LMFDB - Embedded newform 3076.1.bd.a.759.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3076,1,Mod(75,3076)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3076.75"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3076, base_ring=CyclotomicField(192)) chi = DirichletCharacter(H, H._module([96, 131])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3076 = 2^{2} \cdot 769 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3076.bd (of order $192$, degree $64$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$1.53512397886$
Analytic rank:	$0$
Dimension:	$64$
Coefficient field:	$\Q(\zeta_{192})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{64} - x^{32} + 1 $ x^64 - x^32 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{192}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{192} - \cdots)$

Embedding invariants

Embedding label			759.1
Root			$-0.910864 + 0.412707i$ of defining polynomial
Character	$\chi$	$=$	3076.759
Dual form			3076.1.bd.a.2999.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.0327191 - 0.999465i) q^{2} +(-0.997859 + 0.0654031i) q^{4} +(1.75535 + 0.172887i) q^{5} +(0.0980171 + 0.995185i) q^{8} +(0.866025 + 0.500000i) q^{9} +(0.115361 - 1.76007i) q^{10} +(1.32090 - 0.914211i) q^{13} +(0.991445 - 0.130526i) q^{16} +(-1.33889 + 1.09880i) q^{17} +(0.471397 - 0.881921i) q^{18} +(-1.76290 - 0.0577113i) q^{20} +(2.07058 + 0.411863i) q^{25} +(-0.956940 - 1.29028i) q^{26} +(-1.60032 + 1.03183i) q^{29} +(-0.162895 - 0.986643i) q^{32} +(1.14202 + 1.30222i) q^{34} +(-0.896873 - 0.442289i) q^{36} +(1.02234 - 0.707574i) q^{37} +1.76384i q^{40} +(-1.78990 + 0.882683i) q^{41} +(1.43373 + 1.02740i) q^{45} +(-0.881921 - 0.471397i) q^{49} +(0.343895 - 2.08294i) q^{50} +(-1.25828 + 0.998645i) q^{52} +(1.24546 + 0.803030i) q^{53} +(1.08364 + 1.56570i) q^{58} +(0.0227573 - 0.0235145i) q^{61} +(-0.980785 + 0.195090i) q^{64} +(2.47670 - 1.37639i) q^{65} +(1.26416 - 1.18402i) q^{68} +(-0.412707 + 0.910864i) q^{72} +(-0.682320 - 1.57392i) q^{73} +(-0.740646 - 0.998645i) q^{74} +(1.76290 - 0.0577113i) q^{80} +(0.500000 + 0.866025i) q^{81} +(0.940775 + 1.76007i) q^{82} +(-2.54019 + 1.69730i) q^{85} +(-0.953924 - 1.16236i) q^{89} +(0.979938 - 1.46658i) q^{90} +(-1.12175 - 0.401370i) q^{97} +(-0.442289 + 0.896873i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q + 32 q^{81}+O(q^{100}) $ 64 * q + 32 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3076\mathbb{Z}\right)^\times$.

$n$	$1539$	$1549$
$\chi(n)$	$-1$	$e\left(\frac{173}{192}\right)$

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 $

$2$	−0.0327191	−	0.999465i	−0.0327191	−	0.999465i
$2$	−0.0327191	−	0.999465i	−0.0327191	−	0.999465i
$3$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$3$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$4$	−0.997859	+	0.0654031i	−0.997859	+	0.0654031i
$5$	1.75535	+	0.172887i	1.75535	+	0.172887i	0.923880	−	0.382683i	$-0.125000\pi$
$5$	1.75535	+	0.172887i	1.75535	+	0.172887i	0.831470	+	0.555570i	$0.187500\pi$
$6$		0			0
$7$		0			0		0.242980	−	0.970031i	$-0.421875\pi$
$7$		0			0		−0.242980	+	0.970031i	$0.578125\pi$
$8$	0.0980171	+	0.995185i	0.0980171	+	0.995185i
$9$	0.866025	+	0.500000i	0.866025	+	0.500000i
$10$	0.115361	−	1.76007i	0.115361	−	1.76007i
$11$		0			0		0.952063	−	0.305903i	$-0.0989583\pi$
$11$		0			0		−0.952063	+	0.305903i	$0.901042\pi$
$12$		0			0
$13$	1.32090	−	0.914211i	1.32090	−	0.914211i	0.321439	−	0.946930i	$-0.395833\pi$
$13$	1.32090	−	0.914211i	1.32090	−	0.914211i	0.999465	+	0.0327191i	$0.0104167\pi$
$14$		0			0
$15$		0			0
$16$	0.991445	−	0.130526i	0.991445	−	0.130526i
$17$	−1.33889	+	1.09880i	−1.33889	+	1.09880i	−0.352250	+	0.935906i	$0.614583\pi$
$17$	−1.33889	+	1.09880i	−1.33889	+	1.09880i	−0.986643	+	0.162895i	$0.947917\pi$
$18$	0.471397	−	0.881921i	0.471397	−	0.881921i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$	−1.76290	−	0.0577113i	−1.76290	−	0.0577113i
$21$		0			0
$22$		0			0
$23$		0			0		0.114287	−	0.993448i	$-0.463542\pi$
$23$		0			0		−0.114287	+	0.993448i	$0.536458\pi$
$24$		0			0
$25$	2.07058	+	0.411863i	2.07058	+	0.411863i
$26$	−0.956940	−	1.29028i	−0.956940	−	1.29028i
$27$		0			0
$28$		0			0
$29$	−1.60032	+	1.03183i	−1.60032	+	1.03183i	−0.634393	+	0.773010i	$0.718750\pi$
$29$	−1.60032	+	1.03183i	−1.60032	+	1.03183i	−0.965926	+	0.258819i	$0.916667\pi$
$30$		0			0
$31$		0			0		0.0327191	−	0.999465i	$-0.489583\pi$
$31$		0			0		−0.0327191	+	0.999465i	$0.510417\pi$
$32$	−0.162895	−	0.986643i	−0.162895	−	0.986643i
$33$		0			0
$34$	1.14202	+	1.30222i	1.14202	+	1.30222i
$35$		0			0
$36$	−0.896873	−	0.442289i	−0.896873	−	0.442289i
$37$	1.02234	−	0.707574i	1.02234	−	0.707574i	0.0654031	−	0.997859i	$-0.479167\pi$
$37$	1.02234	−	0.707574i	1.02234	−	0.707574i	0.956940	+	0.290285i	$0.0937500\pi$
$38$		0			0
$39$		0			0
$40$			1.76384i			1.76384i
$41$	−1.78990	+	0.882683i	−1.78990	+	0.882683i	−0.866025	+	0.500000i	$0.833333\pi$
$41$	−1.78990	+	0.882683i	−1.78990	+	0.882683i	−0.923880	+	0.382683i	$0.875000\pi$
$42$		0			0
$43$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$43$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$44$		0			0
$45$	1.43373	+	1.02740i	1.43373	+	1.02740i
$46$		0			0
$47$		0			0		0.849202	−	0.528068i	$-0.177083\pi$
$47$		0			0		−0.849202	+	0.528068i	$0.822917\pi$
$48$		0			0
$49$	−0.881921	−	0.471397i	−0.881921	−	0.471397i
$50$	0.343895	−	2.08294i	0.343895	−	2.08294i
$51$		0			0
$52$	−1.25828	+	0.998645i	−1.25828	+	0.998645i
$53$	1.24546	+	0.803030i	1.24546	+	0.803030i	0.986643	−	0.162895i	$-0.0520833\pi$
$53$	1.24546	+	0.803030i	1.24546	+	0.803030i	0.258819	+	0.965926i	$0.416667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.08364	+	1.56570i	1.08364	+	1.56570i
$59$		0			0		0.541892	−	0.840448i	$-0.317708\pi$
$59$		0			0		−0.541892	+	0.840448i	$0.682292\pi$
$60$		0			0
$61$	0.0227573	−	0.0235145i	0.0227573	−	0.0235145i	−0.707107	−	0.707107i	$-0.750000\pi$
$61$	0.0227573	−	0.0235145i	0.0227573	−	0.0235145i	0.729864	+	0.683592i	$0.239583\pi$
$62$		0			0
$63$		0			0
$64$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$65$	2.47670	−	1.37639i	2.47670	−	1.37639i
$66$		0			0
$67$		0			0		0.146730	−	0.989177i	$-0.453125\pi$
$67$		0			0		−0.146730	+	0.989177i	$0.546875\pi$
$68$	1.26416	−	1.18402i	1.26416	−	1.18402i
$69$		0			0
$70$		0			0
$71$		0			0		−0.874090	−	0.485763i	$-0.838542\pi$
$71$		0			0		0.874090	+	0.485763i	$0.161458\pi$
$72$	−0.412707	+	0.910864i	−0.412707	+	0.910864i
$73$	−0.682320	−	1.57392i	−0.682320	−	1.57392i	−0.812847	−	0.582478i	$-0.802083\pi$
$73$	−0.682320	−	1.57392i	−0.682320	−	1.57392i	0.130526	−	0.991445i	$-0.458333\pi$
$74$	−0.740646	−	0.998645i	−0.740646	−	0.998645i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.903989	−	0.427555i	$-0.859375\pi$
$79$		0			0		0.903989	+	0.427555i	$0.140625\pi$
$80$	1.76290	−	0.0577113i	1.76290	−	0.0577113i
$81$	0.500000	+	0.866025i	0.500000	+	0.866025i
$82$	0.940775	+	1.76007i	0.940775	+	1.76007i
$83$		0			0		−0.762527	−	0.646956i	$-0.776042\pi$
$83$		0			0		0.762527	+	0.646956i	$0.223958\pi$
$84$		0			0
$85$	−2.54019	+	1.69730i	−2.54019	+	1.69730i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.953924	−	1.16236i	−0.953924	−	1.16236i	−0.986643	−	0.162895i	$-0.947917\pi$
$89$	−0.953924	−	1.16236i	−0.953924	−	1.16236i	0.0327191	−	0.999465i	$-0.489583\pi$
$90$	0.979938	−	1.46658i	0.979938	−	1.46658i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.12175	−	0.401370i	−1.12175	−	0.401370i	−0.290285	−	0.956940i	$-0.593750\pi$
$97$	−1.12175	−	0.401370i	−1.12175	−	0.401370i	−0.831470	+	0.555570i	$0.812500\pi$
$98$	−0.442289	+	0.896873i	−0.442289	+	0.896873i
$99$		0			0
$100$	−2.09308	−	0.275559i	−2.09308	−	0.275559i
$101$	−0.432270	−	1.72572i	−0.432270	−	1.72572i	−0.659346	−	0.751840i	$-0.729167\pi$
$101$	−0.432270	−	1.72572i	−0.432270	−	1.72572i	0.227076	−	0.973877i	$-0.427083\pi$
$102$		0			0
$103$		0			0		−0.114287	−	0.993448i	$-0.536458\pi$
$103$		0			0		0.114287	+	0.993448i	$0.463542\pi$
$104$	1.03928	+	1.22494i	1.03928	+	1.22494i
$105$		0			0
$106$	0.761850	−	1.27107i	0.761850	−	1.27107i
$107$		0			0		−0.211112	−	0.977462i	$-0.567708\pi$
$107$		0			0		0.211112	+	0.977462i	$0.432292\pi$
$108$		0			0
$109$	1.29374	−	0.0211706i	1.29374	−	0.0211706i	0.634393	−	0.773010i	$-0.281250\pi$
$109$	1.29374	−	0.0211706i	1.29374	−	0.0211706i	0.659346	+	0.751840i	$0.270833\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.783904	−	0.523788i	0.783904	−	0.523788i	−0.0980171	−	0.995185i	$-0.531250\pi$
$113$	0.783904	−	0.523788i	0.783904	−	0.523788i	0.881921	+	0.471397i	$0.156250\pi$
$114$		0			0
$115$		0			0
$116$	1.52941	−	1.13429i	1.52941	−	1.13429i
$117$	1.60104	−	0.131278i	1.60104	−	0.131278i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.812847	−	0.582478i	0.812847	−	0.582478i
$122$	−0.0242465	−	0.0219757i	−0.0242465	−	0.0219757i
$123$		0			0
$124$		0			0
$125$	1.87549	+	0.568922i	1.87549	+	0.568922i
$126$		0			0
$127$		0			0		−0.999866	−	0.0163617i	$-0.994792\pi$
$127$		0			0		0.999866	+	0.0163617i	$0.00520833\pi$
$128$	0.227076	+	0.973877i	0.227076	+	0.973877i
$129$		0			0
$130$	−1.45669	−	2.43034i	−1.45669	−	2.43034i
$131$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$131$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$137$	−0.588944	+	1.64599i	−0.588944	+	1.64599i	0.162895	+	0.986643i	$0.447917\pi$
$137$	−0.588944	+	1.64599i	−0.588944	+	1.64599i	−0.751840	+	0.659346i	$0.770833\pi$
$138$		0			0
$139$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$139$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.923880	+	0.382683i	0.923880	+	0.382683i
$145$	−2.98751	+	1.53455i	−2.98751	+	1.53455i
$146$	−1.55075	+	0.733452i	−1.55075	+	0.733452i
$147$		0			0
$148$	−0.973877	+	0.772924i	−0.973877	+	0.772924i
$149$	0.101115	−	1.54271i	0.101115	−	1.54271i	−0.582478	−	0.812847i	$-0.697917\pi$
$149$	0.101115	−	1.54271i	0.101115	−	1.54271i	0.683592	−	0.729864i	$-0.260417\pi$
$150$		0			0
$151$		0			0		−0.993448	−	0.114287i	$-0.963542\pi$
$151$		0			0		0.993448	+	0.114287i	$0.0364583\pi$
$152$		0			0
$153$	−1.70892	+	0.282143i	−1.70892	+	0.282143i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.33147	+	0.310455i	−1.33147	+	0.310455i	−0.831470	−	0.555570i	$-0.812500\pi$
$157$	−1.33147	+	0.310455i	−1.33147	+	0.310455i	−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$	−0.115361	−	1.76007i	−0.115361	−	1.76007i
$161$		0			0
$162$	0.849202	−	0.528068i	0.849202	−	0.528068i
$163$		0			0		0.999866	−	0.0163617i	$-0.00520833\pi$
$163$		0			0		−0.999866	+	0.0163617i	$0.994792\pi$
$164$	1.72834	−	0.997859i	1.72834	−	0.997859i
$165$		0			0
$166$		0			0
$167$		0			0		−0.179017	−	0.983846i	$-0.557292\pi$
$167$		0			0		0.179017	+	0.983846i	$0.442708\pi$
$168$		0			0
$169$	0.556756	−	1.47926i	0.556756	−	1.47926i
$170$	1.77951	+	2.48330i	1.77951	+	2.48330i
$171$		0			0
$172$		0			0
$173$		−	1.26879i		−	1.26879i	−0.773010	−	0.634393i	$-0.781250\pi$
$173$		−	1.26879i		−	1.26879i	0.773010	−	0.634393i	$-0.218750\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−1.13053	+	0.991445i	−1.13053	+	0.991445i
$179$		0			0		−0.803208	−	0.595699i	$-0.796875\pi$
$179$		0			0		0.803208	+	0.595699i	$0.203125\pi$
$180$	−1.49786	−	0.931429i	−1.49786	−	0.931429i
$181$	−1.53279	−	1.25793i	−1.53279	−	1.25793i	−0.849202	−	0.528068i	$-0.822917\pi$
$181$	−1.53279	−	1.25793i	−1.53279	−	1.25793i	−0.683592	−	0.729864i	$-0.739583\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.91690	−	1.06529i	1.91690	−	1.06529i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.986643	−	0.162895i	$-0.947917\pi$
$191$		0			0		0.986643	+	0.162895i	$0.0520833\pi$
$192$		0			0
$193$	−1.50250	−	0.510030i	−1.50250	−	0.510030i	−0.555570	−	0.831470i	$-0.687500\pi$
$193$	−1.50250	−	0.510030i	−1.50250	−	0.510030i	−0.946930	+	0.321439i	$0.895833\pi$
$194$	−0.364452	+	1.13429i	−0.364452	+	1.13429i
$195$		0			0
$196$	0.910864	+	0.412707i	0.910864	+	0.412707i
$197$	0.357164	−	0.534534i	0.357164	−	0.534534i	−0.608761	−	0.793353i	$-0.708333\pi$
$197$	0.357164	−	0.534534i	0.357164	−	0.534534i	0.965926	+	0.258819i	$0.0833333\pi$
$198$		0			0
$199$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$199$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$200$	−0.206928	+	2.10097i	−0.206928	+	2.10097i
$201$		0			0
$202$	−1.71065	+	0.488502i	−1.71065	+	0.488502i
$203$		0			0
$204$		0			0
$205$	−3.29451	+	1.23997i	−3.29451	+	1.23997i
$206$		0			0
$207$		0			0
$208$	1.19028	−	1.07880i	1.19028	−	1.07880i
$209$		0			0
$210$		0			0
$211$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$211$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$212$	−1.29532	−	0.719854i	−1.29532	−	0.719854i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.0634893	−	1.29235i	−0.0634893	−	1.29235i
$219$		0			0
$220$		0			0
$221$	−0.764013	+	2.67544i	−0.764013	+	2.67544i
$222$		0			0
$223$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$223$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$224$		0			0
$225$	1.58724	+	1.39197i	1.58724	+	1.39197i
$226$	−0.549156	−	0.766347i	−0.549156	−	0.766347i
$227$		0			0		−0.621661	−	0.783287i	$-0.713542\pi$
$227$		0			0		0.621661	+	0.783287i	$0.286458\pi$
$228$		0			0
$229$	0.901447	+	0.962466i	0.901447	+	0.962466i	0.999465	−	0.0327191i	$-0.0104167\pi$
$229$	0.901447	+	0.962466i	0.901447	+	0.962466i	−0.0980171	+	0.995185i	$0.531250\pi$
$230$		0			0
$231$		0			0
$232$	−1.18372	−	1.49148i	−1.18372	−	1.49148i
$233$	0.734146	+	0.0360663i	0.734146	+	0.0360663i	0.412707	−	0.910864i	$-0.364583\pi$
$233$	0.734146	+	0.0360663i	0.734146	+	0.0360663i	0.321439	+	0.946930i	$0.395833\pi$
$234$	−0.183592	−	1.59589i	−0.183592	−	1.59589i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.595699	−	0.803208i	$-0.296875\pi$
$239$		0			0		−0.595699	+	0.803208i	$0.703125\pi$
$240$		0			0
$241$	1.89206	−	0.344272i	1.89206	−	0.344272i	0.896873	−	0.442289i	$-0.145833\pi$
$241$	1.89206	−	0.344272i	1.89206	−	0.344272i	0.995185	+	0.0980171i	$0.0312500\pi$
$242$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$243$		0			0
$244$	−0.0211706	+	0.0249525i	−0.0211706	+	0.0249525i
$245$	−1.46658	−	0.979938i	−1.46658	−	0.979938i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.507253	−	1.89310i	0.507253	−	1.89310i
$251$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$251$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.965926	−	0.258819i	0.965926	−	0.258819i
$257$	0.256779	+	0.846488i	0.256779	+	0.846488i	0.986643	+	0.162895i	$0.0520833\pi$
$257$	0.256779	+	0.846488i	0.256779	+	0.846488i	−0.729864	+	0.683592i	$0.760417\pi$
$258$		0			0
$259$		0			0
$260$	−2.38138	+	1.53543i	−2.38138	+	1.53543i
$261$	−1.90183	+	0.0934310i	−1.90183	+	0.0934310i
$262$		0			0
$263$		0			0		0.973877	−	0.227076i	$-0.0729167\pi$
$263$		0			0		−0.973877	+	0.227076i	$0.927083\pi$
$264$		0			0
$265$	2.04739	+	1.62492i	2.04739	+	1.62492i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.442289	+	0.103127i	0.442289	+	0.103127i	0.442289	−	0.896873i	$-0.354167\pi$
$269$	0.442289	+	0.103127i	0.442289	+	0.103127i			1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		0.803208	−	0.595699i	$-0.203125\pi$
$271$		0			0		−0.803208	+	0.595699i	$0.796875\pi$
$272$	−1.18402	+	1.26416i	−1.18402	+	1.26416i
$273$		0			0
$274$	1.66438	+	0.534774i	1.66438	+	0.534774i
$275$		0			0
$276$		0			0
$277$	−0.252157	+	0.0675653i	−0.252157	+	0.0675653i	−0.382683	−	0.923880i	$-0.625000\pi$
$277$	−0.252157	+	0.0675653i	−0.252157	+	0.0675653i	0.130526	+	0.991445i	$0.458333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.12571	−	0.893428i	−1.12571	−	0.893428i	−0.130526	−	0.991445i	$-0.541667\pi$
$281$	−1.12571	−	0.893428i	−1.12571	−	0.893428i	−0.995185	+	0.0980171i	$0.968750\pi$
$282$		0			0
$283$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$283$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.352250	−	0.935906i	0.352250	−	0.935906i
$289$	0.390181	−	1.96157i	0.390181	−	1.96157i
$290$	1.63147	+	2.93570i	1.63147	+	2.93570i
$291$		0			0
$292$	0.783799	+	1.52593i	0.783799	+	1.52593i
$293$	−0.0793942	−	0.142863i	−0.0793942	−	0.142863i	0.831470	−	0.555570i	$-0.187500\pi$
$293$	−0.0793942	−	0.142863i	−0.0793942	−	0.142863i	−0.910864	+	0.412707i	$0.864583\pi$
$294$		0			0
$295$		0			0
$296$	0.804374	+	0.948066i	0.804374	+	0.948066i
$297$		0			0
$298$	−1.54519	−	0.0505844i	−1.54519	−	0.0505844i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.0440123	−	0.0373417i	0.0440123	−	0.0373417i
$306$	0.337906	+	1.69877i	0.337906	+	1.69877i
$307$		0			0		−0.986643	−	0.162895i	$-0.947917\pi$
$307$		0			0		0.986643	+	0.162895i	$0.0520833\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$		0			0
$313$	1.47945	+	1.21415i	1.47945	+	1.21415i	0.923880	+	0.382683i	$0.125000\pi$
$313$	1.47945	+	1.21415i	1.47945	+	1.21415i	0.555570	+	0.831470i	$0.312500\pi$
$314$	0.353853	+	1.32060i	0.353853	+	1.32060i
$315$		0			0
$316$		0			0
$317$	−1.56551	+	1.20126i	−1.56551	+	1.20126i	−0.683592	+	0.729864i	$0.739583\pi$
$317$	−1.56551	+	1.20126i	−1.56551	+	1.20126i	−0.881921	+	0.471397i	$0.843750\pi$
$318$		0			0
$319$		0			0
$320$	−1.75535	+	0.172887i	−1.75535	+	0.172887i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$325$	3.11156	−	1.34891i	3.11156	−	1.34891i
$326$		0			0
$327$		0			0
$328$	−1.05387	−	1.69477i	−1.05387	−	1.69477i
$329$		0			0
$330$		0			0
$331$		0			0		0.983846	−	0.179017i	$-0.0572917\pi$
$331$		0			0		−0.983846	+	0.179017i	$0.942708\pi$
$332$		0			0
$333$	1.23916	−	0.101606i	1.23916	−	0.101606i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.654845	+	0.502480i	0.654845	+	0.502480i	0.881921	−	0.471397i	$-0.156250\pi$
$337$	0.654845	+	0.502480i	0.654845	+	0.502480i	−0.227076	+	0.973877i	$0.572917\pi$
$338$	−1.49669	−	0.508057i	−1.49669	−	0.508057i
$339$		0			0
$340$	2.42375	−	1.85981i	2.42375	−	1.85981i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.26811	+	0.0415135i	−1.26811	+	0.0415135i
$347$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$347$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$348$		0			0
$349$	−1.02477	+	1.70972i	−1.02477	+	1.70972i	−0.442289	+	0.896873i	$0.645833\pi$
$349$	−1.02477	+	1.70972i	−1.02477	+	1.70972i	−0.582478	+	0.812847i	$0.697917\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.337958	+	0.349202i	0.337958	+	0.349202i	0.866025	−	0.500000i	$-0.166667\pi$
$353$	0.337958	+	0.349202i	0.337958	+	0.349202i	−0.528068	+	0.849202i	$0.677083\pi$
$354$		0			0
$355$		0			0
$356$	1.02790	+	1.09748i	1.02790	+	1.09748i
$357$		0			0
$358$		0			0
$359$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$359$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$360$	−0.881921	+	1.52753i	−0.881921	+	1.52753i
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$	−1.20711	+	1.57313i	−1.20711	+	1.57313i
$363$		0			0
$364$		0			0
$365$	−0.925600	−	2.88075i	−0.925600	−	2.88075i
$366$		0			0
$367$		0			0		−0.930017	−	0.367516i	$-0.880208\pi$
$367$		0			0		0.930017	+	0.367516i	$0.119792\pi$
$368$		0			0
$369$	−1.99144	−	0.130526i	−1.99144	−	0.130526i
$370$	−1.12744	−	1.88102i	−1.12744	−	1.88102i
$371$		0			0
$372$		0			0
$373$	0.226100	+	0.0335388i	0.226100	+	0.0335388i	0.258819	−	0.965926i	$-0.416667\pi$
$373$	0.226100	+	0.0335388i	0.226100	+	0.0335388i	−0.0327191	+	0.999465i	$0.510417\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.17056	+	2.82598i	−1.17056	+	2.82598i
$378$		0			0
$379$		0			0		−0.274589	−	0.961562i	$-0.588542\pi$
$379$		0			0		0.274589	+	0.961562i	$0.411458\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$383$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$384$		0			0
$385$		0			0
$386$	−0.460597	+	1.51838i	−0.460597	+	1.51838i
$387$		0			0
$388$	1.14560	+	0.327144i	1.14560	+	0.327144i
$389$	0.746801	−	1.20095i	0.746801	−	1.20095i	−0.227076	−	0.973877i	$-0.572917\pi$
$389$	0.746801	−	1.20095i	0.746801	−	1.20095i	0.973877	−	0.227076i	$-0.0729167\pi$
$390$		0			0
$391$		0			0
$392$	0.382683	−	0.923880i	0.382683	−	0.923880i
$393$		0			0
$394$	−0.545934	−	0.339484i	−0.545934	−	0.339484i
$395$		0			0
$396$		0			0
$397$	−0.445426	+	1.91033i	−0.445426	+	1.91033i	−0.0327191	+	0.999465i	$0.510417\pi$
$397$	−0.445426	+	1.91033i	−0.445426	+	1.91033i	−0.412707	+	0.910864i	$0.635417\pi$
$398$		0			0
$399$		0			0
$400$	2.10662	+	0.138075i	2.10662	+	0.138075i
$401$	0.0269075	+	1.64432i	0.0269075	+	1.64432i	0.582478	+	0.812847i	$0.302083\pi$
$401$	0.0269075	+	1.64432i	0.0269075	+	1.64432i	−0.555570	+	0.831470i	$0.687500\pi$
$402$		0			0
$403$		0			0
$404$	0.544211	+	1.69375i	0.544211	+	1.69375i
$405$	0.727950	+	1.60662i	0.727950	+	1.60662i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.729864	−	1.26416i	0.729864	−	1.26416i	−0.227076	−	0.973877i	$-0.572917\pi$
$409$	0.729864	−	1.26416i	0.729864	−	1.26416i	0.956940	−	0.290285i	$-0.0937500\pi$
$410$	1.34710	+	3.25218i	1.34710	+	3.25218i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.11717	−	1.15434i	−1.11717	−	1.15434i
$417$		0			0
$418$		0			0
$419$		0			0		−0.917494	−	0.397748i	$-0.869792\pi$
$419$		0			0		0.917494	+	0.397748i	$0.130208\pi$
$420$		0			0
$421$	−0.568990	+	0.224848i	−0.568990	+	0.224848i	−0.634393	−	0.773010i	$-0.718750\pi$
$421$	−0.568990	+	0.224848i	−0.568990	+	0.224848i	0.0654031	+	0.997859i	$0.479167\pi$
$422$		0			0
$423$		0			0
$424$	−0.677087	+	1.31818i	−0.677087	+	1.31818i
$425$	−3.22484	+	1.72371i	−3.22484	+	1.72371i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$431$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$432$		0			0
$433$	−0.398308	+	1.00794i	−0.398308	+	1.00794i	0.582478	+	0.812847i	$0.302083\pi$
$433$	−0.398308	+	1.00794i	−0.398308	+	1.00794i	−0.980785	+	0.195090i	$0.937500\pi$
$434$		0			0
$435$		0			0
$436$	−1.28958	+	0.105740i	−1.28958	+	0.105740i
$437$		0			0
$438$		0			0
$439$		0			0		0.0163617	−	0.999866i	$-0.494792\pi$
$439$		0			0		−0.0163617	+	0.999866i	$0.505208\pi$
$440$		0			0
$441$	−0.528068	−	0.849202i	−0.528068	−	0.849202i
$442$	2.69901	+	0.676066i	2.69901	+	0.676066i
$443$		0			0		−0.0490677	−	0.998795i	$-0.515625\pi$
$443$		0			0		0.0490677	+	0.998795i	$0.484375\pi$
$444$		0			0
$445$	−1.47351	−	2.20527i	−1.47351	−	2.20527i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.515146	−	0.0507374i	0.515146	−	0.0507374i	0.162895	−	0.986643i	$-0.447917\pi$
$449$	0.515146	−	0.0507374i	0.515146	−	0.0507374i	0.352250	+	0.935906i	$0.385417\pi$
$450$	1.33929	−	1.63193i	1.33929	−	1.63193i
$451$		0			0
$452$	−0.747968	+	0.573936i	−0.747968	+	0.573936i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.669796	+	1.41617i	0.669796	+	1.41617i	0.896873	+	0.442289i	$0.145833\pi$
$457$	0.669796	+	1.41617i	0.669796	+	1.41617i	−0.227076	+	0.973877i	$0.572917\pi$
$458$	0.932456	−	0.932456i	0.932456	−	0.932456i
$459$		0			0
$460$		0			0
$461$	0.212196	+	1.61179i	0.212196	+	1.61179i	0.683592	+	0.729864i	$0.260417\pi$
$461$	0.212196	+	1.61179i	0.212196	+	1.61179i	−0.471397	+	0.881921i	$0.656250\pi$
$462$		0			0
$463$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$463$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$464$	−1.45195	+	1.23189i	−1.45195	+	1.23189i
$465$		0			0
$466$	0.0120264	−	0.734933i	0.0120264	−	0.734933i
$467$		0			0		0.0817211	−	0.996655i	$-0.473958\pi$
$467$		0			0		−0.0817211	+	0.996655i	$0.526042\pi$
$468$	−1.58903	+	0.235710i	−1.58903	+	0.235710i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.677087	+	1.31818i	0.677087	+	1.31818i
$478$		0			0
$479$		0			0		−0.485763	−	0.874090i	$-0.661458\pi$
$479$		0			0		0.485763	+	0.874090i	$0.338542\pi$
$480$		0			0
$481$	0.703545	−	1.86928i	0.703545	−	1.86928i
$482$	−0.405994	−	1.87978i	−0.405994	−	1.87978i
$483$		0			0
$484$	−0.773010	+	0.634393i	−0.773010	+	0.634393i
$485$	−1.89968	−	0.898481i	−1.89968	−	0.898481i
$486$		0			0
$487$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$487$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$488$	0.0256319	+	0.0203429i	0.0256319	+	0.0203429i
$489$		0			0
$490$	−0.931429	+	1.49786i	−0.931429	+	1.49786i
$491$		0			0		0.718582	−	0.695443i	$-0.244792\pi$
$491$		0			0		−0.718582	+	0.695443i	$0.755208\pi$
$492$		0			0
$493$	1.00888	−	3.13994i	1.00888	−	3.13994i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$499$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$500$	−1.90868	−	0.445042i	−1.90868	−	0.445042i
$501$		0			0
$502$		0			0
$503$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$503$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$504$		0			0
$505$	−0.460430	−	3.10397i	−0.460430	−	3.10397i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.461555	+	0.297595i	−0.461555	+	0.297595i	−0.751840	−	0.659346i	$-0.770833\pi$
$509$	−0.461555	+	0.297595i	−0.461555	+	0.297595i	0.290285	+	0.956940i	$0.406250\pi$
$510$		0			0
$511$		0			0
$512$	−0.290285	−	0.956940i	−0.290285	−	0.956940i
$513$		0			0
$514$	0.837633	−	0.284338i	0.837633	−	0.284338i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.61252	+	2.32987i	1.61252	+	2.32987i
$521$	−1.86780	+	0.122422i	−1.86780	+	0.122422i	−0.956940	−	0.290285i	$-0.906250\pi$
$521$	−1.86780	+	0.122422i	−1.86780	+	0.122422i	−0.910864	+	0.412707i	$0.864583\pi$
$522$	0.155607	+	1.89776i	0.155607	+	1.89776i
$523$		0			0		−0.227076	−	0.973877i	$-0.572917\pi$
$523$		0			0		0.227076	+	0.973877i	$0.427083\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.973877	−	0.227076i	−0.973877	−	0.227076i
$530$	1.55706	−	2.09946i	1.55706	−	2.09946i
$531$		0			0
$532$		0			0
$533$	−1.55733	+	2.80229i	−1.55733	+	2.80229i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.0886008	−	0.445426i	0.0886008	−	0.445426i
$539$		0			0
$540$		0			0
$541$	−1.21831	+	1.38922i	−1.21831	+	1.38922i	−0.321439	+	0.946930i	$0.604167\pi$
$541$	−1.21831	+	1.38922i	−1.21831	+	1.38922i	−0.896873	+	0.442289i	$0.854167\pi$
$542$		0			0
$543$		0			0
$544$	1.30222	+	1.14202i	1.30222	+	1.14202i
$545$	2.27462	+	0.186509i	2.27462	+	0.186509i
$546$		0			0
$547$		0			0		−0.367516	−	0.930017i	$-0.619792\pi$
$547$		0			0		0.367516	+	0.930017i	$0.380208\pi$
$548$	0.480031	−	1.68098i	0.480031	−	1.68098i
$549$	0.0314656	−	0.00898549i	0.0314656	−	0.00898549i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.0757795	+	0.249812i	0.0757795	+	0.249812i
$555$		0			0
$556$		0			0
$557$	1.46926	+	0.816518i	1.46926	+	0.816518i	0.997859	−	0.0654031i	$-0.0208333\pi$
$557$	1.46926	+	0.816518i	1.46926	+	0.816518i	0.471397	+	0.881921i	$0.343750\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.856117	+	1.15434i	−0.856117	+	1.15434i
$563$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$563$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$564$		0			0
$565$	1.46658	−	0.783904i	1.46658	−	0.783904i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.195090	−	1.98079i	0.195090	−	1.98079i		−	1.00000i	$-0.5\pi$
$569$	0.195090	−	1.98079i	0.195090	−	1.98079i	0.195090	−	0.980785i	$-0.437500\pi$
$570$		0			0
$571$		0			0		0.179017	−	0.983846i	$-0.442708\pi$
$571$		0			0		−0.179017	+	0.983846i	$0.557292\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.946930	−	0.321439i	−0.946930	−	0.321439i
$577$	1.21492	+	0.0796298i	1.21492	+	0.0796298i	0.659346	−	0.751840i	$-0.270833\pi$
$577$	1.21492	+	0.0796298i	1.21492	+	0.0796298i	0.555570	+	0.831470i	$0.312500\pi$
$578$	−1.97329	−	0.325791i	−1.97329	−	0.325791i
$579$		0			0
$580$	2.88075	−	1.72665i	2.88075	−	1.72665i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.49946	−	0.833306i	1.49946	−	0.833306i
$585$	2.83308	+	0.0463604i	2.83308	+	0.0463604i
$586$	−0.140189	+	0.0840261i	−0.140189	+	0.0840261i
$587$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$587$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.921240	−	0.834963i	0.921240	−	0.834963i
$593$	−1.02892	+	0.486643i	−1.02892	+	0.486643i	−0.866025	−	0.500000i	$-0.833333\pi$
$593$	−1.02892	+	0.486643i	−1.02892	+	0.486643i	−0.162895	+	0.986643i	$0.552083\pi$
$594$		0			0
$595$		0			0
$596$			1.54602i			1.54602i
$597$		0			0
$598$		0			0
$599$		0			0		−0.582478	−	0.812847i	$-0.697917\pi$
$599$		0			0		0.582478	+	0.812847i	$0.302083\pi$
$600$		0			0
$601$	1.23230	+	0.534223i	1.23230	+	0.534223i	0.910864	−	0.412707i	$-0.135417\pi$
$601$	1.23230	+	0.534223i	1.23230	+	0.534223i	0.321439	+	0.946930i	$0.395833\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.52753	−	0.881921i	1.52753	−	0.881921i
$606$		0			0
$607$		0			0		0.849202	−	0.528068i	$-0.177083\pi$
$607$		0			0		−0.849202	+	0.528068i	$0.822917\pi$
$608$		0			0
$609$		0			0
$610$	−0.0387617	−	0.0427670i	−0.0387617	−	0.0427670i
$611$		0			0
$612$	1.68680	−	0.393308i	1.68680	−	0.393308i
$613$	−1.65881	+	0.719121i	−1.65881	+	0.719121i	−0.999465	−	0.0327191i	$-0.989583\pi$
$613$	−1.65881	+	0.719121i	−1.65881	+	0.719121i	−0.659346	+	0.751840i	$0.729167\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.296751	+	0.134456i	0.296751	+	0.134456i	0.555570	−	0.831470i	$-0.312500\pi$
$617$	0.296751	+	0.134456i	0.296751	+	0.134456i	−0.258819	+	0.965926i	$0.583333\pi$
$618$		0			0
$619$		0			0		0.412707	−	0.910864i	$-0.364583\pi$
$619$		0			0		−0.412707	+	0.910864i	$0.635417\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.24333	+	0.515005i	1.24333	+	0.515005i
$626$	1.16510	−	1.51838i	1.16510	−	1.51838i
$627$		0			0
$628$	1.30831	−	0.396873i	1.30831	−	0.396873i
$629$	−0.591325	+	2.07072i	−0.591325	+	2.07072i
$630$		0			0
$631$		0			0		−0.941544	−	0.336890i	$-0.890625\pi$
$631$		0			0		0.941544	+	0.336890i	$0.109375\pi$
$632$		0			0
$633$		0			0
$634$	1.25184	+	1.52537i	1.25184	+	1.52537i
$635$		0			0
$636$		0			0
$637$	−1.59589	+	0.183592i	−1.59589	+	0.183592i
$638$		0			0
$639$		0			0
$640$	0.230228	+	1.74875i	0.230228	+	1.74875i
$641$	−0.0148595	−	0.0637287i	−0.0148595	−	0.0637287i	0.965926	−	0.258819i	$-0.0833333\pi$
$641$	−0.0148595	−	0.0637287i	−0.0148595	−	0.0637287i	−0.980785	+	0.195090i	$0.937500\pi$
$642$		0			0
$643$		0			0		−0.740951	−	0.671559i	$-0.765625\pi$
$643$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.740951	−	0.671559i	$-0.765625\pi$
$647$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$648$	−0.812847	+	0.582478i	−0.812847	+	0.582478i
$649$		0			0
$650$	−1.45000	−	3.06576i	−1.45000	−	3.06576i
$651$		0			0
$652$		0			0
$653$	0.339133	−	0.251518i	0.339133	−	0.251518i	−0.412707	−	0.910864i	$-0.635417\pi$
$653$	0.339133	−	0.251518i	0.339133	−	0.251518i	0.751840	+	0.659346i	$0.229167\pi$
$654$		0			0
$655$		0			0
$656$	−1.65938	+	1.10876i	−1.65938	+	1.10876i
$657$	0.196054	−	1.70422i	0.196054	−	1.70422i
$658$		0			0
$659$		0			0		−0.242980	−	0.970031i	$-0.578125\pi$
$659$		0			0		0.242980	+	0.970031i	$0.421875\pi$
$660$		0			0
$661$	−1.45244	−	1.27376i	−1.45244	−	1.27376i	−0.896873	−	0.442289i	$-0.854167\pi$
$661$	−1.45244	−	1.27376i	−1.45244	−	1.27376i	−0.555570	−	0.831470i	$-0.687500\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.142095	−	1.23517i	−0.142095	−	1.23517i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.0319331	+	0.324222i	−0.0319331	+	0.324222i	0.965926	+	0.258819i	$0.0833333\pi$
$673$	−0.0319331	+	0.324222i	−0.0319331	+	0.324222i	−0.997859	+	0.0654031i	$0.979167\pi$
$674$	0.480785	−	0.670935i	0.480785	−	0.670935i
$675$		0			0
$676$	−0.458815	+	1.51251i	−0.458815	+	1.51251i
$677$	−0.0748309	+	1.52322i	−0.0748309	+	1.52322i	0.608761	+	0.793353i	$0.291667\pi$
$677$	−0.0748309	+	1.52322i	−0.0748309	+	1.52322i	−0.683592	+	0.729864i	$0.739583\pi$
$678$		0			0
$679$		0			0
$680$	−1.93811	−	2.36160i	−1.93811	−	2.36160i
$681$		0			0
$682$		0			0
$683$		0			0		0.910864	−	0.412707i	$-0.135417\pi$
$683$		0			0		−0.910864	+	0.412707i	$0.864583\pi$
$684$		0			0
$685$	−1.31837	+	2.78746i	−1.31837	+	2.78746i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.37928	−	0.0778894i	2.37928	−	0.0778894i
$690$		0			0
$691$		0			0		−0.367516	−	0.930017i	$-0.619792\pi$
$691$		0			0		0.367516	+	0.930017i	$0.380208\pi$
$692$	0.0829826	+	1.26607i	0.0829826	+	1.26607i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.42660	−	3.14857i	1.42660	−	3.14857i
$698$	1.74233	+	0.968277i	1.74233	+	0.968277i
$699$		0			0
$700$		0			0
$701$	−0.0954708	+	0.0894182i	−0.0954708	+	0.0894182i	−0.729864	−	0.683592i	$-0.760417\pi$
$701$	−0.0954708	+	0.0894182i	−0.0954708	+	0.0894182i	0.634393	+	0.773010i	$0.281250\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.337958	−	0.349202i	0.337958	−	0.349202i
$707$		0			0
$708$		0			0
$709$	0.373380	+	0.113263i	0.373380	+	0.113263i	0.471397	−	0.881921i	$-0.343750\pi$
$709$	0.373380	+	0.113263i	0.373380	+	0.113263i	−0.0980171	+	0.995185i	$0.531250\pi$
$710$		0			0
$711$		0			0
$712$	1.06326	−	1.06326i	1.06326	−	1.06326i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.162895	−	0.986643i	$-0.447917\pi$
$719$		0			0		−0.162895	+	0.986643i	$0.552083\pi$
$720$	1.55557	+	0.831470i	1.55557	+	0.831470i
$721$		0			0
$722$	−0.849202	+	0.528068i	−0.849202	+	0.528068i
$723$		0			0
$724$	1.61179	+	1.15499i	1.61179	+	1.15499i
$725$	−3.73855	+	1.47737i	−3.73855	+	1.47737i
$726$		0			0
$727$		0			0		−0.889516	−	0.456904i	$-0.848958\pi$
$727$		0			0		0.889516	+	0.456904i	$0.151042\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$	−2.84892	+	1.01936i	−2.84892	+	1.01936i
$731$		0			0
$732$		0			0
$733$	1.74689	+	0.861470i	1.74689	+	0.861470i	0.973877	+	0.227076i	$0.0729167\pi$
$733$	1.74689	+	0.861470i	1.74689	+	0.861470i	0.773010	+	0.634393i	$0.218750\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.0652981	+	1.99465i	−0.0652981	+	1.99465i
$739$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$739$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$740$	−1.84312	+	1.18838i	−1.84312	+	1.18838i
$741$		0			0
$742$		0			0
$743$		0			0		−0.595699	−	0.803208i	$-0.703125\pi$
$743$		0			0		0.595699	+	0.803208i	$0.296875\pi$
$744$		0			0
$745$	0.444206	−	2.69051i	0.444206	−	2.69051i
$746$	0.0261230	−	0.227076i	0.0261230	−	0.227076i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$751$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$752$		0			0
$753$		0			0
$754$	2.86276	+	1.07747i	2.86276	+	1.07747i
$755$		0			0
$756$		0			0
$757$	0.690963	+	0.137441i	0.690963	+	0.137441i	0.528068	−	0.849202i	$-0.322917\pi$
$757$	0.690963	+	0.137441i	0.690963	+	0.137441i	0.162895	+	0.986643i	$0.447917\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.183470	−	1.86280i	−0.183470	−	1.86280i	−0.442289	−	0.896873i	$-0.645833\pi$
$761$	−0.183470	−	1.86280i	−0.183470	−	1.86280i	0.258819	−	0.965926i	$-0.416667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−3.04852	+	0.199811i	−3.04852	+	0.199811i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.946930	−	0.321439i	−0.946930	−	0.321439i
$769$	−0.946930	−	0.321439i	−0.946930	−	0.321439i
$770$		0			0
$771$		0			0
$772$	1.53264	+	0.410670i	1.53264	+	0.410670i
$773$	1.84380	−	0.120849i	1.84380	−	0.120849i	0.896873	−	0.442289i	$-0.145833\pi$
$773$	1.84380	−	0.120849i	1.84380	−	0.120849i	0.946930	+	0.321439i	$0.104167\pi$
$774$		0			0
$775$		0			0
$776$	0.289486	−	1.15569i	0.289486	−	1.15569i
$777$		0			0
$778$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.935906	−	0.352250i	−0.935906	−	0.352250i
$785$	−2.39087	+	0.314764i	−2.39087	+	0.314764i
$786$		0			0
$787$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$787$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$788$	−0.321439	+	0.556749i	−0.321439	+	0.556749i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.00856300	−	0.0518653i	0.00856300	−	0.0518653i
$794$	1.92388	+	0.382683i	1.92388	+	0.382683i
$795$		0			0
$796$		0			0
$797$	0.349964	−	1.92334i	0.349964	−	1.92334i	−0.0327191	−	0.999465i	$-0.510417\pi$
$797$	0.349964	−	1.92334i	0.349964	−	1.92334i	0.382683	−	0.923880i	$-0.375000\pi$
$798$		0			0
$799$		0			0
$800$	0.0690746	−	2.11001i	0.0690746	−	2.11001i
$801$	−0.244943	−	1.48360i	−0.244943	−	1.48360i
$802$	1.64256	−	0.0806936i	1.64256	−	0.0806936i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.67504	−	0.599338i	1.67504	−	0.599338i
$809$			1.21752i			1.21752i	0.793353	+	0.608761i	$0.208333\pi$
$809$			1.21752i			1.21752i	−0.793353	+	0.608761i	$0.791667\pi$
$810$	1.58194	−	0.780128i	1.58194	−	0.780128i
$811$		0			0		−0.889516	−	0.456904i	$-0.848958\pi$
$811$		0			0		0.889516	+	0.456904i	$0.151042\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.28737	−	0.688111i	−1.28737	−	0.688111i
$819$		0			0
$820$	3.20636	−	1.45278i	3.20636	−	1.45278i
$821$	−1.28814	+	1.02234i	−1.28814	+	1.02234i	−0.290285	+	0.956940i	$0.593750\pi$
$821$	−1.28814	+	1.02234i	−1.28814	+	1.02234i	−0.997859	+	0.0654031i	$0.979167\pi$
$822$		0			0
$823$		0			0		0.683592	−	0.729864i	$-0.260417\pi$
$823$		0			0		−0.683592	+	0.729864i	$0.739583\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.569100	−	0.822268i	$-0.692708\pi$
$827$		0			0		0.569100	+	0.822268i	$0.307292\pi$
$828$		0			0
$829$	0.732410	+	0.222174i	0.732410	+	0.222174i	0.634393	−	0.773010i	$-0.281250\pi$
$829$	0.732410	+	0.222174i	0.732410	+	0.222174i	0.0980171	+	0.995185i	$0.468750\pi$
$830$		0			0
$831$		0			0
$832$	−1.11717	+	1.15434i	−1.11717	+	1.15434i
$833$	1.69877	−	0.337906i	1.69877	−	0.337906i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.952063	−	0.305903i	$-0.901042\pi$
$839$		0			0		0.952063	+	0.305903i	$0.0989583\pi$
$840$		0			0
$841$	1.08364	−	2.39165i	1.08364	−	2.39165i
$842$	0.243345	+	0.561329i	0.243345	+	0.561329i
$843$		0			0
$844$		0			0
$845$	1.23305	−	2.50037i	1.23305	−	2.50037i
$846$		0			0
$847$		0			0
$848$	1.33962	+	0.633595i	1.33962	+	0.633595i
$849$		0			0
$850$	1.82830	+	3.16671i	1.82830	+	3.16671i
$851$		0			0
$852$		0			0
$853$	−0.854080	+	1.80580i	−0.854080	+	1.80580i	−0.382683	+	0.923880i	$0.625000\pi$
$853$	−0.854080	+	1.80580i	−0.854080	+	1.80580i	−0.471397	+	0.881921i	$0.656250\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.0909149	−	0.209715i	−0.0909149	−	0.209715i	0.866025	−	0.500000i	$-0.166667\pi$
$857$	−0.0909149	−	0.209715i	−0.0909149	−	0.209715i	−0.956940	+	0.290285i	$0.906250\pi$
$858$		0			0
$859$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$859$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$863$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$864$		0			0
$865$	0.219356	−	2.22716i	0.219356	−	2.22716i
$866$	1.02043	+	0.365116i	1.02043	+	0.365116i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.147877	+	1.28543i	0.147877	+	1.28543i
$873$	−0.770783	−	0.908474i	−0.770783	−	0.908474i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.09748	−	0.962466i	−1.09748	−	0.962466i	−0.0980171	−	0.995185i	$-0.531250\pi$
$877$	−1.09748	−	0.962466i	−1.09748	−	0.962466i	−0.999465	+	0.0327191i	$0.989583\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.0555389	−	0.482776i	0.0555389	−	0.482776i	−0.935906	−	0.352250i	$-0.885417\pi$
$881$	0.0555389	−	0.482776i	0.0555389	−	0.482776i	0.991445	−	0.130526i	$-0.0416667\pi$
$882$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$883$		0			0		−0.729864	−	0.683592i	$-0.760417\pi$
$883$		0			0		0.729864	+	0.683592i	$0.239583\pi$
$884$	0.587395	−	2.71968i	0.587395	−	2.71968i
$885$		0			0
$886$		0			0
$887$		0			0		−0.857729	−	0.514103i	$-0.828125\pi$
$887$		0			0		0.857729	+	0.514103i	$0.171875\pi$
$888$		0			0
$889$		0			0
$890$	−2.15588	+	1.54488i	−2.15588	+	1.54488i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.0675653	−	0.513210i	−0.0675653	−	0.513210i
$899$		0			0
$900$	−1.67488	−	1.28518i	−1.67488	−	1.28518i
$901$	−2.54991	+	0.293344i	−2.54991	+	0.293344i
$902$		0			0
$903$		0			0
$904$	0.598102	+	0.728789i	0.598102	+	0.728789i
$905$	−2.47311	−	2.47311i	−2.47311	−	2.47311i
$906$		0			0
$907$		0			0		−0.941544	−	0.336890i	$-0.890625\pi$
$907$		0			0		0.941544	+	0.336890i	$0.109375\pi$
$908$		0			0
$909$	0.488502	−	1.71065i	0.488502	−	1.71065i
$910$		0			0
$911$		0			0		0.456904	−	0.889516i	$-0.348958\pi$
$911$		0			0		−0.456904	+	0.889516i	$0.651042\pi$
$912$		0			0
$913$		0			0
$914$	1.39349	−	0.715774i	1.39349	−	0.715774i
$915$		0			0
$916$	−0.962466	−	0.901447i	−0.962466	−	0.901447i
$917$		0			0
$918$		0			0
$919$		0			0		0.412707	−	0.910864i	$-0.364583\pi$
$919$		0			0		−0.412707	+	0.910864i	$0.635417\pi$
$920$		0			0
$921$		0			0
$922$	1.60398	−	0.264818i	1.60398	−	0.264818i
$923$		0			0
$924$		0			0
$925$	2.40826	−	1.04402i	2.40826	−	1.04402i
$926$		0			0
$927$		0			0
$928$	1.27873	+	1.41086i	1.27873	+	1.41086i
$929$	−0.0213077	−	0.325093i	−0.0213077	−	0.325093i	−0.995185	−	0.0980171i	$-0.968750\pi$
$929$	−0.0213077	−	0.325093i	−0.0213077	−	0.325093i	0.973877	−	0.227076i	$-0.0729167\pi$
$930$		0			0
$931$		0			0
$932$	−0.734933	+	0.0120264i	−0.734933	+	0.0120264i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.287575	+	1.58047i	0.287575	+	1.58047i
$937$	−1.72772	−	0.748995i	−1.72772	−	0.748995i	−0.997859	−	0.0654031i	$-0.979167\pi$
$937$	−1.72772	−	0.748995i	−1.72772	−	0.748995i	−0.729864	−	0.683592i	$-0.760417\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.735499	−	1.49144i	−0.735499	−	1.49144i	−0.866025	−	0.500000i	$-0.833333\pi$
$941$	−0.735499	−	1.49144i	−0.735499	−	1.49144i	0.130526	−	0.991445i	$-0.458333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$947$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$948$		0			0
$949$	−2.34018	−	1.45522i	−2.34018	−	1.45522i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.02807	+	0.0168232i	1.02807	+	0.0168232i	0.528068	−	0.849202i	$-0.322917\pi$
$953$	1.02807	+	0.0168232i	1.02807	+	0.0168232i	0.500000	+	0.866025i	$0.333333\pi$
$954$	1.29532	−	0.719854i	1.29532	−	0.719854i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.997859	−	0.0654031i	−0.997859	−	0.0654031i
$962$	−1.89129	−	0.642008i	−1.89129	−	0.642008i
$963$		0			0
$964$	−1.86549	+	0.467281i	−1.86549	+	0.467281i
$965$	−2.54924	−	1.15504i	−2.54924	−	1.15504i
$966$		0			0
$967$		0			0		0.179017	−	0.983846i	$-0.442708\pi$
$967$		0			0		−0.179017	+	0.983846i	$0.557292\pi$
$968$	0.659346	+	0.751840i	0.659346	+	0.751840i
$969$		0			0
$970$	−0.835844	+	1.92806i	−0.835844	+	1.92806i
$971$		0			0		0.961562	−	0.274589i	$-0.0885417\pi$
$971$		0			0		−0.961562	+	0.274589i	$0.911458\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.0194933	−	0.0262837i	0.0194933	−	0.0262837i
$977$	0.406913	−	0.368805i	0.406913	−	0.368805i	−0.442289	−	0.896873i	$-0.645833\pi$
$977$	0.406913	−	0.368805i	0.406913	−	0.368805i	0.849202	+	0.528068i	$0.177083\pi$
$978$		0			0
$979$		0			0
$980$	1.52753	+	0.881921i	1.52753	+	0.881921i
$981$	1.13100	+	0.628535i	1.13100	+	0.628535i
$982$		0			0
$983$		0			0		0.970031	−	0.242980i	$-0.0781250\pi$
$983$		0			0		−0.970031	+	0.242980i	$0.921875\pi$
$984$		0			0
$985$	0.719362	−	0.876545i	0.719362	−	0.876545i
$986$	−3.17127	−	0.905605i	−3.17127	−	0.905605i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.367516	−	0.930017i	$-0.619792\pi$
$991$		0			0		0.367516	+	0.930017i	$0.380208\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.26191	+	1.43893i	−1.26191	+	1.43893i	−0.412707	+	0.910864i	$0.635417\pi$
$997$	−1.26191	+	1.43893i	−1.26191	+	1.43893i	−0.849202	+	0.528068i	$0.822917\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3076.1.bd.a.759.1	✓	64
4.3	odd	2	CM	3076.1.bd.a.759.1	✓	64
769.692	even	192	inner	3076.1.bd.a.2999.1	yes	64
3076.2999	odd	192	inner	3076.1.bd.a.2999.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3076.1.bd.a.759.1	✓	64	1.1	even	1	trivial
3076.1.bd.a.759.1	✓	64	4.3	odd	2	CM
3076.1.bd.a.2999.1	yes	64	769.692	even	192	inner
3076.1.bd.a.2999.1	yes	64	3076.2999	odd	192	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3076 = 2^{2} \cdot 769 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3076.bd (of order \(192\), degree \(64\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.0327191 - 0.999465i) q^{2} +(-0.997859 + 0.0654031i) q^{4} +(1.75535 + 0.172887i) q^{5} +(0.0980171 + 0.995185i) q^{8} +(0.866025 + 0.500000i) q^{9} +(0.115361 - 1.76007i) q^{10} +(1.32090 - 0.914211i) q^{13} +(0.991445 - 0.130526i) q^{16} +(-1.33889 + 1.09880i) q^{17} +(0.471397 - 0.881921i) q^{18} +(-1.76290 - 0.0577113i) q^{20} +(2.07058 + 0.411863i) q^{25} +(-0.956940 - 1.29028i) q^{26} +(-1.60032 + 1.03183i) q^{29} +(-0.162895 - 0.986643i) q^{32} +(1.14202 + 1.30222i) q^{34} +(-0.896873 - 0.442289i) q^{36} +(1.02234 - 0.707574i) q^{37} +1.76384i q^{40} +(-1.78990 + 0.882683i) q^{41} +(1.43373 + 1.02740i) q^{45} +(-0.881921 - 0.471397i) q^{49} +(0.343895 - 2.08294i) q^{50} +(-1.25828 + 0.998645i) q^{52} +(1.24546 + 0.803030i) q^{53} +(1.08364 + 1.56570i) q^{58} +(0.0227573 - 0.0235145i) q^{61} +(-0.980785 + 0.195090i) q^{64} +(2.47670 - 1.37639i) q^{65} +(1.26416 - 1.18402i) q^{68} +(-0.412707 + 0.910864i) q^{72} +(-0.682320 - 1.57392i) q^{73} +(-0.740646 - 0.998645i) q^{74} +(1.76290 - 0.0577113i) q^{80} +(0.500000 + 0.866025i) q^{81} +(0.940775 + 1.76007i) q^{82} +(-2.54019 + 1.69730i) q^{85} +(-0.953924 - 1.16236i) q^{89} +(0.979938 - 1.46658i) q^{90} +(-1.12175 - 0.401370i) q^{97} +(-0.442289 + 0.896873i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q + 32 q^{81}+O(q^{100}) \) 64 * q + 32 * q^81

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