LMFDB - Embedded newform 225.4.f.d.107.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(107,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("225.107");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$13.2754297513$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.11007531417600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.6
Root			$-1.56290 + 0.418778i$ of defining polynomial
Character	$\chi$	$=$	225.107
Dual form			225.4.f.d.143.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.07047 - 1.07047i) q^{2} +5.70820i q^{4} +(7.85846 + 7.85846i) q^{7} +(14.6742 + 14.6742i) q^{8} +O(q^{10})$	\(q+(1.07047 - 1.07047i) q^{2} +5.70820i q^{4} +(7.85846 + 7.85846i) q^{7} +(14.6742 + 14.6742i) q^{8} -33.7047i q^{11} +(-25.7196 + 25.7196i) q^{13} +16.8244 q^{14} -14.2492 q^{16} +(-66.3461 + 66.3461i) q^{17} +128.915i q^{19} +(-36.0797 - 36.0797i) q^{22} +(110.948 + 110.948i) q^{23} +55.0640i q^{26} +(-44.8577 + 44.8577i) q^{28} +268.817 q^{29} -2.91486 q^{31} +(-132.647 + 132.647i) q^{32} +142.043i q^{34} +(-115.736 - 115.736i) q^{37} +(137.999 + 137.999i) q^{38} -251.610i q^{41} +(-340.781 + 340.781i) q^{43} +192.393 q^{44} +237.532 q^{46} +(126.292 - 126.292i) q^{47} -219.489i q^{49} +(-146.813 - 146.813i) q^{52} +(254.947 + 254.947i) q^{53} +230.633i q^{56} +(287.759 - 287.759i) q^{58} +131.161 q^{59} +225.170 q^{61} +(-3.12025 + 3.12025i) q^{62} +169.994i q^{64} +(236.271 + 236.271i) q^{67} +(-378.717 - 378.717i) q^{68} -29.8233i q^{71} +(-41.4328 + 41.4328i) q^{73} -247.784 q^{74} -735.872 q^{76} +(264.867 - 264.867i) q^{77} -450.170i q^{79} +(-269.340 - 269.340i) q^{82} +(-729.807 - 729.807i) q^{83} +729.588i q^{86} +(494.588 - 494.588i) q^{88} -1478.30 q^{89} -404.234 q^{91} +(-633.313 + 633.313i) q^{92} -270.383i q^{94} +(1291.69 + 1291.69i) q^{97} +(-234.956 - 234.956i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 416 q^{16} + 1456 q^{31} - 4464 q^{46} + 6608 q^{61} - 9520 q^{76} + 7056 q^{91}+O(q^{100}) $ 16 * q + 416 * q^16 + 1456 * q^31 - 4464 * q^46 + 6608 * q^61 - 9520 * q^76 + 7056 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.07047	−	1.07047i	0.378467	−	0.378467i	−0.492082	−	0.870549i	$-0.663764\pi$
$2$	1.07047	−	1.07047i	0.378467	−	0.378467i	0.870549	+	0.492082i	$0.163764\pi$
$3$		0			0
$3$		0			0
$4$			5.70820i			0.713525i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.85846	+	7.85846i	0.424317	+	0.424317i	0.886687	−	0.462370i	$-0.153001\pi$
$7$	7.85846	+	7.85846i	0.424317	+	0.424317i	−0.462370	+	0.886687i	$0.653001\pi$
$8$	14.6742	+	14.6742i	0.648513	+	0.648513i
$9$		0			0
$10$		0			0
$11$		−	33.7047i		−	0.923850i	−0.886919	−	0.461925i	$-0.847159\pi$
$11$		−	33.7047i		−	0.923850i	0.886919	−	0.461925i	$-0.152841\pi$
$12$		0			0
$13$	−25.7196	+	25.7196i	−0.548719	+	0.548719i	−0.926070	−	0.377351i	$-0.876835\pi$
$13$	−25.7196	+	25.7196i	−0.548719	+	0.548719i	0.377351	+	0.926070i	$0.376835\pi$
$14$	16.8244			0.321180
$15$		0			0
$16$	−14.2492			−0.222644
$17$	−66.3461	+	66.3461i	−0.946547	+	0.946547i	−0.998642	−	0.0520952i	$-0.983410\pi$
$17$	−66.3461	+	66.3461i	−0.946547	+	0.946547i	0.0520952	+	0.998642i	$0.483410\pi$
$18$		0			0
$19$			128.915i			1.55658i	0.627903	+	0.778292i	$0.283914\pi$
$19$			128.915i			1.55658i	−0.627903	+	0.778292i	$0.716086\pi$
$20$		0			0
$21$		0			0
$22$	−36.0797	−	36.0797i	−0.349647	−	0.349647i
$23$	110.948	+	110.948i	1.00584	+	1.00584i	0.999983	+	0.00585236i	$0.00186287\pi$
$23$	110.948	+	110.948i	1.00584	+	1.00584i	0.00585236	+	0.999983i	$0.498137\pi$
$24$		0			0
$25$		0			0
$26$			55.0640i			0.415344i
$27$		0			0
$28$	−44.8577	+	44.8577i	−0.302761	+	0.302761i
$29$	268.817			1.72131			0.860655	−	0.509189i	$-0.170054\pi$
$29$	268.817			1.72131			0.860655	+	0.509189i	$0.170054\pi$
$30$		0			0
$31$	−2.91486			−0.0168879			−0.00844393	−	0.999964i	$-0.502688\pi$
$31$	−2.91486			−0.0168879			−0.00844393	+	0.999964i	$0.502688\pi$
$32$	−132.647	+	132.647i	−0.732776	+	0.732776i
$33$		0			0
$34$			142.043i			0.716474i
$35$		0			0
$36$		0			0
$37$	−115.736	−	115.736i	−0.514242	−	0.514242i	0.401581	−	0.915823i	$-0.368461\pi$
$37$	−115.736	−	115.736i	−0.514242	−	0.514242i	−0.915823	+	0.401581i	$0.868461\pi$
$38$	137.999	+	137.999i	0.589116	+	0.589116i
$39$		0			0
$40$		0			0
$41$		−	251.610i		−	0.958410i	−0.877703	−	0.479205i	$-0.840925\pi$
$41$		−	251.610i		−	0.958410i	0.877703	−	0.479205i	$-0.159075\pi$
$42$		0			0
$43$	−340.781	+	340.781i	−1.20857	+	1.20857i	−0.237081	+	0.971490i	$0.576191\pi$
$43$	−340.781	+	340.781i	−1.20857	+	1.20857i	−0.971490	+	0.237081i	$0.923809\pi$
$44$	192.393			0.659190
$45$		0			0
$46$	237.532			0.761351
$47$	126.292	−	126.292i	0.391949	−	0.391949i	−0.483432	−	0.875382i	$-0.660610\pi$
$47$	126.292	−	126.292i	0.391949	−	0.391949i	0.875382	+	0.483432i	$0.160610\pi$
$48$		0			0
$49$		−	219.489i		−	0.639910i
$50$		0			0
$51$		0			0
$52$	−146.813	−	146.813i	−0.391525	−	0.391525i
$53$	254.947	+	254.947i	0.660747	+	0.660747i	0.955556	−	0.294809i	$-0.0952560\pi$
$53$	254.947	+	254.947i	0.660747	+	0.660747i	−0.294809	+	0.955556i	$0.595256\pi$
$54$		0			0
$55$		0			0
$56$			230.633i			0.550350i
$57$		0			0
$58$	287.759	−	287.759i	0.651459	−	0.651459i
$59$	131.161			0.289418			0.144709	−	0.989474i	$-0.453775\pi$
$59$	131.161			0.289418			0.144709	+	0.989474i	$0.453775\pi$
$60$		0			0
$61$	225.170			0.472625			0.236312	−	0.971677i	$-0.424061\pi$
$61$	225.170			0.472625			0.236312	+	0.971677i	$0.424061\pi$
$62$	−3.12025	+	3.12025i	−0.00639149	+	0.00639149i
$63$		0			0
$64$			169.994i			0.332019i
$65$		0			0
$66$		0			0
$67$	236.271	+	236.271i	0.430823	+	0.430823i	0.888908	−	0.458085i	$-0.151465\pi$
$67$	236.271	+	236.271i	0.430823	+	0.430823i	−0.458085	+	0.888908i	$0.651465\pi$
$68$	−378.717	−	378.717i	−0.675385	−	0.675385i
$69$		0			0
$70$		0			0
$71$		−	29.8233i		−	0.0498503i	−0.999689	−	0.0249252i	$-0.992065\pi$
$71$		−	29.8233i		−	0.0498503i	0.999689	−	0.0249252i	$-0.00793475\pi$
$72$		0			0
$73$	−41.4328	+	41.4328i	−0.0664293	+	0.0664293i	−0.739541	−	0.673112i	$-0.764957\pi$
$73$	−41.4328	+	41.4328i	−0.0664293	+	0.0664293i	0.673112	+	0.739541i	$0.264957\pi$
$74$	−247.784			−0.389247
$75$		0			0
$76$	−735.872			−1.11066
$77$	264.867	−	264.867i	0.392005	−	0.392005i
$78$		0			0
$79$		−	450.170i		−	0.641115i	−0.947229	−	0.320558i	$-0.896130\pi$
$79$		−	450.170i		−	0.641115i	0.947229	−	0.320558i	$-0.103870\pi$
$80$		0			0
$81$		0			0
$82$	−269.340	−	269.340i	−0.362727	−	0.362727i
$83$	−729.807	−	729.807i	−0.965142	−	0.965142i	0.0342709	−	0.999413i	$-0.489089\pi$
$83$	−729.807	−	729.807i	−0.965142	−	0.965142i	−0.999413	+	0.0342709i	$0.989089\pi$
$84$		0			0
$85$		0			0
$86$			729.588i			0.914808i
$87$		0			0
$88$	494.588	−	494.588i	0.599128	−	0.599128i
$89$	−1478.30			−1.76067			−0.880337	−	0.474349i	$-0.842683\pi$
$89$	−1478.30			−1.76067			−0.880337	+	0.474349i	$0.842683\pi$
$90$		0			0
$91$	−404.234			−0.465662
$92$	−633.313	+	633.313i	−0.717689	+	0.717689i
$93$		0			0
$94$		−	270.383i		−	0.296680i
$95$		0			0
$96$		0			0
$97$	1291.69	+	1291.69i	1.35207	+	1.35207i	0.883339	+	0.468734i	$0.155290\pi$
$97$	1291.69	+	1291.69i	1.35207	+	1.35207i	0.468734	+	0.883339i	$0.344710\pi$
$98$	−234.956	−	234.956i	−0.242185	−	0.242185i
$99$		0			0
$100$		0			0
$101$		−	1342.18i		−	1.32229i	−0.750256	−	0.661147i	$-0.770070\pi$
$101$		−	1342.18i		−	1.32229i	0.750256	−	0.661147i	$-0.229930\pi$
$102$		0			0
$103$	888.748	−	888.748i	0.850203	−	0.850203i	−0.139955	−	0.990158i	$-0.544696\pi$
$103$	888.748	−	888.748i	0.850203	−	0.850203i	0.990158	+	0.139955i	$0.0446958\pi$
$104$	−754.829			−0.711702
$105$		0			0
$106$	545.823			0.500142
$107$	1367.01	−	1367.01i	1.23509	−	1.23509i	0.273101	−	0.961985i	$-0.411951\pi$
$107$	1367.01	−	1367.01i	1.23509	−	1.23509i	0.961985	−	0.273101i	$-0.0880494\pi$
$108$		0			0
$109$		−	155.177i		−	0.136360i	−0.997673	−	0.0681799i	$-0.978281\pi$
$109$		−	155.177i		−	0.136360i	0.997673	−	0.0681799i	$-0.0217192\pi$
$110$		0			0
$111$		0			0
$112$	−111.977	−	111.977i	−0.0944717	−	0.0944717i
$113$	−241.613	−	241.613i	−0.201142	−	0.201142i	0.599347	−	0.800489i	$-0.295427\pi$
$113$	−241.613	−	241.613i	−0.201142	−	0.201142i	−0.800489	+	0.599347i	$0.795427\pi$
$114$		0			0
$115$		0			0
$116$			1534.46i			1.22820i
$117$		0			0
$118$	140.403	−	140.403i	0.109535	−	0.109535i
$119$	−1042.76			−0.803272
$120$		0			0
$121$	194.994			0.146502
$122$	241.037	−	241.037i	0.178873	−	0.178873i
$123$		0			0
$124$		−	16.6386i		−	0.0120499i
$125$		0			0
$126$		0			0
$127$	597.467	+	597.467i	0.417454	+	0.417454i	0.884325	−	0.466871i	$-0.154619\pi$
$127$	597.467	+	597.467i	0.417454	+	0.417454i	−0.466871	+	0.884325i	$0.654619\pi$
$128$	−879.201	−	879.201i	−0.607118	−	0.607118i
$129$		0			0
$130$		0			0
$131$		−	1289.40i		−	0.859966i	−0.902837	−	0.429983i	$-0.858520\pi$
$131$		−	1289.40i		−	0.859966i	0.902837	−	0.429983i	$-0.141480\pi$
$132$		0			0
$133$	−1013.07	+	1013.07i	−0.660485	+	0.660485i
$134$	505.841			0.326105
$135$		0			0
$136$	−1947.15			−1.22770
$137$	672.223	−	672.223i	0.419211	−	0.419211i	−0.465721	−	0.884932i	$-0.654205\pi$
$137$	672.223	−	672.223i	0.419211	−	0.419211i	0.884932	+	0.465721i	$0.154205\pi$
$138$		0			0
$139$		−	1028.98i		−	0.627891i	−0.949441	−	0.313945i	$-0.898349\pi$
$139$		−	1028.98i		−	0.627891i	0.949441	−	0.313945i	$-0.101651\pi$
$140$		0			0
$141$		0			0
$142$	−31.9248	−	31.9248i	−0.0188667	−	0.0188667i
$143$	866.873	+	866.873i	0.506934	+	0.506934i
$144$		0			0
$145$		0			0
$146$			88.7048i			0.0502826i
$147$		0			0
$148$	660.647	−	660.647i	0.366925	−	0.366925i
$149$	1475.95			0.811509			0.405754	−	0.913982i	$-0.367009\pi$
$149$	1475.95			0.811509			0.405754	+	0.913982i	$0.367009\pi$
$150$		0			0
$151$	3253.22			1.75327			0.876633	−	0.481160i	$-0.159784\pi$
$151$	3253.22			1.75327			0.876633	+	0.481160i	$0.159784\pi$
$152$	−1891.72	+	1891.72i	−1.00946	+	1.00946i
$153$		0			0
$154$		−	567.062i		−	0.296722i
$155$		0			0
$156$		0			0
$157$	−2284.72	−	2284.72i	−1.16141	−	1.16141i	−0.984168	−	0.177237i	$-0.943284\pi$
$157$	−2284.72	−	2284.72i	−1.16141	−	1.16141i	−0.177237	−	0.984168i	$-0.556716\pi$
$158$	−481.892	−	481.892i	−0.242641	−	0.242641i
$159$		0			0
$160$		0			0
$161$			1743.76i			0.853586i
$162$		0			0
$163$	−65.0158	+	65.0158i	−0.0312419	+	0.0312419i	−0.722555	−	0.691313i	$-0.757032\pi$
$163$	−65.0158	+	65.0158i	−0.0312419	+	0.0312419i	0.691313	+	0.722555i	$0.257032\pi$
$164$	1436.24			0.683850
$165$		0			0
$166$	−1562.47			−0.730549
$167$	−111.465	+	111.465i	−0.0516493	+	0.0516493i	−0.732460	−	0.680810i	$-0.761628\pi$
$167$	−111.465	+	111.465i	−0.0516493	+	0.0516493i	0.680810	+	0.732460i	$0.261628\pi$
$168$		0			0
$169$			874.000i			0.397815i
$170$		0			0
$171$		0			0
$172$	−1945.24	−	1945.24i	−0.862346	−	0.862346i
$173$	2261.53	+	2261.53i	0.993878	+	0.993878i	0.999981	−	0.00610303i	$-0.00194267\pi$
$173$	2261.53	+	2261.53i	0.993878	+	0.993878i	−0.00610303	+	0.999981i	$0.501943\pi$
$174$		0			0
$175$		0			0
$176$			480.266i			0.205690i
$177$		0			0
$178$	−1582.47	+	1582.47i	−0.666357	+	0.666357i
$179$	771.589			0.322186			0.161093	−	0.986939i	$-0.448498\pi$
$179$	771.589			0.322186			0.161093	+	0.986939i	$0.448498\pi$
$180$		0			0
$181$	1010.32			0.414897			0.207449	−	0.978246i	$-0.433484\pi$
$181$	1010.32			0.414897			0.207449	+	0.978246i	$0.433484\pi$
$182$	−432.719	+	432.719i	−0.176238	+	0.176238i
$183$		0			0
$184$			3256.13i			1.30459i
$185$		0			0
$186$		0			0
$187$	2236.18	+	2236.18i	0.874467	+	0.874467i
$188$	720.902	+	720.902i	0.279666	+	0.279666i
$189$		0			0
$190$		0			0
$191$		−	1285.52i		−	0.487000i	−0.969901	−	0.243500i	$-0.921704\pi$
$191$		−	1285.52i		−	0.487000i	0.969901	−	0.243500i	$-0.0782956\pi$
$192$		0			0
$193$	−1987.54	+	1987.54i	−0.741275	+	0.741275i	−0.972823	−	0.231548i	$-0.925621\pi$
$193$	−1987.54	+	1987.54i	−0.741275	+	0.741275i	0.231548	+	0.972823i	$0.425621\pi$
$194$	2765.42			1.02343
$195$		0			0
$196$	1252.89			0.456592
$197$	−91.4449	+	91.4449i	−0.0330720	+	0.0330720i	−0.723449	−	0.690377i	$-0.757444\pi$
$197$	−91.4449	+	91.4449i	−0.0330720	+	0.0330720i	0.690377	+	0.723449i	$0.257444\pi$
$198$		0			0
$199$			1163.72i			0.414543i	0.978283	+	0.207272i	$0.0664584\pi$
$199$			1163.72i			0.414543i	−0.978283	+	0.207272i	$0.933542\pi$
$200$		0			0
$201$		0			0
$202$	−1436.76	−	1436.76i	−0.500445	−	0.500445i
$203$	2112.49	+	2112.49i	0.730381	+	0.730381i
$204$		0			0
$205$		0			0
$206$		−	1902.75i		−	0.643548i
$207$		0			0
$208$	366.485	−	366.485i	0.122169	−	0.122169i
$209$	4345.04			1.43805
$210$		0			0
$211$	−2121.60			−0.692212			−0.346106	−	0.938195i	$-0.612496\pi$
$211$	−2121.60			−0.692212			−0.346106	+	0.938195i	$0.612496\pi$
$212$	−1455.29	+	1455.29i	−0.471460	+	0.471460i
$213$		0			0
$214$		−	2926.69i		−	0.934879i
$215$		0			0
$216$		0			0
$217$	−22.9063	−	22.9063i	−0.00716580	−	0.00716580i
$218$	−166.111	−	166.111i	−0.0516077	−	0.0516077i
$219$		0			0
$220$		0			0
$221$		−	3412.80i		−	1.03878i
$222$		0			0
$223$	−1439.60	+	1439.60i	−0.432298	+	0.432298i	−0.889410	−	0.457111i	$-0.848884\pi$
$223$	−1439.60	+	1439.60i	−0.432298	+	0.432298i	0.457111	+	0.889410i	$0.348884\pi$
$224$	−2084.80			−0.621859
$225$		0			0
$226$	−517.277			−0.152251
$227$	−501.434	+	501.434i	−0.146614	+	0.146614i	−0.776604	−	0.629990i	$-0.783059\pi$
$227$	−501.434	+	501.434i	−0.146614	+	0.146614i	0.629990	+	0.776604i	$0.283059\pi$
$228$		0			0
$229$			6371.08i			1.83848i	0.393692	+	0.919242i	$0.371198\pi$
$229$			6371.08i			1.83848i	−0.393692	+	0.919242i	$0.628802\pi$
$230$		0			0
$231$		0			0
$232$	3944.66	+	3944.66i	1.11629	+	1.11629i
$233$	3586.88	+	3586.88i	1.00852	+	1.00852i	0.999963	+	0.00855336i	$0.00272265\pi$
$233$	3586.88	+	3586.88i	1.00852	+	1.00852i	0.00855336	+	0.999963i	$0.497277\pi$
$234$		0			0
$235$		0			0
$236$			748.692i			0.206507i
$237$		0			0
$238$	−1116.24	+	1116.24i	−0.304012	+	0.304012i
$239$	−5726.18			−1.54977			−0.774886	−	0.632100i	$-0.782193\pi$
$239$	−5726.18			−1.54977			−0.774886	+	0.632100i	$0.782193\pi$
$240$		0			0
$241$	4533.68			1.21178			0.605892	−	0.795547i	$-0.292816\pi$
$241$	4533.68			1.21178			0.605892	+	0.795547i	$0.292816\pi$
$242$	208.734	−	208.734i	0.0554461	−	0.0554461i
$243$		0			0
$244$			1285.32i			0.337230i
$245$		0			0
$246$		0			0
$247$	−3315.64	−	3315.64i	−0.854127	−	0.854127i
$248$	−42.7731	−	42.7731i	−0.0109520	−	0.0109520i
$249$		0			0
$250$		0			0
$251$			2022.81i			0.508679i	0.967115	+	0.254340i	$0.0818581\pi$
$251$			2022.81i			0.508679i	−0.967115	+	0.254340i	$0.918142\pi$
$252$		0			0
$253$	3739.46	−	3739.46i	0.929241	−	0.929241i
$254$	1279.14			0.315985
$255$		0			0
$256$	−3242.26			−0.791567
$257$	−2225.48	+	2225.48i	−0.540161	+	0.540161i	−0.923576	−	0.383415i	$-0.874748\pi$
$257$	−2225.48	+	2225.48i	−0.540161	+	0.540161i	0.383415	+	0.923576i	$0.374748\pi$
$258$		0			0
$259$		−	1819.02i		−	0.436404i
$260$		0			0
$261$		0			0
$262$	−1380.26	−	1380.26i	−0.325469	−	0.325469i
$263$	−5187.30	−	5187.30i	−1.21621	−	1.21621i	−0.968950	−	0.247257i	$-0.920471\pi$
$263$	−5187.30	−	5187.30i	−1.21621	−	1.21621i	−0.247257	−	0.968950i	$-0.579529\pi$
$264$		0			0
$265$		0			0
$266$			2168.92i			0.499944i
$267$		0			0
$268$	−1348.69	+	1348.69i	−0.307403	+	0.307403i
$269$	1505.47			0.341228			0.170614	−	0.985338i	$-0.445425\pi$
$269$	1505.47			0.341228			0.170614	+	0.985338i	$0.445425\pi$
$270$		0			0
$271$	−3810.34			−0.854102			−0.427051	−	0.904227i	$-0.640448\pi$
$271$	−3810.34			−0.854102			−0.427051	+	0.904227i	$0.640448\pi$
$272$	945.381	−	945.381i	0.210743	−	0.210743i
$273$		0			0
$274$		−	1439.18i		−	0.317315i
$275$		0			0
$276$		0			0
$277$	4620.91	+	4620.91i	1.00232	+	1.00232i	0.999997	+	0.00232589i	$0.000740355\pi$
$277$	4620.91	+	4620.91i	1.00232	+	1.00232i	0.00232589	+	0.999997i	$0.499260\pi$
$278$	−1101.49	−	1101.49i	−0.237636	−	0.237636i
$279$		0			0
$280$		0			0
$281$			1168.20i			0.248004i	0.992282	+	0.124002i	$0.0395730\pi$
$281$			1168.20i			0.248004i	−0.992282	+	0.124002i	$0.960427\pi$
$282$		0			0
$283$	4505.94	−	4505.94i	0.946468	−	0.946468i	−0.0521705	−	0.998638i	$-0.516614\pi$
$283$	4505.94	−	4505.94i	0.946468	−	0.946468i	0.998638	+	0.0521705i	$0.0166139\pi$
$284$	170.237			0.0355695
$285$		0			0
$286$	1855.92			0.383715
$287$	1977.26	−	1977.26i	0.406670	−	0.406670i
$288$		0			0
$289$		−	3890.62i		−	0.791902i
$290$		0			0
$291$		0			0
$292$	−236.507	−	236.507i	−0.0473990	−	0.0473990i
$293$	−3623.76	−	3623.76i	−0.722533	−	0.722533i	0.246588	−	0.969120i	$-0.420691\pi$
$293$	−3623.76	−	3623.76i	−0.722533	−	0.722533i	−0.969120	+	0.246588i	$0.920691\pi$
$294$		0			0
$295$		0			0
$296$		−	3396.67i		−	0.666985i
$297$		0			0
$298$	1579.96	−	1579.96i	0.307129	−	0.307129i
$299$	−5707.07			−1.10384
$300$		0			0
$301$	−5356.02			−1.02563
$302$	3482.46	−	3482.46i	0.663553	−	0.663553i
$303$		0			0
$304$		−	1836.94i		−	0.346564i
$305$		0			0
$306$		0			0
$307$	−1210.91	−	1210.91i	−0.225114	−	0.225114i	0.585534	−	0.810648i	$-0.300885\pi$
$307$	−1210.91	−	1210.91i	−0.225114	−	0.225114i	−0.810648	+	0.585534i	$0.800885\pi$
$308$	1511.92	+	1511.92i	0.279706	+	0.279706i
$309$		0			0
$310$		0			0
$311$			6608.74i			1.20498i	0.798128	+	0.602488i	$0.205824\pi$
$311$			6608.74i			1.20498i	−0.798128	+	0.602488i	$0.794176\pi$
$312$		0			0
$313$	−4399.41	+	4399.41i	−0.794471	+	0.794471i	−0.982218	−	0.187746i	$-0.939882\pi$
$313$	−4399.41	+	4399.41i	−0.794471	+	0.794471i	0.187746	+	0.982218i	$0.439882\pi$
$314$	−4891.43			−0.879107
$315$		0			0
$316$	2569.66			0.457452
$317$	484.443	−	484.443i	0.0858329	−	0.0858329i	−0.662887	−	0.748720i	$-0.730669\pi$
$317$	484.443	−	484.443i	0.0858329	−	0.0858329i	0.748720	+	0.662887i	$0.230669\pi$
$318$		0			0
$319$		−	9060.38i		−	1.59023i
$320$		0			0
$321$		0			0
$322$	1866.63	+	1866.63i	0.323054	+	0.323054i
$323$	−8553.00	−	8553.00i	−1.47338	−	1.47338i
$324$		0			0
$325$		0			0
$326$			139.194i			0.0236480i
$327$		0			0
$328$	3692.16	−	3692.16i	0.621541	−	0.621541i
$329$	1984.93			0.332622
$330$		0			0
$331$	9323.92			1.54830			0.774152	−	0.632999i	$-0.218176\pi$
$331$	9323.92			1.54830			0.774152	+	0.632999i	$0.218176\pi$
$332$	4165.89	−	4165.89i	0.688653	−	0.688653i
$333$		0			0
$334$			238.639i			0.0390951i
$335$		0			0
$336$		0			0
$337$	−4993.89	−	4993.89i	−0.807224	−	0.807224i	0.176989	−	0.984213i	$-0.443364\pi$
$337$	−4993.89	−	4993.89i	−0.807224	−	0.807224i	−0.984213	+	0.176989i	$0.943364\pi$
$338$	935.588	+	935.588i	0.150560	+	0.150560i
$339$		0			0
$340$		0			0
$341$			98.2443i			0.0156018i
$342$		0			0
$343$	4420.30	−	4420.30i	0.695842	−	0.695842i
$344$	−10001.3			−1.56755
$345$		0			0
$346$	4841.78			0.752300
$347$	4772.75	−	4772.75i	0.738370	−	0.738370i	−0.233892	−	0.972263i	$-0.575146\pi$
$347$	4772.75	−	4772.75i	0.738370	−	0.738370i	0.972263	+	0.233892i	$0.0751463\pi$
$348$		0			0
$349$		−	2130.21i		−	0.326726i	−0.986566	−	0.163363i	$-0.947766\pi$
$349$		−	2130.21i		−	0.326726i	0.986566	−	0.163363i	$-0.0522342\pi$
$350$		0			0
$351$		0			0
$352$	4470.82	+	4470.82i	0.676975	+	0.676975i
$353$	3735.22	+	3735.22i	0.563189	+	0.563189i	0.930212	−	0.367023i	$-0.119623\pi$
$353$	3735.22	+	3735.22i	0.563189	+	0.563189i	−0.367023	+	0.930212i	$0.619623\pi$
$354$		0			0
$355$		0			0
$356$		−	8438.46i		−	1.25629i
$357$		0			0
$358$	825.960	−	825.960i	0.121937	−	0.121937i
$359$	818.350			0.120309			0.0601544	−	0.998189i	$-0.480841\pi$
$359$	818.350			0.120309			0.0601544	+	0.998189i	$0.480841\pi$
$360$		0			0
$361$	−9760.04			−1.42295
$362$	1081.51	−	1081.51i	0.157025	−	0.157025i
$363$		0			0
$364$		−	2307.45i		−	0.332261i
$365$		0			0
$366$		0			0
$367$	6106.19	+	6106.19i	0.868502	+	0.868502i	0.992307	−	0.123805i	$-0.0395096\pi$
$367$	6106.19	+	6106.19i	0.868502	+	0.868502i	−0.123805	+	0.992307i	$0.539510\pi$
$368$	−1580.92	−	1580.92i	−0.223943	−	0.223943i
$369$		0			0
$370$		0			0
$371$			4006.98i			0.560733i
$372$		0			0
$373$	741.335	−	741.335i	0.102908	−	0.102908i	−0.653778	−	0.756686i	$-0.726817\pi$
$373$	741.335	−	741.335i	0.102908	−	0.102908i	0.756686	+	0.653778i	$0.226817\pi$
$374$	4787.50			0.661914
$375$		0			0
$376$	3706.47			0.508368
$377$	−6913.87	+	6913.87i	−0.944515	+	0.944515i
$378$		0			0
$379$		−	56.6115i		−	0.00767266i	−0.999993	−	0.00383633i	$-0.998779\pi$
$379$		−	56.6115i		−	0.00767266i	0.999993	−	0.00383633i	$-0.00122115\pi$
$380$		0			0
$381$		0			0
$382$	−1376.11	−	1376.11i	−0.184313	−	0.184313i
$383$	−4830.26	−	4830.26i	−0.644425	−	0.644425i	0.307215	−	0.951640i	$-0.400603\pi$
$383$	−4830.26	−	4830.26i	−0.644425	−	0.644425i	−0.951640	+	0.307215i	$0.900603\pi$
$384$		0			0
$385$		0			0
$386$			4255.18i			0.561096i
$387$		0			0
$388$	−7373.22	+	7373.22i	−0.964739	+	0.964739i
$389$	1186.46			0.154643			0.0773213	−	0.997006i	$-0.475363\pi$
$389$	1186.46			0.154643			0.0773213	+	0.997006i	$0.475363\pi$
$390$		0			0
$391$	−14721.9			−1.90414
$392$	3220.82	−	3220.82i	0.414990	−	0.414990i
$393$		0			0
$394$			195.777i			0.0250333i
$395$		0			0
$396$		0			0
$397$	−8913.20	−	8913.20i	−1.12680	−	1.12680i	−0.990694	−	0.136108i	$-0.956540\pi$
$397$	−8913.20	−	8913.20i	−1.12680	−	1.12680i	−0.136108	−	0.990694i	$-0.543460\pi$
$398$	1245.73	+	1245.73i	0.156891	+	0.156891i
$399$		0			0
$400$		0			0
$401$			5637.87i			0.702099i	0.936357	+	0.351050i	$0.114175\pi$
$401$			5637.87i			0.702099i	−0.936357	+	0.351050i	$0.885825\pi$
$402$		0			0
$403$	74.9690	−	74.9690i	0.00926668	−	0.00926668i
$404$	7661.43			0.943491
$405$		0			0
$406$	4522.69			0.552850
$407$	−3900.86	+	3900.86i	−0.475083	+	0.475083i
$408$		0			0
$409$			12265.1i			1.48282i	0.671055	+	0.741408i	$0.265841\pi$
$409$			12265.1i			1.48282i	−0.671055	+	0.741408i	$0.734159\pi$
$410$		0			0
$411$		0			0
$412$	5073.15	+	5073.15i	0.606642	+	0.606642i
$413$	1030.72	+	1030.72i	0.122805	+	0.122805i
$414$		0			0
$415$		0			0
$416$		−	6823.25i		−	0.804176i
$417$		0			0
$418$	4651.21	−	4651.21i	0.544254	−	0.544254i
$419$	−9519.10			−1.10988			−0.554938	−	0.831891i	$-0.687258\pi$
$419$	−9519.10			−1.10988			−0.554938	+	0.831891i	$0.687258\pi$
$420$		0			0
$421$	4787.83			0.554263			0.277131	−	0.960832i	$-0.410616\pi$
$421$	4787.83			0.554263			0.277131	+	0.960832i	$0.410616\pi$
$422$	−2271.10	+	2271.10i	−0.261979	+	0.261979i
$423$		0			0
$424$			7482.26i			0.857006i
$425$		0			0
$426$		0			0
$427$	1769.49	+	1769.49i	0.200543	+	0.200543i
$428$	7803.20	+	7803.20i	0.881266	+	0.881266i
$429$		0			0
$430$		0			0
$431$		−	8308.56i		−	0.928560i	−0.885689	−	0.464280i	$-0.846313\pi$
$431$		−	8308.56i		−	0.928560i	0.885689	−	0.464280i	$-0.153687\pi$
$432$		0			0
$433$	−4885.19	+	4885.19i	−0.542188	+	0.542188i	−0.924170	−	0.381982i	$-0.875242\pi$
$433$	−4885.19	+	4885.19i	−0.542188	+	0.542188i	0.381982	+	0.924170i	$0.375242\pi$
$434$	−49.0408			−0.00542404
$435$		0			0
$436$	885.779			0.0972962
$437$	−14302.8	+	14302.8i	−1.56567	+	1.56567i
$438$		0			0
$439$			4105.93i			0.446391i	0.974774	+	0.223195i	$0.0716488\pi$
$439$			4105.93i			0.446391i	−0.974774	+	0.223195i	$0.928351\pi$
$440$		0			0
$441$		0			0
$442$	−3653.28	−	3653.28i	−0.393143	−	0.393143i
$443$	1722.96	+	1722.96i	0.184786	+	0.184786i	0.793438	−	0.608652i	$-0.208289\pi$
$443$	1722.96	+	1722.96i	0.184786	+	0.184786i	−0.608652	+	0.793438i	$0.708289\pi$
$444$		0			0
$445$		0			0
$446$			3082.08i			0.327221i
$447$		0			0
$448$	−1335.89	+	1335.89i	−0.140881	+	0.140881i
$449$	15259.1			1.60383			0.801917	−	0.597436i	$-0.203814\pi$
$449$	15259.1			1.60383			0.801917	+	0.597436i	$0.203814\pi$
$450$		0			0
$451$	−8480.42			−0.885427
$452$	1379.18	−	1379.18i	0.143520	−	0.143520i
$453$		0			0
$454$			1073.54i			0.110977i
$455$		0			0
$456$		0			0
$457$	−5301.02	−	5301.02i	−0.542607	−	0.542607i	0.381685	−	0.924292i	$-0.375344\pi$
$457$	−5301.02	−	5301.02i	−0.542607	−	0.542607i	−0.924292	+	0.381685i	$0.875344\pi$
$458$	6820.03	+	6820.03i	0.695806	+	0.695806i
$459$		0			0
$460$		0			0
$461$			4117.47i			0.415986i	0.978130	+	0.207993i	$0.0666932\pi$
$461$			4117.47i			0.415986i	−0.978130	+	0.207993i	$0.933307\pi$
$462$		0			0
$463$	3550.72	−	3550.72i	0.356406	−	0.356406i	−0.506081	−	0.862486i	$-0.668906\pi$
$463$	3550.72	−	3550.72i	0.356406	−	0.356406i	0.862486	+	0.506081i	$0.168906\pi$
$464$	−3830.43			−0.383240
$465$		0			0
$466$	7679.27			0.763381
$467$	−2539.40	+	2539.40i	−0.251626	+	0.251626i	−0.821637	−	0.570011i	$-0.806939\pi$
$467$	−2539.40	+	2539.40i	−0.251626	+	0.251626i	0.570011	+	0.821637i	$0.306939\pi$
$468$		0			0
$469$			3713.46i			0.365611i
$470$		0			0
$471$		0			0
$472$	1924.67	+	1924.67i	0.187691	+	0.187691i
$473$	11485.9	+	11485.9i	1.11654	+	1.11654i
$474$		0			0
$475$		0			0
$476$		−	5952.27i		−	0.573155i
$477$		0			0
$478$	−6129.68	+	6129.68i	−0.586538	+	0.586538i
$479$	15920.6			1.51865			0.759324	−	0.650712i	$-0.225530\pi$
$479$	15920.6			1.51865			0.759324	+	0.650712i	$0.225530\pi$
$480$		0			0
$481$	5953.40			0.564349
$482$	4853.15	−	4853.15i	0.458620	−	0.458620i
$483$		0			0
$484$			1113.06i			0.104533i
$485$		0			0
$486$		0			0
$487$	2724.10	+	2724.10i	0.253472	+	0.253472i	0.822393	−	0.568920i	$-0.192639\pi$
$487$	2724.10	+	2724.10i	0.253472	+	0.253472i	−0.568920	+	0.822393i	$0.692639\pi$
$488$	3304.19	+	3304.19i	0.306503	+	0.306503i
$489$		0			0
$490$		0			0
$491$		−	4422.67i		−	0.406502i	−0.979127	−	0.203251i	$-0.934849\pi$
$491$		−	4422.67i		−	0.406502i	0.979127	−	0.203251i	$-0.0651507\pi$
$492$		0			0
$493$	−17834.9	+	17834.9i	−1.62930	+	1.62930i
$494$	−7098.57			−0.646518
$495$		0			0
$496$	41.5344			0.00375998
$497$	234.365	−	234.365i	0.0211523	−	0.0211523i
$498$		0			0
$499$		−	653.980i		−	0.0586697i	−0.999570	−	0.0293348i	$-0.990661\pi$
$499$		−	653.980i		−	0.0586697i	0.999570	−	0.0293348i	$-0.00933891\pi$
$500$		0			0
$501$		0			0
$502$	2165.35	+	2165.35i	0.192518	+	0.192518i
$503$	−7109.17	−	7109.17i	−0.630183	−	0.630183i	0.317931	−	0.948114i	$-0.397012\pi$
$503$	−7109.17	−	7109.17i	−0.630183	−	0.630183i	−0.948114	+	0.317931i	$0.897012\pi$
$504$		0			0
$505$		0			0
$506$		−	8005.93i		−	0.703374i
$507$		0			0
$508$	−3410.46	+	3410.46i	−0.297864	+	0.297864i
$509$	15370.7			1.33849			0.669246	−	0.743041i	$-0.266617\pi$
$509$	15370.7			1.33849			0.669246	+	0.743041i	$0.266617\pi$
$510$		0			0
$511$	−651.196			−0.0563742
$512$	3562.88	−	3562.88i	0.307536	−	0.307536i
$513$		0			0
$514$			4764.59i			0.408866i
$515$		0			0
$516$		0			0
$517$	−4256.64	−	4256.64i	−0.362102	−	0.362102i
$518$	−1947.20	−	1947.20i	−0.165164	−	0.165164i
$519$		0			0
$520$		0			0
$521$		−	8014.41i		−	0.673930i	−0.941517	−	0.336965i	$-0.890600\pi$
$521$		−	8014.41i		−	0.673930i	0.941517	−	0.336965i	$-0.109400\pi$
$522$		0			0
$523$	8470.99	−	8470.99i	0.708242	−	0.708242i	−0.257924	−	0.966165i	$-0.583038\pi$
$523$	8470.99	−	8470.99i	0.708242	−	0.708242i	0.966165	+	0.257924i	$0.0830383\pi$
$524$	7360.17			0.613608
$525$		0			0
$526$	−11105.7			−0.920588
$527$	193.389	−	193.389i	0.0159851	−	0.0159851i
$528$		0			0
$529$			12451.8i			1.02341i
$530$		0			0
$531$		0			0
$532$	−5782.82	−	5782.82i	−0.471273	−	0.471273i
$533$	6471.31	+	6471.31i	0.525898	+	0.525898i
$534$		0			0
$535$		0			0
$536$			6934.18i			0.558789i
$537$		0			0
$538$	1611.56	−	1611.56i	0.129144	−	0.129144i
$539$	−7397.81			−0.591181
$540$		0			0
$541$	16590.7			1.31846			0.659231	−	0.751941i	$-0.270882\pi$
$541$	16590.7			1.31846			0.659231	+	0.751941i	$0.270882\pi$
$542$	−4078.84	+	4078.84i	−0.323249	+	0.323249i
$543$		0			0
$544$		−	17601.2i		−	1.38721i
$545$		0			0
$546$		0			0
$547$	−7105.34	−	7105.34i	−0.555397	−	0.555397i	0.372596	−	0.927994i	$-0.378468\pi$
$547$	−7105.34	−	7105.34i	−0.555397	−	0.555397i	−0.927994	+	0.372596i	$0.878468\pi$
$548$	3837.19	+	3837.19i	0.299118	+	0.299118i
$549$		0			0
$550$		0			0
$551$			34654.5i			2.67936i
$552$		0			0
$553$	3537.65	−	3537.65i	0.272036	−	0.272036i
$554$	9893.05			0.758692
$555$		0			0
$556$	5873.62			0.448016
$557$	1686.40	−	1686.40i	0.128286	−	0.128286i	−0.640049	−	0.768334i	$-0.721086\pi$
$557$	1686.40	−	1686.40i	0.128286	−	0.128286i	0.768334	+	0.640049i	$0.221086\pi$
$558$		0			0
$559$		−	17529.5i		−	1.32633i
$560$		0			0
$561$		0			0
$562$	1250.52	+	1250.52i	0.0938614	+	0.0938614i
$563$	−4011.17	−	4011.17i	−0.300268	−	0.300268i	0.540851	−	0.841119i	$-0.318102\pi$
$563$	−4011.17	−	4011.17i	−0.300268	−	0.300268i	−0.841119	+	0.540851i	$0.818102\pi$
$564$		0			0
$565$		0			0
$566$		−	9646.91i		−	0.716414i
$567$		0			0
$568$	437.632	−	437.632i	0.0323286	−	0.0323286i
$569$	−11917.1			−0.878012			−0.439006	−	0.898484i	$-0.644669\pi$
$569$	−11917.1			−0.878012			−0.439006	+	0.898484i	$0.644669\pi$
$570$		0			0
$571$	5233.00			0.383528			0.191764	−	0.981441i	$-0.438579\pi$
$571$	5233.00			0.383528			0.191764	+	0.981441i	$0.438579\pi$
$572$	−4948.29	+	4948.29i	−0.361710	+	0.361710i
$573$		0			0
$574$		−	4233.19i		−	0.307822i
$575$		0			0
$576$		0			0
$577$	−471.420	−	471.420i	−0.0340130	−	0.0340130i	0.689896	−	0.723909i	$-0.257656\pi$
$577$	−471.420	−	471.420i	−0.0340130	−	0.0340130i	−0.723909	+	0.689896i	$0.757656\pi$
$578$	−4164.77	−	4164.77i	−0.299709	−	0.299709i
$579$		0			0
$580$		0			0
$581$		−	11470.3i		−	0.819052i
$582$		0			0
$583$	8592.90	−	8592.90i	0.610431	−	0.610431i
$584$	−1215.98			−0.0861605
$585$		0			0
$586$	−7758.22			−0.546910
$587$	11088.0	−	11088.0i	0.779645	−	0.779645i	−0.200125	−	0.979770i	$-0.564135\pi$
$587$	11088.0	−	11088.0i	0.779645	−	0.779645i	0.979770	+	0.200125i	$0.0641349\pi$
$588$		0			0
$589$		−	375.768i		−	0.0262874i
$590$		0			0
$591$		0			0
$592$	1649.16	+	1649.16i	0.114493	+	0.114493i
$593$	−4249.89	−	4249.89i	−0.294303	−	0.294303i	0.544474	−	0.838778i	$-0.316729\pi$
$593$	−4249.89	−	4249.89i	−0.294303	−	0.294303i	−0.838778	+	0.544474i	$0.816729\pi$
$594$		0			0
$595$		0			0
$596$			8425.04i			0.579032i
$597$		0			0
$598$	−6109.23	+	6109.23i	−0.417768	+	0.417768i
$599$	−25049.7			−1.70868			−0.854341	−	0.519712i	$-0.826039\pi$
$599$	−25049.7			−1.70868			−0.854341	+	0.519712i	$0.826039\pi$
$600$		0			0
$601$	−23667.8			−1.60637			−0.803186	−	0.595728i	$-0.796864\pi$
$601$	−23667.8			−1.60637			−0.803186	+	0.595728i	$0.796864\pi$
$602$	−5733.44	+	5733.44i	−0.388169	+	0.388169i
$603$		0			0
$604$			18570.0i			1.25100i
$605$		0			0
$606$		0			0
$607$	−1377.34	−	1377.34i	−0.0920997	−	0.0920997i	0.659556	−	0.751656i	$-0.270744\pi$
$607$	−1377.34	−	1377.34i	−0.0920997	−	0.0920997i	−0.751656	+	0.659556i	$0.770744\pi$
$608$	−17100.1	−	17100.1i	−1.14063	−	1.14063i
$609$		0			0
$610$		0			0
$611$			6496.38i			0.430140i
$612$		0			0
$613$	6308.16	−	6308.16i	0.415635	−	0.415635i	−0.468061	−	0.883696i	$-0.655047\pi$
$613$	6308.16	−	6308.16i	0.415635	−	0.415635i	0.883696	+	0.468061i	$0.155047\pi$
$614$	−2592.47			−0.170397
$615$		0			0
$616$	7773.41			0.508441
$617$	−699.955	+	699.955i	−0.0456712	+	0.0456712i	−0.729574	−	0.683902i	$-0.760281\pi$
$617$	−699.955	+	699.955i	−0.0456712	+	0.0456712i	0.683902	+	0.729574i	$0.260281\pi$
$618$		0			0
$619$		−	19461.7i		−	1.26370i	−0.775089	−	0.631852i	$-0.782295\pi$
$619$		−	19461.7i		−	1.26370i	0.775089	−	0.631852i	$-0.217705\pi$
$620$		0			0
$621$		0			0
$622$	7074.43	+	7074.43i	0.456043	+	0.456043i
$623$	−11617.2	−	11617.2i	−0.747084	−	0.747084i
$624$		0			0
$625$		0			0
$626$			9418.84i			0.601362i
$627$		0			0
$628$	13041.7	−	13041.7i	0.828692	−	0.828692i
$629$	15357.3			0.973509
$630$		0			0
$631$	7257.30			0.457858			0.228929	−	0.973443i	$-0.426478\pi$
$631$	7257.30			0.457858			0.228929	+	0.973443i	$0.426478\pi$
$632$	6605.88	−	6605.88i	0.415772	−	0.415772i
$633$		0			0
$634$		−	1037.16i		−	0.0649699i
$635$		0			0
$636$		0			0
$637$	5645.18	+	5645.18i	0.351131	+	0.351131i
$638$	−9698.83	−	9698.83i	−0.601850	−	0.601850i
$639$		0			0
$640$		0			0
$641$		−	21156.1i		−	1.30361i	−0.758385	−	0.651807i	$-0.774011\pi$
$641$		−	21156.1i		−	1.30361i	0.758385	−	0.651807i	$-0.225989\pi$
$642$		0			0
$643$	−8931.56	+	8931.56i	−0.547786	+	0.547786i	−0.925800	−	0.378014i	$-0.876607\pi$
$643$	−8931.56	+	8931.56i	−0.547786	+	0.547786i	0.378014	+	0.925800i	$0.376607\pi$
$644$	−9953.73			−0.609055
$645$		0			0
$646$	−18311.4			−1.11525
$647$	16740.1	−	16740.1i	1.01719	−	1.01719i	0.0173412	−	0.999850i	$-0.494480\pi$
$647$	16740.1	−	16740.1i	1.01719	−	1.01719i	0.999850	−	0.0173412i	$-0.00552015\pi$
$648$		0			0
$649$		−	4420.73i		−	0.267379i
$650$		0			0
$651$		0			0
$652$	−371.123	−	371.123i	−0.0222919	−	0.0222919i
$653$	12157.8	+	12157.8i	0.728592	+	0.728592i	0.970339	−	0.241747i	$-0.0777205\pi$
$653$	12157.8	+	12157.8i	0.728592	+	0.728592i	−0.241747	+	0.970339i	$0.577720\pi$
$654$		0			0
$655$		0			0
$656$			3585.24i			0.213384i
$657$		0			0
$658$	2124.80	−	2124.80i	0.125886	−	0.125886i
$659$	−28300.0			−1.67285			−0.836427	−	0.548079i	$-0.815359\pi$
$659$	−28300.0			−1.67285			−0.836427	+	0.548079i	$0.815359\pi$
$660$		0			0
$661$	4326.04			0.254559			0.127280	−	0.991867i	$-0.459375\pi$
$661$	4326.04			0.254559			0.127280	+	0.991867i	$0.459375\pi$
$662$	9980.94	−	9980.94i	0.585982	−	0.585982i
$663$		0			0
$664$		−	21418.6i		−	1.25181i
$665$		0			0
$666$		0			0
$667$	29824.6	+	29824.6i	1.73135	+	1.73135i
$668$	−636.266	−	636.266i	−0.0368531	−	0.0368531i
$669$		0			0
$670$		0			0
$671$		−	7589.30i		−	0.436634i
$672$		0			0
$673$	11089.2	−	11089.2i	0.635153	−	0.635153i	−0.314203	−	0.949356i	$-0.601737\pi$
$673$	11089.2	−	11089.2i	0.635153	−	0.635153i	0.949356	+	0.314203i	$0.101737\pi$
$674$	−10691.6			−0.611015
$675$		0			0
$676$	−4988.97			−0.283851
$677$	−3104.83	+	3104.83i	−0.176260	+	0.176260i	−0.789723	−	0.613463i	$-0.789776\pi$
$677$	−3104.83	+	3104.83i	−0.176260	+	0.176260i	0.613463	+	0.789723i	$0.289776\pi$
$678$		0			0
$679$			20301.4i			1.14742i
$680$		0			0
$681$		0			0
$682$	105.167	+	105.167i	0.00590478	+	0.00590478i
$683$	−2278.90	−	2278.90i	−0.127672	−	0.127672i	0.640384	−	0.768055i	$-0.278775\pi$
$683$	−2278.90	−	2278.90i	−0.127672	−	0.127672i	−0.768055	+	0.640384i	$0.778775\pi$
$684$		0			0
$685$		0			0
$686$		−	9463.56i		−	0.526706i
$687$		0			0
$688$	4855.86	−	4855.86i	0.269081	−	0.269081i
$689$	−13114.3			−0.725129
$690$		0			0
$691$	6466.55			0.356004			0.178002	−	0.984030i	$-0.443037\pi$
$691$	6466.55			0.356004			0.178002	+	0.984030i	$0.443037\pi$
$692$	−12909.3	+	12909.3i	−0.709158	+	0.709158i
$693$		0			0
$694$		−	10218.1i		−	0.558897i
$695$		0			0
$696$		0			0
$697$	16693.3	+	16693.3i	0.907180	+	0.907180i
$698$	−2280.32	−	2280.32i	−0.123655	−	0.123655i
$699$		0			0
$700$		0			0
$701$			31472.6i			1.69573i	0.530214	+	0.847864i	$0.322112\pi$
$701$			31472.6i			1.69573i	−0.530214	+	0.847864i	$0.677888\pi$
$702$		0			0
$703$	14920.2	−	14920.2i	0.800461	−	0.800461i
$704$	5729.59			0.306736
$705$		0			0
$706$	7996.86			0.426297
$707$	10547.5	−	10547.5i	0.561072	−	0.561072i
$708$		0			0
$709$			21493.4i			1.13851i	0.822162	+	0.569254i	$0.192768\pi$
$709$			21493.4i			1.13851i	−0.822162	+	0.569254i	$0.807232\pi$
$710$		0			0
$711$		0			0
$712$	−21692.9	−	21692.9i	−1.14182	−	1.14182i
$713$	−323.397	−	323.397i	−0.0169864	−	0.0169864i
$714$		0			0
$715$		0			0
$716$			4404.39i			0.229888i
$717$		0			0
$718$	876.016	−	876.016i	0.0455329	−	0.0455329i
$719$	−34918.3			−1.81117			−0.905587	−	0.424160i	$-0.860569\pi$
$719$	−34918.3			−1.81117			−0.905587	+	0.424160i	$0.860569\pi$
$720$		0			0
$721$	13968.4			0.721511
$722$	−10447.8	+	10447.8i	−0.538541	+	0.538541i
$723$		0			0
$724$			5767.11i			0.296040i
$725$		0			0
$726$		0			0
$727$	−14767.9	−	14767.9i	−0.753388	−	0.753388i	0.221722	−	0.975110i	$-0.428832\pi$
$727$	−14767.9	−	14767.9i	−0.753388	−	0.753388i	−0.975110	+	0.221722i	$0.928832\pi$
$728$	−5931.79	−	5931.79i	−0.301988	−	0.301988i
$729$		0			0
$730$		0			0
$731$		−	45218.9i		−	2.28794i
$732$		0			0
$733$	−15017.4	+	15017.4i	−0.756726	+	0.756726i	−0.975725	−	0.218999i	$-0.929721\pi$
$733$	−15017.4	+	15017.4i	−0.756726	+	0.756726i	0.218999	+	0.975725i	$0.429721\pi$
$734$	13072.9			0.657399
$735$		0			0
$736$	−29433.7			−1.47410
$737$	7963.46	−	7963.46i	0.398016	−	0.398016i
$738$		0			0
$739$		−	30967.6i		−	1.54149i	−0.637143	−	0.770746i	$-0.719884\pi$
$739$		−	30967.6i		−	1.54149i	0.637143	−	0.770746i	$-0.280116\pi$
$740$		0			0
$741$		0			0
$742$	4289.33	+	4289.33i	0.212219	+	0.212219i
$743$	−9588.32	−	9588.32i	−0.473434	−	0.473434i	0.429590	−	0.903024i	$-0.358658\pi$
$743$	−9588.32	−	9588.32i	−0.473434	−	0.473434i	−0.903024	+	0.429590i	$0.858658\pi$
$744$		0			0
$745$		0			0
$746$		−	1587.15i		−	0.0778949i
$747$		0			0
$748$	−12764.5	+	12764.5i	−0.623955	+	0.623955i
$749$	21485.3			1.04814
$750$		0			0
$751$	8514.08			0.413693			0.206846	−	0.978373i	$-0.433680\pi$
$751$	8514.08			0.413693			0.206846	+	0.978373i	$0.433680\pi$
$752$	−1799.57	+	1799.57i	−0.0872652	+	0.0872652i
$753$		0			0
$754$			14802.1i			0.714936i
$755$		0			0
$756$		0			0
$757$	7031.43	+	7031.43i	0.337598	+	0.337598i	0.855463	−	0.517865i	$-0.173273\pi$
$757$	7031.43	+	7031.43i	0.337598	+	0.337598i	−0.517865	+	0.855463i	$0.673273\pi$
$758$	−60.6007	−	60.6007i	−0.00290385	−	0.00290385i
$759$		0			0
$760$		0			0
$761$			15458.2i			0.736346i	0.929757	+	0.368173i	$0.120017\pi$
$761$			15458.2i			0.736346i	−0.929757	+	0.368173i	$0.879983\pi$
$762$		0			0
$763$	1219.45	−	1219.45i	0.0578598	−	0.0578598i
$764$	7338.01			0.347487
$765$		0			0
$766$	−10341.3			−0.487787
$767$	−3373.40	+	3373.40i	−0.158809	+	0.158809i
$768$		0			0
$769$		−	10361.1i		−	0.485865i	−0.970043	−	0.242933i	$-0.921891\pi$
$769$		−	10361.1i		−	0.485865i	0.970043	−	0.242933i	$-0.0781094\pi$
$770$		0			0
$771$		0			0
$772$	−11345.3	−	11345.3i	−0.528919	−	0.528919i
$773$	18766.0	+	18766.0i	0.873179	+	0.873179i	0.992818	−	0.119638i	$-0.0381734\pi$
$773$	18766.0	+	18766.0i	0.873179	+	0.873179i	−0.119638	+	0.992818i	$0.538173\pi$
$774$		0			0
$775$		0			0
$776$			37908.9i			1.75367i
$777$		0			0
$778$	1270.07	−	1270.07i	0.0585271	−	0.0585271i
$779$	32436.2			1.49185
$780$		0			0
$781$	−1005.18			−0.0460542
$782$	−15759.3	+	15759.3i	−0.720654	+	0.720654i
$783$		0			0
$784$			3127.55i			0.142472i
$785$		0			0
$786$		0			0
$787$	7235.25	+	7235.25i	0.327711	+	0.327711i	0.851716	−	0.524004i	$-0.175562\pi$
$787$	7235.25	+	7235.25i	0.327711	+	0.327711i	−0.524004	+	0.851716i	$0.675562\pi$
$788$	−521.986	−	521.986i	−0.0235977	−	0.0235977i
$789$		0			0
$790$		0			0
$791$		−	3797.41i		−	0.170696i
$792$		0			0
$793$	−5791.30	+	5791.30i	−0.259338	+	0.259338i
$794$	−19082.6			−0.852915
$795$		0			0
$796$	−6642.77			−0.295787
$797$	−1551.35	+	1551.35i	−0.0689483	+	0.0689483i	−0.740740	−	0.671792i	$-0.765525\pi$
$797$	−1551.35	+	1551.35i	−0.0689483	+	0.0689483i	0.671792	+	0.740740i	$0.265525\pi$
$798$		0			0
$799$			16758.0i			0.741997i
$800$		0			0
$801$		0			0
$802$	6035.15	+	6035.15i	0.265721	+	0.265721i
$803$	1396.48	+	1396.48i	0.0613707	+	0.0613707i
$804$		0			0
$805$		0			0
$806$		−	160.504i		−	0.00701427i
$807$		0			0
$808$	19695.4	−	19695.4i	0.857525	−	0.857525i
$809$	35332.5			1.53551			0.767753	−	0.640746i	$-0.221375\pi$
$809$	35332.5			1.53551			0.767753	+	0.640746i	$0.221375\pi$
$810$		0			0
$811$	−3294.91			−0.142663			−0.0713316	−	0.997453i	$-0.522725\pi$
$811$	−3294.91			−0.142663			−0.0713316	+	0.997453i	$0.522725\pi$
$812$	−12058.5	+	12058.5i	−0.521146	+	0.521146i
$813$		0			0
$814$			8351.48i			0.359606i
$815$		0			0
$816$		0			0
$817$	−43931.7	−	43931.7i	−1.88124	−	1.88124i
$818$	13129.4	+	13129.4i	0.561197	+	0.561197i
$819$		0			0
$820$		0			0
$821$		−	23285.6i		−	0.989856i	−0.868934	−	0.494928i	$-0.835194\pi$
$821$		−	23285.6i		−	0.989856i	0.868934	−	0.494928i	$-0.164806\pi$
$822$		0			0
$823$	−21867.6	+	21867.6i	−0.926193	+	0.926193i	−0.997457	−	0.0712645i	$-0.977297\pi$
$823$	−21867.6	+	21867.6i	−0.926193	+	0.926193i	0.0712645	+	0.997457i	$0.477297\pi$
$824$	26083.3			1.10274
$825$		0			0
$826$	2206.70			0.0929553
$827$	9797.82	−	9797.82i	0.411975	−	0.411975i	−0.470451	−	0.882426i	$-0.655909\pi$
$827$	9797.82	−	9797.82i	0.411975	−	0.411975i	0.882426	+	0.470451i	$0.155909\pi$
$828$		0			0
$829$			553.239i			0.0231783i	0.999933	+	0.0115891i	$0.00368902\pi$
$829$			553.239i			0.0231783i	−0.999933	+	0.0115891i	$0.996311\pi$
$830$		0			0
$831$		0			0
$832$	−4372.18	−	4372.18i	−0.182185	−	0.182185i
$833$	14562.3	+	14562.3i	0.605705	+	0.605705i
$834$		0			0
$835$		0			0
$836$			24802.3i			1.02609i
$837$		0			0
$838$	−10189.9	+	10189.9i	−0.420052	+	0.420052i
$839$	11448.4			0.471086			0.235543	−	0.971864i	$-0.424313\pi$
$839$	11448.4			0.471086			0.235543	+	0.971864i	$0.424313\pi$
$840$		0			0
$841$	47873.4			1.96291
$842$	5125.21	−	5125.21i	0.209770	−	0.209770i
$843$		0			0
$844$		−	12110.5i		−	0.493911i
$845$		0			0
$846$		0			0
$847$	1532.35	+	1532.35i	0.0621632	+	0.0621632i
$848$	−3632.79	−	3632.79i	−0.147112	−	0.147112i
$849$		0			0
$850$		0			0
$851$		−	25681.4i		−	1.03449i
$852$		0			0
$853$	−21294.0	+	21294.0i	−0.854737	+	0.854737i	−0.990712	−	0.135975i	$-0.956583\pi$
$853$	−21294.0	+	21294.0i	−0.854737	+	0.854737i	0.135975	+	0.990712i	$0.456583\pi$
$854$	3788.36			0.151798
$855$		0			0
$856$	40119.6			1.60194
$857$	7601.22	−	7601.22i	0.302979	−	0.302979i	−0.539199	−	0.842178i	$-0.681273\pi$
$857$	7601.22	−	7601.22i	0.302979	−	0.302979i	0.842178	+	0.539199i	$0.181273\pi$
$858$		0			0
$859$			8245.11i			0.327497i	0.986502	+	0.163748i	$0.0523585\pi$
$859$			8245.11i			0.327497i	−0.986502	+	0.163748i	$0.947642\pi$
$860$		0			0
$861$		0			0
$862$	−8894.03	−	8894.03i	−0.351429	−	0.351429i
$863$	28577.1	+	28577.1i	1.12720	+	1.12720i	0.990630	+	0.136571i	$0.0436082\pi$
$863$	28577.1	+	28577.1i	1.12720	+	1.12720i	0.136571	+	0.990630i	$0.456392\pi$
$864$		0			0
$865$		0			0
$866$			10458.9i			0.410400i
$867$		0			0
$868$	130.754	−	130.754i	0.00511298	−	0.00511298i
$869$	−15172.9			−0.592294
$870$		0			0
$871$	−12153.6			−0.472802
$872$	2277.09	−	2277.09i	0.0884310	−	0.0884310i
$873$		0			0
$874$			30621.4i			1.18511i
$875$		0			0
$876$		0			0
$877$	7711.86	+	7711.86i	0.296934	+	0.296934i	0.839812	−	0.542878i	$-0.182665\pi$
$877$	7711.86	+	7711.86i	0.296934	+	0.296934i	−0.542878	+	0.839812i	$0.682665\pi$
$878$	4395.26	+	4395.26i	0.168944	+	0.168944i
$879$		0			0
$880$		0			0
$881$		−	34974.5i		−	1.33748i	−0.743496	−	0.668741i	$-0.766834\pi$
$881$		−	34974.5i		−	1.33748i	0.743496	−	0.668741i	$-0.233166\pi$
$882$		0			0
$883$	−7417.63	+	7417.63i	−0.282699	+	0.282699i	−0.834184	−	0.551486i	$-0.814061\pi$
$883$	−7417.63	+	7417.63i	−0.282699	+	0.282699i	0.551486	+	0.834184i	$0.314061\pi$
$884$	19480.9			0.741193
$885$		0			0
$886$	3688.73			0.139871
$887$	−27827.0	+	27827.0i	−1.05337	+	1.05337i	−0.0548766	+	0.998493i	$0.517477\pi$
$887$	−27827.0	+	27827.0i	−1.05337	+	1.05337i	−0.998493	+	0.0548766i	$0.982523\pi$
$888$		0			0
$889$			9390.34i			0.354265i
$890$		0			0
$891$		0			0
$892$	−8217.51	−	8217.51i	−0.308456	−	0.308456i
$893$	16280.9	+	16280.9i	0.610102	+	0.610102i
$894$		0			0
$895$		0			0
$896$		−	13818.3i		−	0.515221i
$897$		0			0
$898$	16334.3	−	16334.3i	0.606998	−	0.606998i
$899$	−783.561			−0.0290692
$900$		0			0
$901$	−33829.4			−1.25086
$902$	−9078.01	+	9078.01i	−0.335105	+	0.335105i
$903$		0			0
$904$		−	7090.94i		−	0.260886i
$905$		0			0
$906$		0			0
$907$	5203.68	+	5203.68i	0.190502	+	0.190502i	0.795913	−	0.605411i	$-0.206991\pi$
$907$	5203.68	+	5203.68i	0.190502	+	0.190502i	−0.605411	+	0.795913i	$0.706991\pi$
$908$	−2862.29	−	2862.29i	−0.104613	−	0.104613i
$909$		0			0
$910$		0			0
$911$			32968.5i			1.19900i	0.800373	+	0.599502i	$0.204635\pi$
$911$			32968.5i			1.19900i	−0.800373	+	0.599502i	$0.795365\pi$
$912$		0			0
$913$	−24597.9	+	24597.9i	−0.891646	+	0.891646i
$914$	−11349.1			−0.410718
$915$		0			0
$916$	−36367.4			−1.31181
$917$	10132.7	−	10132.7i	0.364898	−	0.364898i
$918$		0			0
$919$		−	27700.9i		−	0.994306i	−0.867663	−	0.497153i	$-0.834379\pi$
$919$		−	27700.9i		−	0.994306i	0.867663	−	0.497153i	$-0.165621\pi$
$920$		0			0
$921$		0			0
$922$	4407.61	+	4407.61i	0.157437	+	0.157437i
$923$	767.044	+	767.044i	0.0273538	+	0.0273538i
$924$		0			0
$925$		0			0
$926$		−	7601.85i		−	0.269776i
$927$		0			0
$928$	−35657.6	+	35657.6i	−1.26134	+	1.26134i
$929$	43665.6			1.54211			0.771057	−	0.636767i	$-0.219728\pi$
$929$	43665.6			1.54211			0.771057	+	0.636767i	$0.219728\pi$
$930$		0			0
$931$	28295.4			0.996074
$932$	−20474.7	+	20474.7i	−0.719602	+	0.719602i
$933$		0			0
$934$			5436.67i			0.190464i
$935$		0			0
$936$		0			0
$937$	26320.8	+	26320.8i	0.917676	+	0.917676i	0.996860	−	0.0791842i	$-0.0252315\pi$
$937$	26320.8	+	26320.8i	0.917676	+	0.917676i	−0.0791842	+	0.996860i	$0.525232\pi$
$938$	3975.13	+	3975.13i	0.138372	+	0.138372i
$939$		0			0
$940$		0			0
$941$		−	24448.5i		−	0.846969i	−0.905903	−	0.423484i	$-0.860807\pi$
$941$		−	24448.5i		−	0.846969i	0.905903	−	0.423484i	$-0.139193\pi$
$942$		0			0
$943$	27915.5	−	27915.5i	0.964003	−	0.964003i
$944$	−1868.94			−0.0644372
$945$		0			0
$946$	24590.5			0.845145
$947$	18184.0	−	18184.0i	0.623971	−	0.623971i	−0.322573	−	0.946544i	$-0.604548\pi$
$947$	18184.0	−	18184.0i	0.623971	−	0.623971i	0.946544	+	0.322573i	$0.104548\pi$
$948$		0			0
$949$		−	2131.27i		−	0.0729020i
$950$		0			0
$951$		0			0
$952$	−15301.6	−	15301.6i	−0.520932	−	0.520932i
$953$	−3808.31	−	3808.31i	−0.129447	−	0.129447i	0.639415	−	0.768862i	$-0.279177\pi$
$953$	−3808.31	−	3808.31i	−0.129447	−	0.129447i	−0.768862	+	0.639415i	$0.779177\pi$
$954$		0			0
$955$		0			0
$956$		−	32686.2i		−	1.10580i
$957$		0			0
$958$	17042.5	−	17042.5i	0.574758	−	0.574758i
$959$	10565.3			0.355757
$960$		0			0
$961$	−29782.5			−0.999715
$962$	6372.92	−	6372.92i	0.213587	−	0.213587i
$963$		0			0
$964$			25879.2i			0.864639i
$965$		0			0
$966$		0			0
$967$	4380.51	+	4380.51i	0.145675	+	0.145675i	0.776183	−	0.630508i	$-0.217153\pi$
$967$	4380.51	+	4380.51i	0.145675	+	0.145675i	−0.630508	+	0.776183i	$0.717153\pi$
$968$	2861.37	+	2861.37i	0.0950082	+	0.0950082i
$969$		0			0
$970$		0			0
$971$			32754.2i			1.08252i	0.840854	+	0.541262i	$0.182053\pi$
$971$			32754.2i			1.08252i	−0.840854	+	0.541262i	$0.817947\pi$
$972$		0			0
$973$	8086.19	−	8086.19i	0.266425	−	0.266425i
$974$	5832.12			0.191862
$975$		0			0
$976$	−3208.50			−0.105227
$977$	−705.759	+	705.759i	−0.0231108	+	0.0231108i	−0.718568	−	0.695457i	$-0.755202\pi$
$977$	−705.759	+	705.759i	−0.0231108	+	0.0231108i	0.695457	+	0.718568i	$0.255202\pi$
$978$		0			0
$979$			49825.8i			1.62660i
$980$		0			0
$981$		0			0
$982$	−4734.32	−	4734.32i	−0.153847	−	0.153847i
$983$	−20743.7	−	20743.7i	−0.673064	−	0.673064i	0.285357	−	0.958421i	$-0.407888\pi$
$983$	−20743.7	−	20743.7i	−0.673064	−	0.673064i	−0.958421	+	0.285357i	$0.907888\pi$
$984$		0			0
$985$		0			0
$986$			38183.4i			1.23327i
$987$		0			0
$988$	18926.4	−	18926.4i	0.609441	−	0.609441i
$989$	−75617.7			−2.43125
$990$		0			0
$991$	−5799.55			−0.185902			−0.0929509	−	0.995671i	$-0.529630\pi$
$991$	−5799.55			−0.185902			−0.0929509	+	0.995671i	$0.529630\pi$
$992$	386.646	−	386.646i	0.0123750	−	0.0123750i
$993$		0			0
$994$		−	501.760i		−	0.0160109i
$995$		0			0
$996$		0			0
$997$	−10653.7	−	10653.7i	−0.338420	−	0.338420i	0.517352	−	0.855773i	$-0.326918\pi$
$997$	−10653.7	−	10653.7i	−0.338420	−	0.338420i	−0.855773	+	0.517352i	$0.826918\pi$
$998$	−700.064	−	700.064i	−0.0222045	−	0.0222045i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.f.d.107.6	yes	16
3.2	odd	2	inner	225.4.f.d.107.4	yes	16
5.2	odd	4	inner	225.4.f.d.143.5	yes	16
5.3	odd	4	inner	225.4.f.d.143.4	yes	16
5.4	even	2	inner	225.4.f.d.107.3	✓	16
15.2	even	4	inner	225.4.f.d.143.3	yes	16
15.8	even	4	inner	225.4.f.d.143.6	yes	16
15.14	odd	2	inner	225.4.f.d.107.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.f.d.107.3	✓	16	5.4	even	2	inner
225.4.f.d.107.4	yes	16	3.2	odd	2	inner
225.4.f.d.107.5	yes	16	15.14	odd	2	inner
225.4.f.d.107.6	yes	16	1.1	even	1	trivial
225.4.f.d.143.3	yes	16	15.2	even	4	inner
225.4.f.d.143.4	yes	16	5.3	odd	4	inner
225.4.f.d.143.5	yes	16	5.2	odd	4	inner
225.4.f.d.143.6	yes	16	15.8	even	4	inner

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.07047 - 1.07047i) q^{2} +5.70820i q^{4} +(7.85846 + 7.85846i) q^{7} +(14.6742 + 14.6742i) q^{8} +O(q^{10})\)	\(q+(1.07047 - 1.07047i) q^{2} +5.70820i q^{4} +(7.85846 + 7.85846i) q^{7} +(14.6742 + 14.6742i) q^{8} -33.7047i q^{11} +(-25.7196 + 25.7196i) q^{13} +16.8244 q^{14} -14.2492 q^{16} +(-66.3461 + 66.3461i) q^{17} +128.915i q^{19} +(-36.0797 - 36.0797i) q^{22} +(110.948 + 110.948i) q^{23} +55.0640i q^{26} +(-44.8577 + 44.8577i) q^{28} +268.817 q^{29} -2.91486 q^{31} +(-132.647 + 132.647i) q^{32} +142.043i q^{34} +(-115.736 - 115.736i) q^{37} +(137.999 + 137.999i) q^{38} -251.610i q^{41} +(-340.781 + 340.781i) q^{43} +192.393 q^{44} +237.532 q^{46} +(126.292 - 126.292i) q^{47} -219.489i q^{49} +(-146.813 - 146.813i) q^{52} +(254.947 + 254.947i) q^{53} +230.633i q^{56} +(287.759 - 287.759i) q^{58} +131.161 q^{59} +225.170 q^{61} +(-3.12025 + 3.12025i) q^{62} +169.994i q^{64} +(236.271 + 236.271i) q^{67} +(-378.717 - 378.717i) q^{68} -29.8233i q^{71} +(-41.4328 + 41.4328i) q^{73} -247.784 q^{74} -735.872 q^{76} +(264.867 - 264.867i) q^{77} -450.170i q^{79} +(-269.340 - 269.340i) q^{82} +(-729.807 - 729.807i) q^{83} +729.588i q^{86} +(494.588 - 494.588i) q^{88} -1478.30 q^{89} -404.234 q^{91} +(-633.313 + 633.313i) q^{92} -270.383i q^{94} +(1291.69 + 1291.69i) q^{97} +(-234.956 - 234.956i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 416 q^{16} + 1456 q^{31} - 4464 q^{46} + 6608 q^{61} - 9520 q^{76} + 7056 q^{91}+O(q^{100}) \) 16 * q + 416 * q^16 + 1456 * q^31 - 4464 * q^46 + 6608 * q^61 - 9520 * q^76 + 7056 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more