LMFDB - Embedded newform 968.2.i.q.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,2,Mod(9,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 6]))

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

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

chi := DirichletCharacter("968.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	968.i (of order $5$, degree $4$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.72951891566$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	8.0.1305015625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 5x^{6} - 9x^{5} + 29x^{4} + 36x^{3} + 80x^{2} + 64x + 256 $ x^8 - x^7 + 5x^6 - 9x^5 + 29x^4 + 36x^3 + 80x^2 + 64x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			81.1
Root			$-1.26332 - 0.917858i$ of defining polynomial
Character	$\chi$	$=$	968.81
Dual form			968.2.i.q.729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.482546 + 1.48512i) q^{3} +(-2.88136 + 2.09343i) q^{5} +(-0.965093 - 2.97025i) q^{7} +(0.454306 + 0.330072i) q^{9} +O(q^{10})$	\(q+(-0.482546 + 1.48512i) q^{3} +(-2.88136 + 2.09343i) q^{5} +(-0.965093 - 2.97025i) q^{7} +(0.454306 + 0.330072i) q^{9} +(-4.14468 - 3.01129i) q^{13} +(-1.71861 - 5.28935i) q^{15} +(1.61803 - 1.17557i) q^{17} +(1.23607 - 3.80423i) q^{19} +4.87689 q^{21} +2.43845 q^{23} +(2.37469 - 7.30854i) q^{25} +(-4.49939 + 3.26900i) q^{27} +(1.58313 + 4.87236i) q^{29} +(4.49939 + 3.26900i) q^{31} +(8.99878 + 6.53800i) q^{35} +(-2.33665 - 7.19146i) q^{37} +(6.47214 - 4.70228i) q^{39} +(0.347059 - 1.06814i) q^{41} +7.12311 q^{43} -2.00000 q^{45} +(2.47214 - 7.60845i) q^{47} +(-2.22786 + 1.61864i) q^{49} +(0.965093 + 2.97025i) q^{51} +(-9.90739 - 7.19814i) q^{53} +(5.05329 + 3.67143i) q^{57} +(2.41273 + 7.42562i) q^{59} +(0.908612 - 0.660145i) q^{61} +(0.541951 - 1.66795i) q^{63} +18.2462 q^{65} +9.56155 q^{67} +(-1.17666 + 3.62140i) q^{69} +(7.02604 - 5.10471i) q^{71} +(-1.58313 - 4.87236i) q^{73} +(9.70820 + 7.05342i) q^{75} +(-8.99878 - 6.53800i) q^{79} +(-2.16312 - 6.65740i) q^{81} +(0.709422 - 0.515426i) q^{83} +(-2.20116 + 6.77448i) q^{85} -8.00000 q^{87} +2.68466 q^{89} +(-4.94427 + 15.2169i) q^{91} +(-7.02604 + 5.10471i) q^{93} +(4.40232 + 13.5490i) q^{95} +(-12.5896 - 9.14685i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 8 * q - q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9	$ 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9} - 2 q^{13} + 7 q^{15} + 4 q^{17} - 8 q^{19} + 72 q^{21} + 36 q^{23} - 3 q^{25} - 7 q^{27} - 2 q^{29} + 7 q^{31} + 14 q^{35} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 24 q^{43} - 16 q^{45} - 16 q^{47} - 22 q^{49} + 2 q^{51} - 8 q^{53} - 4 q^{57} + 5 q^{59} - 6 q^{61} - 20 q^{63} + 80 q^{65} + 60 q^{67} - 13 q^{69} + 5 q^{71} + 2 q^{73} + 24 q^{75} - 14 q^{79} + 14 q^{81} + 10 q^{83} + 6 q^{85} - 64 q^{87} - 28 q^{89} + 32 q^{91} - 5 q^{93} - 12 q^{95} - 27 q^{97}+O(q^{100}) $ 8 * q - q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9 - 2 * q^13 + 7 * q^15 + 4 * q^17 - 8 * q^19 + 72 * q^21 + 36 * q^23 - 3 * q^25 - 7 * q^27 - 2 * q^29 + 7 * q^31 + 14 * q^35 + 11 * q^37 + 16 * q^39 + 6 * q^41 + 24 * q^43 - 16 * q^45 - 16 * q^47 - 22 * q^49 + 2 * q^51 - 8 * q^53 - 4 * q^57 + 5 * q^59 - 6 * q^61 - 20 * q^63 + 80 * q^65 + 60 * q^67 - 13 * q^69 + 5 * q^71 + 2 * q^73 + 24 * q^75 - 14 * q^79 + 14 * q^81 + 10 * q^83 + 6 * q^85 - 64 * q^87 - 28 * q^89 + 32 * q^91 - 5 * q^93 - 12 * q^95 - 27 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$849$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{5}\right)$

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
$3$	−0.482546	+	1.48512i	−0.278598	+	0.857437i	0.709647	+	0.704558i	$0.248855\pi$
$3$	−0.482546	+	1.48512i	−0.278598	+	0.857437i	−0.988245	+	0.152879i	$0.951145\pi$
$4$		0			0
$5$	−2.88136	+	2.09343i	−1.28858	+	0.936210i	−0.999776	−	0.0211780i	$-0.993258\pi$
$5$	−2.88136	+	2.09343i	−1.28858	+	0.936210i	−0.288806	+	0.957388i	$0.593258\pi$
$6$		0			0
$7$	−0.965093	−	2.97025i	−0.364771	−	1.12265i	−0.950124	−	0.311871i	$-0.899044\pi$
$7$	−0.965093	−	2.97025i	−0.364771	−	1.12265i	0.585354	−	0.810778i	$-0.300956\pi$
$8$		0			0
$9$	0.454306	+	0.330072i	0.151435	+	0.110024i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$	−4.14468	−	3.01129i	−1.14953	−	0.835180i	−0.161109	−	0.986937i	$-0.551507\pi$
$13$	−4.14468	−	3.01129i	−1.14953	−	0.835180i	−0.988418	+	0.151756i	$0.951507\pi$
$14$		0			0
$15$	−1.71861	−	5.28935i	−0.443744	−	1.36570i
$16$		0			0
$17$	1.61803	−	1.17557i	0.392431	−	0.285118i	−0.374020	−	0.927421i	$-0.622021\pi$
$17$	1.61803	−	1.17557i	0.392431	−	0.285118i	0.766451	+	0.642303i	$0.222021\pi$
$18$		0			0
$19$	1.23607	−	3.80423i	0.283573	−	0.872749i	−0.703249	−	0.710943i	$-0.748268\pi$
$19$	1.23607	−	3.80423i	0.283573	−	0.872749i	0.986823	−	0.161806i	$-0.0517318\pi$
$20$		0			0
$21$	4.87689			1.06423
$22$		0			0
$23$	2.43845			0.508451			0.254226	−	0.967145i	$-0.418179\pi$
$23$	2.43845			0.508451			0.254226	+	0.967145i	$0.418179\pi$
$24$		0			0
$25$	2.37469	−	7.30854i	0.474938	−	1.46171i
$26$		0			0
$27$	−4.49939	+	3.26900i	−0.865908	+	0.629119i
$28$		0			0
$29$	1.58313	+	4.87236i	0.293979	+	0.904775i	0.983562	+	0.180569i	$0.0577938\pi$
$29$	1.58313	+	4.87236i	0.293979	+	0.904775i	−0.689583	+	0.724207i	$0.742206\pi$
$30$		0			0
$31$	4.49939	+	3.26900i	0.808114	+	0.587130i	0.913283	−	0.407325i	$-0.133538\pi$
$31$	4.49939	+	3.26900i	0.808114	+	0.587130i	−0.105169	+	0.994454i	$0.533538\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.99878	+	6.53800i	1.52107	+	1.10512i
$36$		0			0
$37$	−2.33665	−	7.19146i	−0.384143	−	1.18227i	−0.937100	−	0.349060i	$-0.886501\pi$
$37$	−2.33665	−	7.19146i	−0.384143	−	1.18227i	0.552958	−	0.833209i	$-0.313499\pi$
$38$		0			0
$39$	6.47214	−	4.70228i	1.03637	−	0.752968i
$40$		0			0
$41$	0.347059	−	1.06814i	0.0542015	−	0.166815i	−0.920291	−	0.391234i	$-0.872048\pi$
$41$	0.347059	−	1.06814i	0.0542015	−	0.166815i	0.974493	+	0.224419i	$0.0720484\pi$
$42$		0			0
$43$	7.12311			1.08626			0.543132	−	0.839648i	$-0.317238\pi$
$43$	7.12311			1.08626			0.543132	+	0.839648i	$0.317238\pi$
$44$		0			0
$45$	−2.00000			−0.298142
$46$		0			0
$47$	2.47214	−	7.60845i	0.360598	−	1.10981i	−0.592094	−	0.805869i	$-0.701699\pi$
$47$	2.47214	−	7.60845i	0.360598	−	1.10981i	0.952692	−	0.303938i	$-0.0983015\pi$
$48$		0			0
$49$	−2.22786	+	1.61864i	−0.318266	+	0.231234i
$50$		0			0
$51$	0.965093	+	2.97025i	0.135140	+	0.415918i
$52$		0			0
$53$	−9.90739	−	7.19814i	−1.36089	−	0.988741i	−0.998388	−	0.0567560i	$-0.981924\pi$
$53$	−9.90739	−	7.19814i	−1.36089	−	0.988741i	−0.362497	−	0.931985i	$-0.618076\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	5.05329	+	3.67143i	0.669325	+	0.486293i
$58$		0			0
$59$	2.41273	+	7.42562i	0.314111	+	0.966734i	0.976119	+	0.217237i	$0.0697045\pi$
$59$	2.41273	+	7.42562i	0.314111	+	0.966734i	−0.662008	+	0.749497i	$0.730295\pi$
$60$		0			0
$61$	0.908612	−	0.660145i	0.116336	−	0.0845229i	−0.528096	−	0.849185i	$-0.677094\pi$
$61$	0.908612	−	0.660145i	0.116336	−	0.0845229i	0.644432	+	0.764662i	$0.277094\pi$
$62$		0			0
$63$	0.541951	−	1.66795i	0.0682793	−	0.210142i
$64$		0			0
$65$	18.2462			2.26316
$66$		0			0
$67$	9.56155			1.16813			0.584065	−	0.811707i	$-0.301461\pi$
$67$	9.56155			1.16813			0.584065	+	0.811707i	$0.301461\pi$
$68$		0			0
$69$	−1.17666	+	3.62140i	−0.141654	+	0.435965i
$70$		0			0
$71$	7.02604	−	5.10471i	0.833837	−	0.605818i	−0.0868052	−	0.996225i	$-0.527666\pi$
$71$	7.02604	−	5.10471i	0.833837	−	0.605818i	0.920642	+	0.390407i	$0.127666\pi$
$72$		0			0
$73$	−1.58313	−	4.87236i	−0.185291	−	0.570267i	0.814662	−	0.579936i	$-0.196922\pi$
$73$	−1.58313	−	4.87236i	−0.185291	−	0.570267i	−0.999953	+	0.00966869i	$0.996922\pi$
$74$		0			0
$75$	9.70820	+	7.05342i	1.12101	+	0.814459i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.99878	−	6.53800i	−1.01244	−	0.735582i	−0.0477221	−	0.998861i	$-0.515196\pi$
$79$	−8.99878	−	6.53800i	−1.01244	−	0.735582i	−0.964720	+	0.263278i	$0.915196\pi$
$80$		0			0
$81$	−2.16312	−	6.65740i	−0.240347	−	0.739711i
$82$		0			0
$83$	0.709422	−	0.515426i	0.0778692	−	0.0565753i	−0.548170	−	0.836367i	$-0.684675\pi$
$83$	0.709422	−	0.515426i	0.0778692	−	0.0565753i	0.626039	+	0.779792i	$0.284675\pi$
$84$		0			0
$85$	−2.20116	+	6.77448i	−0.238749	+	0.734795i
$86$		0			0
$87$	−8.00000			−0.857690
$88$		0			0
$89$	2.68466			0.284573			0.142287	−	0.989825i	$-0.454555\pi$
$89$	2.68466			0.284573			0.142287	+	0.989825i	$0.454555\pi$
$90$		0			0
$91$	−4.94427	+	15.2169i	−0.518301	+	1.59516i
$92$		0			0
$93$	−7.02604	+	5.10471i	−0.728566	+	0.529334i
$94$		0			0
$95$	4.40232	+	13.5490i	0.451669	+	1.39009i
$96$		0			0
$97$	−12.5896	−	9.14685i	−1.27828	−	0.928722i	−0.278777	−	0.960356i	$-0.589929\pi$
$97$	−12.5896	−	9.14685i	−1.27828	−	0.928722i	−0.999500	+	0.0316339i	$0.989929\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.61803	−	1.17557i	−0.161000	−	0.116974i	0.504368	−	0.863489i	$-0.331726\pi$
$101$	−1.61803	−	1.17557i	−0.161000	−	0.116974i	−0.665368	+	0.746515i	$0.731726\pi$
$102$		0			0
$103$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$103$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$104$		0			0
$105$	−14.0521	+	10.2094i	−1.37134	+	0.996338i
$106$		0			0
$107$	4.13135	−	12.7150i	0.399392	−	1.22920i	−0.526095	−	0.850426i	$-0.676345\pi$
$107$	4.13135	−	12.7150i	0.399392	−	1.22920i	0.925488	−	0.378778i	$-0.123655\pi$
$108$		0			0
$109$	−12.2462			−1.17297			−0.586487	−	0.809959i	$-0.699490\pi$
$109$	−12.2462			−1.17297			−0.586487	+	0.809959i	$0.699490\pi$
$110$		0			0
$111$	11.8078			1.12074
$112$		0			0
$113$	−0.135488	+	0.416988i	−0.0127456	+	0.0392269i	−0.957227	−	0.289338i	$-0.906565\pi$
$113$	−0.135488	+	0.416988i	−0.0127456	+	0.0392269i	0.944481	+	0.328565i	$0.106565\pi$
$114$		0			0
$115$	−7.02604	+	5.10471i	−0.655181	+	0.476017i
$116$		0			0
$117$	−0.889009	−	2.73609i	−0.0821889	−	0.252952i
$118$		0			0
$119$	−5.05329	−	3.67143i	−0.463234	−	0.336560i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	1.41884	+	1.03085i	0.127933	+	0.0929487i
$124$		0			0
$125$	2.95468	+	9.09358i	0.264275	+	0.813354i
$126$		0			0
$127$	5.05329	−	3.67143i	0.448407	−	0.325787i	−0.340559	−	0.940223i	$-0.610616\pi$
$127$	5.05329	−	3.67143i	0.448407	−	0.325787i	0.788967	+	0.614436i	$0.210616\pi$
$128$		0			0
$129$	−3.43723	+	10.5787i	−0.302631	+	0.931403i
$130$		0			0
$131$	−13.3693			−1.16808			−0.584041	−	0.811724i	$-0.698529\pi$
$131$	−13.3693			−1.16808			−0.584041	+	0.811724i	$0.698529\pi$
$132$		0			0
$133$	−12.4924			−1.08323
$134$		0			0
$135$	6.12094	−	18.8383i	0.526806	−	1.62134i
$136$		0			0
$137$	6.82685	−	4.95999i	0.583257	−	0.423761i	−0.256640	−	0.966507i	$-0.582615\pi$
$137$	6.82685	−	4.95999i	0.583257	−	0.423761i	0.839897	+	0.542746i	$0.182615\pi$
$138$		0			0
$139$	−4.67330	−	14.3829i	−0.396384	−	1.21994i	−0.927879	−	0.372882i	$-0.878369\pi$
$139$	−4.67330	−	14.3829i	−0.396384	−	1.21994i	0.531495	−	0.847062i	$-0.321631\pi$
$140$		0			0
$141$	10.1066	+	7.34286i	0.851128	+	0.618381i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−14.7615	−	10.7249i	−1.22588	−	0.890651i
$146$		0			0
$147$	−1.32883	−	4.08972i	−0.109600	−	0.337314i
$148$		0			0
$149$	3.43526	−	2.49586i	0.281427	−	0.204469i	−0.438112	−	0.898920i	$-0.644353\pi$
$149$	3.43526	−	2.49586i	0.281427	−	0.204469i	0.719540	+	0.694451i	$0.244353\pi$
$150$		0			0
$151$	2.89528	−	8.91075i	0.235614	−	0.725147i	−0.761425	−	0.648253i	$-0.775500\pi$
$151$	2.89528	−	8.91075i	0.235614	−	0.725147i	0.997039	−	0.0768934i	$-0.0245001\pi$
$152$		0			0
$153$	1.12311			0.0907977
$154$		0			0
$155$	−19.8078			−1.59100
$156$		0			0
$157$	−1.37156	+	4.22121i	−0.109462	+	0.336890i	−0.990752	−	0.135687i	$-0.956676\pi$
$157$	−1.37156	+	4.22121i	−0.109462	+	0.336890i	0.881290	+	0.472576i	$0.156676\pi$
$158$		0			0
$159$	15.4709	−	11.2403i	1.22692	−	0.891412i
$160$		0			0
$161$	−2.35333	−	7.24280i	−0.185468	−	0.570812i
$162$		0			0
$163$	3.23607	+	2.35114i	0.253468	+	0.184156i	0.707263	−	0.706951i	$-0.249930\pi$
$163$	3.23607	+	2.35114i	0.253468	+	0.184156i	−0.453794	+	0.891107i	$0.649930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.47214	+	4.70228i	0.500829	+	0.363874i	0.809334	−	0.587349i	$-0.199829\pi$
$167$	6.47214	+	4.70228i	0.500829	+	0.363874i	−0.308505	+	0.951223i	$0.599829\pi$
$168$		0			0
$169$	4.09330	+	12.5979i	0.314870	+	0.969069i
$170$		0			0
$171$	1.81722	−	1.32029i	0.138967	−	0.100965i
$172$		0			0
$173$	−3.78429	+	11.6468i	−0.287714	+	0.885493i	0.697858	+	0.716236i	$0.254137\pi$
$173$	−3.78429	+	11.6468i	−0.287714	+	0.885493i	−0.985572	+	0.169257i	$0.945863\pi$
$174$		0			0
$175$	−24.0000			−1.81423
$176$		0			0
$177$	−12.1922			−0.916425
$178$		0			0
$179$	−1.98959	+	6.12333i	−0.148709	+	0.457679i	−0.997469	−	0.0710987i	$-0.977349\pi$
$179$	−1.98959	+	6.12333i	−0.148709	+	0.457679i	0.848760	+	0.528778i	$0.177349\pi$
$180$		0			0
$181$	1.06413	−	0.773138i	0.0790964	−	0.0574669i	−0.547534	−	0.836783i	$-0.684433\pi$
$181$	1.06413	−	0.773138i	0.0790964	−	0.0574669i	0.626631	+	0.779316i	$0.284433\pi$
$182$		0			0
$183$	0.541951	+	1.66795i	0.0400621	+	0.123299i
$184$		0			0
$185$	21.7875	+	15.8296i	1.60185	+	1.16381i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	14.0521	+	10.2094i	1.02214	+	0.742627i
$190$		0			0
$191$	−3.22566	−	9.92755i	−0.233400	−	0.718333i	−0.997330	−	0.0730324i	$-0.976732\pi$
$191$	−3.22566	−	9.92755i	−0.233400	−	0.718333i	0.763929	−	0.645300i	$-0.223268\pi$
$192$		0			0
$193$	−7.38075	+	5.36243i	−0.531278	+	0.385996i	−0.820836	−	0.571165i	$-0.806492\pi$
$193$	−7.38075	+	5.36243i	−0.531278	+	0.385996i	0.289558	+	0.957161i	$0.406492\pi$
$194$		0			0
$195$	−8.80464	+	27.0979i	−0.630514	+	1.94052i
$196$		0			0
$197$	14.4924			1.03254			0.516271	−	0.856425i	$-0.327320\pi$
$197$	14.4924			1.03254			0.516271	+	0.856425i	$0.327320\pi$
$198$		0			0
$199$	−12.4924			−0.885564			−0.442782	−	0.896629i	$-0.646009\pi$
$199$	−12.4924			−0.885564			−0.442782	+	0.896629i	$0.646009\pi$
$200$		0			0
$201$	−4.61389	+	14.2001i	−0.325439	+	1.00160i
$202$		0			0
$203$	12.9443	−	9.40456i	0.908510	−	0.660071i
$204$		0			0
$205$	1.23607	+	3.80423i	0.0863307	+	0.265699i
$206$		0			0
$207$	1.10780	+	0.804864i	0.0769975	+	0.0559419i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	6.87051	+	4.99172i	0.472986	+	0.343644i	0.798604	−	0.601857i	$-0.205572\pi$
$211$	6.87051	+	4.99172i	0.472986	+	0.343644i	−0.325618	+	0.945501i	$0.605572\pi$
$212$		0			0
$213$	4.19075	+	12.8978i	0.287145	+	0.883743i
$214$		0			0
$215$	−20.5242	+	14.9117i	−1.39974	+	1.01697i
$216$		0			0
$217$	5.36741	−	16.5192i	0.364364	−	1.12140i
$218$		0			0
$219$	8.00000			0.540590
$220$		0			0
$221$	−10.2462			−0.689235
$222$		0			0
$223$	−3.64880	+	11.2299i	−0.244342	+	0.752006i	0.751402	+	0.659844i	$0.229378\pi$
$223$	−3.64880	+	11.2299i	−0.244342	+	0.752006i	−0.995744	+	0.0921621i	$0.970622\pi$
$224$		0			0
$225$	3.49118	−	2.53649i	0.232746	−	0.169100i
$226$		0			0
$227$	−7.14543	−	21.9914i	−0.474259	−	1.45962i	−0.846954	−	0.531665i	$-0.821566\pi$
$227$	−7.14543	−	21.9914i	−0.474259	−	1.45962i	0.372695	−	0.927954i	$-0.378434\pi$
$228$		0			0
$229$	−11.8801	−	8.63143i	−0.785062	−	0.570381i	0.121432	−	0.992600i	$-0.461251\pi$
$229$	−11.8801	−	8.63143i	−0.785062	−	0.570381i	−0.906494	+	0.422219i	$0.861251\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.96190	−	4.33158i	−0.390577	−	0.283771i	0.375115	−	0.926978i	$-0.377603\pi$
$233$	−5.96190	−	4.33158i	−0.390577	−	0.283771i	−0.765692	+	0.643207i	$0.777603\pi$
$234$		0			0
$235$	8.80464	+	27.0979i	0.574352	+	1.76767i
$236$		0			0
$237$	14.0521	−	10.2094i	0.912780	−	0.663174i
$238$		0			0
$239$	1.50704	−	4.63820i	0.0974825	−	0.300020i	−0.890410	−	0.455159i	$-0.849582\pi$
$239$	1.50704	−	4.63820i	0.0974825	−	0.300020i	0.987893	+	0.155139i	$0.0495824\pi$
$240$		0			0
$241$	−29.1231			−1.87598			−0.937992	−	0.346657i	$-0.887317\pi$
$241$	−29.1231			−1.87598			−0.937992	+	0.346657i	$0.887317\pi$
$242$		0			0
$243$	−5.75379			−0.369106
$244$		0			0
$245$	3.03077	−	9.32774i	0.193629	−	0.595927i
$246$		0			0
$247$	−16.5787	+	12.0451i	−1.05488	+	0.766414i
$248$		0			0
$249$	0.423142	+	1.30230i	0.0268155	+	0.0825298i
$250$		0			0
$251$	−1.26332	−	0.917858i	−0.0797402	−	0.0579347i	0.547201	−	0.837001i	$-0.315693\pi$
$251$	−1.26332	−	0.917858i	−0.0797402	−	0.0579347i	−0.626941	+	0.779067i	$0.715693\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−8.99878	−	6.53800i	−0.563526	−	0.409425i
$256$		0			0
$257$	3.63212	+	11.1785i	0.226565	+	0.697297i	0.998129	+	0.0611444i	$0.0194750\pi$
$257$	3.63212	+	11.1785i	0.226565	+	0.697297i	−0.771563	+	0.636152i	$0.780525\pi$
$258$		0			0
$259$	−19.1054	+	13.8809i	−1.18715	+	0.862515i
$260$		0			0
$261$	−0.889009	+	2.73609i	−0.0550283	+	0.169360i
$262$		0			0
$263$	−19.1231			−1.17918			−0.589591	−	0.807702i	$-0.700711\pi$
$263$	−19.1231			−1.17918			−0.589591	+	0.807702i	$0.700711\pi$
$264$		0			0
$265$	43.6155			2.67928
$266$		0			0
$267$	−1.29547	+	3.98705i	−0.0792816	+	0.244004i
$268$		0			0
$269$	16.7779	−	12.1899i	1.02297	−	0.743229i	0.0560781	−	0.998426i	$-0.482140\pi$
$269$	16.7779	−	12.1899i	1.02297	−	0.743229i	0.966889	+	0.255197i	$0.0821404\pi$
$270$		0			0
$271$	8.80464	+	27.0979i	0.534844	+	1.64608i	0.743986	+	0.668195i	$0.232933\pi$
$271$	8.80464	+	27.0979i	0.534844	+	1.64608i	−0.209142	+	0.977885i	$0.567067\pi$
$272$		0			0
$273$	−20.2132	−	14.6857i	−1.22336	−	0.888820i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−14.5623	−	10.5801i	−0.874964	−	0.635699i	0.0569502	−	0.998377i	$-0.481862\pi$
$277$	−14.5623	−	10.5801i	−0.874964	−	0.635699i	−0.931914	+	0.362678i	$0.881862\pi$
$278$		0			0
$279$	0.965093	+	2.97025i	0.0577786	+	0.177824i
$280$		0			0
$281$	13.1435	−	9.54928i	0.784073	−	0.569662i	−0.122126	−	0.992515i	$-0.538971\pi$
$281$	13.1435	−	9.54928i	0.784073	−	0.569662i	0.906199	+	0.422852i	$0.138971\pi$
$282$		0			0
$283$	−6.18034	+	19.0211i	−0.367383	+	1.13069i	0.581092	+	0.813838i	$0.302626\pi$
$283$	−6.18034	+	19.0211i	−0.367383	+	1.13069i	−0.948475	+	0.316851i	$0.897374\pi$
$284$		0			0
$285$	−22.2462			−1.31775
$286$		0			0
$287$	−3.50758			−0.207046
$288$		0			0
$289$	−4.01722	+	12.3637i	−0.236307	+	0.727279i
$290$		0			0
$291$	19.6593	−	14.2833i	1.15245	−	0.837301i
$292$		0			0
$293$	1.04118	+	3.20441i	0.0608262	+	0.187204i	0.976852	−	0.213915i	$-0.0686215\pi$
$293$	1.04118	+	3.20441i	0.0608262	+	0.187204i	−0.916026	+	0.401119i	$0.868622\pi$
$294$		0			0
$295$	−22.4970	−	16.3450i	−1.30982	−	0.951642i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.1066	−	7.34286i	−0.584479	−	0.424649i
$300$		0			0
$301$	−6.87446	−	21.1574i	−0.396237	−	1.21949i
$302$		0			0
$303$	2.52665	−	1.83572i	0.145152	−	0.105459i
$304$		0			0
$305$	−1.23607	+	3.80423i	−0.0707770	+	0.217829i
$306$		0			0
$307$	32.4924			1.85444			0.927220	−	0.374516i	$-0.122191\pi$
$307$	32.4924			1.85444			0.927220	+	0.374516i	$0.122191\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.01409	−	9.27640i	0.170913	−	0.526017i	−0.828510	−	0.559974i	$-0.810811\pi$
$311$	3.01409	−	9.27640i	0.170913	−	0.526017i	0.999423	+	0.0339575i	$0.0108111\pi$
$312$		0			0
$313$	7.93465	−	5.76486i	0.448493	−	0.325849i	−0.340508	−	0.940242i	$-0.610599\pi$
$313$	7.93465	−	5.76486i	0.448493	−	0.325849i	0.789000	+	0.614393i	$0.210599\pi$
$314$		0			0
$315$	1.93019	+	5.94050i	0.108754	+	0.334709i
$316$		0			0
$317$	11.4818	+	8.34199i	0.644880	+	0.468533i	0.861523	−	0.507718i	$-0.169511\pi$
$317$	11.4818	+	8.34199i	0.644880	+	0.468533i	−0.216643	+	0.976251i	$0.569511\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	16.8898	+	12.2711i	0.942695	+	0.684908i
$322$		0			0
$323$	−2.47214	−	7.60845i	−0.137553	−	0.423346i
$324$		0			0
$325$	−31.8504	+	23.1407i	−1.76674	+	1.28362i
$326$		0			0
$327$	5.90936	−	18.1872i	0.326789	−	1.00575i
$328$		0			0
$329$	−24.9848			−1.37746
$330$		0			0
$331$	34.9309			1.91997			0.959987	−	0.280044i	$-0.0903491\pi$
$331$	34.9309			1.91997			0.959987	+	0.280044i	$0.0903491\pi$
$332$		0			0
$333$	1.31215	−	4.03839i	0.0719055	−	0.221302i
$334$		0			0
$335$	−27.5502	+	20.0164i	−1.50523	+	1.09361i
$336$		0			0
$337$	5.17252	+	15.9194i	0.281765	+	0.867184i	0.987350	+	0.158558i	$0.0506846\pi$
$337$	5.17252	+	15.9194i	0.281765	+	0.867184i	−0.705584	+	0.708626i	$0.749315\pi$
$338$		0			0
$339$	−0.553900	−	0.402432i	−0.0300837	−	0.0218571i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.7287	−	7.79484i	−0.579294	−	0.420882i
$344$		0			0
$345$	−4.19075	−	12.8978i	−0.225622	−	0.694394i
$346$		0			0
$347$	18.3959	−	13.3654i	0.987546	−	0.717494i	0.0281636	−	0.999603i	$-0.491034\pi$
$347$	18.3959	−	13.3654i	0.987546	−	0.717494i	0.959382	+	0.282109i	$0.0910341\pi$
$348$		0			0
$349$	9.96463	−	30.6680i	0.533394	−	1.64162i	−0.213699	−	0.976900i	$-0.568551\pi$
$349$	9.96463	−	30.6680i	0.533394	−	1.64162i	0.747093	−	0.664719i	$-0.231449\pi$
$350$		0			0
$351$	28.4924			1.52081
$352$		0			0
$353$	−24.0540			−1.28026			−0.640132	−	0.768265i	$-0.721120\pi$
$353$	−24.0540			−1.28026			−0.640132	+	0.768265i	$0.721120\pi$
$354$		0			0
$355$	−9.55816	+	29.4170i	−0.507295	+	1.56129i
$356$		0			0
$357$	7.89098	−	5.73313i	0.417635	−	0.303430i
$358$		0			0
$359$	−1.38823	−	4.27255i	−0.0732682	−	0.225496i	0.907716	−	0.419586i	$-0.137825\pi$
$359$	−1.38823	−	4.27255i	−0.0732682	−	0.225496i	−0.980984	+	0.194090i	$0.937825\pi$
$360$		0			0
$361$	2.42705	+	1.76336i	0.127740	+	0.0928082i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.7615	+	10.7249i	0.772652	+	0.561365i
$366$		0			0
$367$	7.08603	+	21.8086i	0.369888	+	1.13840i	0.946863	+	0.321636i	$0.104233\pi$
$367$	7.08603	+	21.8086i	0.369888	+	1.13840i	−0.576976	+	0.816761i	$0.695767\pi$
$368$		0			0
$369$	0.510233	−	0.370706i	0.0265617	−	0.0192982i
$370$		0			0
$371$	−11.8187	+	36.3743i	−0.613598	+	1.88846i
$372$		0			0
$373$	8.24621			0.426973			0.213486	−	0.976946i	$-0.431518\pi$
$373$	8.24621			0.426973			0.213486	+	0.976946i	$0.431518\pi$
$374$		0			0
$375$	−14.9309			−0.771027
$376$		0			0
$377$	8.11053	−	24.9616i	0.417713	−	1.28559i
$378$		0			0
$379$	−0.155522	+	0.112993i	−0.00798864	+	0.00580408i	−0.591772	−	0.806105i	$-0.701572\pi$
$379$	−0.155522	+	0.112993i	−0.00798864	+	0.00580408i	0.583784	+	0.811909i	$0.301572\pi$
$380$		0			0
$381$	3.01409	+	9.27640i	0.154416	+	0.475245i
$382$		0			0
$383$	1.66170	+	1.20730i	0.0849090	+	0.0616900i	0.629430	−	0.777057i	$-0.283288\pi$
$383$	1.66170	+	1.20730i	0.0849090	+	0.0616900i	−0.544521	+	0.838747i	$0.683288\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.23607	+	2.35114i	0.164499	+	0.119515i
$388$		0			0
$389$	1.10058	+	3.38724i	0.0558016	+	0.171740i	0.975073	−	0.221885i	$-0.0712209\pi$
$389$	1.10058	+	3.38724i	0.0558016	+	0.171740i	−0.919271	+	0.393625i	$0.871221\pi$
$390$		0			0
$391$	3.94549	−	2.86657i	0.199532	−	0.144969i
$392$		0			0
$393$	6.45132	−	19.8551i	0.325426	−	1.00156i
$394$		0			0
$395$	39.6155			1.99327
$396$		0			0
$397$	10.4924			0.526600			0.263300	−	0.964714i	$-0.415189\pi$
$397$	10.4924			0.526600			0.263300	+	0.964714i	$0.415189\pi$
$398$		0			0
$399$	6.02817	−	18.5528i	0.301786	−	0.928802i
$400$		0			0
$401$	−24.6689	+	17.9230i	−1.23191	+	0.895032i	−0.997032	−	0.0769941i	$-0.975468\pi$
$401$	−24.6689	+	17.9230i	−1.23191	+	0.895032i	−0.234874	+	0.972026i	$0.575468\pi$
$402$		0			0
$403$	−8.80464	−	27.0979i	−0.438590	−	1.34984i
$404$		0			0
$405$	20.1695	+	14.6540i	1.00223	+	0.728163i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	18.1968	+	13.2207i	0.899771	+	0.653722i	0.938407	−	0.345531i	$-0.112301\pi$
$409$	18.1968	+	13.2207i	0.899771	+	0.653722i	−0.0386358	+	0.999253i	$0.512301\pi$
$410$		0			0
$411$	4.07194	+	12.5321i	0.200854	+	0.618165i
$412$		0			0
$413$	19.7275	−	14.3328i	0.970724	−	0.705273i
$414$		0			0
$415$	−0.965093	+	2.97025i	−0.0473745	+	0.145804i
$416$		0			0
$417$	23.6155			1.15646
$418$		0			0
$419$	−32.4924			−1.58736			−0.793679	−	0.608336i	$-0.791837\pi$
$419$	−32.4924			−1.58736			−0.793679	+	0.608336i	$0.791837\pi$
$420$		0			0
$421$	0.770201	−	2.37043i	0.0375373	−	0.115528i	−0.930532	−	0.366210i	$-0.880655\pi$
$421$	0.770201	−	2.37043i	0.0375373	−	0.115528i	0.968069	+	0.250683i	$0.0806550\pi$
$422$		0			0
$423$	3.63445	−	2.64058i	0.176713	−	0.128389i
$424$		0			0
$425$	−4.74938	−	14.6171i	−0.230379	−	0.709033i
$426$		0			0
$427$	−2.83769	−	2.06170i	−0.137325	−	0.0997728i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	21.9431	+	15.9426i	1.05696	+	0.767926i	0.973523	−	0.228587i	$-0.0734105\pi$
$431$	21.9431	+	15.9426i	1.05696	+	0.767926i	0.0834362	+	0.996513i	$0.473411\pi$
$432$		0			0
$433$	−7.00994	−	21.5744i	−0.336876	−	1.03680i	−0.965790	−	0.259324i	$-0.916500\pi$
$433$	−7.00994	−	21.5744i	−0.336876	−	1.03680i	0.628914	−	0.777475i	$-0.283500\pi$
$434$		0			0
$435$	23.0509	−	16.7474i	1.10520	−	0.802978i
$436$		0			0
$437$	3.01409	−	9.27640i	0.144183	−	0.443751i
$438$		0			0
$439$	−4.49242			−0.214412			−0.107206	−	0.994237i	$-0.534190\pi$
$439$	−4.49242			−0.214412			−0.107206	+	0.994237i	$0.534190\pi$
$440$		0			0
$441$	−1.54640			−0.0736380
$442$		0			0
$443$	3.49663	−	10.7615i	0.166130	−	0.511296i	−0.832988	−	0.553291i	$-0.813372\pi$
$443$	3.49663	−	10.7615i	0.166130	−	0.511296i	0.999118	+	0.0419957i	$0.0133716\pi$
$444$		0			0
$445$	−7.73546	+	5.62014i	−0.366696	+	0.266420i
$446$		0			0
$447$	2.04899	+	6.30615i	0.0969141	+	0.298271i
$448$		0			0
$449$	29.5667	+	21.4814i	1.39534	+	1.01377i	0.995256	+	0.0972917i	$0.0310180\pi$
$449$	29.5667	+	21.4814i	1.39534	+	1.01377i	0.400081	+	0.916480i	$0.368982\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	11.8365	+	8.59970i	0.556126	+	0.404049i
$454$		0			0
$455$	−17.6093	−	54.1958i	−0.825536	−	2.54074i
$456$		0			0
$457$	19.3046	−	14.0256i	0.903029	−	0.656089i	−0.0362133	−	0.999344i	$-0.511530\pi$
$457$	19.3046	−	14.0256i	0.903029	−	0.656089i	0.939242	+	0.343255i	$0.111530\pi$
$458$		0			0
$459$	−3.43723	+	10.5787i	−0.160436	+	0.493772i
$460$		0			0
$461$	−1.12311			−0.0523082			−0.0261541	−	0.999658i	$-0.508326\pi$
$461$	−1.12311			−0.0523082			−0.0261541	+	0.999658i	$0.508326\pi$
$462$		0			0
$463$	15.3153			0.711764			0.355882	−	0.934531i	$-0.384180\pi$
$463$	15.3153			0.711764			0.355882	+	0.934531i	$0.384180\pi$
$464$		0			0
$465$	9.55816	−	29.4170i	0.443249	−	1.36418i
$466$		0			0
$467$	−22.8953	+	16.6344i	−1.05947	+	0.769750i	−0.973990	−	0.226590i	$-0.927242\pi$
$467$	−22.8953	+	16.6344i	−1.05947	+	0.769750i	−0.0854794	+	0.996340i	$0.527242\pi$
$468$		0			0
$469$	−9.22778	−	28.4002i	−0.426100	−	1.31140i
$470$		0			0
$471$	−5.60719	−	4.07386i	−0.258366	−	0.187714i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−24.8681	−	18.0677i	−1.14103	−	0.829004i
$476$		0			0
$477$	−2.12508	−	6.54032i	−0.0973006	−	0.299460i
$478$		0			0
$479$	−12.9443	+	9.40456i	−0.591439	+	0.429705i	−0.842830	−	0.538180i	$-0.819112\pi$
$479$	−12.9443	+	9.40456i	−0.591439	+	0.429705i	0.251391	+	0.967886i	$0.419112\pi$
$480$		0			0
$481$	−11.9709	+	36.8426i	−0.545826	+	1.67988i
$482$		0			0
$483$	11.8920			0.541107
$484$		0			0
$485$	55.4233			2.51664
$486$		0			0
$487$	4.61389	−	14.2001i	0.209075	−	0.643468i	−0.790446	−	0.612532i	$-0.790151\pi$
$487$	4.61389	−	14.2001i	0.209075	−	0.643468i	0.999521	−	0.0309362i	$-0.00984886\pi$
$488$		0			0
$489$	−5.05329	+	3.67143i	−0.228518	+	0.166028i
$490$		0			0
$491$	−4.25015	−	13.0806i	−0.191807	−	0.590321i	−0.999999	−	0.00140182i	$-0.999554\pi$
$491$	−4.25015	−	13.0806i	−0.191807	−	0.590321i	0.808192	−	0.588919i	$-0.200446\pi$
$492$		0			0
$493$	8.28936	+	6.02257i	0.373334	+	0.271243i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−21.9431	−	15.9426i	−0.984280	−	0.715122i
$498$		0			0
$499$	−8.95681	−	27.5662i	−0.400962	−	1.23403i	−0.924220	−	0.381860i	$-0.875284\pi$
$499$	−8.95681	−	27.5662i	−0.400962	−	1.23403i	0.523258	−	0.852174i	$-0.324716\pi$
$500$		0			0
$501$	−10.1066	+	7.34286i	−0.451529	+	0.328055i
$502$		0			0
$503$	9.76974	−	30.0682i	0.435611	−	1.34067i	−0.456848	−	0.889545i	$-0.651022\pi$
$503$	9.76974	−	30.0682i	0.435611	−	1.34067i	0.892459	−	0.451128i	$-0.148978\pi$
$504$		0			0
$505$	7.12311			0.316974
$506$		0			0
$507$	−20.6847			−0.918638
$508$		0			0
$509$	−5.65507	+	17.4045i	−0.250657	+	0.771441i	0.743998	+	0.668182i	$0.232927\pi$
$509$	−5.65507	+	17.4045i	−0.250657	+	0.771441i	−0.994654	+	0.103260i	$0.967073\pi$
$510$		0			0
$511$	−12.9443	+	9.40456i	−0.572621	+	0.416033i
$512$		0			0
$513$	6.87446	+	21.1574i	0.303515	+	0.934122i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−15.4709	−	11.2403i	−0.679098	−	0.493394i
$520$		0			0
$521$	0.406463	+	1.25096i	0.0178075	+	0.0548057i	0.959565	−	0.281486i	$-0.0908272\pi$
$521$	0.406463	+	1.25096i	0.0178075	+	0.0548057i	−0.941758	+	0.336292i	$0.890827\pi$
$522$		0			0
$523$	−9.70820	+	7.05342i	−0.424510	+	0.308425i	−0.779450	−	0.626464i	$-0.784501\pi$
$523$	−9.70820	+	7.05342i	−0.424510	+	0.308425i	0.354940	+	0.934889i	$0.384501\pi$
$524$		0			0
$525$	11.5811	−	35.6430i	0.505441	−	1.55559i
$526$		0			0
$527$	11.1231			0.484530
$528$		0			0
$529$	−17.0540			−0.741477
$530$		0			0
$531$	−1.35488	+	4.16988i	−0.0587966	+	0.180957i
$532$		0			0
$533$	−4.65491	+	3.38199i	−0.201627	+	0.146490i
$534$		0			0
$535$	14.7140	+	45.2851i	0.636142	+	1.95784i
$536$		0			0
$537$	−8.13384	−	5.90958i	−0.351001	−	0.255017i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−19.3046	−	14.0256i	−0.829968	−	0.603007i	0.0895826	−	0.995979i	$-0.471447\pi$
$541$	−19.3046	−	14.0256i	−0.829968	−	0.603007i	−0.919550	+	0.392973i	$0.871447\pi$
$542$		0			0
$543$	0.634713	+	1.95345i	0.0272382	+	0.0838304i
$544$		0			0
$545$	35.2857	−	25.6366i	1.51147	−	1.09815i
$546$		0			0
$547$	13.0548	−	40.1785i	0.558183	−	1.71791i	−0.129203	−	0.991618i	$-0.541242\pi$
$547$	13.0548	−	40.1785i	0.558183	−	1.71791i	0.687386	−	0.726292i	$-0.258758\pi$
$548$		0			0
$549$	0.630683			0.0269169
$550$		0			0
$551$	20.4924			0.873007
$552$		0			0
$553$	−10.7348	+	33.0384i	−0.456491	+	1.40494i
$554$		0			0
$555$	−34.0224	+	24.7187i	−1.44417	+	1.04925i
$556$		0			0
$557$	1.15998	+	3.57007i	0.0491501	+	0.151269i	0.972619	−	0.232404i	$-0.0746591\pi$
$557$	1.15998	+	3.57007i	0.0491501	+	0.151269i	−0.923469	+	0.383673i	$0.874659\pi$
$558$		0			0
$559$	−29.5230	−	21.4497i	−1.24869	−	0.907226i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.8148	+	14.3963i	0.835094	+	0.606731i	0.920996	−	0.389573i	$-0.127377\pi$
$563$	19.8148	+	14.3963i	0.835094	+	0.606731i	−0.0859021	+	0.996304i	$0.527377\pi$
$564$		0			0
$565$	−0.482546	−	1.48512i	−0.0203009	−	0.0624797i
$566$		0			0
$567$	−17.6865	+	12.8500i	−0.742764	+	0.539650i
$568$		0			0
$569$	8.30542	−	25.5614i	0.348181	−	1.07159i	−0.611677	−	0.791107i	$-0.709505\pi$
$569$	8.30542	−	25.5614i	0.348181	−	1.07159i	0.959859	−	0.280484i	$-0.0904951\pi$
$570$		0			0
$571$	−16.4924			−0.690186			−0.345093	−	0.938568i	$-0.612153\pi$
$571$	−16.4924			−0.690186			−0.345093	+	0.938568i	$0.612153\pi$
$572$		0			0
$573$	16.3002			0.680950
$574$		0			0
$575$	5.79056	−	17.8215i	0.241483	−	0.743208i
$576$		0			0
$577$	−12.5896	+	9.14685i	−0.524110	+	0.380788i	−0.818150	−	0.575005i	$-0.805000\pi$
$577$	−12.5896	+	9.14685i	−0.524110	+	0.380788i	0.294040	+	0.955793i	$0.405000\pi$
$578$		0			0
$579$	−4.40232	−	13.5490i	−0.182954	−	0.563075i
$580$		0			0
$581$	−2.21560	−	1.60973i	−0.0919186	−	0.0667828i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	8.28936	+	6.02257i	0.342723	+	0.249003i
$586$		0			0
$587$	−7.56857	−	23.2937i	−0.312389	−	0.961433i	−0.976816	−	0.214080i	$-0.931325\pi$
$587$	−7.56857	−	23.2937i	−0.312389	−	0.961433i	0.664428	−	0.747353i	$-0.268675\pi$
$588$		0			0
$589$	17.9976	−	13.0760i	0.741577	−	0.538787i
$590$		0			0
$591$	−6.99327	+	21.5231i	−0.287665	+	0.885340i
$592$		0			0
$593$	−3.36932			−0.138361			−0.0691806	−	0.997604i	$-0.522038\pi$
$593$	−3.36932			−0.138361			−0.0691806	+	0.997604i	$0.522038\pi$
$594$		0			0
$595$	22.2462			0.912006
$596$		0			0
$597$	6.02817	−	18.5528i	0.246717	−	0.759316i
$598$		0			0
$599$	12.9443	−	9.40456i	0.528889	−	0.384260i	−0.291053	−	0.956707i	$-0.594006\pi$
$599$	12.9443	−	9.40456i	0.528889	−	0.384260i	0.819942	+	0.572447i	$0.194006\pi$
$600$		0			0
$601$	−1.15998	−	3.57007i	−0.0473168	−	0.145626i	0.924607	−	0.380923i	$-0.124394\pi$
$601$	−1.15998	−	3.57007i	−0.0473168	−	0.145626i	−0.971924	+	0.235297i	$0.924394\pi$
$602$		0			0
$603$	4.34387	+	3.15601i	0.176896	+	0.128522i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−37.1029	−	26.9569i	−1.50596	−	1.09414i	−0.967930	−	0.251220i	$-0.919168\pi$
$607$	−37.1029	−	26.9569i	−1.50596	−	1.09414i	−0.538031	−	0.842925i	$-0.680832\pi$
$608$		0			0
$609$	7.72074	+	23.7620i	0.312860	+	0.962885i
$610$		0			0
$611$	−33.1574	+	24.0903i	−1.34141	+	0.974589i
$612$		0			0
$613$	−3.66548	+	11.2812i	−0.148047	+	0.455643i	−0.997390	−	0.0721986i	$-0.976998\pi$
$613$	−3.66548	+	11.2812i	−0.148047	+	0.455643i	0.849343	+	0.527841i	$0.176998\pi$
$614$		0			0
$615$	−6.24621			−0.251872
$616$		0			0
$617$	−2.49242			−0.100341			−0.0501706	−	0.998741i	$-0.515976\pi$
$617$	−2.49242			−0.100341			−0.0501706	+	0.998741i	$0.515976\pi$
$618$		0			0
$619$	5.84996	−	18.0043i	0.235130	−	0.723655i	−0.761974	−	0.647607i	$-0.775770\pi$
$619$	5.84996	−	18.0043i	0.235130	−	0.723655i	0.997104	−	0.0760479i	$-0.0242302\pi$
$620$		0			0
$621$	−10.9715	+	7.97128i	−0.440272	+	0.319876i
$622$		0			0
$623$	−2.59094	−	7.97411i	−0.103804	−	0.319476i
$624$		0			0
$625$	3.53485	+	2.56822i	0.141394	+	0.102729i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.2348	−	8.88914i	−0.487835	−	0.354433i
$630$		0			0
$631$	−12.9954	−	39.9957i	−0.517338	−	1.59220i	−0.778987	−	0.627041i	$-0.784266\pi$
$631$	−12.9954	−	39.9957i	−0.517338	−	1.59220i	0.261648	−	0.965163i	$-0.415734\pi$
$632$		0			0
$633$	−10.7287	+	7.79484i	−0.426426	+	0.309817i
$634$		0			0
$635$	−6.87446	+	21.1574i	−0.272805	+	0.839606i
$636$		0			0
$637$	14.1080			0.558977
$638$		0			0
$639$	4.87689			0.192927
$640$		0			0
$641$	−14.3075	+	44.0341i	−0.565114	+	1.73924i	0.102498	+	0.994733i	$0.467316\pi$
$641$	−14.3075	+	44.0341i	−0.565114	+	1.73924i	−0.667612	+	0.744509i	$0.732684\pi$
$642$		0			0
$643$	7.42441	−	5.39415i	0.292790	−	0.212725i	−0.431687	−	0.902024i	$-0.642081\pi$
$643$	7.42441	−	5.39415i	0.292790	−	0.212725i	0.724477	+	0.689299i	$0.242081\pi$
$644$		0			0
$645$	−12.2419	−	37.6766i	−0.482023	−	1.48351i
$646$		0			0
$647$	−10.9715	−	7.97128i	−0.431335	−	0.313383i	0.350847	−	0.936433i	$-0.385894\pi$
$647$	−10.9715	−	7.97128i	−0.431335	−	0.313383i	−0.782183	+	0.623049i	$0.785894\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	21.9431	+	15.9426i	0.860016	+	0.624838i
$652$		0			0
$653$	10.8703	+	33.4554i	0.425388	+	1.30921i	0.902622	+	0.430434i	$0.141640\pi$
$653$	10.8703	+	33.4554i	0.425388	+	1.30921i	−0.477234	+	0.878776i	$0.658360\pi$
$654$		0			0
$655$	38.5218	−	27.9877i	1.50517	−	1.09357i
$656$		0			0
$657$	0.889009	−	2.73609i	0.0346836	−	0.106745i
$658$		0			0
$659$	11.6155			0.452477			0.226238	−	0.974072i	$-0.427357\pi$
$659$	11.6155			0.452477			0.226238	+	0.974072i	$0.427357\pi$
$660$		0			0
$661$	41.8078			1.62613			0.813067	−	0.582170i	$-0.197796\pi$
$661$	41.8078			1.62613			0.813067	+	0.582170i	$0.197796\pi$
$662$		0			0
$663$	4.94427	−	15.2169i	0.192020	−	0.590976i
$664$		0			0
$665$	35.9951	−	26.1520i	1.39583	−	1.01413i
$666$		0			0
$667$	3.86037	+	11.8810i	0.149474	+	0.460034i
$668$		0			0
$669$	−14.9170	−	10.8378i	−0.576725	−	0.419015i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	26.8845	+	19.5327i	1.03632	+	0.752931i	0.969564	−	0.244838i	$-0.0787348\pi$
$673$	26.8845	+	19.5327i	1.03632	+	0.752931i	0.0667569	+	0.997769i	$0.478735\pi$
$674$		0			0
$675$	13.2070	+	40.6469i	0.508336	+	1.56450i
$676$		0			0
$677$	−16.7779	+	12.1899i	−0.644827	+	0.468494i	−0.861505	−	0.507749i	$-0.830478\pi$
$677$	−16.7779	+	12.1899i	−0.644827	+	0.468494i	0.216678	+	0.976243i	$0.430478\pi$
$678$		0			0
$679$	−15.0183	+	46.2217i	−0.576351	+	1.77383i
$680$		0			0
$681$	36.1080			1.38366
$682$		0			0
$683$	6.73863			0.257847			0.128923	−	0.991655i	$-0.458848\pi$
$683$	6.73863			0.257847			0.128923	+	0.991655i	$0.458848\pi$
$684$		0			0
$685$	−9.28719	+	28.5830i	−0.354845	+	1.09210i
$686$		0			0
$687$	18.5515	−	13.4784i	0.707782	−	0.514234i
$688$		0			0
$689$	19.3873	+	59.6680i	0.738597	+	2.27317i
$690$		0			0
$691$	8.04650	+	5.84613i	0.306103	+	0.222397i	0.730223	−	0.683209i	$-0.239416\pi$
$691$	8.04650	+	5.84613i	0.306103	+	0.222397i	−0.424119	+	0.905606i	$0.639416\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	43.5751	+	31.6591i	1.65290	+	1.20090i
$696$		0			0
$697$	−0.694117	−	2.13627i	−0.0262916	−	0.0809171i
$698$		0			0
$699$	9.30983	−	6.76398i	0.352130	−	0.255837i
$700$		0			0
$701$	−15.6030	+	48.0211i	−0.589318	+	1.81373i	−0.00812617	+	0.999967i	$0.502587\pi$
$701$	−15.6030	+	48.0211i	−0.589318	+	1.81373i	−0.581192	+	0.813767i	$0.697413\pi$
$702$		0			0
$703$	−30.2462			−1.14076
$704$		0			0
$705$	−44.4924			−1.67568
$706$		0			0
$707$	−1.93019	+	5.94050i	−0.0725921	+	0.223415i
$708$		0			0
$709$	−1.77356	+	1.28856i	−0.0666073	+	0.0483930i	−0.620591	−	0.784135i	$-0.713107\pi$
$709$	−1.77356	+	1.28856i	−0.0666073	+	0.0483930i	0.553983	+	0.832528i	$0.313107\pi$
$710$		0			0
$711$	−1.93019	−	5.94050i	−0.0723876	−	0.222786i
$712$		0			0
$713$	10.9715	+	7.97128i	0.410887	+	0.298527i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	6.16109	+	4.47630i	0.230090	+	0.167170i
$718$		0			0
$719$	10.9464	+	33.6896i	0.408232	+	1.25641i	0.918166	+	0.396195i	$0.129670\pi$
$719$	10.9464	+	33.6896i	0.408232	+	1.25641i	−0.509935	+	0.860213i	$0.670330\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	14.0532	−	43.2515i	0.522646	−	1.60854i
$724$		0			0
$725$	39.3693			1.46214
$726$		0			0
$727$	−23.3153			−0.864718			−0.432359	−	0.901702i	$-0.642319\pi$
$727$	−23.3153			−0.864718			−0.432359	+	0.901702i	$0.642319\pi$
$728$		0			0
$729$	9.26583	−	28.5173i	0.343179	−	1.05620i
$730$		0			0
$731$	11.5254	−	8.37371i	0.426283	−	0.309713i
$732$		0			0
$733$	−0.347059	−	1.06814i	−0.0128189	−	0.0394525i	0.944442	−	0.328677i	$-0.106603\pi$
$733$	−0.347059	−	1.06814i	−0.0128189	−	0.0394525i	−0.957261	+	0.289224i	$0.906603\pi$
$734$		0			0
$735$	12.3904	+	9.00213i	0.457026	+	0.332049i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−2.12827	−	1.54628i	−0.0782896	−	0.0568807i	0.547952	−	0.836510i	$-0.315408\pi$
$739$	−2.12827	−	1.54628i	−0.0782896	−	0.0568807i	−0.626242	+	0.779629i	$0.715408\pi$
$740$		0			0
$741$	−9.88854	−	30.4338i	−0.363265	−	1.11801i
$742$		0			0
$743$	8.68774	−	6.31201i	0.318722	−	0.231565i	−0.416908	−	0.908949i	$-0.636886\pi$
$743$	8.68774	−	6.31201i	0.318722	−	0.231565i	0.735630	+	0.677384i	$0.236886\pi$
$744$		0			0
$745$	−4.67330	+	14.3829i	−0.171216	+	0.526950i
$746$		0			0
$747$	0.492423			0.0180168
$748$		0			0
$749$	−41.7538			−1.52565
$750$		0			0
$751$	1.71861	−	5.28935i	0.0627131	−	0.193011i	−0.914791	−	0.403928i	$-0.867645\pi$
$751$	1.71861	−	5.28935i	0.0627131	−	0.193011i	0.977504	+	0.210916i	$0.0676448\pi$
$752$		0			0
$753$	1.97275	−	1.43328i	0.0718908	−	0.0522317i
$754$		0			0
$755$	10.3117	+	31.7361i	0.375281	+	1.15500i
$756$		0			0
$757$	−12.7451	−	9.25984i	−0.463228	−	0.336555i	0.331568	−	0.943431i	$-0.392422\pi$
$757$	−12.7451	−	9.25984i	−0.463228	−	0.336555i	−0.794796	+	0.606876i	$0.792422\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.14468	+	3.01129i	0.150245	+	0.109159i	0.660368	−	0.750942i	$-0.270400\pi$
$761$	4.14468	+	3.01129i	0.150245	+	0.109159i	−0.510123	+	0.860101i	$0.670400\pi$
$762$		0			0
$763$	11.8187	+	36.3743i	0.427867	+	1.31684i
$764$		0			0
$765$	−3.23607	+	2.35114i	−0.117000	+	0.0850057i
$766$		0			0
$767$	12.3607	−	38.0423i	0.446318	−	1.37363i
$768$		0			0
$769$	−25.6155			−0.923720			−0.461860	−	0.886953i	$-0.652818\pi$
$769$	−25.6155			−0.923720			−0.461860	+	0.886953i	$0.652818\pi$
$770$		0			0
$771$	−18.3542			−0.661009
$772$		0			0
$773$	12.5889	−	38.7447i	0.452792	−	1.39355i	−0.420915	−	0.907100i	$-0.638291\pi$
$773$	12.5889	−	38.7447i	0.452792	−	1.39355i	0.873708	−	0.486452i	$-0.161709\pi$
$774$		0			0
$775$	34.5763	−	25.1211i	1.24202	−	0.902378i
$776$		0			0
$777$	−11.3956	−	35.0720i	−0.408814	−	1.25820i
$778$		0			0
$779$	−3.63445	−	2.64058i	−0.130218	−	0.0946086i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−23.0509	−	16.7474i	−0.823770	−	0.598504i
$784$		0			0
$785$	−4.88487	−	15.0341i	−0.174348	−	0.536589i
$786$		0			0
$787$	−24.0713	+	17.4888i	−0.858050	+	0.623410i	−0.927354	−	0.374186i	$-0.877922\pi$
$787$	−24.0713	+	17.4888i	−0.858050	+	0.623410i	0.0693038	+	0.997596i	$0.477922\pi$
$788$		0			0
$789$	9.22778	−	28.4002i	0.328518	−	1.01107i
$790$

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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.i (of order \(5\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.482546 + 1.48512i) q^{3} +(-2.88136 + 2.09343i) q^{5} +(-0.965093 - 2.97025i) q^{7} +(0.454306 + 0.330072i) q^{9} +O(q^{10})\)	\(q+(-0.482546 + 1.48512i) q^{3} +(-2.88136 + 2.09343i) q^{5} +(-0.965093 - 2.97025i) q^{7} +(0.454306 + 0.330072i) q^{9} +(-4.14468 - 3.01129i) q^{13} +(-1.71861 - 5.28935i) q^{15} +(1.61803 - 1.17557i) q^{17} +(1.23607 - 3.80423i) q^{19} +4.87689 q^{21} +2.43845 q^{23} +(2.37469 - 7.30854i) q^{25} +(-4.49939 + 3.26900i) q^{27} +(1.58313 + 4.87236i) q^{29} +(4.49939 + 3.26900i) q^{31} +(8.99878 + 6.53800i) q^{35} +(-2.33665 - 7.19146i) q^{37} +(6.47214 - 4.70228i) q^{39} +(0.347059 - 1.06814i) q^{41} +7.12311 q^{43} -2.00000 q^{45} +(2.47214 - 7.60845i) q^{47} +(-2.22786 + 1.61864i) q^{49} +(0.965093 + 2.97025i) q^{51} +(-9.90739 - 7.19814i) q^{53} +(5.05329 + 3.67143i) q^{57} +(2.41273 + 7.42562i) q^{59} +(0.908612 - 0.660145i) q^{61} +(0.541951 - 1.66795i) q^{63} +18.2462 q^{65} +9.56155 q^{67} +(-1.17666 + 3.62140i) q^{69} +(7.02604 - 5.10471i) q^{71} +(-1.58313 - 4.87236i) q^{73} +(9.70820 + 7.05342i) q^{75} +(-8.99878 - 6.53800i) q^{79} +(-2.16312 - 6.65740i) q^{81} +(0.709422 - 0.515426i) q^{83} +(-2.20116 + 6.77448i) q^{85} -8.00000 q^{87} +2.68466 q^{89} +(-4.94427 + 15.2169i) q^{91} +(-7.02604 + 5.10471i) q^{93} +(4.40232 + 13.5490i) q^{95} +(-12.5896 - 9.14685i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 8 * q - q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9	\( 8 q - q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9} - 2 q^{13} + 7 q^{15} + 4 q^{17} - 8 q^{19} + 72 q^{21} + 36 q^{23} - 3 q^{25} - 7 q^{27} - 2 q^{29} + 7 q^{31} + 14 q^{35} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 24 q^{43} - 16 q^{45} - 16 q^{47} - 22 q^{49} + 2 q^{51} - 8 q^{53} - 4 q^{57} + 5 q^{59} - 6 q^{61} - 20 q^{63} + 80 q^{65} + 60 q^{67} - 13 q^{69} + 5 q^{71} + 2 q^{73} + 24 q^{75} - 14 q^{79} + 14 q^{81} + 10 q^{83} + 6 q^{85} - 64 q^{87} - 28 q^{89} + 32 q^{91} - 5 q^{93} - 12 q^{95} - 27 q^{97}+O(q^{100}) \) 8 * q - q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9 - 2 * q^13 + 7 * q^15 + 4 * q^17 - 8 * q^19 + 72 * q^21 + 36 * q^23 - 3 * q^25 - 7 * q^27 - 2 * q^29 + 7 * q^31 + 14 * q^35 + 11 * q^37 + 16 * q^39 + 6 * q^41 + 24 * q^43 - 16 * q^45 - 16 * q^47 - 22 * q^49 + 2 * q^51 - 8 * q^53 - 4 * q^57 + 5 * q^59 - 6 * q^61 - 20 * q^63 + 80 * q^65 + 60 * q^67 - 13 * q^69 + 5 * q^71 + 2 * q^73 + 24 * q^75 - 14 * q^79 + 14 * q^81 + 10 * q^83 + 6 * q^85 - 64 * q^87 - 28 * q^89 + 32 * q^91 - 5 * q^93 - 12 * q^95 - 27 * q^97

Introduction

L-functions

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Database

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