LMFDB - Embedded newform 416.2.w.c.225.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(225,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("416.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.w (of order $6$, 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			225.2
Root			$0.500000 - 1.19293i$ of defining polynomial
Character	$\chi$	$=$	416.225
Dual form			416.2.w.c.257.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.619657 + 1.07328i) q^{3} +2.00000i q^{5} +(4.00552 - 2.31259i) q^{7} +(0.732051 + 1.26795i) q^{9} +O(q^{10})$	\(q+(-0.619657 + 1.07328i) q^{3} +2.00000i q^{5} +(4.00552 - 2.31259i) q^{7} +(0.732051 + 1.26795i) q^{9} +(-1.85897 - 1.07328i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(-2.14655 - 1.23931i) q^{15} +(0.232051 + 0.401924i) q^{17} +(4.00552 - 2.31259i) q^{19} +5.73205i q^{21} +(-2.76621 + 4.79122i) q^{23} +1.00000 q^{25} -5.53242 q^{27} +(-1.50000 + 2.59808i) q^{29} +9.25036i q^{31} +(2.30385 - 1.33013i) q^{33} +(4.62518 + 8.01105i) q^{35} +(6.69615 + 3.86603i) q^{37} +(-3.09828 - 3.21983i) q^{39} +(-6.23205 - 3.59808i) q^{41} +(-1.85897 - 3.21983i) q^{43} +(-2.53590 + 1.46410i) q^{45} -11.7290i q^{47} +(7.19615 - 12.4641i) q^{49} -0.575167 q^{51} +2.53590 q^{53} +(2.14655 - 3.71794i) q^{55} +5.73205i q^{57} +(8.29863 - 4.79122i) q^{59} +(-1.50000 - 2.59808i) q^{61} +(5.86450 + 3.38587i) q^{63} +(-6.92820 - 2.00000i) q^{65} +(-4.00552 - 2.31259i) q^{67} +(-3.42820 - 5.93782i) q^{69} +(-5.57691 + 3.21983i) q^{71} -6.00000i q^{73} +(-0.619657 + 1.07328i) q^{75} -9.92820 q^{77} +4.29311 q^{79} +(1.23205 - 2.13397i) q^{81} -4.29311i q^{83} +(-0.803848 + 0.464102i) q^{85} +(-1.85897 - 3.21983i) q^{87} +(-1.03590 - 0.598076i) q^{89} +(4.00552 + 16.1881i) q^{91} +(-9.92820 - 5.73205i) q^{93} +(4.62518 + 8.01105i) q^{95} +(11.8923 - 6.86603i) q^{97} -3.14277i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 8 q^{13} - 12 q^{17} + 8 q^{25} - 12 q^{29} + 60 q^{33} + 12 q^{37} - 36 q^{41} - 48 q^{45} + 16 q^{49} + 48 q^{53} - 12 q^{61} + 28 q^{69} - 24 q^{77} - 4 q^{81} - 48 q^{85} - 36 q^{89} - 24 q^{93} + 12 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^13 - 12 * q^17 + 8 * q^25 - 12 * q^29 + 60 * q^33 + 12 * q^37 - 36 * q^41 - 48 * q^45 + 16 * q^49 + 48 * q^53 - 12 * q^61 + 28 * q^69 - 24 * q^77 - 4 * q^81 - 48 * q^85 - 36 * q^89 - 24 * q^93 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.619657	+	1.07328i	−0.357759	+	0.619657i	−0.987586	−	0.157079i	$-0.949792\pi$
$3$	−0.619657	+	1.07328i	−0.357759	+	0.619657i	0.629827	+	0.776735i	$0.283126\pi$
$4$		0			0
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$	4.00552	−	2.31259i	1.51395	−	0.874077i	0.514079	−	0.857743i	$-0.328134\pi$
$7$	4.00552	−	2.31259i	1.51395	−	0.874077i	0.999867	−	0.0163346i	$-0.00519970\pi$
$8$		0			0
$9$	0.732051	+	1.26795i	0.244017	+	0.422650i
$10$		0			0
$11$	−1.85897	−	1.07328i	−0.560501	−	0.323605i	0.192846	−	0.981229i	$-0.438228\pi$
$11$	−1.85897	−	1.07328i	−0.560501	−	0.323605i	−0.753346	+	0.657624i	$0.771562\pi$
$12$		0			0
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$14$		0			0
$15$	−2.14655	−	1.23931i	−0.554238	−	0.319989i
$16$		0			0
$17$	0.232051	+	0.401924i	0.0562806	+	0.0974808i	0.892793	−	0.450467i	$-0.148743\pi$
$17$	0.232051	+	0.401924i	0.0562806	+	0.0974808i	−0.836512	+	0.547948i	$0.815409\pi$
$18$		0			0
$19$	4.00552	−	2.31259i	0.918930	−	0.530545i	0.0356367	−	0.999365i	$-0.488654\pi$
$19$	4.00552	−	2.31259i	0.918930	−	0.530545i	0.883294	+	0.468820i	$0.155321\pi$
$20$		0			0
$21$			5.73205i			1.25084i
$22$		0			0
$23$	−2.76621	+	4.79122i	−0.576795	+	0.999038i	0.419049	+	0.907964i	$0.362363\pi$
$23$	−2.76621	+	4.79122i	−0.576795	+	0.999038i	−0.995844	+	0.0910745i	$0.970970\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−5.53242			−1.06472
$28$		0			0
$29$	−1.50000	+	2.59808i	−0.278543	+	0.482451i	−0.971023	−	0.238987i	$-0.923185\pi$
$29$	−1.50000	+	2.59808i	−0.278543	+	0.482451i	0.692480	+	0.721437i	$0.256518\pi$
$30$		0			0
$31$			9.25036i			1.66141i	0.556710	+	0.830707i	$0.312064\pi$
$31$			9.25036i			1.66141i	−0.556710	+	0.830707i	$0.687936\pi$
$32$		0			0
$33$	2.30385	−	1.33013i	0.401048	−	0.231545i
$34$		0			0
$35$	4.62518	+	8.01105i	0.781798	+	1.35411i
$36$		0			0
$37$	6.69615	+	3.86603i	1.10084	+	0.635571i	0.936442	−	0.350823i	$-0.114098\pi$
$37$	6.69615	+	3.86603i	1.10084	+	0.635571i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$	−3.09828	−	3.21983i	−0.496123	−	0.515586i
$40$		0			0
$41$	−6.23205	−	3.59808i	−0.973283	−	0.561925i	−0.0730473	−	0.997328i	$-0.523272\pi$
$41$	−6.23205	−	3.59808i	−0.973283	−	0.561925i	−0.900235	+	0.435403i	$0.856606\pi$
$42$		0			0
$43$	−1.85897	−	3.21983i	−0.283490	−	0.491020i	0.688752	−	0.724997i	$-0.258159\pi$
$43$	−1.85897	−	3.21983i	−0.283490	−	0.491020i	−0.972242	+	0.233978i	$0.924826\pi$
$44$		0			0
$45$	−2.53590	+	1.46410i	−0.378029	+	0.218255i
$46$		0			0
$47$		−	11.7290i		−	1.71085i	−0.517927	−	0.855425i	$-0.673296\pi$
$47$		−	11.7290i		−	1.71085i	0.517927	−	0.855425i	$-0.326704\pi$
$48$		0			0
$49$	7.19615	−	12.4641i	1.02802	−	1.78059i
$50$		0			0
$51$	−0.575167			−0.0805396
$52$		0			0
$53$	2.53590			0.348332			0.174166	−	0.984716i	$-0.444277\pi$
$53$	2.53590			0.348332			0.174166	+	0.984716i	$0.444277\pi$
$54$		0			0
$55$	2.14655	−	3.71794i	0.289441	−	0.501327i
$56$		0			0
$57$			5.73205i			0.759229i
$58$		0			0
$59$	8.29863	−	4.79122i	1.08039	−	0.623763i	0.149388	−	0.988779i	$-0.452270\pi$
$59$	8.29863	−	4.79122i	1.08039	−	0.623763i	0.931002	+	0.365015i	$0.118936\pi$
$60$		0			0
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	0.753876	−	0.657017i	$-0.228182\pi$
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	−0.945931	+	0.324367i	$0.894849\pi$
$62$		0			0
$63$	5.86450	+	3.38587i	0.738857	+	0.426579i
$64$		0			0
$65$	−6.92820	−	2.00000i	−0.859338	−	0.248069i
$66$		0			0
$67$	−4.00552	−	2.31259i	−0.489353	−	0.282528i	0.234953	−	0.972007i	$-0.424506\pi$
$67$	−4.00552	−	2.31259i	−0.489353	−	0.282528i	−0.724306	+	0.689479i	$0.757840\pi$
$68$		0			0
$69$	−3.42820	−	5.93782i	−0.412707	−	0.714830i
$70$		0			0
$71$	−5.57691	+	3.21983i	−0.661858	+	0.382124i	−0.792984	−	0.609242i	$-0.791474\pi$
$71$	−5.57691	+	3.21983i	−0.661858	+	0.382124i	0.131127	+	0.991366i	$0.458141\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$	−0.619657	+	1.07328i	−0.0715518	+	0.123931i
$76$		0			0
$77$	−9.92820			−1.13142
$78$		0			0
$79$	4.29311			0.483012			0.241506	−	0.970399i	$-0.422359\pi$
$79$	4.29311			0.483012			0.241506	+	0.970399i	$0.422359\pi$
$80$		0			0
$81$	1.23205	−	2.13397i	0.136895	−	0.237108i
$82$		0			0
$83$		−	4.29311i		−	0.471230i	−0.971846	−	0.235615i	$-0.924290\pi$
$83$		−	4.29311i		−	0.471230i	0.971846	−	0.235615i	$-0.0757104\pi$
$84$		0			0
$85$	−0.803848	+	0.464102i	−0.0871895	+	0.0503389i
$86$		0			0
$87$	−1.85897	−	3.21983i	−0.199303	−	0.345202i
$88$		0			0
$89$	−1.03590	−	0.598076i	−0.109805	−	0.0633960i	0.444092	−	0.895981i	$-0.353526\pi$
$89$	−1.03590	−	0.598076i	−0.109805	−	0.0633960i	−0.553897	+	0.832585i	$0.686860\pi$
$90$		0			0
$91$	4.00552	+	16.1881i	0.419893	+	1.69698i
$92$		0			0
$93$	−9.92820	−	5.73205i	−1.02951	−	0.594386i
$94$		0			0
$95$	4.62518	+	8.01105i	0.474534	+	0.821916i
$96$		0			0
$97$	11.8923	−	6.86603i	1.20748	−	0.697139i	0.245272	−	0.969454i	$-0.421123\pi$
$97$	11.8923	−	6.86603i	1.20748	−	0.697139i	0.962208	+	0.272315i	$0.0877893\pi$
$98$		0			0
$99$		−	3.14277i		−	0.315861i
$100$		0			0
$101$	9.69615	−	16.7942i	0.964803	−	1.67109i	0.254660	−	0.967031i	$-0.418036\pi$
$101$	9.69615	−	16.7942i	0.964803	−	1.67109i	0.710143	−	0.704058i	$-0.248630\pi$
$102$		0			0
$103$	−8.58622			−0.846025			−0.423013	−	0.906124i	$-0.639027\pi$
$103$	−8.58622			−0.846025			−0.423013	+	0.906124i	$0.639027\pi$
$104$		0			0
$105$	−11.4641			−1.11878
$106$		0			0
$107$	1.85897	−	3.21983i	0.179713	−	0.311273i	−0.762069	−	0.647496i	$-0.775816\pi$
$107$	1.85897	−	3.21983i	0.179713	−	0.311273i	0.941782	+	0.336223i	$0.109150\pi$
$108$		0			0
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$	−8.29863	+	4.79122i	−0.787671	+	0.454762i
$112$		0			0
$113$	6.23205	+	10.7942i	0.586262	+	1.01544i	0.994717	+	0.102657i	$0.0327344\pi$
$113$	6.23205	+	10.7942i	0.586262	+	1.01544i	−0.408455	+	0.912779i	$0.633932\pi$
$114$		0			0
$115$	−9.58244	−	5.53242i	−0.893567	−	0.515901i
$116$		0			0
$117$	−5.12436	+	1.26795i	−0.473747	+	0.117222i
$118$		0			0
$119$	1.85897	+	1.07328i	0.170412	+	0.0983872i
$120$		0			0
$121$	−3.19615	−	5.53590i	−0.290559	−	0.503263i
$122$		0			0
$123$	7.72347	−	4.45915i	0.696401	−	0.402068i
$124$		0			0
$125$			12.0000i			1.07331i
$126$		0			0
$127$	1.85897	−	3.21983i	0.164957	−	0.285714i	−0.771683	−	0.636007i	$-0.780585\pi$
$127$	1.85897	−	3.21983i	0.164957	−	0.285714i	0.936640	+	0.350293i	$0.113918\pi$
$128$		0			0
$129$	4.60770			0.405685
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	10.6962	−	18.5263i	0.927474	−	1.60643i
$134$		0			0
$135$		−	11.0648i		−	0.952310i
$136$		0			0
$137$	−11.4282	+	6.59808i	−0.976377	+	0.563712i	−0.901174	−	0.433457i	$-0.857294\pi$
$137$	−11.4282	+	6.59808i	−0.976377	+	0.563712i	−0.0752028	+	0.997168i	$0.523960\pi$
$138$		0			0
$139$	−4.00552	−	6.93777i	−0.339744	−	0.588454i	0.644640	−	0.764486i	$-0.277007\pi$
$139$	−4.00552	−	6.93777i	−0.339744	−	0.588454i	−0.984384	+	0.176032i	$0.943674\pi$
$140$		0			0
$141$	12.5885	+	7.26795i	1.06014	+	0.612072i
$142$		0			0
$143$	5.57691	−	5.36639i	0.466365	−	0.448760i
$144$		0			0
$145$	−5.19615	−	3.00000i	−0.431517	−	0.249136i
$146$		0			0
$147$	8.91829	+	15.4469i	0.735568	+	1.27404i
$148$		0			0
$149$	10.0359	−	5.79423i	0.822173	−	0.474682i	−0.0289923	−	0.999580i	$-0.509230\pi$
$149$	10.0359	−	5.79423i	0.822173	−	0.474682i	0.851165	+	0.524898i	$0.175897\pi$
$150$		0			0
$151$			9.25036i			0.752784i	0.926461	+	0.376392i	$0.122835\pi$
$151$			9.25036i			0.752784i	−0.926461	+	0.376392i	$0.877165\pi$
$152$		0			0
$153$	−0.339746	+	0.588457i	−0.0274668	+	0.0475740i
$154$		0			0
$155$	−18.5007			−1.48601
$156$		0			0
$157$	4.39230			0.350544			0.175272	−	0.984520i	$-0.443920\pi$
$157$	4.39230			0.350544			0.175272	+	0.984520i	$0.443920\pi$
$158$		0			0
$159$	−1.57139	+	2.72172i	−0.124619	+	0.215847i
$160$		0			0
$161$			25.5885i			2.01665i
$162$		0			0
$163$	−21.5990	+	12.4702i	−1.69177	+	0.976741i	−0.738671	+	0.674066i	$0.764546\pi$
$163$	−21.5990	+	12.4702i	−1.69177	+	0.976741i	−0.953094	+	0.302675i	$0.902120\pi$
$164$		0			0
$165$	2.66025	+	4.60770i	0.207100	+	0.358709i
$166$		0			0
$167$	19.4525	+	11.2309i	1.50528	+	0.869072i	0.999981	+	0.00612500i	$0.00194966\pi$
$167$	19.4525	+	11.2309i	1.50528	+	0.869072i	0.505295	+	0.862947i	$0.331384\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$	5.86450	+	3.38587i	0.448469	+	0.258924i
$172$		0			0
$173$	−4.96410	−	8.59808i	−0.377414	−	0.653700i	0.613271	−	0.789872i	$-0.289853\pi$
$173$	−4.96410	−	8.59808i	−0.377414	−	0.653700i	−0.990685	+	0.136173i	$0.956520\pi$
$174$		0			0
$175$	4.00552	−	2.31259i	0.302789	−	0.174815i
$176$		0			0
$177$			11.8756i			0.892628i
$178$		0			0
$179$	1.85897	−	3.21983i	0.138946	−	0.240661i	−0.788152	−	0.615481i	$-0.788962\pi$
$179$	1.85897	−	3.21983i	0.138946	−	0.240661i	0.927098	+	0.374819i	$0.122295\pi$
$180$		0			0
$181$	−16.3923			−1.21843			−0.609215	−	0.793005i	$-0.708515\pi$
$181$	−16.3923			−1.21843			−0.609215	+	0.793005i	$0.708515\pi$
$182$		0			0
$183$	3.71794			0.274838
$184$		0			0
$185$	−7.73205	+	13.3923i	−0.568472	+	0.984622i
$186$		0			0
$187$		−	0.996219i		−	0.0728508i
$188$		0			0
$189$	−22.1603	+	12.7942i	−1.61192	+	0.930643i
$190$		0			0
$191$	−6.48415	−	11.2309i	−0.469177	−	0.812638i	0.530202	−	0.847871i	$-0.322116\pi$
$191$	−6.48415	−	11.2309i	−0.469177	−	0.812638i	−0.999379	+	0.0352331i	$0.988783\pi$
$192$		0			0
$193$	17.0885	+	9.86603i	1.23005	+	0.710172i	0.967041	−	0.254620i	$-0.0819505\pi$
$193$	17.0885	+	9.86603i	1.23005	+	0.710172i	0.263013	+	0.964792i	$0.415284\pi$
$194$		0			0
$195$	6.43966	−	6.19657i	0.461154	−	0.443745i
$196$		0			0
$197$	20.4282	+	11.7942i	1.45545	+	0.840304i	0.998782	−	0.0493338i	$-0.0157098\pi$
$197$	20.4282	+	11.7942i	1.45545	+	0.840304i	0.456667	+	0.889638i	$0.349043\pi$
$198$		0			0
$199$	−6.15208	−	10.6557i	−0.436109	−	0.755363i	0.561276	−	0.827629i	$-0.310311\pi$
$199$	−6.15208	−	10.6557i	−0.436109	−	0.755363i	−0.997385	+	0.0722651i	$0.976977\pi$
$200$		0			0
$201$	4.96410	−	2.86603i	0.350141	−	0.202154i
$202$		0			0
$203$			13.8755i			0.973872i
$204$		0			0
$205$	7.19615	−	12.4641i	0.502601	−	0.870531i
$206$		0			0
$207$	−8.10003			−0.562991
$208$		0			0
$209$	−9.92820			−0.686748
$210$		0			0
$211$	13.5880	−	23.5350i	0.935434	−	1.62022i	0.161575	−	0.986860i	$-0.448343\pi$
$211$	13.5880	−	23.5350i	0.935434	−	1.62022i	0.773859	−	0.633359i	$-0.218324\pi$
$212$		0			0
$213$		−	7.98076i		−	0.546833i
$214$		0			0
$215$	6.43966	−	3.71794i	0.439181	−	0.253561i
$216$		0			0
$217$	21.3923	+	37.0526i	1.45220	+	2.51529i
$218$		0			0
$219$	6.43966	+	3.71794i	0.435152	+	0.251235i
$220$		0			0
$221$	−1.62436	+	0.401924i	−0.109266	+	0.0270363i
$222$		0			0
$223$	−4.00552	−	2.31259i	−0.268230	−	0.154863i	0.359853	−	0.933009i	$-0.382827\pi$
$223$	−4.00552	−	2.31259i	−0.268230	−	0.154863i	−0.628083	+	0.778146i	$0.716160\pi$
$224$		0			0
$225$	0.732051	+	1.26795i	0.0488034	+	0.0845299i
$226$		0			0
$227$	−12.0166	+	6.93777i	−0.797568	+	0.460476i	−0.842620	−	0.538509i	$-0.818988\pi$
$227$	−12.0166	+	6.93777i	−0.797568	+	0.460476i	0.0450520	+	0.998985i	$0.485655\pi$
$228$		0			0
$229$			0.928203i			0.0613374i	0.999530	+	0.0306687i	$0.00976368\pi$
$229$			0.928203i			0.0613374i	−0.999530	+	0.0306687i	$0.990236\pi$
$230$		0			0
$231$	6.15208	−	10.6557i	0.404777	−	0.701094i
$232$		0			0
$233$	−14.5359			−0.952278			−0.476139	−	0.879370i	$-0.657964\pi$
$233$	−14.5359			−0.952278			−0.476139	+	0.879370i	$0.657964\pi$
$234$		0			0
$235$	23.4580			1.53023
$236$		0			0
$237$	−2.66025	+	4.60770i	−0.172802	+	0.299302i
$238$		0			0
$239$			4.29311i			0.277698i	0.990314	+	0.138849i	$0.0443403\pi$
$239$			4.29311i			0.277698i	−0.990314	+	0.138849i	$0.955660\pi$
$240$		0			0
$241$	18.6962	−	10.7942i	1.20433	−	0.695317i	0.242811	−	0.970074i	$-0.421930\pi$
$241$	18.6962	−	10.7942i	1.20433	−	0.695317i	0.961514	+	0.274756i	$0.0885971\pi$
$242$		0			0
$243$	−6.77174	−	11.7290i	−0.434407	−	0.752415i
$244$		0			0
$245$	24.9282	+	14.3923i	1.59260	+	0.919491i
$246$		0			0
$247$	4.00552	+	16.1881i	0.254865	+	1.03003i
$248$		0			0
$249$	4.60770	+	2.66025i	0.292001	+	0.168587i
$250$		0			0
$251$	−12.9238	−	22.3847i	−0.815744	−	1.41291i	−0.908792	−	0.417248i	$-0.862995\pi$
$251$	−12.9238	−	22.3847i	−0.815744	−	1.41291i	0.0930485	−	0.995662i	$-0.470339\pi$
$252$		0			0
$253$	10.2846	−	5.93782i	0.646588	−	0.373308i
$254$		0			0
$255$		−	1.15033i		−	0.0720368i
$256$		0			0
$257$	0.232051	−	0.401924i	0.0144749	−	0.0250713i	−0.858697	−	0.512483i	$-0.828726\pi$
$257$	0.232051	−	0.401924i	0.0144749	−	0.0250713i	0.873172	+	0.487412i	$0.162059\pi$
$258$		0			0
$259$	35.7621			2.22215
$260$		0			0
$261$	−4.39230			−0.271877
$262$		0			0
$263$	−12.0166	+	20.8133i	−0.740974	+	1.28340i	0.211079	+	0.977469i	$0.432302\pi$
$263$	−12.0166	+	20.8133i	−0.740974	+	1.28340i	−0.952052	+	0.305935i	$0.901031\pi$
$264$		0			0
$265$			5.07180i			0.311558i
$266$		0			0
$267$	1.28380	−	0.741204i	0.0785675	−	0.0453609i
$268$		0			0
$269$	−2.30385	−	3.99038i	−0.140468	−	0.243298i	0.787205	−	0.616692i	$-0.211527\pi$
$269$	−2.30385	−	3.99038i	−0.140468	−	0.243298i	−0.927673	+	0.373394i	$0.878194\pi$
$270$		0			0
$271$	−20.0276	−	11.5630i	−1.21659	−	0.702399i	−0.252404	−	0.967622i	$-0.581221\pi$
$271$	−20.0276	−	11.5630i	−1.21659	−	0.702399i	−0.964187	+	0.265222i	$0.914555\pi$
$272$		0			0
$273$	−19.8564	−	5.73205i	−1.20176	−	0.346919i
$274$		0			0
$275$	−1.85897	−	1.07328i	−0.112100	−	0.0647210i
$276$		0			0
$277$	1.69615	+	2.93782i	0.101912	+	0.176517i	0.912472	−	0.409138i	$-0.134171\pi$
$277$	1.69615	+	2.93782i	0.101912	+	0.176517i	−0.810560	+	0.585655i	$0.800837\pi$
$278$		0			0
$279$	−11.7290	+	6.77174i	−0.702196	+	0.405413i
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$	−9.87002	+	17.0954i	−0.586712	+	1.01621i	0.407948	+	0.913005i	$0.366245\pi$
$283$	−9.87002	+	17.0954i	−0.586712	+	1.01621i	−0.994660	+	0.103209i	$0.967089\pi$
$284$		0			0
$285$	−11.4641			−0.679075
$286$		0			0
$287$	−33.2835			−1.96466
$288$		0			0
$289$	8.39230	−	14.5359i	0.493665	−	0.855053i
$290$		0			0
$291$			17.0183i			0.997631i
$292$		0			0
$293$	−21.8205	+	12.5981i	−1.27477	+	0.735987i	−0.975881	−	0.218301i	$-0.929948\pi$
$293$	−21.8205	+	12.5981i	−1.27477	+	0.735987i	−0.298886	+	0.954289i	$0.596615\pi$
$294$		0			0
$295$	9.58244	+	16.5973i	0.557911	+	0.966330i
$296$		0			0
$297$	10.2846	+	5.93782i	0.596774	+	0.344547i
$298$		0			0
$299$	−13.8311	−	14.3737i	−0.799871	−	0.831250i
$300$		0			0
$301$	−14.8923	−	8.59808i	−0.858378	−	0.495585i
$302$		0			0
$303$	12.0166	+	20.8133i	0.690334	+	1.19569i
$304$		0			0
$305$	5.19615	−	3.00000i	0.297531	−	0.171780i
$306$		0			0
$307$			5.62140i			0.320830i	0.987050	+	0.160415i	$0.0512833\pi$
$307$			5.62140i			0.320830i	−0.987050	+	0.160415i	$0.948717\pi$
$308$		0			0
$309$	5.32051	−	9.21539i	0.302673	−	0.524245i
$310$		0			0
$311$	18.5007			1.04908			0.524540	−	0.851386i	$-0.324237\pi$
$311$	18.5007			1.04908			0.524540	+	0.851386i	$0.324237\pi$
$312$		0			0
$313$	−8.39230			−0.474361			−0.237181	−	0.971466i	$-0.576223\pi$
$313$	−8.39230			−0.474361			−0.237181	+	0.971466i	$0.576223\pi$
$314$		0			0
$315$	−6.77174	+	11.7290i	−0.381544	+	0.660854i
$316$		0			0
$317$			4.00000i			0.224662i	0.993671	+	0.112331i	$0.0358318\pi$
$317$			4.00000i			0.224662i	−0.993671	+	0.112331i	$0.964168\pi$
$318$		0			0
$319$	5.57691	−	3.21983i	0.312247	−	0.180276i
$320$		0			0
$321$	2.30385	+	3.99038i	0.128588	+	0.222721i
$322$		0			0
$323$	1.85897	+	1.07328i	0.103436	+	0.0597187i
$324$		0			0
$325$	−1.00000	+	3.46410i	−0.0554700	+	0.192154i
$326$		0			0
$327$	6.43966	+	3.71794i	0.356114	+	0.205603i
$328$		0			0
$329$	−27.1244	−	46.9808i	−1.49541	−	2.59013i
$330$		0			0
$331$	−2.43414	+	1.40535i	−0.133792	+	0.0772450i	−0.565402	−	0.824815i	$-0.691279\pi$
$331$	−2.43414	+	1.40535i	−0.133792	+	0.0772450i	0.431610	+	0.902060i	$0.357946\pi$
$332$		0			0
$333$			11.3205i			0.620360i
$334$		0			0
$335$	4.62518	−	8.01105i	0.252701	−	0.437690i
$336$		0			0
$337$	−0.392305			−0.0213702			−0.0106851	−	0.999943i	$-0.503401\pi$
$337$	−0.392305			−0.0213702			−0.0106851	+	0.999943i	$0.503401\pi$
$338$		0			0
$339$	−15.4469			−0.838962
$340$		0			0
$341$	9.92820	−	17.1962i	0.537642	−	0.931224i
$342$		0			0
$343$		−	34.1908i		−	1.84613i
$344$		0			0
$345$	11.8756	−	6.85641i	0.639363	−	0.369137i
$346$		0			0
$347$	7.39139	+	12.8023i	0.396791	+	0.687262i	0.993328	−	0.115324i	$-0.0367905\pi$
$347$	7.39139	+	12.8023i	0.396791	+	0.687262i	−0.596537	+	0.802585i	$0.703457\pi$
$348$		0			0
$349$	8.30385	+	4.79423i	0.444495	+	0.256629i	0.705502	−	0.708708i	$-0.250721\pi$
$349$	8.30385	+	4.79423i	0.444495	+	0.256629i	−0.261008	+	0.965337i	$0.584055\pi$
$350$		0			0
$351$	5.53242	−	19.1649i	0.295299	−	1.02295i
$352$		0			0
$353$	−7.96410	−	4.59808i	−0.423886	−	0.244731i	0.272852	−	0.962056i	$-0.412033\pi$
$353$	−7.96410	−	4.59808i	−0.423886	−	0.244731i	−0.696739	+	0.717325i	$0.745366\pi$
$354$		0			0
$355$	−6.43966	−	11.1538i	−0.341782	−	0.591983i
$356$		0			0
$357$	−2.30385	+	1.33013i	−0.121933	+	0.0703978i
$358$		0			0
$359$		−	4.29311i		−	0.226582i	−0.993562	−	0.113291i	$-0.963861\pi$
$359$		−	4.29311i		−	0.226582i	0.993562	−	0.113291i	$-0.0361392\pi$
$360$		0			0
$361$	1.19615	−	2.07180i	0.0629554	−	0.109042i
$362$		0			0
$363$	7.92207			0.415801
$364$		0			0
$365$	12.0000			0.628109
$366$		0			0
$367$	−0.287584	+	0.498110i	−0.0150117	+	0.0260011i	−0.873434	−	0.486943i	$-0.838112\pi$
$367$	−0.287584	+	0.498110i	−0.0150117	+	0.0260011i	0.858422	+	0.512944i	$0.171445\pi$
$368$		0			0
$369$		−	10.5359i		−	0.548477i
$370$		0			0
$371$	10.1576	−	5.86450i	0.527357	−	0.304469i
$372$		0			0
$373$	−4.69615	−	8.13397i	−0.243158	−	0.421161i	0.718454	−	0.695574i	$-0.244850\pi$
$373$	−4.69615	−	8.13397i	−0.243158	−	0.421161i	−0.961612	+	0.274413i	$0.911517\pi$
$374$		0			0
$375$	−12.8793	−	7.43588i	−0.665086	−	0.383987i
$376$		0			0
$377$	−7.50000	−	7.79423i	−0.386270	−	0.401423i
$378$		0			0
$379$	2.43414	+	1.40535i	0.125033	+	0.0721880i	0.561212	−	0.827672i	$-0.310335\pi$
$379$	2.43414	+	1.40535i	0.125033	+	0.0721880i	−0.436179	+	0.899860i	$0.643669\pi$
$380$		0			0
$381$	2.30385	+	3.99038i	0.118030	+	0.204433i
$382$		0			0
$383$	23.1704	−	13.3774i	1.18395	−	0.683555i	0.227026	−	0.973889i	$-0.427100\pi$
$383$	23.1704	−	13.3774i	1.18395	−	0.683555i	0.956926	+	0.290334i	$0.0937663\pi$
$384$		0			0
$385$		−	19.8564i		−	1.01198i
$386$		0			0
$387$	2.72172	−	4.71416i	0.138353	−	0.239634i
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	−2.56761			−0.129849
$392$		0			0
$393$		0			0
$394$		0			0
$395$			8.58622i			0.432019i
$396$		0			0
$397$	−21.6962	+	12.5263i	−1.08890	+	0.628676i	−0.933283	−	0.359142i	$-0.883069\pi$
$397$	−21.6962	+	12.5263i	−1.08890	+	0.628676i	−0.155616	+	0.987818i	$0.549736\pi$
$398$		0			0
$399$	13.2559	+	22.9599i	0.663624	+	1.14943i
$400$		0			0
$401$	−31.1603	−	17.9904i	−1.55607	−	0.898397i	−0.997627	−	0.0688532i	$-0.978066\pi$
$401$	−31.1603	−	17.9904i	−1.55607	−	0.898397i	−0.558442	−	0.829544i	$-0.688601\pi$
$402$		0			0
$403$	−32.0442	−	9.25036i	−1.59624	−	0.460793i
$404$		0			0
$405$	4.26795	+	2.46410i	0.212076	+	0.122442i
$406$		0			0
$407$	−8.29863	−	14.3737i	−0.411348	−	0.712476i
$408$		0			0
$409$	24.6962	−	14.2583i	1.22115	−	0.705029i	0.255984	−	0.966681i	$-0.417601\pi$
$409$	24.6962	−	14.2583i	1.22115	−	0.705029i	0.965162	+	0.261652i	$0.0842673\pi$
$410$		0			0
$411$		−	16.3542i		−	0.806692i
$412$		0			0
$413$	22.1603	−	38.3827i	1.09043	−	1.88869i
$414$		0			0
$415$	8.58622			0.421481
$416$		0			0
$417$	9.92820			0.486186
$418$		0			0
$419$	−12.0166	+	20.8133i	−0.587048	+	1.01680i	0.407569	+	0.913175i	$0.366377\pi$
$419$	−12.0166	+	20.8133i	−0.587048	+	1.01680i	−0.994617	+	0.103623i	$0.966957\pi$
$420$		0			0
$421$		−	0.928203i		−	0.0452379i	−0.999744	−	0.0226189i	$-0.992800\pi$
$421$		−	0.928203i		−	0.0452379i	0.999744	−	0.0226189i	$-0.00720044\pi$
$422$		0			0
$423$	14.8718	−	8.58622i	0.723090	−	0.417476i
$424$		0			0
$425$	0.232051	+	0.401924i	0.0112561	+	0.0194962i
$426$		0			0
$427$	−12.0166	−	6.93777i	−0.581523	−	0.335742i
$428$		0			0
$429$	2.30385	+	9.31089i	0.111231	+	0.449534i
$430$		0			0
$431$	18.4562	+	10.6557i	0.889006	+	0.513268i	0.873617	−	0.486614i	$-0.161768\pi$
$431$	18.4562	+	10.6557i	0.889006	+	0.513268i	0.0153885	+	0.999882i	$0.495101\pi$
$432$		0			0
$433$	−6.69615	−	11.5981i	−0.321797	−	0.557368i	0.659062	−	0.752088i	$-0.270953\pi$
$433$	−6.69615	−	11.5981i	−0.321797	−	0.557368i	−0.980859	+	0.194720i	$0.937620\pi$
$434$		0			0
$435$	6.43966	−	3.71794i	0.308758	−	0.178262i
$436$		0			0
$437$			25.5885i			1.22406i
$438$		0			0
$439$	−9.87002	+	17.0954i	−0.471070	+	0.815918i	−0.999452	−	0.0330889i	$-0.989466\pi$
$439$	−9.87002	+	17.0954i	−0.471070	+	0.815918i	0.528382	+	0.849007i	$0.322799\pi$
$440$		0			0
$441$	21.0718			1.00342
$442$		0			0
$443$	40.6304			1.93041			0.965205	−	0.261496i	$-0.0842157\pi$
$443$	40.6304			1.93041			0.965205	+	0.261496i	$0.0842157\pi$
$444$		0			0
$445$	1.19615	−	2.07180i	0.0567031	−	0.0982126i
$446$		0			0
$447$			14.3617i			0.679287i
$448$		0			0
$449$	32.5526	−	18.7942i	1.53625	−	0.886954i	0.537197	−	0.843457i	$-0.319483\pi$
$449$	32.5526	−	18.7942i	1.53625	−	0.886954i	0.999054	−	0.0434975i	$-0.0138501\pi$
$450$		0			0
$451$	7.72347	+	13.3774i	0.363684	+	0.629919i
$452$		0			0
$453$	−9.92820	−	5.73205i	−0.466468	−	0.269315i
$454$		0			0
$455$	−32.3763	+	8.01105i	−1.51782	+	0.375564i
$456$		0			0
$457$	−20.8923	−	12.0622i	−0.977301	−	0.564245i	−0.0758467	−	0.997119i	$-0.524166\pi$
$457$	−20.8923	−	12.0622i	−0.977301	−	0.564245i	−0.901454	+	0.432875i	$0.857499\pi$
$458$		0			0
$459$	−1.28380	−	2.22361i	−0.0599228	−	0.103789i
$460$		0			0
$461$	−2.76795	+	1.59808i	−0.128916	+	0.0744298i	−0.563071	−	0.826408i	$-0.690380\pi$
$461$	−2.76795	+	1.59808i	−0.128916	+	0.0744298i	0.434155	+	0.900838i	$0.357047\pi$
$462$		0			0
$463$			31.3801i			1.45835i	0.684325	+	0.729177i	$0.260097\pi$
$463$			31.3801i			1.45835i	−0.684325	+	0.729177i	$0.739903\pi$
$464$		0			0
$465$	11.4641	−	19.8564i	0.531635	−	0.920819i
$466$		0			0
$467$	9.25036			0.428056			0.214028	−	0.976828i	$-0.431342\pi$
$467$	9.25036			0.428056			0.214028	+	0.976828i	$0.431342\pi$
$468$		0			0
$469$	−21.3923			−0.987805
$470$		0			0
$471$	−2.72172	+	4.71416i	−0.125410	+	0.217217i
$472$		0			0
$473$			7.98076i			0.366956i
$474$		0			0
$475$	4.00552	−	2.31259i	0.183786	−	0.106109i
$476$		0			0
$477$	1.85641	+	3.21539i	0.0849990	+	0.147223i
$478$		0			0
$479$	−9.29485	−	5.36639i	−0.424693	−	0.245196i	0.272390	−	0.962187i	$-0.412186\pi$
$479$	−9.29485	−	5.36639i	−0.424693	−	0.245196i	−0.697083	+	0.716990i	$0.745519\pi$
$480$		0			0
$481$	−20.0885	+	19.3301i	−0.915955	+	0.881378i
$482$		0			0
$483$	−27.4635	−	15.8561i	−1.24963	−	0.721476i
$484$		0			0
$485$	13.7321	+	23.7846i	0.623540	+	1.08000i
$486$		0			0
$487$	−2.43414	+	1.40535i	−0.110301	+	0.0636825i	−0.554136	−	0.832426i	$-0.686951\pi$
$487$	−2.43414	+	1.40535i	−0.110301	+	0.0636825i	0.443834	+	0.896109i	$0.353618\pi$
$488$		0			0
$489$		−	30.9090i		−	1.39775i
$490$		0			0
$491$	6.48415	−	11.2309i	0.292626	−	0.506843i	−0.681804	−	0.731535i	$-0.738804\pi$
$491$	6.48415	−	11.2309i	0.292626	−	0.506843i	0.974430	+	0.224692i	$0.0721377\pi$
$492$		0			0
$493$	−1.39230			−0.0627063
$494$		0			0
$495$	6.28555			0.282514
$496$		0			0
$497$	−14.8923	+	25.7942i	−0.668011	+	1.15703i
$498$		0			0
$499$			9.25036i			0.414103i	0.978330	+	0.207052i	$0.0663868\pi$
$499$			9.25036i			0.414103i	−0.978330	+	0.207052i	$0.933613\pi$
$500$		0			0
$501$	−24.1077	+	13.9186i	−1.07705	+	0.621836i
$502$		0			0
$503$	−15.7345	−	27.2530i	−0.701567	−	1.21515i	−0.967916	−	0.251274i	$-0.919151\pi$
$503$	−15.7345	−	27.2530i	−0.701567	−	1.21515i	0.266349	−	0.963877i	$-0.414183\pi$
$504$		0			0
$505$	33.5885	+	19.3923i	1.49467	+	0.862946i
$506$		0			0
$507$	14.2521	−	7.51294i	0.632958	−	0.333661i
$508$		0			0
$509$	−23.5526	−	13.5981i	−1.04395	−	0.602724i	−0.123000	−	0.992407i	$-0.539251\pi$
$509$	−23.5526	−	13.5981i	−1.04395	−	0.602724i	−0.920949	+	0.389683i	$0.872585\pi$
$510$		0			0
$511$	−13.8755	−	24.0331i	−0.613818	−	1.06316i
$512$		0			0
$513$	−22.1603	+	12.7942i	−0.978399	+	0.564879i
$514$		0			0
$515$		−	17.1724i		−	0.756708i
$516$		0			0
$517$	−12.5885	+	21.8038i	−0.553640	+	0.958932i
$518$		0			0
$519$	12.3042			0.540093
$520$		0			0
$521$	6.24871			0.273761			0.136881	−	0.990588i	$-0.456292\pi$
$521$	6.24871			0.273761			0.136881	+	0.990588i	$0.456292\pi$
$522$		0			0
$523$	−3.43036	+	5.94155i	−0.149999	+	0.259806i	−0.931227	−	0.364440i	$-0.881260\pi$
$523$	−3.43036	+	5.94155i	−0.149999	+	0.259806i	0.781228	+	0.624246i	$0.214594\pi$
$524$		0			0
$525$			5.73205i			0.250167i
$526$		0			0
$527$	−3.71794	+	2.14655i	−0.161956	+	0.0935054i
$528$		0			0
$529$	−3.80385	−	6.58846i	−0.165385	−	0.286455i
$530$		0			0
$531$	12.1500	+	7.01483i	0.527267	+	0.304418i
$532$		0			0
$533$	18.6962	−	17.9904i	0.809820	−	0.779250i
$534$		0			0
$535$	6.43966	+	3.71794i	0.278411	+	0.160741i
$536$		0			0
$537$	2.30385	+	3.99038i	0.0994184	+	0.172198i
$538$		0			0
$539$	−26.7549	+	15.4469i	−1.15241	+	0.665346i
$540$		0			0
$541$			6.92820i			0.297867i	0.988847	+	0.148933i	$0.0475840\pi$
$541$			6.92820i			0.297867i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$	10.1576	−	17.5935i	0.435905	−	0.755009i
$544$		0			0
$545$	12.0000			0.514024
$546$		0			0
$547$	−8.58622			−0.367120			−0.183560	−	0.983008i	$-0.558762\pi$
$547$	−8.58622			−0.367120			−0.183560	+	0.983008i	$0.558762\pi$
$548$		0			0
$549$	2.19615	−	3.80385i	0.0937295	−	0.162344i
$550$		0			0
$551$			13.8755i			0.591118i
$552$		0			0
$553$	17.1962	−	9.92820i	0.731255	−	0.422190i
$554$		0			0
$555$	−9.58244	−	16.5973i	−0.406752	−	0.704515i
$556$		0			0
$557$	19.7487	+	11.4019i	0.836780	+	0.483115i	0.856168	−	0.516697i	$-0.172839\pi$
$557$	19.7487	+	11.4019i	0.836780	+	0.483115i	−0.0193886	+	0.999812i	$0.506172\pi$
$558$		0			0
$559$	13.0128	−	3.21983i	0.550383	−	0.136184i
$560$		0			0
$561$	1.06922	+	0.617314i	0.0451425	+	0.0260630i
$562$		0			0
$563$	19.4525	+	33.6926i	0.819823	+	1.41998i	0.905812	+	0.423679i	$0.139262\pi$
$563$	19.4525	+	33.6926i	0.819823	+	1.41998i	−0.0859889	+	0.996296i	$0.527405\pi$
$564$		0			0
$565$	−21.5885	+	12.4641i	−0.908233	+	0.524369i
$566$		0			0
$567$		−	11.3969i		−	0.478626i
$568$		0			0
$569$	−4.96410	+	8.59808i	−0.208106	+	0.360450i	−0.951118	−	0.308828i	$-0.900063\pi$
$569$	−4.96410	+	8.59808i	−0.208106	+	0.360450i	0.743012	+	0.669278i	$0.233397\pi$
$570$		0			0
$571$	−23.4580			−0.981686			−0.490843	−	0.871248i	$-0.663311\pi$
$571$	−23.4580			−0.981686			−0.490843	+	0.871248i	$0.663311\pi$
$572$		0			0
$573$	16.0718			0.671409
$574$		0			0
$575$	−2.76621	+	4.79122i	−0.115359	+	0.199808i
$576$		0			0
$577$			43.8564i			1.82577i	0.408221	+	0.912883i	$0.366149\pi$
$577$			43.8564i			1.82577i	−0.408221	+	0.912883i	$0.633851\pi$
$578$		0			0
$579$	−21.1780	+	12.2271i	−0.880126	+	0.508141i
$580$		0			0
$581$	−9.92820	−	17.1962i	−0.411891	−	0.713417i
$582$		0			0
$583$	−4.71416	−	2.72172i	−0.195241	−	0.112722i
$584$		0			0
$585$	−2.53590	−	10.2487i	−0.104846	−	0.423732i
$586$		0			0
$587$	19.4525	+	11.2309i	0.802889	+	0.463548i	0.844480	−	0.535587i	$-0.179909\pi$
$587$	19.4525	+	11.2309i	0.802889	+	0.463548i	−0.0415915	+	0.999135i	$0.513243\pi$
$588$		0			0
$589$	21.3923	+	37.0526i	0.881455	+	1.52672i
$590$		0			0
$591$	−25.3170	+	14.6167i	−1.04140	+	0.601253i
$592$		0			0
$593$			22.7846i			0.935652i	0.883821	+	0.467826i	$0.154963\pi$
$593$			22.7846i			0.935652i	−0.883821	+	0.467826i	$0.845037\pi$
$594$		0			0
$595$	−2.14655	+	3.71794i	−0.0880001	+	0.152421i
$596$		0			0
$597$	15.2487			0.624088
$598$		0			0
$599$	12.8793			0.526235			0.263117	−	0.964764i	$-0.415249\pi$
$599$	12.8793			0.526235			0.263117	+	0.964764i	$0.415249\pi$
$600$		0			0
$601$	−2.69615	+	4.66987i	−0.109978	+	0.190488i	−0.915761	−	0.401723i	$-0.868411\pi$
$601$	−2.69615	+	4.66987i	−0.109978	+	0.190488i	0.805783	+	0.592211i	$0.201745\pi$
$602$		0			0
$603$		−	6.77174i		−	0.275766i
$604$		0			0
$605$	11.0718	−	6.39230i	0.450133	−	0.259884i
$606$		0			0
$607$	2.43414	+	4.21605i	0.0987986	+	0.171124i	0.911188	−	0.411992i	$-0.135167\pi$
$607$	2.43414	+	4.21605i	0.0987986	+	0.171124i	−0.812389	+	0.583116i	$0.801833\pi$
$608$		0			0
$609$	−14.8923	−	8.59808i	−0.603467	−	0.348412i
$610$		0			0
$611$	40.6304	+	11.7290i	1.64373	+	0.474504i
$612$		0			0
$613$	0.696152	+	0.401924i	0.0281173	+	0.0162335i	0.513993	−	0.857795i	$-0.328166\pi$
$613$	0.696152	+	0.401924i	0.0281173	+	0.0162335i	−0.485875	+	0.874028i	$0.661499\pi$
$614$		0			0
$615$	8.91829	+	15.4469i	0.359620	+	0.622880i
$616$		0			0
$617$	10.0359	−	5.79423i	0.404030	−	0.233267i	−0.284191	−	0.958768i	$-0.591725\pi$
$617$	10.0359	−	5.79423i	0.404030	−	0.233267i	0.688221	+	0.725501i	$0.258392\pi$
$618$		0			0
$619$			9.25036i			0.371803i	0.982568	+	0.185902i	$0.0595206\pi$
$619$			9.25036i			0.371803i	−0.982568	+	0.185902i	$0.940479\pi$
$620$		0			0
$621$	15.3038	−	26.5070i	0.614122	−	1.06369i
$622$		0			0
$623$	−5.53242			−0.221652
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	6.15208	−	10.6557i	0.245690	−	0.425548i
$628$		0			0
$629$			3.58846i			0.143081i
$630$		0			0
$631$	−24.7418	+	14.2847i	−0.984955	+	0.568664i	−0.903762	−	0.428034i	$-0.859206\pi$
$631$	−24.7418	+	14.2847i	−0.984955	+	0.568664i	−0.0811925	+	0.996698i	$0.525873\pi$
$632$		0			0
$633$	16.8397	+	29.1673i	0.669320	+	1.15930i
$634$		0			0
$635$	6.43966	+	3.71794i	0.255550	+	0.147542i
$636$		0			0
$637$	35.9808	+	37.3923i	1.42561	+	1.48154i
$638$		0			0
$639$	−8.16517	−	4.71416i	−0.323009	−	0.186489i
$640$		0			0
$641$	9.69615	+	16.7942i	0.382975	+	0.663332i	0.991486	−	0.130213i	$-0.0415662\pi$
$641$	9.69615	+	16.7942i	0.382975	+	0.663332i	−0.608511	+	0.793545i	$0.708233\pi$
$642$		0			0
$643$	26.4673	−	15.2809i	1.04377	−	0.602620i	0.122870	−	0.992423i	$-0.460790\pi$
$643$	26.4673	−	15.2809i	1.04377	−	0.602620i	0.920898	+	0.389803i	$0.127457\pi$
$644$		0			0
$645$			9.21539i			0.362856i
$646$		0			0
$647$	−5.57691	+	9.65949i	−0.219251	+	0.379754i	−0.954579	−	0.297957i	$-0.903695\pi$
$647$	−5.57691	+	9.65949i	−0.219251	+	0.379754i	0.735328	+	0.677711i	$0.237028\pi$
$648$		0			0
$649$	−20.5692			−0.807412
$650$		0			0
$651$	−53.0236			−2.07816
$652$		0			0
$653$	−7.62436	+	13.2058i	−0.298364	+	0.516782i	−0.975762	−	0.218835i	$-0.929774\pi$
$653$	−7.62436	+	13.2058i	−0.298364	+	0.516782i	0.677398	+	0.735617i	$0.263108\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	7.60770	−	4.39230i	0.296804	−	0.171360i
$658$		0			0
$659$	−3.67345	−	6.36260i	−0.143097	−	0.247852i	0.785564	−	0.618780i	$-0.212373\pi$
$659$	−3.67345	−	6.36260i	−0.143097	−	0.247852i	−0.928661	+	0.370928i	$0.879040\pi$
$660$		0			0
$661$	−11.3038	−	6.52628i	−0.439669	−	0.253843i	0.263788	−	0.964581i	$-0.415028\pi$
$661$	−11.3038	−	6.52628i	−0.439669	−	0.253843i	−0.703457	+	0.710738i	$0.748361\pi$
$662$		0			0
$663$	0.575167	−	1.99244i	0.0223377	−	0.0773799i
$664$		0			0
$665$	37.0526	+	21.3923i	1.43684	+	0.829558i
$666$		0			0
$667$	−8.29863	−	14.3737i	−0.321324	−	0.556550i
$668$		0			0
$669$	4.96410	−	2.86603i	0.191923	−	0.110807i
$670$		0			0
$671$			6.43966i			0.248600i
$672$		0			0
$673$	−11.8923	+	20.5981i	−0.458415	+	0.793997i	−0.998877	−	0.0473705i	$-0.984916\pi$
$673$	−11.8923	+	20.5981i	−0.458415	+	0.793997i	0.540463	+	0.841368i	$0.318249\pi$
$674$		0			0
$675$	−5.53242			−0.212943
$676$		0			0
$677$	9.46410			0.363735			0.181867	−	0.983323i	$-0.441786\pi$
$677$	9.46410			0.363735			0.181867	+	0.983323i	$0.441786\pi$
$678$		0			0
$679$	31.7566	−	55.0041i	1.21871	−	2.11086i
$680$		0			0
$681$		−	17.1962i		−	0.658958i
$682$		0			0
$683$	8.29863	−	4.79122i	0.317538	−	0.183331i	−0.332756	−	0.943013i	$-0.607979\pi$
$683$	8.29863	−	4.79122i	0.317538	−	0.183331i	0.650295	+	0.759682i	$0.274645\pi$
$684$		0			0
$685$	−13.1962	−	22.8564i	−0.504199	−	0.873298i
$686$		0			0
$687$	−0.996219	−	0.575167i	−0.0380081	−	0.0219440i
$688$		0			0
$689$	−2.53590	+	8.78461i	−0.0966100	+	0.334667i
$690$		0			0
$691$	34.4783	+	19.9061i	1.31162	+	0.757263i	0.982364	−	0.186979i	$-0.0598695\pi$
$691$	34.4783	+	19.9061i	1.31162	+	0.757263i	0.329254	+	0.944241i	$0.393203\pi$
$692$		0			0
$693$	−7.26795	−	12.5885i	−0.276087	−	0.478196i
$694$		0			0
$695$	13.8755	−	8.01105i	0.526329	−	0.303876i
$696$		0			0
$697$		−	3.33975i		−	0.126502i
$698$		0			0
$699$	9.00727	−	15.6010i	0.340686	−	0.590086i
$700$		0			0
$701$	32.1051			1.21259			0.606297	−	0.795238i	$-0.292654\pi$
$701$	32.1051			1.21259			0.606297	+	0.795238i	$0.292654\pi$
$702$		0			0
$703$	35.7621			1.34879
$704$		0			0
$705$	−14.5359	+	25.1769i	−0.547454	+	0.948217i
$706$		0			0
$707$		−	89.6929i		−	3.37325i
$708$		0			0
$709$	−23.3038	+	13.4545i	−0.875194	+	0.505294i	−0.869071	−	0.494688i	$-0.835283\pi$
$709$	−23.3038	+	13.4545i	−0.875194	+	0.505294i	−0.00612347	+	0.999981i	$0.501949\pi$
$710$		0			0
$711$	3.14277	+	5.44344i	0.117863	+	0.204145i
$712$		0			0
$713$	−44.3205	−	25.5885i	−1.65982	−	0.958295i
$714$		0			0
$715$	10.7328	+	11.1538i	0.401383	+	0.417129i
$716$		0			0
$717$	−4.60770	−	2.66025i	−0.172078	−	0.0993490i
$718$		0			0
$719$	−6.48415	−	11.2309i	−0.241818	−	0.418841i	0.719414	−	0.694581i	$-0.244410\pi$
$719$	−6.48415	−	11.2309i	−0.241818	−	0.418841i	−0.961232	+	0.275740i	$0.911077\pi$
$720$		0			0
$721$	−34.3923	+	19.8564i	−1.28084	+	0.739491i
$722$		0			0
$723$			26.7549i			0.995024i
$724$		0			0
$725$	−1.50000	+	2.59808i	−0.0557086	+	0.0964901i
$726$		0			0
$727$	−17.1724			−0.636890			−0.318445	−	0.947941i	$-0.603161\pi$
$727$	−17.1724			−0.636890			−0.318445	+	0.947941i	$0.603161\pi$
$728$		0			0
$729$	24.1769			0.895441
$730$		0			0
$731$	0.862751	−	1.49433i	0.0319100	−	0.0552698i
$732$		0			0
$733$		−	15.7128i		−	0.580366i	−0.956971	−	0.290183i	$-0.906284\pi$
$733$		−	15.7128i		−	0.580366i	0.956971	−	0.290183i	$-0.0937162\pi$
$734$		0			0
$735$	−30.8939	+	17.8366i	−1.13954	+	0.657912i
$736$		0			0
$737$	4.96410	+	8.59808i	0.182855	+	0.316714i
$738$		0			0
$739$	31.1814	+	18.0026i	1.14703	+	0.662237i	0.948161	−	0.317790i	$-0.102941\pi$
$739$	31.1814	+	18.0026i	1.14703	+	0.662237i	0.198867	+	0.980027i	$0.436274\pi$
$740$		0			0
$741$	−19.8564	−	5.73205i	−0.729443	−	0.210572i
$742$		0			0
$743$	−37.0459	−	21.3885i	−1.35908	−	0.784667i	−0.369583	−	0.929198i	$-0.620500\pi$
$743$	−37.0459	−	21.3885i	−1.35908	−	0.784667i	−0.989500	+	0.144530i	$0.953833\pi$
$744$		0			0
$745$	11.5885	+	20.0718i	0.424568	+	0.735374i
$746$		0			0
$747$	5.44344	−	3.14277i	0.199165	−	0.114988i
$748$		0			0
$749$		−	17.1962i		−	0.628334i
$750$		0			0
$751$	−0.287584	+	0.498110i	−0.0104941	+	0.0181763i	−0.871225	−	0.490884i	$-0.836674\pi$
$751$	−0.287584	+	0.498110i	−0.0104941	+	0.0181763i	0.860731	+	0.509061i	$0.170007\pi$
$752$		0			0
$753$	32.0333			1.16736
$754$		0			0
$755$	−18.5007			−0.673310
$756$		0			0
$757$	11.3038	−	19.5788i	0.410845	−	0.711605i	−0.584137	−	0.811655i	$-0.698567\pi$
$757$	11.3038	−	19.5788i	0.410845	−	0.711605i	0.994982	+	0.100050i	$0.0319003\pi$
$758$		0			0
$759$			14.7176i			0.534217i
$760$		0			0
$761$	−2.76795	+	1.59808i	−0.100338	+	0.0579302i	−0.549329	−	0.835606i	$-0.685117\pi$
$761$	−2.76795	+	1.59808i	−0.100338	+	0.0579302i	0.448991	+	0.893536i	$0.351783\pi$
$762$		0			0
$763$	−13.8755	−	24.0331i	−0.502328	−	0.870058i
$764$		0			0
$765$	−1.17691	−	0.679492i	−0.0425514	−	0.0245671i
$766$		0			0
$767$	8.29863	+	33.5385i	0.299646	+	1.21101i
$768$		0			0
$769$	−36.4808	−	21.0622i	−1.31553	−	0.759522i	−0.332524	−	0.943095i	$-0.607900\pi$
$769$	−36.4808	−	21.0622i	−1.31553	−	0.759522i	−0.983006	+	0.183573i	$0.941234\pi$
$770$		0			0
$771$	0.287584	+	0.498110i	0.0103571	+	0.0179390i
$772$		0			0
$773$	14.5526	−	8.40192i	0.523419	−	0.302196i	−0.214913	−	0.976633i	$-0.568947\pi$
$773$	14.5526	−	8.40192i	0.523419	−	0.302196i	0.738332	+	0.674437i	$0.235614\pi$
$774$		0			0
$775$			9.25036i			0.332283i
$776$		0			0
$777$	−22.1603	+	38.3827i	−0.794995	+	1.37697i
$778$		0			0
$779$	−33.2835			−1.19251
$780$		0			0
$781$	13.8231			0.494629
$782$		0			0
$783$	8.29863	−	14.3737i	0.296569	−	0.513673i
$784$		0			0
$785$			8.78461i			0.313536i
$786$		0			0
$787$	23.3245	−	13.4664i	0.831429	−	0.480026i	−0.0229126	−	0.999737i	$-0.507294\pi$
$787$	23.3245	−	13.4664i	0.831429	−	0.480026i	0.854342	+	0.519712i	$0.173961\pi$
$788$		0			0
$789$	−14.8923	−	25.7942i	−0.530180	−	0.918299i
$790$		0			0

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.w (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.619657 + 1.07328i) q^{3} +2.00000i q^{5} +(4.00552 - 2.31259i) q^{7} +(0.732051 + 1.26795i) q^{9} +O(q^{10})\)	\(q+(-0.619657 + 1.07328i) q^{3} +2.00000i q^{5} +(4.00552 - 2.31259i) q^{7} +(0.732051 + 1.26795i) q^{9} +(-1.85897 - 1.07328i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(-2.14655 - 1.23931i) q^{15} +(0.232051 + 0.401924i) q^{17} +(4.00552 - 2.31259i) q^{19} +5.73205i q^{21} +(-2.76621 + 4.79122i) q^{23} +1.00000 q^{25} -5.53242 q^{27} +(-1.50000 + 2.59808i) q^{29} +9.25036i q^{31} +(2.30385 - 1.33013i) q^{33} +(4.62518 + 8.01105i) q^{35} +(6.69615 + 3.86603i) q^{37} +(-3.09828 - 3.21983i) q^{39} +(-6.23205 - 3.59808i) q^{41} +(-1.85897 - 3.21983i) q^{43} +(-2.53590 + 1.46410i) q^{45} -11.7290i q^{47} +(7.19615 - 12.4641i) q^{49} -0.575167 q^{51} +2.53590 q^{53} +(2.14655 - 3.71794i) q^{55} +5.73205i q^{57} +(8.29863 - 4.79122i) q^{59} +(-1.50000 - 2.59808i) q^{61} +(5.86450 + 3.38587i) q^{63} +(-6.92820 - 2.00000i) q^{65} +(-4.00552 - 2.31259i) q^{67} +(-3.42820 - 5.93782i) q^{69} +(-5.57691 + 3.21983i) q^{71} -6.00000i q^{73} +(-0.619657 + 1.07328i) q^{75} -9.92820 q^{77} +4.29311 q^{79} +(1.23205 - 2.13397i) q^{81} -4.29311i q^{83} +(-0.803848 + 0.464102i) q^{85} +(-1.85897 - 3.21983i) q^{87} +(-1.03590 - 0.598076i) q^{89} +(4.00552 + 16.1881i) q^{91} +(-9.92820 - 5.73205i) q^{93} +(4.62518 + 8.01105i) q^{95} +(11.8923 - 6.86603i) q^{97} -3.14277i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 8 q^{13} - 12 q^{17} + 8 q^{25} - 12 q^{29} + 60 q^{33} + 12 q^{37} - 36 q^{41} - 48 q^{45} + 16 q^{49} + 48 q^{53} - 12 q^{61} + 28 q^{69} - 24 q^{77} - 4 q^{81} - 48 q^{85} - 36 q^{89} - 24 q^{93} + 12 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^13 - 12 * q^17 + 8 * q^25 - 12 * q^29 + 60 * q^33 + 12 * q^37 - 36 * q^41 - 48 * q^45 + 16 * q^49 + 48 * q^53 - 12 * q^61 + 28 * q^69 - 24 * q^77 - 4 * q^81 - 48 * q^85 - 36 * q^89 - 24 * q^93 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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