LMFDB - Embedded newform 225.4.f.d.143.8

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			143.8
Root			$-0.159959 - 0.596975i$ of defining polynomial
Character	$\chi$	$=$	225.143
Dual form			225.4.f.d.107.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.80252 + 2.80252i) q^{2} +7.70820i q^{4} +(25.0049 - 25.0049i) q^{7} +(0.817763 - 0.817763i) q^{8} +O(q^{10})$	\(q+(2.80252 + 2.80252i) q^{2} +7.70820i q^{4} +(25.0049 - 25.0049i) q^{7} +(0.817763 - 0.817763i) q^{8} -42.1900i q^{11} +(25.7196 + 25.7196i) q^{13} +140.153 q^{14} +66.2492 q^{16} +(-42.0974 - 42.0974i) q^{17} +58.9149i q^{19} +(118.238 - 118.238i) q^{22} +(-141.932 + 141.932i) q^{23} +144.159i q^{26} +(192.743 + 192.743i) q^{28} +79.0799 q^{29} +184.915 q^{31} +(179.122 + 179.122i) q^{32} -235.957i q^{34} +(-81.4436 + 81.4436i) q^{37} +(-165.110 + 165.110i) q^{38} -14.0217i q^{41} +(-152.170 - 152.170i) q^{43} +325.209 q^{44} -795.532 q^{46} +(199.038 + 199.038i) q^{47} -907.489i q^{49} +(-198.252 + 198.252i) q^{52} +(-84.5353 + 84.5353i) q^{53} -40.8962i q^{56} +(221.623 + 221.623i) q^{58} -665.733 q^{59} +600.830 q^{61} +(518.227 + 518.227i) q^{62} +473.994i q^{64} +(-6.22801 + 6.22801i) q^{67} +(324.495 - 324.495i) q^{68} +750.823i q^{71} +(-418.654 - 418.654i) q^{73} -456.494 q^{74} -454.128 q^{76} +(-1054.96 - 1054.96i) q^{77} +825.830i q^{79} +(39.2961 - 39.2961i) q^{82} +(-463.072 + 463.072i) q^{83} -852.917i q^{86} +(-34.5014 - 34.5014i) q^{88} +646.747 q^{89} +1286.23 q^{91} +(-1094.04 - 1094.04i) q^{92} +1115.62i q^{94} +(-371.515 + 371.515i) q^{97} +(2543.25 - 2543.25i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	2.80252	+	2.80252i	0.990839	+	0.990839i	0.999958	−	0.00911901i	$-0.00290271\pi$
$2$	2.80252	+	2.80252i	0.990839	+	0.990839i	−0.00911901	+	0.999958i	$0.502903\pi$
$3$		0			0
$3$		0			0
$4$			7.70820i			0.963525i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	25.0049	−	25.0049i	1.35014	−	1.35014i	0.464635	−	0.885502i	$-0.346186\pi$
$7$	25.0049	−	25.0049i	1.35014	−	1.35014i	0.885502	−	0.464635i	$-0.153814\pi$
$8$	0.817763	−	0.817763i	0.0361404	−	0.0361404i
$9$		0			0
$10$		0			0
$11$		−	42.1900i		−	1.15643i	−0.815884	−	0.578216i	$-0.803749\pi$
$11$		−	42.1900i		−	1.15643i	0.815884	−	0.578216i	$-0.196251\pi$
$12$		0			0
$13$	25.7196	+	25.7196i	0.548719	+	0.548719i	0.926070	−	0.377351i	$-0.123165\pi$
$13$	25.7196	+	25.7196i	0.548719	+	0.548719i	−0.377351	+	0.926070i	$0.623165\pi$
$14$	140.153			2.67554
$15$		0			0
$16$	66.2492			1.03514
$17$	−42.0974	−	42.0974i	−0.600595	−	0.600595i	0.339875	−	0.940471i	$-0.389615\pi$
$17$	−42.0974	−	42.0974i	−0.600595	−	0.600595i	−0.940471	+	0.339875i	$0.889615\pi$
$18$		0			0
$19$			58.9149i			0.711368i	0.934606	+	0.355684i	$0.115752\pi$
$19$			58.9149i			0.711368i	−0.934606	+	0.355684i	$0.884248\pi$
$20$		0			0
$21$		0			0
$22$	118.238	−	118.238i	1.14584	−	1.14584i
$23$	−141.932	+	141.932i	−1.28673	+	1.28673i	−0.349969	+	0.936761i	$0.613808\pi$
$23$	−141.932	+	141.932i	−1.28673	+	1.28673i	−0.936761	+	0.349969i	$0.886192\pi$
$24$		0			0
$25$		0			0
$26$			144.159i			1.08738i
$27$		0			0
$28$	192.743	+	192.743i	1.30089	+	1.30089i
$29$	79.0799			0.506372			0.253186	−	0.967418i	$-0.418522\pi$
$29$	79.0799			0.506372			0.253186	+	0.967418i	$0.418522\pi$
$30$		0			0
$31$	184.915			1.07134			0.535672	−	0.844426i	$-0.320058\pi$
$31$	184.915			1.07134			0.535672	+	0.844426i	$0.320058\pi$
$32$	179.122	+	179.122i	0.989521	+	0.989521i
$33$		0			0
$34$		−	235.957i		−	1.19019i
$35$		0			0
$36$		0			0
$37$	−81.4436	+	81.4436i	−0.361872	+	0.361872i	−0.864502	−	0.502630i	$-0.832366\pi$
$37$	−81.4436	+	81.4436i	−0.361872	+	0.361872i	0.502630	+	0.864502i	$0.332366\pi$
$38$	−165.110	+	165.110i	−0.704852	+	0.704852i
$39$		0			0
$40$		0			0
$41$		−	14.0217i		−	0.0534104i	−0.999643	−	0.0267052i	$-0.991498\pi$
$41$		−	14.0217i		−	0.0534104i	0.999643	−	0.0267052i	$-0.00850154\pi$
$42$		0			0
$43$	−152.170	−	152.170i	−0.539667	−	0.539667i	0.383764	−	0.923431i	$-0.374628\pi$
$43$	−152.170	−	152.170i	−0.539667	−	0.539667i	−0.923431	+	0.383764i	$0.874628\pi$
$44$	325.209			1.11425
$45$		0			0
$46$	−795.532			−2.54989
$47$	199.038	+	199.038i	0.617718	+	0.617718i	0.944945	−	0.327228i	$-0.106114\pi$
$47$	199.038	+	199.038i	0.617718	+	0.617718i	−0.327228	+	0.944945i	$0.606114\pi$
$48$		0			0
$49$		−	907.489i		−	2.64574i
$50$		0			0
$51$		0			0
$52$	−198.252	+	198.252i	−0.528705	+	0.528705i
$53$	−84.5353	+	84.5353i	−0.219091	+	0.219091i	−0.808115	−	0.589024i	$-0.799512\pi$
$53$	−84.5353	+	84.5353i	−0.219091	+	0.219091i	0.589024	+	0.808115i	$0.299512\pi$
$54$		0			0
$55$		0			0
$56$		−	40.8962i		−	0.0975889i
$57$		0			0
$58$	221.623	+	221.623i	0.501733	+	0.501733i
$59$	−665.733			−1.46900			−0.734501	−	0.678608i	$-0.762584\pi$
$59$	−665.733			−1.46900			−0.734501	+	0.678608i	$0.762584\pi$
$60$		0			0
$61$	600.830			1.26112			0.630560	−	0.776140i	$-0.282825\pi$
$61$	600.830			1.26112			0.630560	+	0.776140i	$0.282825\pi$
$62$	518.227	+	518.227i	1.06153	+	1.06153i
$63$		0			0
$64$			473.994i			0.925769i
$65$		0			0
$66$		0			0
$67$	−6.22801	+	6.22801i	−0.0113563	+	0.0113563i	−0.712762	−	0.701406i	$-0.752556\pi$
$67$	−6.22801	+	6.22801i	−0.0113563	+	0.0113563i	0.701406	+	0.712762i	$0.252556\pi$
$68$	324.495	−	324.495i	0.578689	−	0.578689i
$69$		0			0
$70$		0			0
$71$			750.823i			1.25502i	0.778609	+	0.627509i	$0.215925\pi$
$71$			750.823i			1.25502i	−0.778609	+	0.627509i	$0.784075\pi$
$72$		0			0
$73$	−418.654	−	418.654i	−0.671230	−	0.671230i	0.286770	−	0.958000i	$-0.407419\pi$
$73$	−418.654	−	418.654i	−0.671230	−	0.671230i	−0.958000	+	0.286770i	$0.907419\pi$
$74$	−456.494			−0.717113
$75$		0			0
$76$	−454.128			−0.685421
$77$	−1054.96	−	1054.96i	−1.56134	−	1.56134i
$78$		0			0
$79$			825.830i			1.17612i	0.808819	+	0.588058i	$0.200107\pi$
$79$			825.830i			1.17612i	−0.808819	+	0.588058i	$0.799893\pi$
$80$		0			0
$81$		0			0
$82$	39.2961	−	39.2961i	0.0529211	−	0.0529211i
$83$	−463.072	+	463.072i	−0.612394	+	0.612394i	−0.943569	−	0.331175i	$-0.892555\pi$
$83$	−463.072	+	463.072i	−0.612394	+	0.612394i	0.331175	+	0.943569i	$0.392555\pi$
$84$		0			0
$85$		0			0
$86$		−	852.917i		−	1.06945i
$87$		0			0
$88$	−34.5014	−	34.5014i	−0.0417939	−	0.0417939i
$89$	646.747			0.770281			0.385140	−	0.922858i	$-0.374153\pi$
$89$	646.747			0.770281			0.385140	+	0.922858i	$0.374153\pi$
$90$		0			0
$91$	1286.23			1.48169
$92$	−1094.04	−	1094.04i	−1.23980	−	1.23980i
$93$		0			0
$94$			1115.62i			1.22412i
$95$		0			0
$96$		0			0
$97$	−371.515	+	371.515i	−0.388883	+	0.388883i	−0.874289	−	0.485406i	$-0.838672\pi$
$97$	−371.515	+	371.515i	−0.388883	+	0.388883i	0.485406	+	0.874289i	$0.338672\pi$
$98$	2543.25	−	2543.25i	2.62150	−	2.62150i
$99$		0			0
$100$		0			0
$101$		−	1579.77i		−	1.55636i	−0.628040	−	0.778181i	$-0.716142\pi$
$101$		−	1579.77i		−	1.55636i	0.628040	−	0.778181i	$-0.283858\pi$
$102$		0			0
$103$	31.4262	+	31.4262i	0.0300633	+	0.0300633i	0.721979	−	0.691915i	$-0.243233\pi$
$103$	31.4262	+	31.4262i	0.0300633	+	0.0300633i	−0.691915	+	0.721979i	$0.743233\pi$
$104$	42.0652			0.0396618
$105$		0			0
$106$	−473.823			−0.434168
$107$	−669.877	−	669.877i	−0.605229	−	0.605229i	0.336467	−	0.941695i	$-0.390768\pi$
$107$	−669.877	−	669.877i	−0.605229	−	0.605229i	−0.941695	+	0.336467i	$0.890768\pi$
$108$		0			0
$109$			1174.82i			1.03236i	0.856479	+	0.516182i	$0.172647\pi$
$109$			1174.82i			1.03236i	−0.856479	+	0.516182i	$0.827353\pi$
$110$		0			0
$111$		0			0
$112$	1656.55	−	1656.55i	1.39759	−	1.39759i
$113$	−594.951	+	594.951i	−0.495295	+	0.495295i	−0.909970	−	0.414675i	$-0.863895\pi$
$113$	−594.951	+	594.951i	−0.495295	+	0.495295i	0.414675	+	0.909970i	$0.363895\pi$
$114$		0			0
$115$		0			0
$116$			609.564i			0.487902i
$117$		0			0
$118$	−1865.73	−	1865.73i	−1.45554	−	1.45554i
$119$	−2105.28			−1.62177
$120$		0			0
$121$	−448.994			−0.337336
$122$	1683.84	+	1683.84i	1.24957	+	1.24957i
$123$		0			0
$124$			1425.36i			1.03227i
$125$		0			0
$126$		0			0
$127$	−137.380	+	137.380i	−0.0959882	+	0.0959882i	−0.753470	−	0.657482i	$-0.771622\pi$
$127$	−137.380	+	137.380i	−0.0959882	+	0.0959882i	0.657482	+	0.753470i	$0.271622\pi$
$128$	104.604	−	104.604i	0.0722327	−	0.0722327i
$129$		0			0
$130$		0			0
$131$			492.508i			0.328478i	0.986421	+	0.164239i	$0.0525168\pi$
$131$			492.508i			0.328478i	−0.986421	+	0.164239i	$0.947483\pi$
$132$		0			0
$133$	1473.16	+	1473.16i	0.960445	+	0.960445i
$134$	−34.9082			−0.0225045
$135$		0			0
$136$	−68.8514			−0.0434115
$137$	−176.482	−	176.482i	−0.110057	−	0.110057i	0.649934	−	0.759991i	$-0.274797\pi$
$137$	−176.482	−	176.482i	−0.110057	−	0.110057i	−0.759991	+	0.649934i	$0.774797\pi$
$138$		0			0
$139$		−	1224.98i		−	0.747491i	−0.927531	−	0.373746i	$-0.878073\pi$
$139$		−	1224.98i		−	0.747491i	0.927531	−	0.373746i	$-0.121927\pi$
$140$		0			0
$141$		0			0
$142$	−2104.19	+	2104.19i	−1.24352	+	1.24352i
$143$	1085.11	−	1085.11i	0.634556	−	0.634556i
$144$		0			0
$145$		0			0
$146$		−	2346.57i		−	1.33016i
$147$		0			0
$148$	−627.784	−	627.784i	−0.348673	−	0.348673i
$149$	−41.9404			−0.0230597			−0.0115298	−	0.999934i	$-0.503670\pi$
$149$	−41.9404			−0.0230597			−0.0115298	+	0.999934i	$0.503670\pi$
$150$		0			0
$151$	−47.2182			−0.0254474			−0.0127237	−	0.999919i	$-0.504050\pi$
$151$	−47.2182			−0.0254474			−0.0127237	+	0.999919i	$0.504050\pi$
$152$	48.1784	+	48.1784i	0.0257091	+	0.0257091i
$153$		0			0
$154$		−	5913.06i		−	3.09408i
$155$		0			0
$156$		0			0
$157$	−1856.06	+	1856.06i	−0.943502	+	0.943502i	−0.998487	−	0.0549853i	$-0.982489\pi$
$157$	−1856.06	+	1856.06i	−0.943502	+	0.943502i	0.0549853	+	0.998487i	$0.482489\pi$
$158$	−2314.40	+	2314.40i	−1.16534	+	1.16534i
$159$		0			0
$160$		0			0
$161$			7097.97i			3.47452i
$162$		0			0
$163$	295.059	+	295.059i	0.141784	+	0.141784i	0.774436	−	0.632652i	$-0.218034\pi$
$163$	295.059	+	295.059i	0.141784	+	0.141784i	−0.632652	+	0.774436i	$0.718034\pi$
$164$	108.082			0.0514623
$165$		0			0
$166$	−2595.53			−1.21357
$167$	−1081.41	−	1081.41i	−0.501092	−	0.501092i	0.410685	−	0.911777i	$-0.365289\pi$
$167$	−1081.41	−	1081.41i	−0.501092	−	0.501092i	−0.911777	+	0.410685i	$0.865289\pi$
$168$		0			0
$169$		−	874.000i		−	0.397815i
$170$		0			0
$171$		0			0
$172$	1172.96	−	1172.96i	0.519983	−	0.519983i
$173$	−2369.97	+	2369.97i	−1.04154	+	1.04154i	−0.0424371	+	0.999099i	$0.513512\pi$
$173$	−2369.97	+	2369.97i	−1.04154	+	1.04154i	−0.999099	+	0.0424371i	$0.986488\pi$
$174$		0			0
$175$		0			0
$176$		−	2795.05i		−	1.19707i
$177$		0			0
$178$	1812.52	+	1812.52i	0.763225	+	0.763225i
$179$	2631.01			1.09861			0.549304	−	0.835623i	$-0.314893\pi$
$179$	2631.01			1.09861			0.549304	+	0.835623i	$0.314893\pi$
$180$		0			0
$181$	−492.319			−0.202176			−0.101088	−	0.994878i	$-0.532232\pi$
$181$	−492.319			−0.202176			−0.101088	+	0.994878i	$0.532232\pi$
$182$	3604.69	+	3604.69i	1.46812	+	1.46812i
$183$		0			0
$184$			232.133i			0.0930058i
$185$		0			0
$186$		0			0
$187$	−1776.09	+	1776.09i	−0.694548	+	0.694548i
$188$	−1534.23	+	1534.23i	−0.595187	+	0.595187i
$189$		0			0
$190$		0			0
$191$			1285.52i			0.487000i	0.969901	+	0.243500i	$0.0782956\pi$
$191$			1285.52i			0.487000i	−0.969901	+	0.243500i	$0.921704\pi$
$192$		0			0
$193$	−49.9908	−	49.9908i	−0.0186446	−	0.0186446i	0.697723	−	0.716368i	$-0.254197\pi$
$193$	−49.9908	−	49.9908i	−0.0186446	−	0.0186446i	−0.716368	+	0.697723i	$0.754197\pi$
$194$	−2082.35			−0.770640
$195$		0			0
$196$	6995.11			2.54924
$197$	3313.77	+	3313.77i	1.19846	+	1.19846i	0.974628	+	0.223829i	$0.0718557\pi$
$197$	3313.77	+	3313.77i	1.19846	+	1.19846i	0.223829	+	0.974628i	$0.428144\pi$
$198$		0			0
$199$			1653.72i			0.589092i	0.955637	+	0.294546i	$0.0951684\pi$
$199$			1653.72i			0.589092i	−0.955637	+	0.294546i	$0.904832\pi$
$200$		0			0
$201$		0			0
$202$	4427.32	−	4427.32i	1.54211	−	1.54211i
$203$	1977.39	−	1977.39i	0.683671	−	0.683671i
$204$		0			0
$205$		0			0
$206$			176.145i			0.0595758i
$207$		0			0
$208$	1703.91	+	1703.91i	0.568003	+	0.568003i
$209$	2485.62			0.822649
$210$		0			0
$211$	−3436.40			−1.12119			−0.560597	−	0.828089i	$-0.689428\pi$
$211$	−3436.40			−1.12119			−0.560597	+	0.828089i	$0.689428\pi$
$212$	−651.616	−	651.616i	−0.211100	−	0.211100i
$213$		0			0
$214$		−	3754.69i		−	1.19937i
$215$		0			0
$216$		0			0
$217$	4623.78	−	4623.78i	1.44646	−	1.44646i
$218$	−3292.46	+	3292.46i	−1.02291	+	1.02291i
$219$		0			0
$220$		0			0
$221$		−	2165.46i		−	0.659116i
$222$		0			0
$223$	2589.81	+	2589.81i	0.777699	+	0.777699i	0.979439	−	0.201740i	$-0.0646598\pi$
$223$	2589.81	+	2589.81i	0.777699	+	0.777699i	−0.201740	+	0.979439i	$0.564660\pi$
$224$	8957.88			2.67198
$225$		0			0
$226$	−3334.72			−0.981515
$227$	−3944.75	−	3944.75i	−1.15340	−	1.15340i	−0.985866	−	0.167536i	$-0.946419\pi$
$227$	−3944.75	−	3944.75i	−1.15340	−	1.15340i	−0.167536	−	0.985866i	$-0.553581\pi$
$228$		0			0
$229$			2269.08i			0.654783i	0.944889	+	0.327391i	$0.106170\pi$
$229$			2269.08i			0.654783i	−0.944889	+	0.327391i	$0.893830\pi$
$230$		0			0
$231$		0			0
$232$	64.6687	−	64.6687i	0.0183005	−	0.0183005i
$233$	1757.84	−	1757.84i	0.494247	−	0.494247i	−0.415394	−	0.909642i	$-0.636356\pi$
$233$	1757.84	−	1757.84i	0.494247	−	0.494247i	0.909642	+	0.415394i	$0.136356\pi$
$234$		0			0
$235$		0			0
$236$		−	5131.61i		−	1.41542i
$237$		0			0
$238$	−5900.09	−	5900.09i	−1.60692	−	1.60692i
$239$	3570.92			0.966458			0.483229	−	0.875494i	$-0.339464\pi$
$239$	3570.92			0.966458			0.483229	+	0.875494i	$0.339464\pi$
$240$		0			0
$241$	6036.32			1.61342			0.806708	−	0.590950i	$-0.201247\pi$
$241$	6036.32			1.61342			0.806708	+	0.590950i	$0.201247\pi$
$242$	−1258.31	−	1258.31i	−0.334245	−	0.334245i
$243$		0			0
$244$			4631.32i			1.21512i
$245$		0			0
$246$		0			0
$247$	−1515.27	+	1515.27i	−0.390341	+	0.390341i
$248$	151.217	−	151.217i	0.0387188	−	0.0387188i
$249$		0			0
$250$		0			0
$251$		−	3085.33i		−	0.775874i	−0.921686	−	0.387937i	$-0.873188\pi$
$251$		−	3085.33i		−	0.775874i	0.921686	−	0.387937i	$-0.126812\pi$
$252$		0			0
$253$	5988.09	+	5988.09i	1.48802	+	1.48802i
$254$	−770.020			−0.190218
$255$		0			0
$256$	4378.26			1.06891
$257$	490.380	+	490.380i	0.119023	+	0.119023i	0.764110	−	0.645086i	$-0.223179\pi$
$257$	490.380	+	490.380i	0.119023	+	0.119023i	−0.645086	+	0.764110i	$0.723179\pi$
$258$		0			0
$259$			4072.98i			0.977153i
$260$		0			0
$261$		0			0
$262$	−1380.26	+	1380.26i	−0.325469	+	0.325469i
$263$	−157.421	+	157.421i	−0.0369087	+	0.0369087i	−0.725320	−	0.688412i	$-0.758308\pi$
$263$	−157.421	+	157.421i	−0.0369087	+	0.0369087i	0.688412	+	0.725320i	$0.258308\pi$
$264$		0			0
$265$		0			0
$266$			8257.11i			1.90329i
$267$		0			0
$268$	−48.0067	−	48.0067i	−0.0109421	−	0.0109421i
$269$	−6197.83			−1.40479			−0.702395	−	0.711787i	$-0.747886\pi$
$269$	−6197.83			−1.40479			−0.702395	+	0.711787i	$0.747886\pi$
$270$		0			0
$271$	−4561.66			−1.02251			−0.511257	−	0.859428i	$-0.670820\pi$
$271$	−4561.66			−1.02251			−0.511257	+	0.859428i	$0.670820\pi$
$272$	−2788.92	−	2788.92i	−0.621703	−	0.621703i
$273$		0			0
$274$		−	989.185i		−	0.218098i
$275$		0			0
$276$		0			0
$277$	900.134	−	900.134i	0.195248	−	0.195248i	−0.602711	−	0.797960i	$-0.705913\pi$
$277$	900.134	−	900.134i	0.195248	−	0.195248i	0.797960	+	0.602711i	$0.205913\pi$
$278$	3433.02	−	3433.02i	0.740644	−	0.740644i
$279$		0			0
$280$		0			0
$281$			4485.95i			0.952346i	0.879352	+	0.476173i	$0.157976\pi$
$281$			4485.95i			0.952346i	−0.879352	+	0.476173i	$0.842024\pi$
$282$		0			0
$283$	−4735.98	−	4735.98i	−0.994788	−	0.994788i	0.00519838	−	0.999986i	$-0.498345\pi$
$283$	−4735.98	−	4735.98i	−0.994788	−	0.994788i	−0.999986	+	0.00519838i	$0.998345\pi$
$284$	−5787.49			−1.20924
$285$		0			0
$286$	6082.08			1.25749
$287$	−350.612	−	350.612i	−0.0721113	−	0.0721113i
$288$		0			0
$289$		−	1368.62i		−	0.278570i
$290$		0			0
$291$		0			0
$292$	3227.07	−	3227.07i	0.646747	−	0.646747i
$293$	−4617.95	+	4617.95i	−0.920763	+	0.920763i	−0.997083	−	0.0763199i	$-0.975683\pi$
$293$	−4617.95	+	4617.95i	−0.920763	+	0.920763i	0.0763199	+	0.997083i	$0.475683\pi$
$294$		0			0
$295$		0			0
$296$			133.203i			0.0261564i
$297$		0			0
$298$	−117.539	−	117.539i	−0.0228484	−	0.0228484i
$299$	−7300.86			−1.41211
$300$		0			0
$301$	−7609.98			−1.45725
$302$	−132.330	−	132.330i	−0.0252143	−	0.0252143i
$303$		0			0
$304$			3903.06i			0.736369i
$305$		0			0
$306$		0			0
$307$	−5000.27	+	5000.27i	−0.929578	+	0.929578i	−0.997678	−	0.0681008i	$-0.978306\pi$
$307$	−5000.27	+	5000.27i	−0.929578	+	0.929578i	0.0681008	+	0.997678i	$0.478306\pi$
$308$	8131.81	−	8131.81i	1.50439	−	1.50439i
$309$		0			0
$310$		0			0
$311$		−	6874.37i		−	1.25341i	−0.779257	−	0.626704i	$-0.784404\pi$
$311$		−	6874.37i		−	1.25341i	0.779257	−	0.626704i	$-0.215596\pi$
$312$		0			0
$313$	−3422.07	−	3422.07i	−0.617976	−	0.617976i	0.327036	−	0.945012i	$-0.393950\pi$
$313$	−3422.07	−	3422.07i	−0.617976	−	0.617976i	−0.945012	+	0.327036i	$0.893950\pi$
$314$	−10403.3			−1.86972
$315$		0			0
$316$	−6365.66			−1.13322
$317$	4689.86	+	4689.86i	0.830943	+	0.830943i	0.987646	−	0.156703i	$-0.0500865\pi$
$317$	4689.86	+	4689.86i	0.830943	+	0.830943i	−0.156703	+	0.987646i	$0.550086\pi$
$318$		0			0
$319$		−	3336.38i		−	0.585584i
$320$		0			0
$321$		0			0
$322$	−19892.2	+	19892.2i	−3.44270	+	3.44270i
$323$	2480.16	−	2480.16i	0.427244	−	0.427244i
$324$		0			0
$325$		0			0
$326$			1653.82i			0.280971i
$327$		0			0
$328$	−11.4665	−	11.4665i	−0.00193027	−	0.00193027i
$329$	9953.87			1.66801
$330$		0			0
$331$	952.081			0.158100			0.0790500	−	0.996871i	$-0.474811\pi$
$331$	952.081			0.158100			0.0790500	+	0.996871i	$0.474811\pi$
$332$	−3569.45	−	3569.45i	−0.590057	−	0.590057i
$333$		0			0
$334$		−	6061.36i		−	0.993003i
$335$		0			0
$336$		0			0
$337$	4007.99	−	4007.99i	0.647860	−	0.647860i	−0.304615	−	0.952476i	$-0.598528\pi$
$337$	4007.99	−	4007.99i	0.647860	−	0.647860i	0.952476	+	0.304615i	$0.0985278\pi$
$338$	2449.40	−	2449.40i	0.394171	−	0.394171i
$339$		0			0
$340$		0			0
$341$		−	7801.55i		−	1.23894i
$342$		0			0
$343$	−14115.0	−	14115.0i	−2.22198	−	2.22198i
$344$	−248.878			−0.0390075
$345$		0			0
$346$	−13283.8			−2.06399
$347$	1873.29	+	1873.29i	0.289809	+	0.289809i	0.837005	−	0.547196i	$-0.184305\pi$
$347$	1873.29	+	1873.29i	0.289809	+	0.289809i	−0.547196	+	0.837005i	$0.684305\pi$
$348$		0			0
$349$		−	10642.2i		−	1.63228i	−0.577857	−	0.816138i	$-0.696111\pi$
$349$		−	10642.2i		−	1.63228i	0.577857	−	0.816138i	$-0.303889\pi$
$350$		0			0
$351$		0			0
$352$	7557.17	−	7557.17i	1.14431	−	1.14431i
$353$	5699.37	−	5699.37i	0.859339	−	0.859339i	−0.131921	−	0.991260i	$-0.542115\pi$
$353$	5699.37	−	5699.37i	0.859339	−	0.859339i	0.991260	+	0.131921i	$0.0421146\pi$
$354$		0			0
$355$		0			0
$356$			4985.25i			0.742185i
$357$		0			0
$358$	7373.45	+	7373.45i	1.08854	+	1.08854i
$359$	−12273.5			−1.80437			−0.902186	−	0.431347i	$-0.858038\pi$
$359$	−12273.5			−1.80437			−0.902186	+	0.431347i	$0.858038\pi$
$360$		0			0
$361$	3388.04			0.493955
$362$	−1379.73	−	1379.73i	−0.200323	−	0.200323i
$363$		0			0
$364$			9914.55i			1.42765i
$365$		0			0
$366$		0			0
$367$	2865.51	−	2865.51i	0.407571	−	0.407571i	−0.473320	−	0.880891i	$-0.656945\pi$
$367$	2865.51	−	2865.51i	0.407571	−	0.407571i	0.880891	+	0.473320i	$0.156945\pi$
$368$	−9402.86	+	9402.86i	−1.33195	+	1.33195i
$369$		0			0
$370$		0			0
$371$			4227.59i			0.591606i
$372$		0			0
$373$	3859.53	+	3859.53i	0.535762	+	0.535762i	0.922281	−	0.386519i	$-0.126323\pi$
$373$	3859.53	+	3859.53i	0.535762	+	0.535762i	−0.386519	+	0.922281i	$0.626323\pi$
$374$	−9955.04			−1.37637
$375$		0			0
$376$	325.533			0.0446491
$377$	2033.91	+	2033.91i	0.277856	+	0.277856i
$378$		0			0
$379$			2981.39i			0.404073i	0.979378	+	0.202036i	$0.0647560\pi$
$379$			2981.39i			0.404073i	−0.979378	+	0.202036i	$0.935244\pi$
$380$		0			0
$381$		0			0
$382$	−3602.69	+	3602.69i	−0.482539	+	0.482539i
$383$	10252.4	−	10252.4i	1.36782	−	1.36782i	0.504280	−	0.863540i	$-0.331758\pi$
$383$	10252.4	−	10252.4i	1.36782	−	1.36782i	0.863540	−	0.504280i	$-0.168242\pi$
$384$		0			0
$385$		0			0
$386$		−	280.200i		−	0.0369477i
$387$		0			0
$388$	−2863.71	−	2863.71i	−0.374698	−	0.374698i
$389$	6840.61			0.891601			0.445801	−	0.895132i	$-0.352919\pi$
$389$	6840.61			0.891601			0.445801	+	0.895132i	$0.352919\pi$
$390$		0			0
$391$	11949.9			1.54561
$392$	−742.111	−	742.111i	−0.0956181	−	0.0956181i
$393$		0			0
$394$			18573.8i			2.37496i
$395$		0			0
$396$		0			0
$397$	−288.543	+	288.543i	−0.0364774	+	0.0364774i	−0.725110	−	0.688633i	$-0.758211\pi$
$397$	−288.543	+	288.543i	−0.0364774	+	0.0364774i	0.688633	+	0.725110i	$0.258211\pi$
$398$	−4634.59	+	4634.59i	−0.583696	+	0.583696i
$399$		0			0
$400$		0			0
$401$		−	4765.08i		−	0.593409i	−0.954969	−	0.296704i	$-0.904112\pi$
$401$		−	4765.08i		−	0.593409i	0.954969	−	0.296704i	$-0.0958876\pi$
$402$		0			0
$403$	4755.94	+	4755.94i	0.587867	+	0.587867i
$404$	12177.2			1.49959
$405$		0			0
$406$	11083.3			1.35482
$407$	3436.10	+	3436.10i	0.418480	+	0.418480i
$408$		0			0
$409$		−	8132.87i		−	0.983239i	−0.870810	−	0.491619i	$-0.836405\pi$
$409$		−	8132.87i		−	0.983239i	0.870810	−	0.491619i	$-0.163595\pi$
$410$		0			0
$411$		0			0
$412$	−242.240	+	242.240i	−0.0289668	+	0.0289668i
$413$	−16646.6	+	16646.6i	−1.98335	+	1.98335i
$414$		0			0
$415$		0			0
$416$			9213.93i			1.08594i
$417$		0			0
$418$	6965.98	+	6965.98i	0.815113	+	0.815113i
$419$	−222.003			−0.0258844			−0.0129422	−	0.999916i	$-0.504120\pi$
$419$	−222.003			−0.0258844			−0.0129422	+	0.999916i	$0.504120\pi$
$420$		0			0
$421$	4412.17			0.510774			0.255387	−	0.966839i	$-0.417797\pi$
$421$	4412.17			0.510774			0.255387	+	0.966839i	$0.417797\pi$
$422$	−9630.58	−	9630.58i	−1.11092	−	1.11092i
$423$		0			0
$424$			138.260i			0.0158361i
$425$		0			0
$426$		0			0
$427$	15023.7	−	15023.7i	1.70269	−	1.70269i
$428$	5163.55	−	5163.55i	0.583153	−	0.583153i
$429$		0			0
$430$		0			0
$431$		−	10285.6i		−	1.14952i	−0.818323	−	0.574758i	$-0.805096\pi$
$431$		−	10285.6i		−	1.14952i	0.818323	−	0.574758i	$-0.194904\pi$
$432$		0			0
$433$	−6616.98	−	6616.98i	−0.734392	−	0.734392i	0.237094	−	0.971487i	$-0.423805\pi$
$433$	−6616.98	−	6616.98i	−0.734392	−	0.734392i	−0.971487	+	0.237094i	$0.923805\pi$
$434$	25916.4			2.86642
$435$		0			0
$436$	−9055.78			−0.994709
$437$	−8361.88	−	8361.88i	−0.915339	−	0.915339i
$438$		0			0
$439$			11483.9i			1.24852i	0.781219	+	0.624258i	$0.214598\pi$
$439$			11483.9i			1.24852i	−0.781219	+	0.624258i	$0.785402\pi$
$440$		0			0
$441$		0			0
$442$	6068.74	−	6068.74i	0.653078	−	0.653078i
$443$	3420.37	−	3420.37i	0.366832	−	0.366832i	−0.499489	−	0.866320i	$-0.666479\pi$
$443$	3420.37	−	3420.37i	0.366832	−	0.366832i	0.866320	+	0.499489i	$0.166479\pi$
$444$		0			0
$445$		0			0
$446$			14516.0i			1.54115i
$447$		0			0
$448$	11852.2	+	11852.2i	1.24992	+	1.24992i
$449$	11995.6			1.26082			0.630410	−	0.776262i	$-0.282887\pi$
$449$	11995.6			1.26082			0.630410	+	0.776262i	$0.282887\pi$
$450$		0			0
$451$	−591.576			−0.0617655
$452$	−4586.01	−	4586.01i	−0.477229	−	0.477229i
$453$		0			0
$454$		−	22110.5i		−	2.28567i
$455$		0			0
$456$		0			0
$457$	−3243.45	+	3243.45i	−0.331996	+	0.331996i	−0.853344	−	0.521348i	$-0.825429\pi$
$457$	−3243.45	+	3243.45i	−0.331996	+	0.331996i	0.521348	+	0.853344i	$0.325429\pi$
$458$	−6359.14	+	6359.14i	−0.648785	+	0.648785i
$459$		0			0
$460$		0			0
$461$		−	15008.4i		−	1.51629i	−0.652087	−	0.758144i	$-0.726106\pi$
$461$		−	15008.4i		−	1.51629i	0.652087	−	0.758144i	$-0.273894\pi$
$462$		0			0
$463$	−1250.28	−	1250.28i	−0.125498	−	0.125498i	0.641568	−	0.767066i	$-0.278284\pi$
$463$	−1250.28	−	1250.28i	−0.125498	−	0.125498i	−0.767066	+	0.641568i	$0.778284\pi$
$464$	5238.98			0.524168
$465$		0			0
$466$	9852.73			0.979440
$467$	4274.49	+	4274.49i	0.423554	+	0.423554i	0.886426	−	0.462871i	$-0.153181\pi$
$467$	4274.49	+	4274.49i	0.423554	+	0.423554i	−0.462871	+	0.886426i	$0.653181\pi$
$468$		0			0
$469$			311.461i			0.0306651i
$470$		0			0
$471$		0			0
$472$	−544.412	+	544.412i	−0.0530903	+	0.0530903i
$473$	−6420.04	+	6420.04i	−0.624088	+	0.624088i
$474$		0			0
$475$		0			0
$476$		−	16227.9i		−	1.56262i
$477$		0			0
$478$	10007.6	+	10007.6i	0.957605	+	0.957605i
$479$	−7189.29			−0.685776			−0.342888	−	0.939376i	$-0.611405\pi$
$479$	−7189.29			−0.685776			−0.342888	+	0.939376i	$0.611405\pi$
$480$		0			0
$481$	−4189.40			−0.397132
$482$	16916.9	+	16916.9i	1.59864	+	1.59864i
$483$		0			0
$484$		−	3460.94i		−	0.325032i
$485$		0			0
$486$		0			0
$487$	923.729	−	923.729i	0.0859510	−	0.0859510i	−0.662824	−	0.748775i	$-0.730642\pi$
$487$	923.729	−	923.729i	0.0859510	−	0.0859510i	0.748775	+	0.662824i	$0.230642\pi$
$488$	491.337	−	491.337i	0.0455774	−	0.0455774i
$489$		0			0
$490$		0			0
$491$		−	434.588i		−	0.0399444i	−0.999801	−	0.0199722i	$-0.993642\pi$
$491$		−	434.588i		−	0.0399444i	0.999801	−	0.0199722i	$-0.00635777\pi$
$492$		0			0
$493$	−3329.06	−	3329.06i	−0.304124	−	0.304124i
$494$	−8493.13			−0.773531
$495$		0			0
$496$	12250.5			1.10900
$497$	18774.2	+	18774.2i	1.69445	+	1.69445i
$498$		0			0
$499$			7228.02i			0.648438i	0.945982	+	0.324219i	$0.105101\pi$
$499$			7228.02i			0.648438i	−0.945982	+	0.324219i	$0.894899\pi$
$500$		0			0
$501$		0			0
$502$	8646.70	−	8646.70i	0.768767	−	0.768767i
$503$	−1240.98	+	1240.98i	−0.110005	+	0.110005i	−0.759967	−	0.649962i	$-0.774785\pi$
$503$	−1240.98	+	1240.98i	−0.110005	+	0.110005i	0.649962	+	0.759967i	$0.274785\pi$
$504$		0			0
$505$		0			0
$506$			33563.5i			2.94877i
$507$		0			0
$508$	−1058.95	−	1058.95i	−0.0924871	−	0.0924871i
$509$	20683.3			1.80112			0.900561	−	0.434730i	$-0.143156\pi$
$509$	20683.3			1.80112			0.900561	+	0.434730i	$0.143156\pi$
$510$		0			0
$511$	−20936.8			−1.81250
$512$	11433.3	+	11433.3i	0.986887	+	0.986887i
$513$		0			0
$514$			2748.59i			0.235866i
$515$		0			0
$516$		0			0
$517$	8397.42	−	8397.42i	0.714349	−	0.714349i
$518$	−11414.6	+	11414.6i	−0.968201	+	0.968201i
$519$		0			0
$520$		0			0
$521$			1904.89i			0.160182i	0.996788	+	0.0800908i	$0.0255210\pi$
$521$			1904.89i			0.160182i	−0.996788	+	0.0800908i	$0.974479\pi$
$522$		0			0
$523$	−1799.72	−	1799.72i	−0.150471	−	0.150471i	0.627857	−	0.778329i	$-0.283932\pi$
$523$	−1799.72	−	1799.72i	−0.150471	−	0.150471i	−0.778329	+	0.627857i	$0.783932\pi$
$524$	−3796.35			−0.316497
$525$		0			0
$526$	−882.349			−0.0731411
$527$	−7784.44	−	7784.44i	−0.643445	−	0.643445i
$528$		0			0
$529$		−	28122.2i		−	2.31135i
$530$		0			0
$531$		0			0
$532$	−11355.4	+	11355.4i	−0.925413	+	0.925413i
$533$	360.634	−	360.634i	0.0293073	−	0.0293073i
$534$		0			0
$535$		0			0
$536$			10.1861i			0.000820842i
$537$		0			0
$538$	−17369.5	−	17369.5i	−1.39192	−	1.39192i
$539$	−38286.9			−3.05962
$540$		0			0
$541$	−19472.7			−1.54749			−0.773747	−	0.633494i	$-0.781620\pi$
$541$	−19472.7			−1.54749			−0.773747	+	0.633494i	$0.781620\pi$
$542$	−12784.1	−	12784.1i	−1.01315	−	1.01315i
$543$		0			0
$544$		−	15081.2i		−	1.18860i
$545$		0			0
$546$		0			0
$547$	−15899.0	+	15899.0i	−1.24277	+	1.24277i	−0.283916	+	0.958849i	$0.591634\pi$
$547$	−15899.0	+	15899.0i	−1.24277	+	1.24277i	−0.958849	+	0.283916i	$0.908366\pi$
$548$	1360.36	−	1360.36i	0.106043	−	0.106043i
$549$		0			0
$550$		0			0
$551$			4658.98i			0.360217i
$552$		0			0
$553$	20649.8	+	20649.8i	1.58792	+	1.58792i
$554$	5045.28			0.386920
$555$		0			0
$556$	9442.38			0.720227
$557$	2403.47	+	2403.47i	0.182834	+	0.182834i	0.792589	−	0.609756i	$-0.208732\pi$
$557$	2403.47	+	2403.47i	0.182834	+	0.182834i	−0.609756	+	0.792589i	$0.708732\pi$
$558$		0			0
$559$		−	7827.51i		−	0.592251i
$560$		0			0
$561$		0			0
$562$	−12571.9	+	12571.9i	−0.943622	+	0.943622i
$563$	7264.48	−	7264.48i	0.543803	−	0.543803i	−0.380838	−	0.924642i	$-0.624365\pi$
$563$	7264.48	−	7264.48i	0.543803	−	0.543803i	0.924642	+	0.380838i	$0.124365\pi$
$564$		0			0
$565$		0			0
$566$		−	26545.4i		−	1.97135i
$567$		0			0
$568$	613.995	+	613.995i	0.0453568	+	0.0453568i
$569$	5273.08			0.388505			0.194252	−	0.980952i	$-0.437772\pi$
$569$	5273.08			0.388505			0.194252	+	0.980952i	$0.437772\pi$
$570$		0			0
$571$	14061.0			1.03053			0.515267	−	0.857030i	$-0.327693\pi$
$571$	14061.0			1.03053			0.515267	+	0.857030i	$0.327693\pi$
$572$	8364.26	+	8364.26i	0.611411	+	0.611411i
$573$		0			0
$574$		−	1965.19i		−	0.142901i
$575$		0			0
$576$		0			0
$577$	−10570.7	+	10570.7i	−0.762673	+	0.762673i	−0.976805	−	0.214131i	$-0.931308\pi$
$577$	−10570.7	+	10570.7i	−0.762673	+	0.762673i	0.214131	+	0.976805i	$0.431308\pi$
$578$	3835.57	−	3835.57i	0.276018	−	0.276018i
$579$		0			0
$580$		0			0
$581$			23158.1i			1.65363i
$582$		0			0
$583$	3566.54	+	3566.54i	0.253364	+	0.253364i
$584$	−684.720			−0.0485170
$585$		0			0
$586$	−25883.8			−1.82466
$587$	−7292.50	−	7292.50i	−0.512766	−	0.512766i	0.402607	−	0.915373i	$-0.368104\pi$
$587$	−7292.50	−	7292.50i	−0.512766	−	0.512766i	−0.915373	+	0.402607i	$0.868104\pi$
$588$		0			0
$589$			10894.2i			0.762121i
$590$		0			0
$591$		0			0
$592$	−5395.58	+	5395.58i	−0.374589	+	0.374589i
$593$	454.363	−	454.363i	0.0314645	−	0.0314645i	−0.691199	−	0.722664i	$-0.742917\pi$
$593$	454.363	−	454.363i	0.0314645	−	0.0314645i	0.722664	+	0.691199i	$0.242917\pi$
$594$		0			0
$595$		0			0
$596$		−	323.285i		−	0.0222186i
$597$		0			0
$598$	−20460.8	−	20460.8i	−1.39917	−	1.39917i
$599$	7281.47			0.496683			0.248341	−	0.968673i	$-0.420115\pi$
$599$	7281.47			0.496683			0.248341	+	0.968673i	$0.420115\pi$
$600$		0			0
$601$	14273.8			0.968786			0.484393	−	0.874850i	$-0.339040\pi$
$601$	14273.8			0.968786			0.484393	+	0.874850i	$0.339040\pi$
$602$	−21327.1	−	21327.1i	−1.44390	−	1.44390i
$603$		0			0
$604$		−	363.967i		−	0.0245192i
$605$		0			0
$606$		0			0
$607$	−7824.40	+	7824.40i	−0.523200	+	0.523200i	−0.918536	−	0.395336i	$-0.870628\pi$
$607$	−7824.40	+	7824.40i	−0.523200	+	0.523200i	0.395336	+	0.918536i	$0.370628\pi$
$608$	−10553.0	+	10553.0i	−0.703914	+	0.703914i
$609$		0			0
$610$		0			0
$611$			10238.4i			0.677907i
$612$		0			0
$613$	2893.58	+	2893.58i	0.190653	+	0.190653i	0.795978	−	0.605325i	$-0.206957\pi$
$613$	2893.58	+	2893.58i	0.190653	+	0.190653i	−0.605325	+	0.795978i	$0.706957\pi$
$614$	−28026.7			−1.84212
$615$		0			0
$616$	−1725.41			−0.112855
$617$	−2847.70	−	2847.70i	−0.185809	−	0.185809i	0.608073	−	0.793881i	$-0.291943\pi$
$617$	−2847.70	−	2847.70i	−0.185809	−	0.185809i	−0.793881	+	0.608073i	$0.791943\pi$
$618$		0			0
$619$			16644.3i			1.08076i	0.841421	+	0.540380i	$0.181719\pi$
$619$			16644.3i			1.08076i	−0.841421	+	0.540380i	$0.818281\pi$
$620$		0			0
$621$		0			0
$622$	19265.5	−	19265.5i	1.24193	−	1.24193i
$623$	16171.8	−	16171.8i	1.03998	−	1.03998i
$624$		0			0
$625$		0			0
$626$		−	19180.8i		−	1.22463i
$627$		0			0
$628$	−14306.9	−	14306.9i	−0.909088	−	0.909088i
$629$	6857.13			0.434677
$630$		0			0
$631$	−5327.30			−0.336096			−0.168048	−	0.985779i	$-0.553746\pi$
$631$	−5327.30			−0.336096			−0.168048	+	0.985779i	$0.553746\pi$
$632$	675.333	+	675.333i	0.0425053	+	0.0425053i
$633$		0			0
$634$			26286.8i			1.64666i
$635$		0			0
$636$		0			0
$637$	23340.3	−	23340.3i	1.45177	−	1.45177i
$638$	9350.26	−	9350.26i	0.580220	−	0.580220i
$639$		0			0
$640$		0			0
$641$			16450.6i			1.01367i	0.862044	+	0.506834i	$0.169184\pi$
$641$			16450.6i			1.01367i	−0.862044	+	0.506834i	$0.830816\pi$
$642$		0			0
$643$	−21894.3	−	21894.3i	−1.34281	−	1.34281i	−0.893252	−	0.449555i	$-0.851582\pi$
$643$	−21894.3	−	21894.3i	−1.34281	−	1.34281i	−0.449555	−	0.893252i	$-0.648418\pi$
$644$	−54712.6			−3.34779
$645$		0			0
$646$	13901.4			0.846661
$647$	7768.11	+	7768.11i	0.472018	+	0.472018i	0.902567	−	0.430549i	$-0.141680\pi$
$647$	7768.11	+	7768.11i	0.472018	+	0.472018i	−0.430549	+	0.902567i	$0.641680\pi$
$648$		0			0
$649$			28087.3i			1.69880i
$650$		0			0
$651$		0			0
$652$	−2274.38	+	2274.38i	−0.136613	+	0.136613i
$653$	14364.4	−	14364.4i	0.860831	−	0.860831i	−0.130603	−	0.991435i	$-0.541691\pi$
$653$	14364.4	−	14364.4i	0.860831	−	0.860831i	0.991435	+	0.130603i	$0.0416914\pi$
$654$		0			0
$655$		0			0
$656$		−	928.928i		−	0.0552874i
$657$		0			0
$658$	27895.9	+	27895.9i	1.65273	+	1.65273i
$659$	18185.5			1.07497			0.537486	−	0.843272i	$-0.319374\pi$
$659$	18185.5			1.07497			0.537486	+	0.843272i	$0.319374\pi$
$660$		0			0
$661$	8833.96			0.519820			0.259910	−	0.965633i	$-0.416307\pi$
$661$	8833.96			0.519820			0.259910	+	0.965633i	$0.416307\pi$
$662$	2668.22	+	2668.22i	0.156652	+	0.156652i
$663$		0			0
$664$			757.366i			0.0442643i
$665$		0			0
$666$		0			0
$667$	−11223.9	+	11223.9i	−0.651564	+	0.651564i
$668$	8335.76	−	8335.76i	0.482815	−	0.482815i
$669$		0			0
$670$		0			0
$671$		−	25349.0i		−	1.45840i
$672$		0			0
$673$	12769.6	+	12769.6i	0.731398	+	0.731398i	0.970897	−	0.239499i	$-0.0769831\pi$
$673$	12769.6	+	12769.6i	0.731398	+	0.731398i	−0.239499	+	0.970897i	$0.576983\pi$
$674$	22464.9			1.28385
$675$		0			0
$676$	6736.97			0.383305
$677$	1478.18	+	1478.18i	0.0839157	+	0.0839157i	0.747819	−	0.663903i	$-0.231101\pi$
$677$	1478.18	+	1478.18i	0.0839157	+	0.0839157i	−0.663903	+	0.747819i	$0.731101\pi$
$678$		0			0
$679$			18579.4i			1.05009i
$680$		0			0
$681$		0			0
$682$	21864.0	−	21864.0i	1.22759	−	1.22759i
$683$	2681.69	−	2681.69i	0.150237	−	0.150237i	−0.627987	−	0.778224i	$-0.716121\pi$
$683$	2681.69	−	2681.69i	0.150237	−	0.150237i	0.778224	+	0.627987i	$0.216121\pi$
$684$		0			0
$685$		0			0
$686$		−	79115.0i		−	4.40324i
$687$		0			0
$688$	−10081.1	−	10081.1i	−0.558633	−	0.558633i
$689$	−4348.44			−0.240439
$690$		0			0
$691$	−23210.5			−1.27782			−0.638908	−	0.769283i	$-0.720614\pi$
$691$	−23210.5			−1.27782			−0.638908	+	0.769283i	$0.720614\pi$
$692$	−18268.2	−	18268.2i	−1.00355	−	1.00355i
$693$		0			0
$694$			10499.9i			0.574308i
$695$		0			0
$696$		0			0
$697$	−590.278	+	590.278i	−0.0320780	+	0.0320780i
$698$	29825.0	−	29825.0i	1.61732	−	1.61732i
$699$		0			0
$700$		0			0
$701$			26169.3i			1.40999i	0.709213	+	0.704995i	$0.249051\pi$
$701$			26169.3i			1.40999i	−0.709213	+	0.704995i	$0.750949\pi$
$702$		0			0
$703$	−4798.24	−	4798.24i	−0.257424	−	0.257424i
$704$	19997.8			1.07059
$705$		0			0
$706$	31945.1			1.70293
$707$	−39501.9	−	39501.9i	−2.10130	−	2.10130i
$708$		0			0
$709$		−	13604.6i		−	0.720636i	−0.932830	−	0.360318i	$-0.882668\pi$
$709$		−	13604.6i		−	0.720636i	0.932830	−	0.360318i	$-0.117332\pi$
$710$		0			0
$711$		0			0
$712$	528.886	−	528.886i	0.0278382	−	0.0278382i
$713$	−26245.3	+	26245.3i	−1.37853	+	1.37853i
$714$		0			0
$715$		0			0
$716$			20280.4i			1.05854i
$717$		0			0
$718$	−34396.6	−	34396.6i	−1.78784	−	1.78784i
$719$	15020.4			0.779088			0.389544	−	0.921008i	$-0.372633\pi$
$719$	15020.4			0.779088			0.389544	+	0.921008i	$0.372633\pi$
$720$		0			0
$721$	1571.62			0.0811792
$722$	9495.04	+	9495.04i	0.489430	+	0.489430i
$723$		0			0
$724$		−	3794.89i		−	0.194801i
$725$		0			0
$726$		0			0
$727$	1195.38	−	1195.38i	0.0609824	−	0.0609824i	−0.675958	−	0.736940i	$-0.736270\pi$
$727$	1195.38	−	1195.38i	0.0609824	−	0.0609824i	0.736940	+	0.675958i	$0.236270\pi$
$728$	1051.83	−	1051.83i	0.0535489	−	0.0535489i
$729$		0			0
$730$		0			0
$731$			12811.9i			0.648243i
$732$		0			0
$733$	14097.2	+	14097.2i	0.710359	+	0.710359i	0.966610	−	0.256251i	$-0.0824875\pi$
$733$	14097.2	+	14097.2i	0.710359	+	0.710359i	−0.256251	+	0.966610i	$0.582487\pi$
$734$	16061.3			0.807674
$735$		0			0
$736$	−50846.3			−2.54649
$737$	262.759	+	262.759i	0.0131328	+	0.0131328i
$738$		0			0
$739$			8052.39i			0.400828i	0.979711	+	0.200414i	$0.0642287\pi$
$739$			8052.39i			0.400828i	−0.979711	+	0.200414i	$0.935771\pi$
$740$		0			0
$741$		0			0
$742$	−11847.9	+	11847.9i	−0.586186	+	0.586186i
$743$	6706.82	−	6706.82i	0.331156	−	0.331156i	−0.521869	−	0.853026i	$-0.674765\pi$
$743$	6706.82	−	6706.82i	0.331156	−	0.331156i	0.853026	+	0.521869i	$0.174765\pi$
$744$		0			0
$745$		0			0
$746$			21632.8i			1.06171i
$747$		0			0
$748$	−13690.5	−	13690.5i	−0.669215	−	0.669215i
$749$	−33500.4			−1.63428
$750$		0			0
$751$	−17782.1			−0.864018			−0.432009	−	0.901869i	$-0.642195\pi$
$751$	−17782.1			−0.864018			−0.432009	+	0.901869i	$0.642195\pi$
$752$	13186.1	+	13186.1i	0.639427	+	0.639427i
$753$		0			0
$754$			11400.1i			0.550621i
$755$		0			0
$756$		0			0
$757$	−3547.92	+	3547.92i	−0.170345	+	0.170345i	−0.787131	−	0.616786i	$-0.788434\pi$
$757$	−3547.92	+	3547.92i	−0.170345	+	0.170345i	0.616786	+	0.787131i	$0.288434\pi$
$758$	−8355.39	+	8355.39i	−0.400371	+	0.400371i
$759$		0			0
$760$		0			0
$761$		−	29802.3i		−	1.41962i	−0.704392	−	0.709811i	$-0.748780\pi$
$761$		−	29802.3i		−	1.41962i	0.704392	−	0.709811i	$-0.251220\pi$
$762$		0			0
$763$	29376.3	+	29376.3i	1.39383	+	1.39383i
$764$	−9909.05			−0.469237
$765$		0			0
$766$	57465.3			2.71058
$767$	−17122.4	−	17122.4i	−0.806069	−	0.806069i
$768$		0			0
$769$			1720.92i			0.0806994i	0.999186	+	0.0403497i	$0.0128472\pi$
$769$			1720.92i			0.0806994i	−0.999186	+	0.0403497i	$0.987153\pi$
$770$		0			0
$771$		0			0
$772$	385.339	−	385.339i	0.0179646	−	0.0179646i
$773$	−8247.02	+	8247.02i	−0.383732	+	0.383732i	−0.872445	−	0.488713i	$-0.837467\pi$
$773$	−8247.02	+	8247.02i	−0.383732	+	0.383732i	0.488713	+	0.872445i	$0.337467\pi$
$774$		0			0
$775$		0			0
$776$			607.622i			0.0281087i
$777$		0			0
$778$	19170.9	+	19170.9i	0.883434	+	0.883434i
$779$	826.088			0.0379944
$780$		0			0
$781$	31677.2			1.45134
$782$	33489.8	+	33489.8i	1.53145	+	1.53145i
$783$		0			0
$784$		−	60120.5i		−	2.73872i
$785$		0			0
$786$		0			0
$787$	−24948.6	+	24948.6i	−1.13001	+	1.13001i	−0.139840	+	0.990174i	$0.544659\pi$
$787$	−24948.6	+	24948.6i	−1.13001	+	1.13001i	−0.990174	+	0.139840i	$0.955341\pi$
$788$	−25543.2	+	25543.2i	−1.15474	+	1.15474i
$789$		0			0
$790$		0			0
$791$			29753.4i			1.33743i
$792$		0			0
$793$	15453.1	+	15453.1i	0.692001	+	0.692001i
$794$	−1617.29			−0.0722866
$795$		0			0
$796$	−12747.2			−0.567605
$797$	13046.4	+	13046.4i	0.579832	+	0.579832i	0.934857	−	0.355025i	$-0.115528\pi$
$797$	13046.4	+	13046.4i	0.579832	+	0.579832i	−0.355025	+	0.934857i	$0.615528\pi$
$798$		0			0
$799$		−	16758.0i		−	0.741997i
$800$		0			0
$801$		0			0
$802$	13354.2	−	13354.2i	0.587973	−	0.587973i
$803$	−17663.0	+	17663.0i	−0.776232	+	0.776232i
$804$		0			0
$805$		0			0
$806$			26657.2i			1.16496i
$807$		0			0
$808$	−1291.88	−	1291.88i	−0.0562475	−	0.0562475i
$809$	−23979.2			−1.04211			−0.521053	−	0.853524i	$-0.674461\pi$
$809$	−23979.2			−1.04211			−0.521053	+	0.853524i	$0.674461\pi$
$810$		0			0
$811$	32204.9			1.39441			0.697205	−	0.716871i	$-0.254427\pi$
$811$	32204.9			1.39441			0.697205	+	0.716871i	$0.254427\pi$
$812$	15242.1	+	15242.1i	0.658735	+	0.658735i
$813$		0			0
$814$			19259.5i			0.829293i
$815$		0			0
$816$		0			0
$817$	8965.06	−	8965.06i	0.383902	−	0.383902i
$818$	22792.5	−	22792.5i	0.974232	−	0.974232i
$819$		0			0
$820$		0			0
$821$			30723.2i			1.30603i	0.757346	+	0.653014i	$0.226496\pi$
$821$			30723.2i			1.30603i	−0.757346	+	0.653014i	$0.773504\pi$
$822$		0			0
$823$	−11028.6	−	11028.6i	−0.467112	−	0.467112i	0.433866	−	0.900978i	$-0.357149\pi$
$823$	−11028.6	−	11028.6i	−0.467112	−	0.467112i	−0.900978	+	0.433866i	$0.857149\pi$
$824$	51.3985			0.00217300
$825$		0			0
$826$	−93304.7			−3.93037
$827$	6174.37	+	6174.37i	0.259618	+	0.259618i	0.824898	−	0.565281i	$-0.191232\pi$
$827$	6174.37	+	6174.37i	0.259618	+	0.259618i	−0.565281	+	0.824898i	$0.691232\pi$
$828$		0			0
$829$		−	25346.8i		−	1.06192i	−0.847398	−	0.530959i	$-0.821832\pi$
$829$		−	25346.8i		−	1.06192i	0.847398	−	0.530959i	$-0.178168\pi$
$830$		0			0
$831$		0			0
$832$	−12191.0	+	12191.0i	−0.507987	+	0.507987i
$833$	−38202.9	+	38202.9i	−1.58902	+	1.58902i
$834$		0			0
$835$		0			0
$836$			19159.6i			0.792643i
$837$		0			0
$838$	−622.168	−	622.168i	−0.0256473	−	0.0256473i
$839$	−31583.9			−1.29964			−0.649820	−	0.760088i	$-0.725156\pi$
$839$	−31583.9			−1.29964			−0.649820	+	0.760088i	$0.725156\pi$
$840$		0			0
$841$	−18135.4			−0.743588
$842$	12365.2	+	12365.2i	0.506095	+	0.506095i
$843$		0			0
$844$		−	26488.5i		−	1.08030i
$845$		0			0
$846$		0			0
$847$	−11227.0	+	11227.0i	−0.455449	+	0.455449i
$848$	−5600.40	+	5600.40i	−0.226791	+	0.226791i
$849$		0			0
$850$		0			0
$851$		−	23118.9i		−	0.931262i
$852$		0			0
$853$	−28855.5	−	28855.5i	−1.15826	−	1.15826i	−0.984850	−	0.173408i	$-0.944522\pi$
$853$	−28855.5	−	28855.5i	−1.15826	−	1.15826i	−0.173408	−	0.984850i	$-0.555478\pi$
$854$	84208.2			3.37418
$855$		0			0
$856$	−1095.60			−0.0437464
$857$	423.601	+	423.601i	0.0168844	+	0.0168844i	0.715499	−	0.698614i	$-0.246200\pi$
$857$	423.601	+	423.601i	0.0168844	+	0.0168844i	−0.698614	+	0.715499i	$0.746200\pi$
$858$		0			0
$859$		−	37170.9i		−	1.47643i	−0.674565	−	0.738216i	$-0.735669\pi$
$859$		−	37170.9i		−	1.47643i	0.674565	−	0.738216i	$-0.264331\pi$
$860$		0			0
$861$		0			0
$862$	28825.7	−	28825.7i	1.13899	−	1.13899i
$863$	24808.1	−	24808.1i	0.978538	−	0.978538i	−0.0212362	−	0.999774i	$-0.506760\pi$
$863$	24808.1	−	24808.1i	0.978538	−	0.978538i	0.999774	+	0.0212362i	$0.00676019\pi$
$864$		0			0
$865$		0			0
$866$		−	37088.4i		−	1.45533i
$867$		0			0
$868$	35641.0	+	35641.0i	1.39370	+	1.39370i
$869$	34841.7			1.36010
$870$		0			0
$871$	−320.364			−0.0124628
$872$	960.728	+	960.728i	0.0373100	+	0.0373100i
$873$		0			0
$874$		−	46868.6i		−	1.81391i
$875$		0			0
$876$		0			0
$877$	−33936.8	+	33936.8i	−1.30669	+	1.30669i	−0.382896	+	0.923792i	$0.625073\pi$
$877$	−33936.8	+	33936.8i	−1.30669	+	1.30669i	−0.923792	+	0.382896i	$0.874927\pi$
$878$	−32183.9	+	32183.9i	−1.23708	+	1.23708i
$879$		0			0
$880$		0			0
$881$		−	6463.97i		−	0.247193i	−0.992333	−	0.123596i	$-0.960557\pi$
$881$		−	6463.97i		−	0.247193i	0.992333	−	0.123596i	$-0.0394428\pi$
$882$		0			0
$883$	−27976.2	−	27976.2i	−1.06622	−	1.06622i	−0.997646	−	0.0685762i	$-0.978154\pi$
$883$	−27976.2	−	27976.2i	−1.06622	−	1.06622i	−0.0685762	−	0.997646i	$-0.521846\pi$
$884$	16691.8			0.635075
$885$		0			0
$886$	19171.3			0.726943
$887$	18609.3	+	18609.3i	0.704441	+	0.704441i	0.965360	−	0.260920i	$-0.0840258\pi$
$887$	18609.3	+	18609.3i	0.704441	+	0.704441i	−0.260920	+	0.965360i	$0.584026\pi$
$888$		0			0
$889$			6870.34i			0.259195i
$890$		0			0
$891$		0			0
$892$	−19962.8	+	19962.8i	−0.749332	+	0.749332i
$893$	−11726.3	+	11726.3i	−0.439425	+	0.439425i
$894$		0			0
$895$		0			0
$896$		−	5231.23i		−	0.195048i
$897$		0			0
$898$	33617.9	+	33617.9i	1.24927	+	1.24927i
$899$	14623.1			0.542499
$900$		0			0
$901$	7117.44			0.263170
$902$	−1657.90	−	1657.90i	−0.0611997	−	0.0611997i
$903$		0			0
$904$			973.059i			0.0358003i
$905$		0			0
$906$		0			0
$907$	22007.2	−	22007.2i	0.805663	−	0.805663i	−0.178311	−	0.983974i	$-0.557063\pi$
$907$	22007.2	−	22007.2i	0.805663	−	0.805663i	0.983974	+	0.178311i	$0.0570634\pi$
$908$	30406.9	−	30406.9i	1.11133	−	1.11133i
$909$		0			0
$910$		0			0
$911$		−	27845.6i		−	1.01269i	−0.862330	−	0.506347i	$-0.830995\pi$
$911$		−	27845.6i		−	1.01269i	0.862330	−	0.506347i	$-0.169005\pi$
$912$		0			0
$913$	19537.0	+	19537.0i	0.708192	+	0.708192i
$914$	−18179.6			−0.657910
$915$		0			0
$916$	−17490.6			−0.630900
$917$	12315.1	+	12315.1i	0.443490	+	0.443490i
$918$		0			0
$919$			40017.1i			1.43639i	0.695841	+	0.718196i	$0.255032\pi$
$919$			40017.1i			1.43639i	−0.695841	+	0.718196i	$0.744968\pi$
$920$		0			0
$921$		0			0
$922$	42061.2	−	42061.2i	1.50240	−	1.50240i
$923$	−19310.9	+	19310.9i	−0.688652	+	0.688652i
$924$		0			0
$925$		0			0
$926$		−	7007.88i		−	0.248697i
$927$		0			0
$928$	14165.0	+	14165.0i	0.501065	+	0.501065i
$929$	22680.8			0.801003			0.400502	−	0.916296i	$-0.368836\pi$
$929$	22680.8			0.801003			0.400502	+	0.916296i	$0.368836\pi$
$930$		0			0
$931$	53464.6			1.88210
$932$	13549.8	+	13549.8i	0.476220	+	0.476220i
$933$		0			0
$934$			23958.7i			0.839349i
$935$		0			0
$936$		0			0
$937$	17387.5	−	17387.5i	0.606216	−	0.606216i	−0.335739	−	0.941955i	$-0.608986\pi$
$937$	17387.5	−	17387.5i	0.606216	−	0.606216i	0.941955	+	0.335739i	$0.108986\pi$
$938$	−872.875	+	872.875i	−0.0303842	+	0.0303842i
$939$		0			0
$940$		0			0
$941$			26573.5i			0.920587i	0.887767	+	0.460293i	$0.152256\pi$
$941$			26573.5i			0.920587i	−0.887767	+	0.460293i	$0.847744\pi$
$942$		0			0
$943$	1990.13	+	1990.13i	0.0687247	+	0.0687247i
$944$	−44104.3			−1.52063
$945$		0			0
$946$	−35984.5			−1.23674
$947$	−34605.4	−	34605.4i	−1.18746	−	1.18746i	−0.977767	−	0.209694i	$-0.932753\pi$
$947$	−34605.4	−	34605.4i	−1.18746	−	1.18746i	−0.209694	−	0.977767i	$-0.567247\pi$
$948$		0			0
$949$		−	21535.3i		−	0.736633i
$950$		0			0
$951$		0			0
$952$	−1721.62	+	1721.62i	−0.0586115	+	0.0586115i
$953$	15334.3	−	15334.3i	0.521225	−	0.521225i	−0.396717	−	0.917941i	$-0.629850\pi$
$953$	15334.3	−	15334.3i	0.521225	−	0.521225i	0.917941	+	0.396717i	$0.129850\pi$
$954$		0			0
$955$		0			0
$956$			27525.4i			0.931207i
$957$		0			0
$958$	−20148.1	−	20148.1i	−0.679494	−	0.679494i
$959$	−8825.80			−0.297185
$960$		0			0
$961$	4402.50			0.147780
$962$	−11740.9	−	11740.9i	−0.393494	−	0.393494i
$963$		0			0
$964$			46529.2i			1.55457i
$965$		0			0
$966$		0			0
$967$	37027.3	−	37027.3i	1.23135	−	1.23135i	0.267908	−	0.963444i	$-0.413668\pi$
$967$	37027.3	−	37027.3i	1.23135	−	1.23135i	0.963444	−	0.267908i	$-0.0863324\pi$
$968$	−367.171	+	367.171i	−0.0121914	+	0.0121914i
$969$		0			0
$970$		0			0
$971$			49325.9i			1.63022i	0.579306	+	0.815110i	$0.303324\pi$
$971$			49325.9i			1.63022i	−0.579306	+	0.815110i	$0.696676\pi$
$972$		0			0
$973$	−30630.4	−	30630.4i	−1.00922	−	1.00922i
$974$	5177.53			0.170327
$975$		0			0
$976$	39804.5			1.30544
$977$	−34615.8	−	34615.8i	−1.13353	−	1.13353i	−0.989586	−	0.143945i	$-0.954021\pi$
$977$	−34615.8	−	34615.8i	−1.13353	−	1.13353i	−0.143945	−	0.989586i	$-0.545979\pi$
$978$		0			0
$979$		−	27286.2i		−	0.890778i
$980$		0			0
$981$		0			0
$982$	1217.94	−	1217.94i	0.0395785	−	0.0395785i
$983$	−18512.8	+	18512.8i	−0.600679	+	0.600679i	−0.940493	−	0.339813i	$-0.889636\pi$
$983$	−18512.8	+	18512.8i	−0.600679	+	0.600679i	0.339813	+	0.940493i	$0.389636\pi$
$984$		0			0
$985$		0			0
$986$		−	18659.5i		−	0.602677i
$987$		0			0
$988$	−11680.0	−	11680.0i	−0.376104	−	0.376104i
$989$	43195.4			1.38881
$990$		0			0
$991$	49717.5			1.59367			0.796837	−	0.604195i	$-0.206505\pi$
$991$	49717.5			1.59367			0.796837	+	0.604195i	$0.206505\pi$
$992$	33122.4	+	33122.4i	1.06012	+	1.06012i
$993$		0			0
$994$			105230.i			3.35785i
$995$		0			0
$996$		0			0
$997$	12494.0	−	12494.0i	0.396880	−	0.396880i	−0.480251	−	0.877131i	$-0.659454\pi$
$997$	12494.0	−	12494.0i	0.396880	−	0.396880i	0.877131	+	0.480251i	$0.159454\pi$
$998$	−20256.6	+	20256.6i	−0.642498	+	0.642498i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.f.d.143.8	yes	16
3.2	odd	2	inner	225.4.f.d.143.2	yes	16
5.2	odd	4	inner	225.4.f.d.107.2	yes	16
5.3	odd	4	inner	225.4.f.d.107.7	yes	16
5.4	even	2	inner	225.4.f.d.143.1	yes	16
15.2	even	4	inner	225.4.f.d.107.8	yes	16
15.8	even	4	inner	225.4.f.d.107.1	✓	16
15.14	odd	2	inner	225.4.f.d.143.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.f.d.107.1	✓	16	15.8	even	4	inner
225.4.f.d.107.2	yes	16	5.2	odd	4	inner
225.4.f.d.107.7	yes	16	5.3	odd	4	inner
225.4.f.d.107.8	yes	16	15.2	even	4	inner
225.4.f.d.143.1	yes	16	5.4	even	2	inner
225.4.f.d.143.2	yes	16	3.2	odd	2	inner
225.4.f.d.143.7	yes	16	15.14	odd	2	inner
225.4.f.d.143.8	yes	16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(2.80252 + 2.80252i) q^{2} +7.70820i q^{4} +(25.0049 - 25.0049i) q^{7} +(0.817763 - 0.817763i) q^{8} +O(q^{10})\)	\(q+(2.80252 + 2.80252i) q^{2} +7.70820i q^{4} +(25.0049 - 25.0049i) q^{7} +(0.817763 - 0.817763i) q^{8} -42.1900i q^{11} +(25.7196 + 25.7196i) q^{13} +140.153 q^{14} +66.2492 q^{16} +(-42.0974 - 42.0974i) q^{17} +58.9149i q^{19} +(118.238 - 118.238i) q^{22} +(-141.932 + 141.932i) q^{23} +144.159i q^{26} +(192.743 + 192.743i) q^{28} +79.0799 q^{29} +184.915 q^{31} +(179.122 + 179.122i) q^{32} -235.957i q^{34} +(-81.4436 + 81.4436i) q^{37} +(-165.110 + 165.110i) q^{38} -14.0217i q^{41} +(-152.170 - 152.170i) q^{43} +325.209 q^{44} -795.532 q^{46} +(199.038 + 199.038i) q^{47} -907.489i q^{49} +(-198.252 + 198.252i) q^{52} +(-84.5353 + 84.5353i) q^{53} -40.8962i q^{56} +(221.623 + 221.623i) q^{58} -665.733 q^{59} +600.830 q^{61} +(518.227 + 518.227i) q^{62} +473.994i q^{64} +(-6.22801 + 6.22801i) q^{67} +(324.495 - 324.495i) q^{68} +750.823i q^{71} +(-418.654 - 418.654i) q^{73} -456.494 q^{74} -454.128 q^{76} +(-1054.96 - 1054.96i) q^{77} +825.830i q^{79} +(39.2961 - 39.2961i) q^{82} +(-463.072 + 463.072i) q^{83} -852.917i q^{86} +(-34.5014 - 34.5014i) q^{88} +646.747 q^{89} +1286.23 q^{91} +(-1094.04 - 1094.04i) q^{92} +1115.62i q^{94} +(-371.515 + 371.515i) q^{97} +(2543.25 - 2543.25i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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

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Database

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