LMFDB - Embedded newform 680.2.bd.b.361.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [680,2,Mod(81,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.81"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 680 = 2^{3} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	680.bd (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$5.42982733745$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 38 x^{18} + 597 x^{16} + 5004 x^{14} + 24072 x^{12} + 66452 x^{10} + 99328 x^{8} + 70784 x^{6} + \cdots + 16 $ x^20 + 38x^18 + 597x^16 + 5004x^14 + 24072x^12 + 66452x^10 + 99328x^8 + 70784x^6 + 19972x^4 + 1104*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			361.10
Root			$-2.91112i$ of defining polynomial
Character	$\chi$	$=$	680.361
Dual form			680.2.bd.b.81.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.05847 - 2.05847i) q^{3} +(-0.707107 + 0.707107i) q^{5} +(2.82095 + 2.82095i) q^{7} -5.47464i q^{9} +(4.19898 + 4.19898i) q^{11} -2.09669 q^{13} +2.91112i q^{15} +(3.90859 + 1.31259i) q^{17} -2.76433i q^{19} +11.6137 q^{21} +(-4.94663 - 4.94663i) q^{23} -1.00000i q^{25} +(-5.09398 - 5.09398i) q^{27} +(-3.23769 + 3.23769i) q^{29} +(-2.83189 + 2.83189i) q^{31} +17.2870 q^{33} -3.98942 q^{35} +(1.73551 - 1.73551i) q^{37} +(-4.31599 + 4.31599i) q^{39} +(-6.82233 - 6.82233i) q^{41} -8.10856i q^{43} +(3.87115 + 3.87115i) q^{45} +2.89927 q^{47} +8.91549i q^{49} +(10.7477 - 5.34380i) q^{51} +7.53227i q^{53} -5.93825 q^{55} +(-5.69030 - 5.69030i) q^{57} -5.43616i q^{59} +(-10.1370 - 10.1370i) q^{61} +(15.4437 - 15.4437i) q^{63} +(1.48259 - 1.48259i) q^{65} -10.9061 q^{67} -20.3650 q^{69} +(-2.59393 + 2.59393i) q^{71} +(0.889261 - 0.889261i) q^{73} +(-2.05847 - 2.05847i) q^{75} +23.6902i q^{77} +(-3.27892 - 3.27892i) q^{79} -4.54775 q^{81} +2.84677i q^{83} +(-3.69194 + 1.83565i) q^{85} +13.3294i q^{87} +13.9998 q^{89} +(-5.91466 - 5.91466i) q^{91} +11.6588i q^{93} +(1.95468 + 1.95468i) q^{95} +(1.92396 - 1.92396i) q^{97} +(22.9879 - 22.9879i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} + 4 q^{7} + 12 q^{11} + 4 q^{13} + 4 q^{17} + 16 q^{21} - 4 q^{23} - 16 q^{27} - 4 q^{29} - 4 q^{31} + 16 q^{33} - 8 q^{35} - 8 q^{37} - 8 q^{39} - 8 q^{41} + 4 q^{47} - 4 q^{51} - 8 q^{55}+ \cdots + 52 q^{99}+O(q^{100}) $ 20 * q - 4 * q^3 + 4 * q^7 + 12 * q^11 + 4 * q^13 + 4 * q^17 + 16 * q^21 - 4 * q^23 - 16 * q^27 - 4 * q^29 - 4 * q^31 + 16 * q^33 - 8 * q^35 - 8 * q^37 - 8 * q^39 - 8 * q^41 + 4 * q^47 - 4 * q^51 - 8 * q^55 - 24 * q^57 - 24 * q^61 + 60 * q^63 - 4 * q^65 - 56 * q^67 + 24 * q^69 - 40 * q^71 - 20 * q^73 + 4 * q^75 - 8 * q^79 + 4 * q^81 - 12 * q^85 + 28 * q^89 - 4 * q^95 + 24 * q^97 + 52 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/680\mathbb{Z}\right)^\times$.

$n$	$137$	$241$	$341$	$511$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.05847	−	2.05847i	1.18846	−	1.18846i	0.210968	−	0.977493i	$-0.432338\pi$
$3$	2.05847	−	2.05847i	1.18846	−	1.18846i	0.977493	−	0.210968i	$-0.0676616\pi$
$4$		0			0
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$6$		0			0
$7$	2.82095	+	2.82095i	1.06622	+	1.06622i	0.997646	+	0.0685719i	$0.0218442\pi$
$7$	2.82095	+	2.82095i	1.06622	+	1.06622i	0.0685719	+	0.997646i	$0.478156\pi$
$8$		0			0
$9$		−	5.47464i		−	1.82488i
$10$		0			0
$11$	4.19898	+	4.19898i	1.26604	+	1.26604i	0.948117	+	0.317922i	$0.102985\pi$
$11$	4.19898	+	4.19898i	1.26604	+	1.26604i	0.317922	+	0.948117i	$0.397015\pi$
$12$		0			0
$13$	−2.09669			−0.581518			−0.290759	−	0.956796i	$-0.593908\pi$
$13$	−2.09669			−0.581518			−0.290759	+	0.956796i	$0.593908\pi$
$14$		0			0
$15$			2.91112i			0.751649i
$16$		0			0
$17$	3.90859	+	1.31259i	0.947973	+	0.318351i
$17$	3.90859	+	1.31259i	0.947973	+	0.318351i
$18$		0			0
$19$		−	2.76433i		−	0.634181i	−0.948395	−	0.317090i	$-0.897294\pi$
$19$		−	2.76433i		−	0.634181i	0.948395	−	0.317090i	$-0.102706\pi$
$20$		0			0
$21$	11.6137			2.53432
$22$		0			0
$23$	−4.94663	−	4.94663i	−1.03144	−	1.03144i	−0.999489	−	0.0319542i	$-0.989827\pi$
$23$	−4.94663	−	4.94663i	−1.03144	−	1.03144i	−0.0319542	−	0.999489i	$-0.510173\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$	−5.09398	−	5.09398i	−0.980337	−	0.980337i
$28$		0			0
$29$	−3.23769	+	3.23769i	−0.601224	+	0.601224i	−0.940637	−	0.339413i	$-0.889772\pi$
$29$	−3.23769	+	3.23769i	−0.601224	+	0.601224i	0.339413	+	0.940637i	$0.389772\pi$
$30$		0			0
$31$	−2.83189	+	2.83189i	−0.508623	+	0.508623i	−0.914104	−	0.405481i	$-0.867104\pi$
$31$	−2.83189	+	2.83189i	−0.508623	+	0.508623i	0.405481	+	0.914104i	$0.367104\pi$
$32$		0			0
$33$	17.2870			3.00928
$34$		0			0
$35$	−3.98942			−0.674336
$36$		0			0
$37$	1.73551	−	1.73551i	0.285316	−	0.285316i	−0.549909	−	0.835225i	$-0.685337\pi$
$37$	1.73551	−	1.73551i	0.285316	−	0.285316i	0.835225	+	0.549909i	$0.185337\pi$
$38$		0			0
$39$	−4.31599	+	4.31599i	−0.691112	+	0.691112i
$40$		0			0
$41$	−6.82233	−	6.82233i	−1.06547	−	1.06547i	−0.997701	−	0.0677679i	$-0.978412\pi$
$41$	−6.82233	−	6.82233i	−1.06547	−	1.06547i	−0.0677679	−	0.997701i	$-0.521588\pi$
$42$		0			0
$43$		−	8.10856i		−	1.23654i	−0.785964	−	0.618272i	$-0.787833\pi$
$43$		−	8.10856i		−	1.23654i	0.785964	−	0.618272i	$-0.212167\pi$
$44$		0			0
$45$	3.87115	+	3.87115i	0.577078	+	0.577078i
$46$		0			0
$47$	2.89927			0.422902			0.211451	−	0.977389i	$-0.432181\pi$
$47$	2.89927			0.422902			0.211451	+	0.977389i	$0.432181\pi$
$48$		0			0
$49$			8.91549i			1.27364i
$50$		0			0
$51$	10.7477	−	5.34380i	1.50498	−	0.748281i
$52$		0			0
$53$			7.53227i			1.03464i	0.855793	+	0.517318i	$0.173070\pi$
$53$			7.53227i			1.03464i	−0.855793	+	0.517318i	$0.826930\pi$
$54$		0			0
$55$	−5.93825			−0.800713
$56$		0			0
$57$	−5.69030	−	5.69030i	−0.753699	−	0.753699i
$58$		0			0
$59$		−	5.43616i		−	0.707727i	−0.935297	−	0.353864i	$-0.884868\pi$
$59$		−	5.43616i		−	0.707727i	0.935297	−	0.353864i	$-0.115132\pi$
$60$		0			0
$61$	−10.1370	−	10.1370i	−1.29792	−	1.29792i	−0.929767	−	0.368149i	$-0.879992\pi$
$61$	−10.1370	−	10.1370i	−1.29792	−	1.29792i	−0.368149	−	0.929767i	$-0.620008\pi$
$62$		0			0
$63$	15.4437	−	15.4437i	1.94572	−	1.94572i
$64$		0			0
$65$	1.48259	−	1.48259i	0.183892	−	0.183892i
$66$		0			0
$67$	−10.9061			−1.33240			−0.666198	−	0.745775i	$-0.732080\pi$
$67$	−10.9061			−1.33240			−0.666198	+	0.745775i	$0.732080\pi$
$68$		0			0
$69$	−20.3650			−2.45166
$70$		0			0
$71$	−2.59393	+	2.59393i	−0.307843	+	0.307843i	−0.844072	−	0.536230i	$-0.819848\pi$
$71$	−2.59393	+	2.59393i	−0.307843	+	0.307843i	0.536230	+	0.844072i	$0.319848\pi$
$72$		0			0
$73$	0.889261	−	0.889261i	0.104080	−	0.104080i	−0.653149	−	0.757229i	$-0.726553\pi$
$73$	0.889261	−	0.889261i	0.104080	−	0.104080i	0.757229	+	0.653149i	$0.226553\pi$
$74$		0			0
$75$	−2.05847	−	2.05847i	−0.237692	−	0.237692i
$76$		0			0
$77$			23.6902i			2.69975i
$78$		0			0
$79$	−3.27892	−	3.27892i	−0.368907	−	0.368907i	0.498172	−	0.867078i	$-0.334005\pi$
$79$	−3.27892	−	3.27892i	−0.368907	−	0.368907i	−0.867078	+	0.498172i	$0.834005\pi$
$80$		0			0
$81$	−4.54775			−0.505306
$82$		0			0
$83$			2.84677i			0.312473i	0.987720	+	0.156237i	$0.0499362\pi$
$83$			2.84677i			0.312473i	−0.987720	+	0.156237i	$0.950064\pi$
$84$		0			0
$85$	−3.69194	+	1.83565i	−0.400447	+	0.199104i
$86$		0			0
$87$			13.3294i			1.42906i
$88$		0			0
$89$	13.9998			1.48397			0.741987	−	0.670414i	$-0.233883\pi$
$89$	13.9998			1.48397			0.741987	+	0.670414i	$0.233883\pi$
$90$		0			0
$91$	−5.91466	−	5.91466i	−0.620025	−	0.620025i
$92$		0			0
$93$			11.6588i			1.20896i
$94$		0			0
$95$	1.95468	+	1.95468i	0.200546	+	0.200546i
$96$		0			0
$97$	1.92396	−	1.92396i	0.195349	−	0.195349i	−0.602654	−	0.798003i	$-0.705890\pi$
$97$	1.92396	−	1.92396i	0.195349	−	0.195349i	0.798003	+	0.602654i	$0.205890\pi$
$98$		0			0
$99$	22.9879	−	22.9879i	2.31037	−	2.31037i
$100$		0			0
$101$	11.5378			1.14805			0.574027	−	0.818837i	$-0.305381\pi$
$101$	11.5378			1.14805			0.574027	+	0.818837i	$0.305381\pi$
$102$		0			0
$103$	10.1567			1.00077			0.500385	−	0.865803i	$-0.333192\pi$
$103$	10.1567			1.00077			0.500385	+	0.865803i	$0.333192\pi$
$104$		0			0
$105$	−8.21213	+	8.21213i	−0.801422	+	0.801422i
$106$		0			0
$107$	−5.57763	+	5.57763i	−0.539210	+	0.539210i	−0.923297	−	0.384087i	$-0.874516\pi$
$107$	−5.57763	+	5.57763i	−0.539210	+	0.539210i	0.384087	+	0.923297i	$0.374516\pi$
$108$		0			0
$109$	−1.38413	−	1.38413i	−0.132576	−	0.132576i	0.637705	−	0.770281i	$-0.279884\pi$
$109$	−1.38413	−	1.38413i	−0.132576	−	0.132576i	−0.770281	+	0.637705i	$0.779884\pi$
$110$		0			0
$111$		−	7.14501i		−	0.678174i
$112$		0			0
$113$	14.2440	+	14.2440i	1.33996	+	1.33996i	0.896093	+	0.443866i	$0.146393\pi$
$113$	14.2440	+	14.2440i	1.33996	+	1.33996i	0.443866	+	0.896093i	$0.353607\pi$
$114$		0			0
$115$	6.99559			0.652342
$116$		0			0
$117$			11.4786i			1.06120i
$118$		0			0
$119$	7.32317	+	14.7287i	0.671314	+	1.35018i
$120$		0			0
$121$			24.2628i			2.20571i
$122$		0			0
$123$	−28.0872			−2.53254
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$		−	4.63445i		−	0.411241i	−0.978632	−	0.205620i	$-0.934079\pi$
$127$		−	4.63445i		−	0.411241i	0.978632	−	0.205620i	$-0.0659212\pi$
$128$		0			0
$129$	−16.6913	−	16.6913i	−1.46958	−	1.46958i
$130$		0			0
$131$	−13.3285	+	13.3285i	−1.16452	+	1.16452i	−0.181043	+	0.983475i	$0.557947\pi$
$131$	−13.3285	+	13.3285i	−1.16452	+	1.16452i	−0.983475	+	0.181043i	$0.942053\pi$
$132$		0			0
$133$	7.79803	−	7.79803i	0.676175	−	0.676175i
$134$		0			0
$135$	7.20398			0.620020
$136$		0			0
$137$	0.159296			0.0136096			0.00680479	−	0.999977i	$-0.497834\pi$
$137$	0.159296			0.0136096			0.00680479	+	0.999977i	$0.497834\pi$
$138$		0			0
$139$	−12.1081	+	12.1081i	−1.02699	+	1.02699i	−0.0273687	+	0.999625i	$0.508713\pi$
$139$	−12.1081	+	12.1081i	−1.02699	+	1.02699i	−0.999625	+	0.0273687i	$0.991287\pi$
$140$		0			0
$141$	5.96808	−	5.96808i	0.502603	−	0.502603i
$142$		0			0
$143$	−8.80397	−	8.80397i	−0.736225	−	0.736225i
$144$		0			0
$145$		−	4.57879i		−	0.380248i
$146$		0			0
$147$	18.3523	+	18.3523i	1.51367	+	1.51367i
$148$		0			0
$149$	3.66668			0.300386			0.150193	−	0.988657i	$-0.452010\pi$
$149$	3.66668			0.300386			0.150193	+	0.988657i	$0.452010\pi$
$150$		0			0
$151$		−	12.9239i		−	1.05173i	−0.850568	−	0.525865i	$-0.823742\pi$
$151$		−	12.9239i		−	1.05173i	0.850568	−	0.525865i	$-0.176258\pi$
$152$		0			0
$153$	7.18598	−	21.3981i	0.580952	−	1.72994i
$154$		0			0
$155$		−	4.00490i		−	0.321682i
$156$		0			0
$157$	−4.71806			−0.376542			−0.188271	−	0.982117i	$-0.560288\pi$
$157$	−4.71806			−0.376542			−0.188271	+	0.982117i	$0.560288\pi$
$158$		0			0
$159$	15.5050	+	15.5050i	1.22963	+	1.22963i
$160$		0			0
$161$		−	27.9084i		−	2.19949i
$162$		0			0
$163$	−8.67514	−	8.67514i	−0.679489	−	0.679489i	0.280395	−	0.959885i	$-0.409534\pi$
$163$	−8.67514	−	8.67514i	−0.679489	−	0.679489i	−0.959885	+	0.280395i	$0.909534\pi$
$164$		0			0
$165$	−12.2237	+	12.2237i	−0.951617	+	0.951617i
$166$		0			0
$167$	−5.70359	+	5.70359i	−0.441357	+	0.441357i	−0.892468	−	0.451111i	$-0.851028\pi$
$167$	−5.70359	+	5.70359i	−0.441357	+	0.441357i	0.451111	+	0.892468i	$0.351028\pi$
$168$		0			0
$169$	−8.60388			−0.661837
$170$		0			0
$171$	−15.1337			−1.15730
$172$		0			0
$173$	−13.6664	+	13.6664i	−1.03904	+	1.03904i	−0.0398333	+	0.999206i	$0.512683\pi$
$173$	−13.6664	+	13.6664i	−1.03904	+	1.03904i	−0.999206	+	0.0398333i	$0.987317\pi$
$174$		0			0
$175$	2.82095	−	2.82095i	0.213244	−	0.213244i
$176$		0			0
$177$	−11.1902	−	11.1902i	−0.841106	−	0.841106i
$178$		0			0
$179$		−	14.7409i		−	1.10179i	−0.834575	−	0.550895i	$-0.814287\pi$
$179$		−	14.7409i		−	1.10179i	0.834575	−	0.550895i	$-0.185713\pi$
$180$		0			0
$181$	4.34077	+	4.34077i	0.322647	+	0.322647i	0.849782	−	0.527135i	$-0.176734\pi$
$181$	4.34077	+	4.34077i	0.322647	+	0.322647i	−0.527135	+	0.849782i	$0.676734\pi$
$182$		0			0
$183$	−41.7337			−3.08505
$184$		0			0
$185$			2.45438i			0.180450i
$186$		0			0
$187$	10.9005	+	21.9236i	0.797126	+	1.60322i
$188$		0			0
$189$		−	28.7397i		−	2.09051i
$190$		0			0
$191$	9.49122			0.686761			0.343380	−	0.939196i	$-0.388428\pi$
$191$	9.49122			0.686761			0.343380	+	0.939196i	$0.388428\pi$
$192$		0			0
$193$	3.18113	+	3.18113i	0.228983	+	0.228983i	0.812268	−	0.583285i	$-0.198233\pi$
$193$	3.18113	+	3.18113i	0.228983	+	0.228983i	−0.583285	+	0.812268i	$0.698233\pi$
$194$		0			0
$195$		−	6.10373i		−	0.437097i
$196$		0			0
$197$	8.97616	+	8.97616i	0.639525	+	0.639525i	0.950438	−	0.310913i	$-0.100635\pi$
$197$	8.97616	+	8.97616i	0.639525	+	0.639525i	−0.310913	+	0.950438i	$0.600635\pi$
$198$		0			0
$199$	16.8897	−	16.8897i	1.19728	−	1.19728i	0.222298	−	0.974979i	$-0.428644\pi$
$199$	16.8897	−	16.8897i	1.19728	−	1.19728i	0.974979	−	0.222298i	$-0.0713557\pi$
$200$		0			0
$201$	−22.4500	+	22.4500i	−1.58350	+	1.58350i
$202$		0			0
$203$	−18.2667			−1.28207
$204$		0			0
$205$	9.64823			0.673862
$206$		0			0
$207$	−27.0810	+	27.0810i	−1.88226	+	1.88226i
$208$		0			0
$209$	11.6074	−	11.6074i	0.802897	−	0.802897i
$210$		0			0
$211$	−6.98903	−	6.98903i	−0.481145	−	0.481145i	0.424352	−	0.905497i	$-0.360502\pi$
$211$	−6.98903	−	6.98903i	−0.481145	−	0.481145i	−0.905497	+	0.424352i	$0.860502\pi$
$212$		0			0
$213$			10.6791i			0.731718i
$214$		0			0
$215$	5.73362	+	5.73362i	0.391029	+	0.391029i
$216$		0			0
$217$	−15.9772			−1.08461
$218$		0			0
$219$		−	3.66104i		−	0.247390i
$220$		0			0
$221$	−8.19512	−	2.75211i	−0.551264	−	0.185127i
$222$		0			0
$223$			15.3642i			1.02886i	0.857531	+	0.514432i	$0.171997\pi$
$223$			15.3642i			1.02886i	−0.857531	+	0.514432i	$0.828003\pi$
$224$		0			0
$225$	−5.47464			−0.364976
$226$		0			0
$227$	−12.9872	−	12.9872i	−0.861990	−	0.861990i	0.129579	−	0.991569i	$-0.458637\pi$
$227$	−12.9872	−	12.9872i	−0.861990	−	0.861990i	−0.991569	+	0.129579i	$0.958637\pi$
$228$		0			0
$229$		−	25.2673i		−	1.66971i	−0.550471	−	0.834854i	$-0.685552\pi$
$229$		−	25.2673i		−	1.66971i	0.550471	−	0.834854i	$-0.314448\pi$
$230$		0			0
$231$	48.7656	+	48.7656i	3.20854	+	3.20854i
$232$		0			0
$233$	−6.43800	+	6.43800i	−0.421768	+	0.421768i	−0.885812	−	0.464044i	$-0.846398\pi$
$233$	−6.43800	+	6.43800i	−0.421768	+	0.421768i	0.464044	+	0.885812i	$0.346398\pi$
$234$		0			0
$235$	−2.05010	+	2.05010i	−0.133733	+	0.133733i
$236$		0			0
$237$	−13.4991			−0.876863
$238$		0			0
$239$	−9.90234			−0.640529			−0.320265	−	0.947328i	$-0.603772\pi$
$239$	−9.90234			−0.640529			−0.320265	+	0.947328i	$0.603772\pi$
$240$		0			0
$241$	−0.262472	+	0.262472i	−0.0169073	+	0.0169073i	−0.715510	−	0.698603i	$-0.753806\pi$
$241$	−0.262472	+	0.262472i	−0.0169073	+	0.0169073i	0.698603	+	0.715510i	$0.253806\pi$
$242$		0			0
$243$	5.92051	−	5.92051i	0.379801	−	0.379801i
$244$		0			0
$245$	−6.30421	−	6.30421i	−0.402761	−	0.402761i
$246$		0			0
$247$			5.79595i			0.368788i
$248$		0			0
$249$	5.86000	+	5.86000i	0.371362	+	0.371362i
$250$		0			0
$251$	−24.0817			−1.52002			−0.760011	−	0.649911i	$-0.774806\pi$
$251$	−24.0817			−1.52002			−0.760011	+	0.649911i	$0.774806\pi$
$252$		0			0
$253$		−	41.5416i		−	2.61170i
$254$		0			0
$255$	−3.82112	+	11.3784i	−0.239288	+	0.712543i
$256$		0			0
$257$			5.67250i			0.353841i	0.984225	+	0.176920i	$0.0566135\pi$
$257$			5.67250i			0.353841i	−0.984225	+	0.176920i	$0.943386\pi$
$258$		0			0
$259$	9.79157			0.608418
$260$		0			0
$261$	17.7252	+	17.7252i	1.09716	+	1.09716i
$262$		0			0
$263$			24.0031i			1.48009i	0.672555	+	0.740047i	$0.265197\pi$
$263$			24.0031i			1.48009i	−0.672555	+	0.740047i	$0.734803\pi$
$264$		0			0
$265$	−5.32612	−	5.32612i	−0.327181	−	0.327181i
$266$		0			0
$267$	28.8182	−	28.8182i	1.76365	−	1.76365i
$268$		0			0
$269$	12.6651	−	12.6651i	0.772204	−	0.772204i	−0.206288	−	0.978491i	$-0.566138\pi$
$269$	12.6651	−	12.6651i	0.772204	−	0.772204i	0.978491	+	0.206288i	$0.0661383\pi$
$270$		0			0
$271$	1.45621			0.0884586			0.0442293	−	0.999021i	$-0.485917\pi$
$271$	1.45621			0.0884586			0.0442293	+	0.999021i	$0.485917\pi$
$272$		0			0
$273$	−24.3504			−1.47375
$274$		0			0
$275$	4.19898	−	4.19898i	0.253208	−	0.253208i
$276$		0			0
$277$	−9.01170	+	9.01170i	−0.541460	+	0.541460i	−0.923957	−	0.382497i	$-0.875064\pi$
$277$	−9.01170	+	9.01170i	−0.541460	+	0.541460i	0.382497	+	0.923957i	$0.375064\pi$
$278$		0			0
$279$	15.5036	+	15.5036i	0.928176	+	0.928176i
$280$		0			0
$281$		−	15.8563i		−	0.945908i	−0.881087	−	0.472954i	$-0.843188\pi$
$281$		−	15.8563i		−	0.945908i	0.881087	−	0.472954i	$-0.156812\pi$
$282$		0			0
$283$	2.73612	+	2.73612i	0.162645	+	0.162645i	0.783737	−	0.621092i	$-0.213311\pi$
$283$	2.73612	+	2.73612i	0.162645	+	0.162645i	−0.621092	+	0.783737i	$0.713311\pi$
$284$		0			0
$285$	8.04730			0.476681
$286$		0			0
$287$		−	38.4909i		−	2.27204i
$288$		0			0
$289$	13.5542	+	10.2608i	0.797305	+	0.603576i
$290$		0			0
$291$		−	7.92087i		−	0.464329i
$292$		0			0
$293$	18.3190			1.07021			0.535104	−	0.844786i	$-0.320272\pi$
$293$	18.3190			1.07021			0.535104	+	0.844786i	$0.320272\pi$
$294$		0			0
$295$	3.84394	+	3.84394i	0.223803	+	0.223803i
$296$		0			0
$297$		−	42.7790i		−	2.48229i
$298$		0			0
$299$	10.3716	+	10.3716i	0.599803	+	0.599803i
$300$		0			0
$301$	22.8738	−	22.8738i	1.31843	−	1.31843i
$302$		0			0
$303$	23.7503	−	23.7503i	1.36442	−	1.36442i
$304$		0			0
$305$	14.3360			0.820874
$306$		0			0
$307$	6.82114			0.389303			0.194651	−	0.980872i	$-0.437642\pi$
$307$	6.82114			0.389303			0.194651	+	0.980872i	$0.437642\pi$
$308$		0			0
$309$	20.9073	−	20.9073i	1.18938	−	1.18938i
$310$		0			0
$311$	−9.76974	+	9.76974i	−0.553991	+	0.553991i	−0.927590	−	0.373599i	$-0.878124\pi$
$311$	−9.76974	+	9.76974i	−0.553991	+	0.553991i	0.373599	+	0.927590i	$0.378124\pi$
$312$		0			0
$313$	2.55820	+	2.55820i	0.144598	+	0.144598i	0.775700	−	0.631102i	$-0.217397\pi$
$313$	2.55820	+	2.55820i	0.144598	+	0.144598i	−0.631102	+	0.775700i	$0.717397\pi$
$314$		0			0
$315$			21.8406i			1.23058i
$316$		0			0
$317$	−10.7924	−	10.7924i	−0.606161	−	0.606161i	0.335780	−	0.941940i	$-0.391000\pi$
$317$	−10.7924	−	10.7924i	−0.606161	−	0.606161i	−0.941940	+	0.335780i	$0.891000\pi$
$318$		0			0
$319$	−27.1900			−1.52235
$320$		0			0
$321$			22.9628i			1.28166i
$322$		0			0
$323$	3.62844	−	10.8046i	0.201892	−	0.601186i
$324$		0			0
$325$			2.09669i			0.116304i
$326$		0			0
$327$	−5.69841			−0.315123
$328$		0			0
$329$	8.17870	+	8.17870i	0.450906	+	0.450906i
$330$		0			0
$331$		−	27.3436i		−	1.50294i	−0.659767	−	0.751470i	$-0.729345\pi$
$331$		−	27.3436i		−	1.50294i	0.659767	−	0.751470i	$-0.270655\pi$
$332$		0			0
$333$	−9.50129	−	9.50129i	−0.520668	−	0.520668i
$334$		0			0
$335$	7.71180	−	7.71180i	0.421341	−	0.421341i
$336$		0			0
$337$	−5.41215	+	5.41215i	−0.294818	+	0.294818i	−0.838980	−	0.544162i	$-0.816848\pi$
$337$	−5.41215	+	5.41215i	−0.294818	+	0.294818i	0.544162	+	0.838980i	$0.316848\pi$
$338$		0			0
$339$	58.6417			3.18498
$340$		0			0
$341$	−23.7821			−1.28787
$342$		0			0
$343$	−5.40351	+	5.40351i	−0.291762	+	0.291762i
$344$		0			0
$345$	14.4002	−	14.4002i	0.775283	−	0.775283i
$346$		0			0
$347$	23.3630	+	23.3630i	1.25419	+	1.25419i	0.953824	+	0.300365i	$0.0971085\pi$
$347$	23.3630	+	23.3630i	1.25419	+	1.25419i	0.300365	+	0.953824i	$0.402892\pi$
$348$		0			0
$349$			0.517999i			0.0277278i	0.999904	+	0.0138639i	$0.00441316\pi$
$349$			0.517999i			0.0277278i	−0.999904	+	0.0138639i	$0.995587\pi$
$350$		0			0
$351$	10.6805	+	10.6805i	0.570084	+	0.570084i
$352$		0			0
$353$	31.2253			1.66195			0.830976	−	0.556308i	$-0.187782\pi$
$353$	31.2253			1.66195			0.830976	+	0.556308i	$0.187782\pi$
$354$		0			0
$355$		−	3.66837i		−	0.194697i
$356$		0			0
$357$	45.3932	+	15.2441i	2.40246	+	0.806802i
$358$		0			0
$359$			11.6951i			0.617243i	0.951185	+	0.308622i	$0.0998677\pi$
$359$			11.6951i			0.617243i	−0.951185	+	0.308622i	$0.900132\pi$
$360$		0			0
$361$	11.3585			0.597815
$362$		0			0
$363$	49.9444	+	49.9444i	2.62140	+	2.62140i
$364$		0			0
$365$			1.25760i			0.0658260i
$366$		0			0
$367$	0.454347	+	0.454347i	0.0237167	+	0.0237167i	0.718866	−	0.695149i	$-0.244662\pi$
$367$	0.454347	+	0.454347i	0.0237167	+	0.0237167i	−0.695149	+	0.718866i	$0.744662\pi$
$368$		0			0
$369$	−37.3498	+	37.3498i	−1.94435	+	1.94435i
$370$		0			0
$371$	−21.2481	+	21.2481i	−1.10315	+	1.10315i
$372$		0			0
$373$	−13.9448			−0.722035			−0.361017	−	0.932559i	$-0.617571\pi$
$373$	−13.9448			−0.722035			−0.361017	+	0.932559i	$0.617571\pi$
$374$		0			0
$375$	2.91112			0.150330
$376$		0			0
$377$	6.78845	−	6.78845i	0.349623	−	0.349623i
$378$		0			0
$379$	−15.0574	+	15.0574i	−0.773447	+	0.773447i	−0.978707	−	0.205261i	$-0.934196\pi$
$379$	−15.0574	+	15.0574i	−0.773447	+	0.773447i	0.205261	+	0.978707i	$0.434196\pi$
$380$		0			0
$381$	−9.53989	−	9.53989i	−0.488743	−	0.488743i
$382$		0			0
$383$			27.7632i			1.41863i	0.704889	+	0.709317i	$0.250997\pi$
$383$			27.7632i			1.41863i	−0.704889	+	0.709317i	$0.749003\pi$
$384$		0			0
$385$	−16.7515	−	16.7515i	−0.853735	−	0.853735i
$386$		0			0
$387$	−44.3914			−2.25654
$388$		0			0
$389$		−	15.8342i		−	0.802828i	−0.915897	−	0.401414i	$-0.868519\pi$
$389$		−	15.8342i		−	0.802828i	0.915897	−	0.401414i	$-0.131481\pi$
$390$		0			0
$391$	−12.8414	−	25.8273i	−0.649420	−	1.30614i
$392$		0			0
$393$			54.8729i			2.76797i
$394$		0			0
$395$	4.63709			0.233317
$396$		0			0
$397$	11.4157	+	11.4157i	0.572936	+	0.572936i	0.932948	−	0.360012i	$-0.117227\pi$
$397$	11.4157	+	11.4157i	0.572936	+	0.572936i	−0.360012	+	0.932948i	$0.617227\pi$
$398$		0			0
$399$		−	32.1041i		−	1.60721i
$400$		0			0
$401$	1.96450	+	1.96450i	0.0981022	+	0.0981022i	0.754455	−	0.656352i	$-0.227901\pi$
$401$	1.96450	+	1.96450i	0.0981022	+	0.0981022i	−0.656352	+	0.754455i	$0.727901\pi$
$402$		0			0
$403$	5.93761	−	5.93761i	0.295774	−	0.295774i
$404$		0			0
$405$	3.21574	−	3.21574i	0.159792	−	0.159792i
$406$		0			0
$407$	14.5747			0.722443
$408$		0			0
$409$	−3.92333			−0.193996			−0.0969982	−	0.995285i	$-0.530924\pi$
$409$	−3.92333			−0.193996			−0.0969982	+	0.995285i	$0.530924\pi$
$410$		0			0
$411$	0.327907	−	0.327907i	0.0161745	−	0.0161745i
$412$		0			0
$413$	15.3351	−	15.3351i	0.754592	−	0.754592i
$414$		0			0
$415$	−2.01297	−	2.01297i	−0.0988127	−	0.0988127i
$416$		0			0
$417$			49.8484i			2.44108i
$418$		0			0
$419$	7.79273	+	7.79273i	0.380700	+	0.380700i	0.871354	−	0.490654i	$-0.163242\pi$
$419$	7.79273	+	7.79273i	0.380700	+	0.380700i	−0.490654	+	0.871354i	$0.663242\pi$
$420$		0			0
$421$	−13.7575			−0.670502			−0.335251	−	0.942129i	$-0.608821\pi$
$421$	−13.7575			−0.670502			−0.335251	+	0.942129i	$0.608821\pi$
$422$		0			0
$423$		−	15.8725i		−	0.771746i
$424$		0			0
$425$	1.31259	−	3.90859i	0.0636702	−	0.189595i
$426$		0			0
$427$		−	57.1922i		−	2.76772i
$428$		0			0
$429$	−36.2455			−1.74995
$430$		0			0
$431$	−9.54879	−	9.54879i	−0.459949	−	0.459949i	0.438690	−	0.898639i	$-0.355443\pi$
$431$	−9.54879	−	9.54879i	−0.459949	−	0.459949i	−0.898639	+	0.438690i	$0.855443\pi$
$432$		0			0
$433$		−	3.19820i		−	0.153696i	−0.997043	−	0.0768478i	$-0.975514\pi$
$433$		−	3.19820i		−	0.153696i	0.997043	−	0.0768478i	$-0.0244856\pi$
$434$		0			0
$435$	−9.42532	−	9.42532i	−0.451910	−	0.451910i
$436$		0			0
$437$	−13.6741	+	13.6741i	−0.654122	+	0.654122i
$438$		0			0
$439$	14.2608	−	14.2608i	0.680632	−	0.680632i	−0.279511	−	0.960143i	$-0.590172\pi$
$439$	14.2608	−	14.2608i	0.680632	−	0.680632i	0.960143	+	0.279511i	$0.0901722\pi$
$440$		0			0
$441$	48.8091			2.32424
$442$		0			0
$443$	13.4708			0.640017			0.320009	−	0.947415i	$-0.396314\pi$
$443$	13.4708			0.640017			0.320009	+	0.947415i	$0.396314\pi$
$444$		0			0
$445$	−9.89935	+	9.89935i	−0.469274	+	0.469274i
$446$		0			0
$447$	7.54778	−	7.54778i	0.356998	−	0.356998i
$448$		0			0
$449$	17.4688	+	17.4688i	0.824402	+	0.824402i	0.986736	−	0.162334i	$-0.0519022\pi$
$449$	17.4688	+	17.4688i	0.824402	+	0.824402i	−0.162334	+	0.986736i	$0.551902\pi$
$450$		0			0
$451$		−	57.2936i		−	2.69785i
$452$		0			0
$453$	−26.6035	−	26.6035i	−1.24994	−	1.24994i
$454$		0			0
$455$	8.36460			0.392138
$456$		0			0
$457$			0.458058i			0.0214271i	0.999943	+	0.0107135i	$0.00341029\pi$
$457$			0.458058i			0.0214271i	−0.999943	+	0.0107135i	$0.996590\pi$
$458$		0			0
$459$	−13.2240	−	26.5966i	−0.617242	−	1.24142i
$460$		0			0
$461$		−	32.3110i		−	1.50487i	−0.658664	−	0.752437i	$-0.728878\pi$
$461$		−	32.3110i		−	1.50487i	0.658664	−	0.752437i	$-0.271122\pi$
$462$		0			0
$463$	−11.4592			−0.532555			−0.266278	−	0.963896i	$-0.585794\pi$
$463$	−11.4592			−0.532555			−0.266278	+	0.963896i	$0.585794\pi$
$464$		0			0
$465$	−8.24399	−	8.24399i	−0.382306	−	0.382306i
$466$		0			0
$467$			34.5948i			1.60086i	0.599429	+	0.800428i	$0.295394\pi$
$467$			34.5948i			1.60086i	−0.599429	+	0.800428i	$0.704606\pi$
$468$		0			0
$469$	−30.7656	−	30.7656i	−1.42063	−	1.42063i
$470$		0			0
$471$	−9.71200	+	9.71200i	−0.447505	+	0.447505i
$472$		0			0
$473$	34.0476	−	34.0476i	1.56551	−	1.56551i
$474$		0			0
$475$	−2.76433			−0.126836
$476$		0			0
$477$	41.2365			1.88809
$478$		0			0
$479$	30.3280	−	30.3280i	1.38572	−	1.38572i	0.551641	−	0.834082i	$-0.314002\pi$
$479$	30.3280	−	30.3280i	1.38572	−	1.38572i	0.834082	−	0.551641i	$-0.185998\pi$
$480$		0			0
$481$	−3.63883	+	3.63883i	−0.165917	+	0.165917i
$482$		0			0
$483$	−57.4487	−	57.4487i	−2.61401	−	2.61401i
$484$		0			0
$485$			2.72090i			0.123550i
$486$		0			0
$487$	6.53184	+	6.53184i	0.295986	+	0.295986i	0.839439	−	0.543453i	$-0.182884\pi$
$487$	6.53184	+	6.53184i	0.295986	+	0.295986i	−0.543453	+	0.839439i	$0.682884\pi$
$488$		0			0
$489$	−35.7151			−1.61509
$490$		0			0
$491$		−	16.8643i		−	0.761074i	−0.924766	−	0.380537i	$-0.875739\pi$
$491$		−	16.8643i		−	0.761074i	0.924766	−	0.380537i	$-0.124261\pi$
$492$		0			0
$493$	−16.9046	+	8.40504i	−0.761345	+	0.378544i
$494$		0			0
$495$			32.5098i			1.46121i
$496$		0			0
$497$	−14.6347			−0.656455
$498$		0			0
$499$	−5.12740	−	5.12740i	−0.229534	−	0.229534i	0.582964	−	0.812498i	$-0.301893\pi$
$499$	−5.12740	−	5.12740i	−0.229534	−	0.229534i	−0.812498	+	0.582964i	$0.801893\pi$
$500$		0			0
$501$			23.4814i			1.04907i
$502$		0			0
$503$	−7.43245	−	7.43245i	−0.331396	−	0.331396i	0.521720	−	0.853117i	$-0.325291\pi$
$503$	−7.43245	−	7.43245i	−0.331396	−	0.331396i	−0.853117	+	0.521720i	$0.825291\pi$
$504$		0			0
$505$	−8.15845	+	8.15845i	−0.363046	+	0.363046i
$506$		0			0
$507$	−17.7109	+	17.7109i	−0.786567	+	0.786567i
$508$		0			0
$509$	−22.7759			−1.00953			−0.504763	−	0.863258i	$-0.668420\pi$
$509$	−22.7759			−1.00953			−0.504763	+	0.863258i	$0.668420\pi$
$510$		0			0
$511$	5.01712			0.221944
$512$		0			0
$513$	−14.0814	+	14.0814i	−0.621711	+	0.621711i
$514$		0			0
$515$	−7.18187	+	7.18187i	−0.316471	+	0.316471i
$516$		0			0
$517$	12.1740	+	12.1740i	0.535411	+	0.535411i
$518$		0			0
$519$			56.2640i			2.46972i
$520$		0			0
$521$	−1.57543	−	1.57543i	−0.0690208	−	0.0690208i	0.671754	−	0.740775i	$-0.265541\pi$
$521$	−1.57543	−	1.57543i	−0.0690208	−	0.0690208i	−0.740775	+	0.671754i	$0.765541\pi$
$522$		0			0
$523$	18.2385			0.797513			0.398757	−	0.917057i	$-0.369442\pi$
$523$	18.2385			0.797513			0.398757	+	0.917057i	$0.369442\pi$
$524$		0			0
$525$		−	11.6137i		−	0.506863i
$526$		0			0
$527$	−14.7858	+	7.35159i	−0.644082	+	0.320240i
$528$		0			0
$529$			25.9383i			1.12775i
$530$		0			0
$531$	−29.7610			−1.29152
$532$		0			0
$533$	14.3043	+	14.3043i	0.619590	+	0.619590i
$534$		0			0
$535$		−	7.88796i		−	0.341026i
$536$		0			0
$537$	−30.3439	−	30.3439i	−1.30943	−	1.30943i
$538$		0			0
$539$	−37.4359	+	37.4359i	−1.61248	+	1.61248i
$540$		0			0
$541$	−5.62690	+	5.62690i	−0.241919	+	0.241919i	−0.817644	−	0.575724i	$-0.804720\pi$
$541$	−5.62690	+	5.62690i	−0.241919	+	0.241919i	0.575724	+	0.817644i	$0.304720\pi$
$542$		0			0
$543$	17.8707			0.766906
$544$		0			0
$545$	1.95746			0.0838484
$546$		0			0
$547$	8.47076	−	8.47076i	0.362183	−	0.362183i	−0.502433	−	0.864616i	$-0.667562\pi$
$547$	8.47076	−	8.47076i	0.362183	−	0.362183i	0.864616	+	0.502433i	$0.167562\pi$
$548$		0			0
$549$	−55.4967	+	55.4967i	−2.36854	+	2.36854i
$550$		0			0
$551$	8.95005	+	8.95005i	0.381285	+	0.381285i
$552$		0			0
$553$		−	18.4993i		−	0.786670i
$554$		0			0
$555$	5.05228	+	5.05228i	0.214458	+	0.214458i
$556$		0			0
$557$	47.0602			1.99400			0.997002	−	0.0773775i	$-0.0246547\pi$
$557$	47.0602			1.99400			0.997002	+	0.0773775i	$0.0246547\pi$
$558$		0			0
$559$			17.0012i			0.719073i
$560$		0			0
$561$	67.5677	+	22.6908i	2.85271	+	0.958006i
$562$		0			0
$563$		−	14.2723i		−	0.601505i	−0.953702	−	0.300752i	$-0.902762\pi$
$563$		−	14.2723i		−	0.601505i	0.953702	−	0.300752i	$-0.0972378\pi$
$564$		0			0
$565$	−20.1440			−0.847464
$566$		0			0
$567$	−12.8290	−	12.8290i	−0.538766	−	0.538766i
$568$		0			0
$569$			34.8997i			1.46307i	0.681802	+	0.731537i	$0.261197\pi$
$569$			34.8997i			1.46307i	−0.681802	+	0.731537i	$0.738803\pi$
$570$		0			0
$571$	11.6490	+	11.6490i	0.487494	+	0.487494i	0.907515	−	0.420020i	$-0.137977\pi$
$571$	11.6490	+	11.6490i	0.487494	+	0.487494i	−0.420020	+	0.907515i	$0.637977\pi$
$572$		0			0
$573$	19.5374	−	19.5374i	0.816188	−	0.816188i
$574$		0			0
$575$	−4.94663	+	4.94663i	−0.206289	+	0.206289i
$576$		0			0
$577$	43.2867			1.80205			0.901024	−	0.433768i	$-0.142816\pi$
$577$	43.2867			1.80205			0.901024	+	0.433768i	$0.142816\pi$
$578$		0			0
$579$	13.0966			0.544274
$580$		0			0
$581$	−8.03058	+	8.03058i	−0.333165	+	0.333165i
$582$		0			0
$583$	−31.6278	+	31.6278i	−1.30989	+	1.30989i
$584$		0			0
$585$	−8.11662	−	8.11662i	−0.335581	−	0.335581i
$586$		0			0
$587$		−	15.9381i		−	0.657835i	−0.944359	−	0.328917i	$-0.893316\pi$
$587$		−	15.9381i		−	0.657835i	0.944359	−	0.328917i	$-0.106684\pi$
$588$		0			0
$589$	7.82829	+	7.82829i	0.322559	+	0.322559i
$590$		0			0
$591$	36.9544			1.52010
$592$		0			0
$593$		−	13.4044i		−	0.550454i	−0.961379	−	0.275227i	$-0.911247\pi$
$593$		−	13.4044i		−	0.550454i	0.961379	−	0.275227i	$-0.0887530\pi$
$594$		0			0
$595$	−15.5930	−	5.23649i	−0.639252	−	0.214675i
$596$		0			0
$597$		−	69.5339i		−	2.84583i
$598$		0			0
$599$	−32.0456			−1.30935			−0.654674	−	0.755911i	$-0.727194\pi$
$599$	−32.0456			−1.30935			−0.654674	+	0.755911i	$0.727194\pi$
$600$		0			0
$601$	−6.15901	−	6.15901i	−0.251231	−	0.251231i	0.570244	−	0.821475i	$-0.306849\pi$
$601$	−6.15901	−	6.15901i	−0.251231	−	0.251231i	−0.821475	+	0.570244i	$0.806849\pi$
$602$		0			0
$603$			59.7072i			2.43146i
$604$		0			0
$605$	−17.1564	−	17.1564i	−0.697506	−	0.697506i
$606$		0			0
$607$	4.72727	−	4.72727i	0.191874	−	0.191874i	−0.604631	−	0.796505i	$-0.706680\pi$
$607$	4.72727	−	4.72727i	0.191874	−	0.191874i	0.796505	+	0.604631i	$0.206680\pi$
$608$		0			0
$609$	−37.6016	+	37.6016i	−1.52369	+	1.52369i
$610$		0			0
$611$	−6.07889			−0.245925
$612$		0			0
$613$	−36.4982			−1.47415			−0.737074	−	0.675812i	$-0.763793\pi$
$613$	−36.4982			−1.47415			−0.737074	+	0.675812i	$0.763793\pi$
$614$		0			0
$615$	19.8606	−	19.8606i	0.800858	−	0.800858i
$616$		0			0
$617$	−8.42292	+	8.42292i	−0.339094	+	0.339094i	−0.856026	−	0.516932i	$-0.827074\pi$
$617$	−8.42292	+	8.42292i	−0.339094	+	0.339094i	0.516932	+	0.856026i	$0.327074\pi$
$618$		0			0
$619$	4.71037	+	4.71037i	0.189326	+	0.189326i	0.795405	−	0.606079i	$-0.207258\pi$
$619$	4.71037	+	4.71037i	0.189326	+	0.189326i	−0.606079	+	0.795405i	$0.707258\pi$
$620$		0			0
$621$			50.3961i			2.02232i
$622$		0			0
$623$	39.4927	+	39.4927i	1.58224	+	1.58224i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		−	47.7869i		−	1.90842i
$628$		0			0
$629$	9.06142	−	4.50538i	0.361303	−	0.179641i
$630$		0			0
$631$		−	10.9619i		−	0.436388i	−0.975905	−	0.218194i	$-0.929983\pi$
$631$		−	10.9619i		−	0.436388i	0.975905	−	0.218194i	$-0.0700166\pi$
$632$		0			0
$633$	−28.7735			−1.14364
$634$		0			0
$635$	3.27705	+	3.27705i	0.130046	+	0.130046i
$636$		0			0
$637$		−	18.6931i		−	0.740646i
$638$		0			0
$639$	14.2008	+	14.2008i	0.561776	+	0.561776i
$640$		0			0
$641$	−2.15336	+	2.15336i	−0.0850524	+	0.0850524i	−0.748353	−	0.663301i	$-0.769155\pi$
$641$	−2.15336	+	2.15336i	−0.0850524	+	0.0850524i	0.663301	+	0.748353i	$0.269155\pi$
$642$		0			0
$643$	−3.35423	+	3.35423i	−0.132278	+	0.132278i	−0.770146	−	0.637868i	$-0.779817\pi$
$643$	−3.35423	+	3.35423i	−0.132278	+	0.132278i	0.637868	+	0.770146i	$0.279817\pi$
$644$		0			0
$645$	23.6050			0.929447
$646$		0			0
$647$	−40.1686			−1.57919			−0.789595	−	0.613629i	$-0.789709\pi$
$647$	−40.1686			−1.57919			−0.789595	+	0.613629i	$0.789709\pi$
$648$		0			0
$649$	22.8263	−	22.8263i	0.896010	−	0.896010i
$650$		0			0
$651$	−32.8888	+	32.8888i	−1.28901	+	1.28901i
$652$		0			0
$653$	−6.83933	−	6.83933i	−0.267644	−	0.267644i	0.560506	−	0.828150i	$-0.310607\pi$
$653$	−6.83933	−	6.83933i	−0.267644	−	0.267644i	−0.828150	+	0.560506i	$0.810607\pi$
$654$		0			0
$655$		−	18.8494i		−	0.736506i
$656$		0			0
$657$	−4.86838	−	4.86838i	−0.189934	−	0.189934i
$658$		0			0
$659$	−41.3724			−1.61164			−0.805819	−	0.592161i	$-0.798275\pi$
$659$	−41.3724			−1.61164			−0.805819	+	0.592161i	$0.798275\pi$
$660$		0			0
$661$			38.9890i			1.51650i	0.651966	+	0.758248i	$0.273945\pi$
$661$			38.9890i			1.51650i	−0.651966	+	0.758248i	$0.726055\pi$
$662$		0			0
$663$	−22.5346	+	11.2043i	−0.875171	+	0.435139i
$664$		0			0
$665$			11.0281i			0.427651i
$666$		0			0
$667$	32.0313			1.24026
$668$		0			0
$669$	31.6268	+	31.6268i	1.22276	+	1.22276i
$670$		0			0
$671$		−	85.1304i		−	3.28642i
$672$		0			0
$673$	−15.7018	−	15.7018i	−0.605258	−	0.605258i	0.336445	−	0.941703i	$-0.390775\pi$
$673$	−15.7018	−	15.7018i	−0.605258	−	0.605258i	−0.941703	+	0.336445i	$0.890775\pi$
$674$		0			0
$675$	−5.09398	+	5.09398i	−0.196067	+	0.196067i
$676$		0			0
$677$	8.42888	−	8.42888i	0.323948	−	0.323948i	−0.526331	−	0.850279i	$-0.676433\pi$
$677$	8.42888	−	8.42888i	0.323948	−	0.323948i	0.850279	+	0.526331i	$0.176433\pi$
$678$		0			0
$679$	10.8548			0.416569
$680$		0			0
$681$	−53.4676			−2.04888
$682$		0			0
$683$	14.2584	−	14.2584i	0.545583	−	0.545583i	−0.379577	−	0.925160i	$-0.623931\pi$
$683$	14.2584	−	14.2584i	0.545583	−	0.545583i	0.925160	+	0.379577i	$0.123931\pi$
$684$		0			0
$685$	−0.112639	+	0.112639i	−0.00430373	+	0.00430373i
$686$		0			0
$687$	−52.0121	−	52.0121i	−1.98438	−	1.98438i
$688$		0			0
$689$		−	15.7929i		−	0.601660i
$690$		0			0
$691$	13.0060	+	13.0060i	0.494771	+	0.494771i	0.909806	−	0.415035i	$-0.136231\pi$
$691$	13.0060	+	13.0060i	0.494771	+	0.494771i	−0.415035	+	0.909806i	$0.636231\pi$
$692$		0			0
$693$	129.695			4.92671
$694$		0			0
$695$		−	17.1234i		−	0.649528i
$696$		0			0
$697$	−17.7108	−	35.6207i	−0.670843	−	1.34923i
$698$		0			0
$699$			26.5049i			1.00251i
$700$		0			0
$701$	47.4735			1.79305			0.896525	−	0.442994i	$-0.146084\pi$
$701$	47.4735			1.79305			0.896525	+	0.442994i	$0.146084\pi$
$702$		0			0
$703$	−4.79752	−	4.79752i	−0.180942	−	0.180942i
$704$		0			0
$705$			8.44014i			0.317874i
$706$		0			0
$707$	32.5475	+	32.5475i	1.22408	+	1.22408i
$708$		0			0
$709$	−8.88457	+	8.88457i	−0.333667	+	0.333667i	−0.853977	−	0.520310i	$-0.825816\pi$
$709$	−8.88457	+	8.88457i	−0.333667	+	0.333667i	0.520310	+	0.853977i	$0.325816\pi$
$710$		0			0
$711$	−17.9509	+	17.9509i	−0.673211	+	0.673211i
$712$		0			0
$713$	28.0167			1.04923
$714$		0			0
$715$	12.4507			0.465629
$716$		0			0
$717$	−20.3837	+	20.3837i	−0.761244	+	0.761244i
$718$		0			0
$719$	−12.8819	+	12.8819i	−0.480413	+	0.480413i	−0.905264	−	0.424850i	$-0.860327\pi$
$719$	−12.8819	+	12.8819i	−0.480413	+	0.480413i	0.424850	+	0.905264i	$0.360327\pi$
$720$		0			0
$721$	28.6515	+	28.6515i	1.06704	+	1.06704i
$722$		0			0
$723$			1.08059i			0.0401874i
$724$		0			0
$725$	3.23769	+	3.23769i	0.120245	+	0.120245i
$726$		0			0
$727$	42.0417			1.55924			0.779621	−	0.626252i	$-0.215412\pi$
$727$	42.0417			1.55924			0.779621	+	0.626252i	$0.215412\pi$
$728$		0			0
$729$		−	38.0177i		−	1.40806i
$730$		0			0
$731$	10.6433	−	31.6931i	0.393655	−	1.17221i
$732$		0			0
$733$			5.75421i			0.212537i	0.994337	+	0.106268i	$0.0338902\pi$
$733$			5.75421i			0.212537i	−0.994337	+	0.106268i	$0.966110\pi$
$734$		0			0
$735$	−25.9541			−0.957331
$736$		0			0
$737$	−45.7946	−	45.7946i	−1.68687	−	1.68687i
$738$		0			0
$739$			44.6567i			1.64272i	0.570408	+	0.821361i	$0.306785\pi$
$739$			44.6567i			1.64272i	−0.570408	+	0.821361i	$0.693215\pi$
$740$		0			0
$741$	11.9308	+	11.9308i	0.438290	+	0.438290i
$742$		0			0
$743$	−10.0237	+	10.0237i	−0.367733	+	0.367733i	−0.866650	−	0.498917i	$-0.833731\pi$
$743$	−10.0237	+	10.0237i	−0.367733	+	0.367733i	0.498917	+	0.866650i	$0.333731\pi$
$744$		0			0
$745$	−2.59274	+	2.59274i	−0.0949905	+	0.0949905i
$746$		0			0
$747$	15.5850			0.570226
$748$		0			0
$749$	−31.4684			−1.14983
$750$		0			0
$751$	−0.387773	+	0.387773i	−0.0141500	+	0.0141500i	−0.714146	−	0.699996i	$-0.753185\pi$
$751$	−0.387773	+	0.387773i	−0.0141500	+	0.0141500i	0.699996	+	0.714146i	$0.253185\pi$
$752$		0			0
$753$	−49.5715	+	49.5715i	−1.80649	+	1.80649i
$754$		0			0
$755$	9.13855	+	9.13855i	0.332586	+	0.332586i
$756$		0			0
$757$			5.35045i			0.194466i	0.995262	+	0.0972328i	$0.0309991\pi$
$757$			5.35045i			0.194466i	−0.995262	+	0.0972328i	$0.969001\pi$
$758$		0			0
$759$	−85.5122	−	85.5122i	−3.10390	−	3.10390i
$760$		0			0
$761$	−5.74506			−0.208258			−0.104129	−	0.994564i	$-0.533206\pi$
$761$	−5.74506			−0.208258			−0.104129	+	0.994564i	$0.533206\pi$
$762$		0			0
$763$		−	7.80913i		−	0.282710i
$764$		0			0
$765$	10.0495	+	20.2120i	0.363341	+	0.730767i
$766$		0			0
$767$			11.3980i			0.411556i
$768$		0			0
$769$	19.9110			0.718008			0.359004	−	0.933336i	$-0.383116\pi$
$769$	19.9110			0.718008			0.359004	+	0.933336i	$0.383116\pi$
$770$		0			0
$771$	11.6767	+	11.6767i	0.420526	+	0.420526i
$772$		0			0
$773$			3.80237i			0.136762i	0.997659	+	0.0683808i	$0.0217833\pi$
$773$			3.80237i			0.136762i	−0.997659	+	0.0683808i	$0.978217\pi$
$774$		0			0
$775$	2.83189	+	2.83189i	0.101725	+	0.101725i
$776$		0			0
$777$	20.1557	−	20.1557i	0.723082	−	0.723082i
$778$		0			0
$779$	−18.8592	+	18.8592i	−0.675700	+	0.675700i
$780$		0			0
$781$	−21.7837			−0.779482
$782$		0			0
$783$	32.9855			1.17881
$784$		0			0
$785$	3.33617	−	3.33617i	0.119073	−	0.119073i
$786$		0			0
$787$	−7.00665	+	7.00665i	−0.249760	+	0.249760i	−0.820872	−	0.571112i	$-0.806512\pi$
$787$	−7.00665	+	7.00665i	−0.249760	+	0.249760i	0.571112	+	0.820872i	$0.306512\pi$
$788$		0			0
$789$	49.4098	+	49.4098i	1.75904	+	1.75904i
$790$		0			0
$791$			80.3629i			2.85738i
$792$		0			0
$793$	21.2543	+	21.2543i	0.754762	+	0.754762i
$794$		0			0
$795$	−21.9274			−0.777684
$796$		0			0
$797$			31.9487i			1.13168i	0.824515	+	0.565841i	$0.191448\pi$
$797$			31.9487i			1.13168i	−0.824515	+	0.565841i	$0.808552\pi$
$798$		0			0
$799$	11.3321	+	3.80557i	0.400900	+	0.134631i
$800$		0			0
$801$		−	76.6438i		−	2.70808i
$802$		0			0
$803$	7.46797			0.263539
$804$		0			0
$805$	19.7342	+	19.7342i	0.695539	+	0.695539i
$806$		0			0
$807$		−	52.1415i		−	1.83547i
$808$		0			0
$809$	−11.2944	−	11.2944i	−0.397092	−	0.397092i	0.480114	−	0.877206i	$-0.340595\pi$
$809$	−11.2944	−	11.2944i	−0.397092	−	0.397092i	−0.877206	+	0.480114i	$0.840595\pi$
$810$		0			0
$811$	32.9954	−	32.9954i	1.15863	−	1.15863i	0.173855	−	0.984771i	$-0.444378\pi$
$811$	32.9954	−	32.9954i	1.15863	−	1.15863i	0.984771	−	0.173855i	$-0.0556223\pi$
$812$		0			0
$813$	2.99758	−	2.99758i	0.105130	−	0.105130i
$814$		0			0
$815$	12.2685			0.429747
$816$		0			0
$817$	−22.4147			−0.784192
$818$		0			0
$819$	−32.3806	+	32.3806i	−1.13147	+	1.13147i
$820$		0			0
$821$	−31.4040	+	31.4040i	−1.09601	+	1.09601i	−0.101134	+	0.994873i	$0.532247\pi$
$821$	−31.4040	+	31.4040i	−1.09601	+	1.09601i	−0.994873	+	0.101134i	$0.967753\pi$
$822$		0			0
$823$	−8.35292	−	8.35292i	−0.291165	−	0.291165i	0.546376	−	0.837540i	$-0.316007\pi$
$823$	−8.35292	−	8.35292i	−0.291165	−	0.291165i	−0.837540	+	0.546376i	$0.816007\pi$
$824$		0			0
$825$		−	17.2870i		−	0.601855i
$826$		0			0
$827$	−34.2825	−	34.2825i	−1.19212	−	1.19212i	−0.976471	−	0.215648i	$-0.930814\pi$
$827$	−34.2825	−	34.2825i	−1.19212	−	1.19212i	−0.215648	−	0.976471i	$-0.569186\pi$
$828$		0			0
$829$	32.1189			1.11554			0.557768	−	0.829997i	$-0.311658\pi$
$829$	32.1189			1.11554			0.557768	+	0.829997i	$0.311658\pi$
$830$		0			0
$831$			37.1007i			1.28701i
$832$		0			0
$833$	−11.7024	+	34.8470i	−0.405465	+	1.20738i
$834$		0			0
$835$		−	8.06609i		−	0.279139i
$836$		0			0
$837$	28.8512			0.997244
$838$		0			0
$839$	−5.02151	−	5.02151i	−0.173362	−	0.173362i	0.615093	−	0.788455i	$-0.289118\pi$
$839$	−5.02151	−	5.02151i	−0.173362	−	0.173362i	−0.788455	+	0.615093i	$0.789118\pi$
$840$		0			0
$841$			8.03470i			0.277059i
$842$		0			0
$843$	−32.6398	−	32.6398i	−1.12417	−	1.12417i
$844$		0			0
$845$	6.08386	−	6.08386i	0.209291	−	0.209291i
$846$		0			0
$847$	−68.4441	+	68.4441i	−2.35177	+	2.35177i
$848$		0			0
$849$	11.2645			0.386595
$850$		0			0
$851$	−17.1699			−0.588575
$852$		0			0
$853$	−8.61501	+	8.61501i	−0.294972	+	0.294972i	−0.839041	−	0.544068i	$-0.816883\pi$
$853$	−8.61501	+	8.61501i	−0.294972	+	0.294972i	0.544068	+	0.839041i	$0.316883\pi$
$854$		0			0
$855$	10.7011	−	10.7011i	0.365971	−	0.365971i
$856$		0			0
$857$	14.7705	+	14.7705i	0.504551	+	0.504551i	0.912849	−	0.408298i	$-0.133878\pi$
$857$	14.7705	+	14.7705i	0.504551	+	0.504551i	−0.408298	+	0.912849i	$0.633878\pi$
$858$		0			0
$859$		−	7.89536i		−	0.269386i	−0.990887	−	0.134693i	$-0.956995\pi$
$859$		−	7.89536i		−	0.269386i	0.990887	−	0.134693i	$-0.0430049\pi$
$860$		0			0
$861$	−79.2325	−	79.2325i	−2.70024	−	2.70024i
$862$		0			0
$863$	14.4016			0.490237			0.245119	−	0.969493i	$-0.421173\pi$
$863$	14.4016			0.490237			0.245119	+	0.969493i	$0.421173\pi$
$864$		0			0
$865$		−	19.3273i		−	0.657146i
$866$		0			0
$867$	49.0226	−	6.77937i	1.66489	−	0.230240i
$868$		0			0
$869$		−	27.5362i		−	0.934101i
$870$		0			0
$871$	22.8668			0.774813
$872$		0			0
$873$	−10.5330	−	10.5330i	−0.356488	−	0.356488i
$874$		0			0
$875$			3.98942i			0.134867i
$876$		0			0
$877$	11.0389	+	11.0389i	0.372757	+	0.372757i	0.868480	−	0.495724i	$-0.165097\pi$
$877$	11.0389	+	11.0389i	0.372757	+	0.372757i	−0.495724	+	0.868480i	$0.665097\pi$
$878$		0			0
$879$	37.7092	−	37.7092i	1.27190	−	1.27190i
$880$		0			0
$881$	−11.9814	+	11.9814i	−0.403663	+	0.403663i	−0.879522	−	0.475858i	$-0.842137\pi$
$881$	−11.9814	+	11.9814i	−0.403663	+	0.403663i	0.475858	+	0.879522i	$0.342137\pi$
$882$		0			0
$883$	−23.3753			−0.786642			−0.393321	−	0.919401i	$-0.628674\pi$
$883$	−23.3753			−0.786642			−0.393321	+	0.919401i	$0.628674\pi$
$884$		0			0
$885$	15.8253			0.531962
$886$		0			0
$887$	14.8185	−	14.8185i	0.497556	−	0.497556i	−0.413121	−	0.910676i	$-0.635561\pi$
$887$	14.8185	−	14.8185i	0.497556	−	0.497556i	0.910676	+	0.413121i	$0.135561\pi$
$888$		0			0
$889$	13.0735	−	13.0735i	0.438472	−	0.438472i
$890$		0			0
$891$	−19.0959	−	19.0959i	−0.639736	−	0.639736i
$892$		0			0
$893$		−	8.01454i		−	0.268196i
$894$		0			0
$895$	10.4234	+	10.4234i	0.348417	+	0.348417i
$896$		0			0
$897$	42.6992			1.42569
$898$		0			0
$899$		−	18.3376i		−	0.611593i
$900$		0			0
$901$	−9.88682	+	29.4406i	−0.329378	+	0.980808i
$902$		0			0
$903$		−	94.1704i		−	3.13379i
$904$		0			0
$905$	−6.13877			−0.204060
$906$		0			0
$907$	32.7366	+	32.7366i	1.08700	+	1.08700i	0.995836	+	0.0911645i	$0.0290589\pi$
$907$	32.7366	+	32.7366i	1.08700	+	1.08700i	0.0911645	+	0.995836i	$0.470941\pi$
$908$		0			0
$909$		−	63.1653i		−	2.09506i
$910$		0			0
$911$	−37.7922	−	37.7922i	−1.25211	−	1.25211i	−0.954771	−	0.297341i	$-0.903900\pi$
$911$	−37.7922	−	37.7922i	−1.25211	−	1.25211i	−0.297341	−	0.954771i	$-0.596100\pi$
$912$		0			0
$913$	−11.9535	+	11.9535i	−0.395603	+	0.395603i
$914$		0			0
$915$	29.5102	−	29.5102i	0.975577	−	0.975577i
$916$		0			0
$917$	−75.1981			−2.48326
$918$		0			0
$919$	−5.67563			−0.187222			−0.0936108	−	0.995609i	$-0.529841\pi$
$919$	−5.67563			−0.187222			−0.0936108	+	0.995609i	$0.529841\pi$
$920$		0			0
$921$	14.0411	−	14.0411i	0.462671	−	0.462671i
$922$		0			0
$923$	5.43867	−	5.43867i	0.179016	−	0.179016i
$924$		0			0
$925$	−1.73551	−	1.73551i	−0.0570632	−	0.0570632i
$926$		0			0
$927$		−	55.6043i		−	1.82628i
$928$		0			0
$929$	38.6326	+	38.6326i	1.26750	+	1.26750i	0.947377	+	0.320119i	$0.103723\pi$
$929$	38.6326	+	38.6326i	1.26750	+	1.26750i	0.320119	+	0.947377i	$0.396277\pi$
$930$		0			0
$931$	24.6454			0.807719
$932$		0			0
$933$			40.2215i			1.31679i
$934$		0			0
$935$	−23.2102	−	7.79451i	−0.759054	−	0.254908i
$936$		0			0
$937$			0.717134i			0.0234278i	0.999931	+	0.0117139i	$0.00372873\pi$
$937$			0.717134i			0.0234278i	−0.999931	+	0.0117139i	$0.996271\pi$
$938$		0			0
$939$	10.5320			0.343699
$940$		0			0
$941$	−1.45190	−	1.45190i	−0.0473305	−	0.0473305i	0.683045	−	0.730376i	$-0.260655\pi$
$941$	−1.45190	−	1.45190i	−0.0473305	−	0.0473305i	−0.730376	+	0.683045i	$0.760655\pi$
$942$		0			0
$943$			67.4951i			2.19794i
$944$		0			0
$945$	20.3220	+	20.3220i	0.661076	+	0.661076i
$946$		0			0
$947$	17.4393	−	17.4393i	0.566701	−	0.566701i	−0.364502	−	0.931203i	$-0.618761\pi$
$947$	17.4393	−	17.4393i	0.566701	−	0.566701i	0.931203	+	0.364502i	$0.118761\pi$
$948$		0			0
$949$	−1.86451	+	1.86451i	−0.0605245	+	0.0605245i
$950$		0			0
$951$	−44.4317			−1.44080
$952$		0			0
$953$	−8.89683			−0.288197			−0.144098	−	0.989563i	$-0.546028\pi$
$953$	−8.89683			−0.288197			−0.144098	+	0.989563i	$0.546028\pi$
$954$		0			0
$955$	−6.71131	+	6.71131i	−0.217173	+	0.217173i
$956$		0			0
$957$	−55.9699	+	55.9699i	−1.80925	+	1.80925i
$958$		0			0
$959$	0.449366	+	0.449366i	0.0145108	+	0.0145108i
$960$		0			0
$961$			14.9608i			0.482605i
$962$		0			0
$963$	30.5355	+	30.5355i	0.983992	+	0.983992i
$964$		0			0
$965$	−4.49880			−0.144821
$966$		0			0
$967$		−	11.9808i		−	0.385277i	−0.981270	−	0.192639i	$-0.938295\pi$
$967$		−	11.9808i		−	0.385277i	0.981270	−	0.192639i	$-0.0617045\pi$
$968$		0			0
$969$	−14.7720	−	29.7101i	−0.474545	−	0.954427i
$970$		0			0
$971$		−	6.41275i		−	0.205795i	−0.994692	−	0.102897i	$-0.967189\pi$
$971$		−	6.41275i		−	0.205795i	0.994692	−	0.102897i	$-0.0328114\pi$
$972$		0			0
$973$	−68.3125			−2.19000
$974$		0			0
$975$	4.31599	+	4.31599i	0.138222	+	0.138222i
$976$		0			0
$977$		−	7.73116i		−	0.247342i	−0.992323	−	0.123671i	$-0.960533\pi$
$977$		−	7.73116i		−	0.247342i	0.992323	−	0.123671i	$-0.0394667\pi$
$978$		0			0
$979$	58.7848	+	58.7848i	1.87877	+	1.87877i
$980$		0			0
$981$	−7.57763	+	7.57763i	−0.241935	+	0.241935i
$982$		0			0
$983$	10.6322	−	10.6322i	0.339113	−	0.339113i	−0.516920	−	0.856034i	$-0.672922\pi$
$983$	10.6322	−	10.6322i	0.339113	−	0.339113i	0.856034	+	0.516920i	$0.172922\pi$
$984$		0			0
$985$	−12.6942			−0.404471
$986$		0			0
$987$	33.6713			1.07177
$988$		0			0
$989$	−40.1100	+	40.1100i	−1.27543	+	1.27543i
$990$		0			0
$991$	2.34983	−	2.34983i	0.0746448	−	0.0746448i	−0.668799	−	0.743444i	$-0.733191\pi$
$991$	2.34983	−	2.34983i	0.0746448	−	0.0746448i	0.743444	+	0.668799i	$0.233191\pi$
$992$		0			0
$993$	−56.2861	−	56.2861i	−1.78619	−	1.78619i
$994$		0			0
$995$			23.8856i			0.757224i
$996$		0			0
$997$	26.2242	+	26.2242i	0.830528	+	0.830528i	0.987589	−	0.157061i	$-0.0502018\pi$
$997$	26.2242	+	26.2242i	0.830528	+	0.830528i	−0.157061	+	0.987589i	$0.550202\pi$
$998$		0			0
$999$	−17.6813			−0.559412

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	680.2.bd.b.361.10	yes	20
4.3	odd	2		1360.2.bt.f.1041.1		20
17.13	even	4	inner	680.2.bd.b.81.10	✓	20
68.47	odd	4		1360.2.bt.f.81.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.bd.b.81.10	✓	20	17.13	even	4	inner
680.2.bd.b.361.10	yes	20	1.1	even	1	trivial
1360.2.bt.f.81.1		20	68.47	odd	4
1360.2.bt.f.1041.1		20	4.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.bd (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.05847 - 2.05847i) q^{3} +(-0.707107 + 0.707107i) q^{5} +(2.82095 + 2.82095i) q^{7} -5.47464i q^{9} +(4.19898 + 4.19898i) q^{11} -2.09669 q^{13} +2.91112i q^{15} +(3.90859 + 1.31259i) q^{17} -2.76433i q^{19} +11.6137 q^{21} +(-4.94663 - 4.94663i) q^{23} -1.00000i q^{25} +(-5.09398 - 5.09398i) q^{27} +(-3.23769 + 3.23769i) q^{29} +(-2.83189 + 2.83189i) q^{31} +17.2870 q^{33} -3.98942 q^{35} +(1.73551 - 1.73551i) q^{37} +(-4.31599 + 4.31599i) q^{39} +(-6.82233 - 6.82233i) q^{41} -8.10856i q^{43} +(3.87115 + 3.87115i) q^{45} +2.89927 q^{47} +8.91549i q^{49} +(10.7477 - 5.34380i) q^{51} +7.53227i q^{53} -5.93825 q^{55} +(-5.69030 - 5.69030i) q^{57} -5.43616i q^{59} +(-10.1370 - 10.1370i) q^{61} +(15.4437 - 15.4437i) q^{63} +(1.48259 - 1.48259i) q^{65} -10.9061 q^{67} -20.3650 q^{69} +(-2.59393 + 2.59393i) q^{71} +(0.889261 - 0.889261i) q^{73} +(-2.05847 - 2.05847i) q^{75} +23.6902i q^{77} +(-3.27892 - 3.27892i) q^{79} -4.54775 q^{81} +2.84677i q^{83} +(-3.69194 + 1.83565i) q^{85} +13.3294i q^{87} +13.9998 q^{89} +(-5.91466 - 5.91466i) q^{91} +11.6588i q^{93} +(1.95468 + 1.95468i) q^{95} +(1.92396 - 1.92396i) q^{97} +(22.9879 - 22.9879i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 4 q^{7} + 12 q^{11} + 4 q^{13} + 4 q^{17} + 16 q^{21} - 4 q^{23} - 16 q^{27} - 4 q^{29} - 4 q^{31} + 16 q^{33} - 8 q^{35} - 8 q^{37} - 8 q^{39} - 8 q^{41} + 4 q^{47} - 4 q^{51} - 8 q^{55}+ \cdots + 52 q^{99}+O(q^{100}) \) 20 * q - 4 * q^3 + 4 * q^7 + 12 * q^11 + 4 * q^13 + 4 * q^17 + 16 * q^21 - 4 * q^23 - 16 * q^27 - 4 * q^29 - 4 * q^31 + 16 * q^33 - 8 * q^35 - 8 * q^37 - 8 * q^39 - 8 * q^41 + 4 * q^47 - 4 * q^51 - 8 * q^55 - 24 * q^57 - 24 * q^61 + 60 * q^63 - 4 * q^65 - 56 * q^67 + 24 * q^69 - 40 * q^71 - 20 * q^73 + 4 * q^75 - 8 * q^79 + 4 * q^81 - 12 * q^85 + 28 * q^89 - 4 * q^95 + 24 * q^97 + 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more