0.894857	+	0.446352i	$0.147277\pi$
$739$			17933.9i			0.892705i	−0.894857	+	0.446352i	$0.852723\pi$
$740$		0			0
$741$	−51384.9	−	23006.0i	−2.54747	−	1.14055i
$742$		0			0
$743$			22954.9i			1.13342i	0.823917	+	0.566711i	$0.191784\pi$
$743$			22954.9i			1.13342i	−0.823917	+	0.566711i	$0.808216\pi$
$744$		0			0
$745$	4593.35			0.225889
$746$		0			0
$747$		−	45661.7i		−	2.23651i
$748$		0			0
$749$			45573.2i			2.22324i
$750$		0			0
$751$	−29914.5			−1.45352			−0.726761	−	0.686890i	$-0.758975\pi$
$751$	−29914.5			−1.45352			−0.726761	+	0.686890i	$0.758975\pi$
$752$		0			0
$753$	−28609.6			−1.38459
$754$		0			0
$755$	−887.333			−0.0427726
$756$		0			0
$757$	9422.87			0.452417			0.226209	−	0.974079i	$-0.427367\pi$
$757$	9422.87			0.452417			0.226209	+	0.974079i	$0.427367\pi$
$758$		0			0
$759$			414.608i			0.0198278i
$760$		0			0
$761$			16008.9i			0.762577i	0.924456	+	0.381288i	$0.124520\pi$
$761$			16008.9i			0.762577i	−0.924456	+	0.381288i	$0.875480\pi$
$762$		0			0
$763$	7073.63			0.335626
$764$		0			0
$765$			5521.34i			0.260947i
$766$		0			0
$767$	35947.1	+	16094.2i	1.69228	+	0.757665i
$768$		0			0
$769$			23058.4i			1.08128i	0.841253	+	0.540642i	$0.181818\pi$
$769$			23058.4i			1.08128i	−0.841253	+	0.540642i	$0.818182\pi$
$770$		0			0
$771$	21601.5			1.00903
$772$		0			0
$773$		−	17638.7i		−	0.820726i	−0.911922	−	0.410363i	$-0.865402\pi$
$773$		−	17638.7i		−	0.820726i	0.911922	−	0.410363i	$-0.134598\pi$
$774$		0			0
$775$		−	15116.0i		−	0.700622i
$776$		0			0
$777$	−81944.7			−3.78346
$778$		0			0
$779$	11414.0			0.524966
$780$		0			0
$781$	−307.029			−0.0140671
$782$		0			0
$783$	63779.0			2.91095
$784$		0			0
$785$			3387.55i			0.154021i
$786$		0			0
$787$			2401.87i			0.108789i	0.998520	+	0.0543947i	$0.0173229\pi$
$787$			2401.87i			0.108789i	−0.998520	+	0.0543947i	$0.982677\pi$
$788$		0			0
$789$	35207.4			1.58861
$790$		0			0
$791$			12084.8i			0.543218i
$792$		0			0
$793$	−11258.6	+	25146.5i	−0.504167	+	1.12608i
$794$		0			0
$795$		−	12970.6i		−	0.578643i
$796$		0			0
$797$	−3281.78			−0.145855			−0.0729276	−	0.997337i	$-0.523234\pi$
$797$	−3281.78			−0.145855			−0.0729276	+	0.997337i	$0.523234\pi$
$798$		0			0
$799$			2811.06i			0.124466i
$800$		0			0
$801$		−	82839.6i		−	3.65417i
$802$		0			0
$803$	107.132			0.00470812
$804$		0			0
$805$	−4809.41			−0.210571
$806$		0			0
$807$	57313.7			2.50005
$808$		0			0
$809$	36413.5			1.58249			0.791243	−	0.611501i	$-0.209434\pi$
$809$	36413.5			1.58249			0.791243	+	0.611501i	$0.209434\pi$
$810$		0			0
$811$		−	31898.2i		−	1.38113i	−0.723269	−	0.690566i	$-0.757361\pi$
$811$		−	31898.2i		−	1.38113i	0.723269	−	0.690566i	$-0.242639\pi$
$812$		0			0
$813$			2987.14i			0.128860i
$814$		0			0
$815$	5348.22			0.229865
$816$		0			0
$817$		−	31763.3i		−	1.36017i
$818$		0			0
$819$	−72108.0	−	32284.1i	−3.07650	−	1.37741i
$820$		0			0
$821$			29162.8i			1.23970i	0.784722	+	0.619848i	$0.212806\pi$
$821$			29162.8i			1.23970i	−0.784722	+	0.619848i	$0.787194\pi$
$822$		0			0
$823$	−30040.6			−1.27236			−0.636179	−	0.771541i	$-0.719486\pi$
$823$	−30040.6			−1.27236			−0.636179	+	0.771541i	$0.719486\pi$
$824$		0			0
$825$		−	672.157i		−	0.0283655i
$826$		0			0
$827$		−	6175.43i		−	0.259662i	−0.991536	−	0.129831i	$-0.958556\pi$
$827$		−	6175.43i		−	0.259662i	0.991536	−	0.129831i	$-0.0414435\pi$
$828$		0			0
$829$	46596.2			1.95217			0.976087	−	0.217381i	$-0.0697513\pi$
$829$	46596.2			1.95217			0.976087	+	0.217381i	$0.0697513\pi$
$830$		0			0
$831$	−38804.0			−1.61985
$832$		0			0
$833$	19349.5			0.804828
$834$		0			0
$835$	−4510.35			−0.186931
$836$		0			0
$837$			37511.0i			1.54907i
$838$		0			0
$839$			124.642i			0.00512887i	0.999997	+	0.00256443i	$0.000816286\pi$
$839$			124.642i			0.00512887i	−0.999997	+	0.00256443i	$0.999184\pi$
$840$		0			0
$841$	21635.1			0.887086
$842$		0			0
$843$		−	3636.41i		−	0.148570i
$844$		0			0
$845$	−3736.41	+	3336.28i	−0.152114	+	0.135824i
$846$		0			0
$847$			37982.9i			1.54086i
$848$		0			0
$849$	−38211.1			−1.54464
$850$		0			0
$851$			22869.1i			0.921200i
$852$		0			0
$853$		−	33839.8i		−	1.35833i	−0.733988	−	0.679163i	$-0.762343\pi$
$853$		−	33839.8i		−	1.35833i	0.733988	−	0.679163i	$-0.237657\pi$
$854$		0			0
$855$	−17432.7			−0.697295
$856$		0			0
$857$	−585.021			−0.0233185			−0.0116592	−	0.999932i	$-0.503711\pi$
$857$	−585.021			−0.0233185			−0.0116592	+	0.999932i	$0.503711\pi$
$858$		0			0
$859$	−21226.5			−0.843120			−0.421560	−	0.906800i	$-0.638517\pi$
$859$	−21226.5			−0.843120			−0.421560	+	0.906800i	$0.638517\pi$
$860$		0			0
$861$	23340.9			0.923876
$862$		0			0
$863$			38443.2i			1.51636i	0.652044	+	0.758181i	$0.273912\pi$
$863$			38443.2i			1.51636i	−0.652044	+	0.758181i	$0.726088\pi$
$864$		0			0
$865$			4315.35i			0.169626i
$866$		0			0
$867$	29972.5			1.17407
$868$		0			0
$869$		−	86.5649i		−	0.00337919i
$870$		0			0
$871$	−25948.5	−	11617.6i	−1.00945	−	0.451950i
$872$		0			0
$873$		−	27648.1i		−	1.07188i
$874$		0			0
$875$	15932.2			0.615552
$876$		0			0
$877$			15392.1i			0.592652i	0.955087	+	0.296326i	$0.0957614\pi$
$877$			15392.1i			0.592652i	−0.955087	+	0.296326i	$0.904239\pi$
$878$		0			0
$879$			45321.9i			1.73910i
$880$		0			0
$881$	16605.9			0.635037			0.317518	−	0.948252i	$-0.397150\pi$
$881$	16605.9			0.635037			0.317518	+	0.948252i	$0.397150\pi$
$882$		0			0
$883$	20909.9			0.796913			0.398456	−	0.917187i	$-0.369546\pi$
$883$	20909.9			0.796913			0.398456	+	0.917187i	$0.369546\pi$
$884$		0			0
$885$	17771.6			0.675014
$886$		0			0
$887$	266.852			0.0101015			0.00505074	−	0.999987i	$-0.498392\pi$
$887$	266.852			0.0101015			0.00505074	+	0.999987i	$0.498392\pi$
$888$		0			0
$889$			18293.2i			0.690138i
$890$		0			0
$891$			703.692i			0.0264585i
$892$		0			0
$893$	−8875.46			−0.332594
$894$		0			0
$895$			9125.14i			0.340804i
$896$		0			0
$897$	−13129.6	+	29325.4i	−0.488722	+	1.09158i
$898$		0			0
$899$			27068.6i			1.00422i
$900$		0			0
$901$	25151.0			0.929968
$902$		0			0
$903$		−	64954.2i		−	2.39373i
$904$		0			0
$905$		−	2573.19i		−	0.0945145i
$906$		0			0
$907$	−43847.3			−1.60521			−0.802606	−	0.596510i	$-0.796554\pi$
$907$	−43847.3			−1.60521			−0.802606	+	0.596510i	$0.796554\pi$
$908$		0			0
$909$	−23617.1			−0.861749
$910$		0			0
$911$	−9417.62			−0.342502			−0.171251	−	0.985227i	$-0.554781\pi$
$911$	−9417.62			−0.342502			−0.171251	+	0.985227i	$0.554781\pi$
$912$		0			0
$913$	467.709			0.0169539
$914$		0			0
$915$			12432.0i			0.449169i
$916$		0			0
$917$		−	23102.8i		−	0.831976i
$918$		0			0
$919$	43296.8			1.55411			0.777056	−	0.629431i	$-0.216712\pi$
$919$	43296.8			1.55411			0.777056	+	0.629431i	$0.216712\pi$
$920$		0			0
$921$		−	18775.3i		−	0.671736i
$922$		0			0
$923$	−21716.4	−	9722.84i	−0.774434	−	0.346729i
$924$		0			0
$925$		−	37075.1i		−	1.31786i
$926$		0			0
$927$	−82911.1			−2.93760
$928$		0			0
$929$			31999.6i			1.13011i	0.825053	+	0.565055i	$0.191145\pi$
$929$			31999.6i			1.13011i	−0.825053	+	0.565055i	$0.808855\pi$
$930$		0			0
$931$			61093.0i			2.15064i
$932$		0			0
$933$	−11808.4			−0.414352
$934$		0			0
$935$	−56.5547			−0.00197811
$936$		0			0
$937$	1997.33			0.0696371			0.0348186	−	0.999394i	$-0.488915\pi$
$937$	1997.33			0.0696371			0.0348186	+	0.999394i	$0.488915\pi$
$938$		0			0
$939$	−3485.02			−0.121118
$940$		0			0
$941$			31241.0i			1.08228i	0.840932	+	0.541141i	$0.182007\pi$
$941$			31241.0i			1.08228i	−0.840932	+	0.541141i	$0.817993\pi$
$942$		0			0
$943$		−	6513.97i		−	0.224946i
$944$		0			0
$945$	−19348.5			−0.666039
$946$		0			0
$947$			2759.74i			0.0946986i	0.998878	+	0.0473493i	$0.0150774\pi$
$947$			2759.74i			0.0946986i	−0.998878	+	0.0473493i	$0.984923\pi$
$948$		0			0
$949$	7577.53	+	3392.61i	0.259196	+	0.116047i
$950$		0			0
$951$		−	96191.6i		−	3.27994i
$952$		0			0
$953$	11233.8			0.381846			0.190923	−	0.981605i	$-0.438852\pi$
$953$	11233.8			0.381846			0.190923	+	0.981605i	$0.438852\pi$
$954$		0			0
$955$		−	7145.35i		−	0.242113i
$956$		0			0
$957$			1203.65i			0.0406568i
$958$		0			0
$959$	47564.9			1.60162
$960$		0			0
$961$	13870.8			0.465605
$962$		0			0
$963$	94274.1			3.15466
$964$		0			0
$965$	−4823.75			−0.160914
$966$		0			0
$967$		−	32213.3i		−	1.07126i	−0.844453	−	0.535630i	$-0.820074\pi$
$967$		−	32213.3i		−	1.07126i	0.844453	−	0.535630i	$-0.179926\pi$
$968$		0			0
$969$		−	49259.9i		−	1.63308i
$970$		0			0
$971$	−24606.5			−0.813245			−0.406622	−	0.913596i	$-0.633294\pi$
$971$	−24606.5			−0.813245			−0.406622	+	0.913596i	$0.633294\pi$
$972$		0			0
$973$		−	70599.8i		−	2.32613i
$974$		0			0
$975$	21285.5	−	47542.0i	0.699161	−	1.56160i
$976$		0			0
$977$			32852.8i			1.07580i	0.843010	+	0.537898i	$0.180782\pi$
$977$			32852.8i			1.07580i	−0.843010	+	0.537898i	$0.819218\pi$
$978$		0			0
$979$	848.519			0.0277005
$980$		0			0
$981$		−	14632.7i		−	0.476235i
$982$		0			0
$983$		−	29422.7i		−	0.954666i	−0.878722	−	0.477333i	$-0.841604\pi$
$983$		−	29422.7i		−	0.954666i	0.878722	−	0.477333i	$-0.158396\pi$
$984$		0			0
$985$	32.8475			0.00106255
$986$		0			0
$987$	−18149.8			−0.585324
$988$		0			0
$989$	−18127.4			−0.582828
$990$		0			0
$991$	1107.84			0.0355113			0.0177557	−	0.999842i	$-0.494348\pi$
$991$	1107.84			0.0355113			0.0177557	+	0.999842i	$0.494348\pi$
$992$		0			0
$993$			21075.9i			0.673539i
$994$		0			0
$995$			3219.90i			0.102591i
$996$		0			0
$997$	15369.7			0.488227			0.244114	−	0.969747i	$-0.421503\pi$
$997$	15369.7			0.488227			0.244114	+	0.969747i	$0.421503\pi$
$998$		0			0
$999$			92003.5i			2.91377i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.f.e.129.1		10
4.3	odd	2		104.4.f.a.25.9	✓	10
8.3	odd	2		832.4.f.k.129.2		10
8.5	even	2		832.4.f.l.129.10		10
12.11	even	2		936.4.c.a.649.6		10
13.12	even	2	inner	208.4.f.e.129.2		10
52.31	even	4		1352.4.a.k.1.5		5
52.47	even	4		1352.4.a.l.1.5		5
52.51	odd	2		104.4.f.a.25.10	yes	10
104.51	odd	2		832.4.f.k.129.1		10
104.77	even	2		832.4.f.l.129.9		10
156.155	even	2		936.4.c.a.649.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.f.a.25.9	✓	10	4.3	odd	2
104.4.f.a.25.10	yes	10	52.51	odd	2
208.4.f.e.129.1		10	1.1	even	1	trivial
208.4.f.e.129.2		10	13.12	even	2	inner
832.4.f.k.129.1		10	104.51	odd	2
832.4.f.k.129.2		10	8.3	odd	2
832.4.f.l.129.9		10	104.77	even	2
832.4.f.l.129.10		10	8.5	even	2
936.4.c.a.649.5		10	156.155	even	2
936.4.c.a.649.6		10	12.11	even	2
1352.4.a.k.1.5		5	52.31	even	4
1352.4.a.l.1.5		5	52.47	even	4

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.27625 q^{3} -2.27999i q^{5} +28.5449i q^{7} +59.0489 q^{9} +O(q^{10})\)	\(q-9.27625 q^{3} -2.27999i q^{5} +28.5449i q^{7} +59.0489 q^{9} +0.604833i q^{11} +(-19.1535 + 42.7802i) q^{13} +21.1498i q^{15} -41.0109 q^{17} -129.485i q^{19} -264.790i q^{21} -73.8974 q^{23} +119.802 q^{25} -297.293 q^{27} -214.532 q^{29} -126.175i q^{31} -5.61058i q^{33} +65.0822 q^{35} -309.470i q^{37} +(177.673 - 396.840i) q^{39} +88.1488i q^{41} +245.304 q^{43} -134.631i q^{45} -68.5441i q^{47} -471.814 q^{49} +380.428 q^{51} -613.275 q^{53} +1.37901 q^{55} +1201.14i q^{57} -840.276i q^{59} +587.808 q^{61} +1685.55i q^{63} +(97.5384 + 43.6699i) q^{65} +606.554i q^{67} +685.491 q^{69} +507.627i q^{71} -177.127i q^{73} -1111.31 q^{75} -17.2649 q^{77} -143.122 q^{79} +1163.45 q^{81} -773.287i q^{83} +93.5046i q^{85} +1990.06 q^{87} -1402.90i q^{89} +(-1221.16 - 546.736i) q^{91} +1170.43i q^{93} -295.226 q^{95} -468.225i q^{97} +35.7147i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{3} + 72 q^{9}+O(q^{10}) \) 10 * q - 6 * q^3 + 72 * q^9	\( 10 q - 6 q^{3} + 72 q^{9} - 24 q^{13} + 58 q^{17} - 180 q^{23} + 28 q^{25} - 354 q^{27} - 392 q^{29} - 154 q^{35} + 532 q^{39} + 234 q^{43} - 128 q^{49} + 510 q^{51} - 1244 q^{53} + 576 q^{55} - 56 q^{61} + 566 q^{65} + 1748 q^{69} - 472 q^{75} - 304 q^{77} - 1908 q^{79} + 2282 q^{81} + 64 q^{87} - 582 q^{91} + 2340 q^{95}+O(q^{100}) \) 10 * q - 6 * q^3 + 72 * q^9 - 24 * q^13 + 58 * q^17 - 180 * q^23 + 28 * q^25 - 354 * q^27 - 392 * q^29 - 154 * q^35 + 532 * q^39 + 234 * q^43 - 128 * q^49 + 510 * q^51 - 1244 * q^53 + 576 * q^55 - 56 * q^61 + 566 * q^65 + 1748 * q^69 - 472 * q^75 - 304 * q^77 - 1908 * q^79 + 2282 * q^81 + 64 * q^87 - 582 * q^91 + 2340 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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