LMFDB - Embedded newform 3459.1.bm.a.1406.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3459,1,Mod(176,3459)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3459, base_ring=CyclotomicField(192))

chi = DirichletCharacter(H, H._module([96, 25]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3459.176");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3459 = 3 \cdot 1153 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3459.bm (of order $192$, degree $64$, 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:	$1.72626587870$
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:	$\Z[a_1, a_2, a_3]$
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			1406.1
Root			$-0.412707 + 0.910864i$ of defining polynomial
Character	$\chi$	$=$	3459.1406
Dual form			3459.1.bm.a.278.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.528068 + 0.849202i) q^{3} +(-0.991445 + 0.130526i) q^{4} +(-0.713004 - 0.178598i) q^{7} +(-0.442289 + 0.896873i) q^{9} +O(q^{10})$	\(q+(0.528068 + 0.849202i) q^{3} +(-0.991445 + 0.130526i) q^{4} +(-0.713004 - 0.178598i) q^{7} +(-0.442289 + 0.896873i) q^{9} +(-0.634393 - 0.773010i) q^{12} +(0.131834 - 0.434597i) q^{13} +(0.965926 - 0.258819i) q^{16} +(-1.15096 + 0.591194i) q^{19} +(-0.224848 - 0.699796i) q^{21} +(-0.973877 + 0.227076i) q^{25} +(-0.995185 + 0.0980171i) q^{27} +(0.730216 + 0.0840046i) q^{28} +(-0.588289 + 1.83093i) q^{31} +(0.321439 - 0.946930i) q^{36} +(-1.51251 - 1.12175i) q^{37} +(0.438678 - 0.117543i) q^{39} +(-0.0382444 - 0.388302i) q^{43} +(0.729864 + 0.683592i) q^{48} +(-0.405444 - 0.216714i) q^{49} +(-0.0739794 + 0.448087i) q^{52} +(-1.10983 - 0.665204i) q^{57} +(-1.50855 + 0.223772i) q^{61} +(0.475533 - 0.560482i) q^{63} +(-0.923880 + 0.382683i) q^{64} +(-0.292893 - 0.707107i) q^{67} +(1.82885 - 0.792836i) q^{73} +(-0.707107 - 0.707107i) q^{75} +(1.06394 - 0.736366i) q^{76} +(-0.0110242 + 0.0308106i) q^{79} +(-0.608761 - 0.793353i) q^{81} +(0.314267 + 0.664461i) q^{84} +(-0.171616 + 0.286324i) q^{91} +(-1.86549 + 0.467281i) q^{93} +(-0.973877 - 1.68680i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q+O(q^{10}) $ 64 * q	$ 64 q - 64 q^{67}+O(q^{100}) $ 64 * q - 64 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3459\mathbb{Z}\right)^\times$.

$n$	$1154$	$2311$
$\chi(n)$	$-1$	$e\left(\frac{89}{192}\right)$

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			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$2$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$3$	0.528068	+	0.849202i	0.528068	+	0.849202i
$3$	0.528068	+	0.849202i	0.528068	+	0.849202i
$4$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$5$		0			0		−0.114287	−	0.993448i	$-0.536458\pi$
$5$		0			0		0.114287	+	0.993448i	$0.463542\pi$
$6$		0			0
$7$	−0.713004	−	0.178598i	−0.713004	−	0.178598i	−0.130526	−	0.991445i	$-0.541667\pi$
$7$	−0.713004	−	0.178598i	−0.713004	−	0.178598i	−0.582478	+	0.812847i	$0.697917\pi$
$8$		0			0
$9$	−0.442289	+	0.896873i	−0.442289	+	0.896873i
$10$		0			0
$11$		0			0		0.352250	−	0.935906i	$-0.385417\pi$
$11$		0			0		−0.352250	+	0.935906i	$0.614583\pi$
$12$	−0.634393	−	0.773010i	−0.634393	−	0.773010i
$13$	0.131834	−	0.434597i	0.131834	−	0.434597i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	0.131834	−	0.434597i	0.131834	−	0.434597i	0.997859	+	0.0654031i	$0.0208333\pi$
$14$		0			0
$15$		0			0
$16$	0.965926	−	0.258819i	0.965926	−	0.258819i
$17$		0			0		−0.840448	−	0.541892i	$-0.817708\pi$
$17$		0			0		0.840448	+	0.541892i	$0.182292\pi$
$18$		0			0
$19$	−1.15096	+	0.591194i	−1.15096	+	0.591194i	−0.923880	−	0.382683i	$-0.875000\pi$
$19$	−1.15096	+	0.591194i	−1.15096	+	0.591194i	−0.227076	+	0.973877i	$0.572917\pi$
$20$		0			0
$21$	−0.224848	−	0.699796i	−0.224848	−	0.699796i
$22$		0			0
$23$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$23$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$24$		0			0
$25$	−0.973877	+	0.227076i	−0.973877	+	0.227076i
$26$		0			0
$27$	−0.995185	+	0.0980171i	−0.995185	+	0.0980171i
$28$	0.730216	+	0.0840046i	0.730216	+	0.0840046i
$29$		0			0		−0.849202	−	0.528068i	$-0.822917\pi$
$29$		0			0		0.849202	+	0.528068i	$0.177083\pi$
$30$		0			0
$31$	−0.588289	+	1.83093i	−0.588289	+	1.83093i	−0.0327191	+	0.999465i	$0.510417\pi$
$31$	−0.588289	+	1.83093i	−0.588289	+	1.83093i	−0.555570	+	0.831470i	$0.687500\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.321439	−	0.946930i	0.321439	−	0.946930i
$37$	−1.51251	−	1.12175i	−1.51251	−	1.12175i	−0.956940	−	0.290285i	$-0.906250\pi$
$37$	−1.51251	−	1.12175i	−1.51251	−	1.12175i	−0.555570	−	0.831470i	$-0.687500\pi$
$38$		0			0
$39$	0.438678	−	0.117543i	0.438678	−	0.117543i
$40$		0			0
$41$		0			0		−0.227076	−	0.973877i	$-0.572917\pi$
$41$		0			0		0.227076	+	0.973877i	$0.427083\pi$
$42$		0			0
$43$	−0.0382444	−	0.388302i	−0.0382444	−	0.388302i	−0.995185	−	0.0980171i	$-0.968750\pi$
$43$	−0.0382444	−	0.388302i	−0.0382444	−	0.388302i	0.956940	−	0.290285i	$-0.0937500\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$	0.729864	+	0.683592i	0.729864	+	0.683592i
$49$	−0.405444	−	0.216714i	−0.405444	−	0.216714i
$50$		0			0
$51$		0			0
$52$	−0.0739794	+	0.448087i	−0.0739794	+	0.448087i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.10983	−	0.665204i	−1.10983	−	0.665204i
$58$		0			0
$59$		0			0		0.146730	−	0.989177i	$-0.453125\pi$
$59$		0			0		−0.146730	+	0.989177i	$0.546875\pi$
$60$		0			0
$61$	−1.50855	+	0.223772i	−1.50855	+	0.223772i	−0.849202	−	0.528068i	$-0.822917\pi$
$61$	−1.50855	+	0.223772i	−1.50855	+	0.223772i	−0.659346	+	0.751840i	$0.729167\pi$
$62$		0			0
$63$	0.475533	−	0.560482i	0.475533	−	0.560482i
$64$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$65$		0			0
$66$		0			0
$67$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$67$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$71$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$72$		0			0
$73$	1.82885	−	0.792836i	1.82885	−	0.792836i	0.881921	−	0.471397i	$-0.156250\pi$
$73$	1.82885	−	0.792836i	1.82885	−	0.792836i	0.946930	−	0.321439i	$-0.104167\pi$
$74$		0			0
$75$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$76$	1.06394	−	0.736366i	1.06394	−	0.736366i
$77$		0			0
$78$		0			0
$79$	−0.0110242	+	0.0308106i	−0.0110242	+	0.0308106i	−0.946930	−	0.321439i	$-0.895833\pi$
$79$	−0.0110242	+	0.0308106i	−0.0110242	+	0.0308106i	0.935906	+	0.352250i	$0.114583\pi$
$80$		0			0
$81$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$82$		0			0
$83$		0			0		−0.961562	−	0.274589i	$-0.911458\pi$
$83$		0			0		0.961562	+	0.274589i	$0.0885417\pi$
$84$	0.314267	+	0.664461i	0.314267	+	0.664461i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.412707	−	0.910864i	$-0.635417\pi$
$89$		0			0		0.412707	+	0.910864i	$0.364583\pi$
$90$		0			0
$91$	−0.171616	+	0.286324i	−0.171616	+	0.286324i
$92$		0			0
$93$	−1.86549	+	0.467281i	−1.86549	+	0.467281i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.973877	−	1.68680i	−0.973877	−	1.68680i	−0.683592	−	0.729864i	$-0.739583\pi$
$97$	−0.973877	−	1.68680i	−0.973877	−	1.68680i	−0.290285	−	0.956940i	$-0.593750\pi$
$98$		0			0
$99$		0			0
$100$	0.935906	−	0.352250i	0.935906	−	0.352250i
$101$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$101$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$102$		0			0
$103$	0.654112	−	0.452717i	0.654112	−	0.452717i	−0.195090	−	0.980785i	$-0.562500\pi$
$103$	0.654112	−	0.452717i	0.654112	−	0.452717i	0.849202	+	0.528068i	$0.177083\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$107$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$108$	0.973877	−	0.227076i	0.973877	−	0.227076i
$109$	−1.53858	+	1.26268i	−1.53858	+	1.26268i	−0.707107	+	0.707107i	$0.750000\pi$
$109$	−1.53858	+	1.26268i	−1.53858	+	1.26268i	−0.831470	+	0.555570i	$0.812500\pi$
$110$		0			0
$111$	0.153888	−	1.87679i	0.153888	−	1.87679i
$112$	−0.734933	+	0.0120264i	−0.734933	+	0.0120264i
$113$		0			0		0.996655	−	0.0817211i	$-0.0260417\pi$
$113$		0			0		−0.996655	+	0.0817211i	$0.973958\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.331470	+	0.310455i	0.331470	+	0.310455i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.751840	−	0.659346i	−0.751840	−	0.659346i
$122$		0			0
$123$		0			0
$124$	0.344272	−	1.89206i	0.344272	−	1.89206i
$125$		0			0
$126$		0			0
$127$	1.18996	+	1.60448i	1.18996	+	1.60448i	0.634393	+	0.773010i	$0.281250\pi$
$127$	1.18996	+	1.60448i	1.18996	+	1.60448i	0.555570	+	0.831470i	$0.312500\pi$
$128$		0			0
$129$	0.309551	−	0.237527i	0.309551	−	0.237527i
$130$		0			0
$131$		0			0		−0.227076	−	0.973877i	$-0.572917\pi$
$131$		0			0		0.227076	+	0.973877i	$0.427083\pi$
$132$		0			0
$133$	0.926222	−	0.215965i	0.926222	−	0.215965i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$137$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$138$		0			0
$139$	0.130736	−	0.00427986i	0.130736	−	0.00427986i	0.0327191	−	0.999465i	$-0.489583\pi$
$139$	0.130736	−	0.00427986i	0.130736	−	0.00427986i	0.0980171	+	0.995185i	$0.468750\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$145$		0			0
$146$		0			0
$147$	−0.0300677	−	0.458744i	−0.0300677	−	0.458744i
$148$	1.64599	+	0.914735i	1.64599	+	0.914735i
$149$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$149$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$150$		0			0
$151$	−0.795267	+	1.68145i	−0.795267	+	1.68145i	−0.0654031	+	0.997859i	$0.520833\pi$
$151$	−0.795267	+	1.68145i	−0.795267	+	1.68145i	−0.729864	+	0.683592i	$0.760417\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.419582	+	0.173797i	−0.419582	+	0.173797i
$157$	0.682320	+	1.57392i	0.682320	+	1.57392i	0.812847	+	0.582478i	$0.197917\pi$
$157$	0.682320	+	1.57392i	0.682320	+	1.57392i	−0.130526	+	0.991445i	$0.541667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.11717	+	0.828549i	1.11717	+	0.828549i	0.986643	−	0.162895i	$-0.0520833\pi$
$163$	1.11717	+	0.828549i	1.11717	+	0.828549i	0.130526	+	0.991445i	$0.458333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.874090	−	0.485763i	$-0.161458\pi$
$167$		0			0		−0.874090	+	0.485763i	$0.838542\pi$
$168$		0			0
$169$	0.659975	+	0.440981i	0.659975	+	0.440981i
$170$		0			0
$171$	−0.0211706	−	1.29374i	−0.0211706	−	1.29374i
$172$	0.0886008	+	0.379988i	0.0886008	+	0.379988i
$173$		0			0		0.242980	−	0.970031i	$-0.421875\pi$
$173$		0			0		−0.242980	+	0.970031i	$0.578125\pi$
$174$		0			0
$175$	0.734933	+	0.0120264i	0.734933	+	0.0120264i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.822268	−	0.569100i	$-0.192708\pi$
$179$		0			0		−0.822268	+	0.569100i	$0.807292\pi$
$180$		0			0
$181$	−0.252314	+	1.08211i	−0.252314	+	1.08211i	0.683592	+	0.729864i	$0.260417\pi$
$181$	−0.252314	+	1.08211i	−0.252314	+	1.08211i	−0.935906	+	0.352250i	$0.885417\pi$
$182$		0			0
$183$	−0.986643	−	1.16290i	−0.986643	−	1.16290i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.727076	+	0.107852i	0.727076	+	0.107852i
$190$		0			0
$191$		0			0		−0.646956	−	0.762527i	$-0.723958\pi$
$191$		0			0		0.646956	+	0.762527i	$0.276042\pi$
$192$	−0.812847	−	0.582478i	−0.812847	−	0.582478i
$193$	−0.891370	+	1.60395i	−0.891370	+	1.60395i	−0.0980171	+	0.995185i	$0.531250\pi$
$193$	−0.891370	+	1.60395i	−0.891370	+	1.60395i	−0.793353	+	0.608761i	$0.791667\pi$
$194$		0			0
$195$		0			0
$196$	0.430262	+	0.161939i	0.430262	+	0.161939i
$197$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$197$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$198$		0			0
$199$	0.739288	−	1.78480i	0.739288	−	1.78480i	0.130526	−	0.991445i	$-0.458333\pi$
$199$	0.739288	−	1.78480i	0.739288	−	1.78480i	0.608761	−	0.793353i	$-0.291667\pi$
$200$		0			0
$201$	0.445809	−	0.622126i	0.445809	−	0.622126i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.0148595	−	0.453909i	0.0148595	−	0.453909i
$209$		0			0
$210$		0			0
$211$	−0.577920	−	0.346392i	−0.577920	−	0.346392i	0.195090	−	0.980785i	$-0.437500\pi$
$211$	−0.577920	−	0.346392i	−0.577920	−	0.346392i	−0.773010	+	0.634393i	$0.781250\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.746454	−	1.20040i	0.746454	−	1.20040i
$218$		0			0
$219$	1.63904	+	1.13439i	1.63904	+	1.13439i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.80614	+	0.818353i	−1.80614	+	0.818353i	−0.849202	+	0.528068i	$0.822917\pi$
$223$	−1.80614	+	0.818353i	−1.80614	+	0.818353i	−0.956940	+	0.290285i	$0.906250\pi$
$224$		0			0
$225$	0.227076	−	0.973877i	0.227076	−	0.973877i
$226$		0			0
$227$		0			0		0.485763	−	0.874090i	$-0.338542\pi$
$227$		0			0		−0.485763	+	0.874090i	$0.661458\pi$
$228$	1.18716	+	0.514652i	1.18716	+	0.514652i
$229$	0.484335	−	0.0397132i	0.484335	−	0.0397132i	0.162895	−	0.986643i	$-0.447917\pi$
$229$	0.484335	−	0.0397132i	0.484335	−	0.0397132i	0.321439	+	0.946930i	$0.395833\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.840448	−	0.541892i	$-0.182292\pi$
$233$		0			0		−0.840448	+	0.541892i	$0.817708\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.0319859	+	0.00690830i	−0.0319859	+	0.00690830i
$238$		0			0
$239$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$239$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$240$		0			0
$241$	−0.405994	−	0.115938i	−0.405994	−	0.115938i	0.0654031	−	0.997859i	$-0.479167\pi$
$241$	−0.405994	−	0.115938i	−0.405994	−	0.115938i	−0.471397	+	0.881921i	$0.656250\pi$
$242$		0			0
$243$	0.352250	−	0.935906i	0.352250	−	0.935906i
$244$	1.46643	−	0.418763i	1.46643	−	0.418763i
$245$		0			0
$246$		0			0
$247$	0.105196	+	0.578141i	0.105196	+	0.578141i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$251$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$252$	−0.398308	+	0.617756i	−0.398308	+	0.617756i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.866025	−	0.500000i	0.866025	−	0.500000i
$257$		0			0		−0.211112	−	0.977462i	$-0.567708\pi$
$257$		0			0		0.211112	+	0.977462i	$0.432292\pi$
$258$		0			0
$259$	0.878083	+	1.06995i	0.878083	+	1.06995i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.569100	−	0.822268i	$-0.692708\pi$
$263$		0			0		0.569100	+	0.822268i	$0.307292\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.382683	+	0.662827i	0.382683	+	0.662827i
$269$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$269$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$270$		0			0
$271$	0.980785	+	1.19509i	0.980785	+	1.19509i	0.980785	+	0.195090i	$0.0625000\pi$
$271$	0.980785	+	1.19509i	0.980785	+	1.19509i			1.00000i	$0.5\pi$
$272$		0			0
$273$	−0.333772	+	0.00546182i	−0.333772	+	0.00546182i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.39349	−	0.715774i	−1.39349	−	0.715774i	−0.412707	−	0.910864i	$-0.635417\pi$
$277$	−1.39349	−	0.715774i	−1.39349	−	0.715774i	−0.980785	+	0.195090i	$0.937500\pi$
$278$		0			0
$279$	−1.38192	−	1.33742i	−1.38192	−	1.33742i
$280$		0			0
$281$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$281$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$282$		0			0
$283$	0.265285	+	1.45796i	0.265285	+	1.45796i	0.793353	+	0.608761i	$0.208333\pi$
$283$	0.265285	+	1.45796i	0.265285	+	1.45796i	−0.528068	+	0.849202i	$0.677083\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.412707	+	0.910864i	0.412707	+	0.910864i
$290$		0			0
$291$	0.918165	−	1.71777i	0.918165	−	1.71777i
$292$	−1.70972	+	1.02477i	−1.70972	+	1.02477i
$293$		0			0		0.941544	−	0.336890i	$-0.109375\pi$
$293$		0			0		−0.941544	+	0.336890i	$0.890625\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.793353	+	0.608761i	0.793353	+	0.608761i
$301$	−0.0420816	+	0.283691i	−0.0420816	+	0.283691i
$302$		0			0
$303$		0			0
$304$	−0.958726	+	0.868938i	−0.958726	+	0.868938i
$305$		0			0
$306$		0			0
$307$	−0.709180	−	0.924221i	−0.709180	−	0.924221i	0.290285	−	0.956940i	$-0.406250\pi$
$307$	−0.709180	−	0.924221i	−0.709180	−	0.924221i	−0.999465	+	0.0327191i	$0.989583\pi$
$308$		0			0
$309$	0.729864	+	0.316408i	0.729864	+	0.316408i
$310$		0			0
$311$		0			0		0.621661	−	0.783287i	$-0.286458\pi$
$311$		0			0		−0.621661	+	0.783287i	$0.713542\pi$
$312$		0			0
$313$	−0.152057	−	0.212196i	−0.152057	−	0.212196i	0.729864	−	0.683592i	$-0.239583\pi$
$313$	−0.152057	−	0.212196i	−0.152057	−	0.212196i	−0.881921	+	0.471397i	$0.843750\pi$
$314$		0			0
$315$		0			0
$316$	0.00690830	−	0.0319859i	0.00690830	−	0.0319859i
$317$		0			0		−0.242980	−	0.970031i	$-0.578125\pi$
$317$		0			0		0.242980	+	0.970031i	$0.421875\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.707107	+	0.707107i	0.707107	+	0.707107i
$325$	−0.0297030	+	0.453180i	−0.0297030	+	0.453180i
$326$		0			0
$327$	−1.88474	−	0.639783i	−1.88474	−	0.639783i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.0186793	−	0.162371i	0.0186793	−	0.162371i	−0.980785	−	0.195090i	$-0.937500\pi$
$331$	0.0186793	−	0.162371i	0.0186793	−	0.162371i	0.999465	+	0.0327191i	$0.0104167\pi$
$332$		0			0
$333$	1.67504	−	0.860390i	1.67504	−	0.860390i
$334$		0			0
$335$		0			0
$336$	−0.398308	−	0.617756i	−0.398308	−	0.617756i
$337$	−0.326351	−	1.30287i	−0.326351	−	1.30287i	−0.881921	−	0.471397i	$-0.843750\pi$
$337$	−0.326351	−	1.30287i	−0.326351	−	1.30287i	0.555570	−	0.831470i	$-0.312500\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.795001	+	0.720547i	0.795001	+	0.720547i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.514103	−	0.857729i	$-0.328125\pi$
$347$		0			0		−0.514103	+	0.857729i	$0.671875\pi$
$348$		0			0
$349$	−0.940775	+	1.76007i	−0.940775	+	1.76007i	−0.412707	+	0.910864i	$0.635417\pi$
$349$	−0.940775	+	1.76007i	−0.940775	+	1.76007i	−0.528068	+	0.849202i	$0.677083\pi$
$350$		0			0
$351$	−0.0886008	+	0.445426i	−0.0886008	+	0.445426i
$352$		0			0
$353$		0			0		−0.528068	−	0.849202i	$-0.677083\pi$
$353$		0			0		0.528068	+	0.849202i	$0.322917\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.0490677	−	0.998795i	$-0.515625\pi$
$359$		0			0		0.0490677	+	0.998795i	$0.484375\pi$
$360$		0			0
$361$	0.392712	−	0.548029i	0.392712	−	0.548029i
$362$		0			0
$363$	0.162895	−	0.986643i	0.162895	−	0.986643i
$364$	0.132775	−	0.306275i	0.132775	−	0.306275i
$365$		0			0
$366$		0			0
$367$	1.47479	+	1.29335i	1.47479	+	1.29335i	0.866025	+	0.500000i	$0.166667\pi$
$367$	1.47479	+	1.29335i	1.47479	+	1.29335i	0.608761	+	0.793353i	$0.291667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.78854	−	0.706779i	1.78854	−	0.706779i
$373$	0.965161	+	1.73673i	0.965161	+	1.73673i	0.582478	+	0.812847i	$0.302083\pi$
$373$	0.965161	+	1.73673i	0.965161	+	1.73673i	0.382683	+	0.923880i	$0.375000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.460597	−	1.51838i	0.460597	−	1.51838i	−0.352250	−	0.935906i	$-0.614583\pi$
$379$	0.460597	−	1.51838i	0.460597	−	1.51838i	0.812847	−	0.582478i	$-0.197917\pi$
$380$		0			0
$381$	−0.734146	+	1.85779i	−0.734146	+	1.85779i
$382$		0			0
$383$		0			0		0.412707	−	0.910864i	$-0.364583\pi$
$383$		0			0		−0.412707	+	0.910864i	$0.635417\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.365172	+	0.137441i	0.365172	+	0.137441i
$388$	1.18572	+	1.54526i	1.18572	+	1.54526i
$389$		0			0		−0.740951	−	0.671559i	$-0.765625\pi$
$389$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.95466	+	0.0319859i	1.95466	+	0.0319859i	0.980785	−	0.195090i	$-0.0625000\pi$
$397$	1.95466	+	0.0319859i	1.95466	+	0.0319859i	0.973877	+	0.227076i	$0.0729167\pi$
$398$		0			0
$399$	0.672506	+	0.672506i	0.672506	+	0.672506i
$400$	−0.881921	+	0.471397i	−0.881921	+	0.471397i
$401$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$401$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$402$		0			0
$403$	0.718162	+	0.497047i	0.718162	+	0.497047i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.747444	−	1.58034i	−0.747444	−	1.58034i	−0.812847	−	0.582478i	$-0.802083\pi$
$409$	−0.747444	−	1.58034i	−0.747444	−	1.58034i	0.0654031	−	0.997859i	$-0.479167\pi$
$410$		0			0
$411$		0			0
$412$	−0.589424	+	0.534223i	−0.589424	+	0.534223i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.0726721	+	0.108761i	0.0726721	+	0.108761i
$418$		0			0
$419$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$419$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$420$		0			0
$421$	−1.41086	−	1.27873i	−1.41086	−	1.27873i	−0.910864	−	0.412707i	$-0.864583\pi$
$421$	−1.41086	−	1.27873i	−1.41086	−	1.27873i	−0.500000	−	0.866025i	$-0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.11557	+	0.109874i	1.11557	+	0.109874i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.849202	−	0.528068i	$-0.822917\pi$
$431$		0			0		0.849202	+	0.528068i	$0.177083\pi$
$432$	−0.935906	+	0.352250i	−0.935906	+	0.352250i
$433$	0.852250	+	0.0698805i	0.852250	+	0.0698805i	0.500000	−	0.866025i	$-0.333333\pi$
$433$	0.852250	+	0.0698805i	0.852250	+	0.0698805i	0.352250	+	0.935906i	$0.385417\pi$
$434$		0			0
$435$		0			0
$436$	1.36060	−	1.45270i	1.36060	−	1.45270i
$437$		0			0
$438$		0			0
$439$	−1.34091	−	0.266724i	−1.34091	−	0.266724i	−0.528068	−	0.849202i	$-0.677083\pi$
$439$	−1.34091	−	0.266724i	−1.34091	−	0.266724i	−0.812847	+	0.582478i	$0.802083\pi$
$440$		0			0
$441$	0.373688	−	0.267781i	0.373688	−	0.267781i
$442$		0			0
$443$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$443$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$444$	0.0923988	+	1.88082i	0.0923988	+	1.88082i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.727076	−	0.107852i	0.727076	−	0.107852i
$449$		0			0		−0.0327191	−	0.999465i	$-0.510417\pi$
$449$		0			0		0.0327191	+	0.999465i	$0.489583\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.84785	+	0.212578i	−1.84785	+	0.212578i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.24441	+	0.247528i	−1.24441	+	0.247528i	−0.773010	−	0.634393i	$-0.781250\pi$
$457$	−1.24441	+	0.247528i	−1.24441	+	0.247528i	−0.471397	+	0.881921i	$0.656250\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.114287	−	0.993448i	$-0.463542\pi$
$461$		0			0		−0.114287	+	0.993448i	$0.536458\pi$
$462$		0			0
$463$	−0.461555	−	0.297595i	−0.461555	−	0.297595i	0.290285	−	0.956940i	$-0.406250\pi$
$463$	−0.461555	−	0.297595i	−0.461555	−	0.297595i	−0.751840	+	0.659346i	$0.770833\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.621661	−	0.783287i	$-0.286458\pi$
$467$		0			0		−0.621661	+	0.783287i	$0.713542\pi$
$468$	−0.369156	−	0.264534i	−0.369156	−	0.264534i
$469$	0.0825460	+	0.556480i	0.0825460	+	0.556480i
$470$		0			0
$471$	−0.976267	+	1.41057i	−0.976267	+	1.41057i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.986643	−	0.837105i	0.986643	−	0.837105i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.952063	−	0.305903i	$-0.0989583\pi$
$479$		0			0		−0.952063	+	0.305903i	$0.901042\pi$
$480$		0			0
$481$	−0.686910	+	0.509448i	−0.686910	+	0.509448i
$482$		0			0
$483$		0			0
$484$	0.831470	+	0.555570i	0.831470	+	0.555570i
$485$		0			0
$486$		0			0
$487$	0.997391	+	1.66405i	0.997391	+	1.66405i	0.707107	+	0.707107i	$0.250000\pi$
$487$	0.997391	+	1.66405i	0.997391	+	1.66405i	0.290285	+	0.956940i	$0.406250\pi$
$488$		0			0
$489$	−0.113665	+	1.38623i	−0.113665	+	1.38623i
$490$		0			0
$491$		0			0		−0.998795	−	0.0490677i	$-0.984375\pi$
$491$		0			0		0.998795	+	0.0490677i	$0.0156250\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.0943632	+	1.92081i	−0.0943632	+	1.92081i
$497$		0			0
$498$		0			0
$499$	−0.439579	+	1.64053i	−0.439579	+	1.64053i	0.290285	+	0.956940i	$0.406250\pi$
$499$	−0.439579	+	1.64053i	−0.439579	+	1.64053i	−0.729864	+	0.683592i	$0.760417\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.0259706	+	0.793321i	−0.0259706	+	0.793321i
$508$	−1.38921	−	1.43543i	−1.38921	−	1.43543i
$509$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$509$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$510$		0			0
$511$	−1.44558	+	0.238666i	−1.44558	+	0.238666i
$512$		0			0
$513$	1.08747	−	0.701160i	1.08747	−	0.701160i
$514$		0			0
$515$		0			0
$516$	−0.275899	+	0.275899i	−0.275899	+	0.275899i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.989177	−	0.146730i	$-0.953125\pi$
$521$		0			0		0.989177	+	0.146730i	$0.0468750\pi$
$522$		0			0
$523$	−0.352250	+	0.610115i	−0.352250	+	0.610115i	−0.986643	−	0.162895i	$-0.947917\pi$
$523$	−0.352250	+	0.610115i	−0.352250	+	0.610115i	0.634393	+	0.773010i	$0.281250\pi$
$524$		0			0
$525$	0.377882	+	0.630458i	0.377882	+	0.630458i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.866025	−	0.500000i	0.866025	−	0.500000i
$530$		0			0
$531$		0			0
$532$	−0.890109	+	0.335013i	−0.890109	+	0.335013i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.25828	−	0.998645i	1.25828	−	0.998645i	0.258819	−	0.965926i	$-0.416667\pi$
$541$	1.25828	−	0.998645i	1.25828	−	0.998645i	0.999465	−	0.0327191i	$-0.0104167\pi$
$542$		0			0
$543$	−1.05217	+	0.357164i	−1.05217	+	0.357164i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.598017	−	0.129159i	−0.598017	−	0.129159i	−0.0980171	−	0.995185i	$-0.531250\pi$
$547$	−0.598017	−	0.129159i	−0.598017	−	0.129159i	−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$	0.466519	−	1.45195i	0.466519	−	1.45195i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.0133630	−	0.0199992i	0.0133630	−	0.0199992i
$554$		0			0
$555$		0			0
$556$	−0.129059	+	0.0213077i	−0.129059	+	0.0213077i
$557$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$557$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$558$		0			0
$559$	−0.173797	−	0.0345703i	−0.173797	−	0.0345703i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.0327191	−	0.999465i	$-0.510417\pi$
$563$		0			0		0.0327191	+	0.999465i	$0.489583\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.292358	+	0.674388i	0.292358	+	0.674388i
$568$		0			0
$569$		0			0		−0.569100	−	0.822268i	$-0.692708\pi$
$569$		0			0		0.569100	+	0.822268i	$0.307292\pi$
$570$		0			0
$571$	0.471498	+	0.213633i	0.471498	+	0.213633i	0.634393	−	0.773010i	$-0.281250\pi$
$571$	0.471498	+	0.213633i	0.471498	+	0.213633i	−0.162895	+	0.986643i	$0.552083\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.0654031	−	0.997859i	0.0654031	−	0.997859i
$577$	−0.125174	+	1.90978i	−0.125174	+	1.90978i	0.227076	+	0.973877i	$0.427083\pi$
$577$	−0.125174	+	1.90978i	−0.125174	+	1.90978i	−0.352250	+	0.935906i	$0.614583\pi$
$578$		0			0
$579$	−1.83278	+	0.0900386i	−1.83278	+	0.0900386i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.970031	−	0.242980i	$-0.0781250\pi$
$587$		0			0		−0.970031	+	0.242980i	$0.921875\pi$
$588$	0.0896885	+	0.450895i	0.0896885	+	0.450895i
$589$	−0.405341	−	2.45512i	−0.405341	−	2.45512i
$590$		0			0
$591$		0			0
$592$	−1.75130	−	0.692065i	−1.75130	−	0.692065i
$593$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$593$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.90605	−	0.314690i	1.90605	−	0.314690i
$598$		0			0
$599$		0			0		0.397748	−	0.917494i	$-0.369792\pi$
$599$		0			0		−0.397748	+	0.917494i	$0.630208\pi$
$600$		0			0
$601$	0.988444	−	1.58955i	0.988444	−	1.58955i	0.195090	−	0.980785i	$-0.437500\pi$
$601$	0.988444	−	1.58955i	0.988444	−	1.58955i	0.793353	−	0.608761i	$-0.208333\pi$
$602$		0			0
$603$	0.763728	+	0.0500574i	0.763728	+	0.0500574i
$604$	0.568990	−	1.77087i	0.568990	−	1.77087i
$605$		0			0
$606$		0			0
$607$	−0.101606	−	1.23916i	−0.101606	−	1.23916i	−0.831470	−	0.555570i	$-0.812500\pi$
$607$	−0.101606	−	1.23916i	−0.101606	−	1.23916i	0.729864	−	0.683592i	$-0.239583\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.120618	+	0.337105i	0.120618	+	0.337105i	0.986643	−	0.162895i	$-0.0520833\pi$
$613$	0.120618	+	0.337105i	0.120618	+	0.337105i	−0.866025	+	0.500000i	$0.833333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$617$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$618$		0			0
$619$	1.57766	+	0.910864i	1.57766	+	0.910864i	0.995185	+	0.0980171i	$0.0312500\pi$
$619$	1.57766	+	0.910864i	1.57766	+	0.910864i	0.582478	+	0.812847i	$0.302083\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.393308	−	0.227076i	0.393308	−	0.227076i
$625$	0.896873	−	0.442289i	0.896873	−	0.442289i
$626$		0			0
$627$		0			0
$628$	−0.881921	−	1.47140i	−0.881921	−	1.47140i
$629$		0			0
$630$		0			0
$631$	0.615175	+	0.858476i	0.615175	+	0.858476i	0.997859	−	0.0654031i	$-0.0208333\pi$
$631$	0.615175	+	0.858476i	0.615175	+	0.858476i	−0.382683	+	0.923880i	$0.625000\pi$
$632$		0			0
$633$	−0.0110242	−	0.673690i	−0.0110242	−	0.673690i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.147634	+	0.147634i	−0.147634	+	0.147634i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$641$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$642$		0			0
$643$	−0.264818	−	1.60398i	−0.264818	−	1.60398i	−0.707107	−	0.707107i	$-0.750000\pi$
$643$	−0.264818	−	1.60398i	−0.264818	−	1.60398i	0.442289	−	0.896873i	$-0.354167\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.999465	−	0.0327191i	$-0.989583\pi$
$647$		0			0		0.999465	+	0.0327191i	$0.0104167\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.41356			1.41356
$652$	−1.21576	−	0.675641i	−1.21576	−	0.675641i
$653$		0			0		−0.456904	−	0.889516i	$-0.651042\pi$
$653$		0			0		0.456904	+	0.889516i	$0.348958\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.0978071	+	1.99091i	−0.0978071	+	1.99091i
$658$		0			0
$659$		0			0		0.970031	−	0.242980i	$-0.0781250\pi$
$659$		0			0		−0.970031	+	0.242980i	$0.921875\pi$
$660$		0			0
$661$	0.353428	−	0.460597i	0.353428	−	0.460597i	−0.582478	−	0.812847i	$-0.697917\pi$
$661$	0.353428	−	0.460597i	0.353428	−	0.460597i	0.935906	+	0.352250i	$0.114583\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.64871	−	1.10163i	−1.64871	−	1.10163i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.300909	−	0.194015i	−0.300909	−	0.194015i	0.382683	−	0.923880i	$-0.375000\pi$
$673$	−0.300909	−	0.194015i	−0.300909	−	0.194015i	−0.683592	+	0.729864i	$0.739583\pi$
$674$		0			0
$675$	0.946930	−	0.321439i	0.946930	−	0.321439i
$676$	−0.711889	−	0.351065i	−0.711889	−	0.351065i
$677$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$677$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$678$		0			0
$679$	0.393118	+	1.37663i	0.393118	+	1.37663i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.941544	−	0.336890i	$-0.890625\pi$
$683$		0			0		0.941544	+	0.336890i	$0.109375\pi$
$684$	0.189856	+	1.27991i	0.189856	+	1.27991i
$685$		0			0
$686$		0			0
$687$	0.289486	+	0.390327i	0.289486	+	0.390327i
$688$	−0.137441	−	0.365172i	−0.137441	−	0.365172i
$689$		0			0
$690$		0			0
$691$	−0.0177326	−	1.08364i	−0.0177326	−	1.08364i	−0.849202	−	0.528068i	$-0.822917\pi$
$691$	−0.0177326	−	1.08364i	−0.0177326	−	1.08364i	0.831470	−	0.555570i	$-0.187500\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.730216	+	0.0840046i	−0.730216	+	0.0840046i
$701$		0			0		0.977462	−	0.211112i	$-0.0677083\pi$
$701$		0			0		−0.977462	+	0.211112i	$0.932292\pi$
$702$		0			0
$703$	2.40401	+	0.396903i	2.40401	+	0.396903i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.0705183	−	1.43543i	−0.0705183	−	1.43543i	−0.729864	−	0.683592i	$-0.760417\pi$
$709$	−0.0705183	−	1.43543i	−0.0705183	−	1.43543i	0.659346	−	0.751840i	$-0.270833\pi$
$710$		0			0
$711$	−0.0227573	−	0.0235145i	−0.0227573	−	0.0235145i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$719$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$720$		0			0
$721$	−0.547239	+	0.205966i	−0.547239	+	0.205966i
$722$		0			0
$723$	−0.115938	−	0.405994i	−0.115938	−	0.405994i
$724$	0.108911	−	1.10579i	0.108911	−	1.10579i
$725$		0			0
$726$		0			0
$727$	−1.00503	−	0.217066i	−1.00503	−	0.217066i	−0.321439	−	0.946930i	$-0.604167\pi$
$727$	−1.00503	−	0.217066i	−1.00503	−	0.217066i	−0.683592	+	0.729864i	$0.739583\pi$
$728$		0			0
$729$	0.980785	−	0.195090i	0.980785	−	0.195090i
$730$		0			0
$731$		0			0
$732$	1.12999	+	1.02416i	1.12999	+	1.02416i
$733$	0.439579	−	0.273348i	0.439579	−	0.273348i	−0.290285	−	0.956940i	$-0.593750\pi$
$733$	0.439579	−	0.273348i	0.439579	−	0.273348i	0.729864	+	0.683592i	$0.239583\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.40211	−	0.184592i	−1.40211	−	0.184592i	−0.608761	−	0.793353i	$-0.708333\pi$
$739$	−1.40211	−	0.184592i	−1.40211	−	0.184592i	−0.793353	+	0.608761i	$0.791667\pi$
$740$		0			0
$741$	−0.435408	+	0.394631i	−0.435408	+	0.394631i
$742$		0			0
$743$		0			0		−0.397748	−	0.917494i	$-0.630208\pi$
$743$		0			0		0.397748	+	0.917494i	$0.369792\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.67572	+	1.04203i	1.67572	+	1.04203i	0.923880	+	0.382683i	$0.125000\pi$
$751$	1.67572	+	1.04203i	1.67572	+	1.04203i	0.751840	+	0.659346i	$0.229167\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.734933	−	0.0120264i	−0.734933	−	0.0120264i
$757$	−0.773010	−	0.634393i	−0.773010	−	0.634393i	0.162895	−	0.986643i	$-0.447917\pi$
$757$	−0.773010	−	0.634393i	−0.773010	−	0.634393i	−0.935906	+	0.352250i	$0.885417\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$761$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$762$		0			0
$763$	1.32252	−	0.625507i	1.32252	−	0.625507i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.881921	+	0.471397i	0.881921	+	0.471397i
$769$	0.794045	−	1.27693i	0.794045	−	1.27693i	−0.162895	−	0.986643i	$-0.552083\pi$
$769$	0.794045	−	1.27693i	0.794045	−	1.27693i	0.956940	−	0.290285i	$-0.0937500\pi$
$770$		0			0
$771$		0			0
$772$	0.674388	−	1.70657i	0.674388	−	1.70657i
$773$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$773$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$774$		0			0
$775$	0.157160	−	1.91669i	0.157160	−	1.91669i
$776$		0			0
$777$	−0.444914	+	1.31067i	−0.444914	+	1.31067i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.447719	−	0.104393i	−0.447719	−	0.104393i
$785$		0			0
$786$		0			0
$787$	−0.817890	+	0.791553i	−0.817890	+	0.791553i	−0.980785	−	0.195090i	$-0.937500\pi$
$787$	−0.817890	+	0.791553i	−0.817890	+	0.791553i	0.162895	+	0.986643i	$0.447917\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.101627	+	0.685111i	−0.101627	+	0.685111i
$794$		0			0
$795$		0			0
$796$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$797$		0			0		−0.528068	−	0.849202i	$-0.677083\pi$
$797$		0			0		0.528068	+	0.849202i	$0.322917\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.360791	+	0.674993i	−0.360791	+	0.674993i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$809$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$810$		0			0
$811$	1.99304	−	0.163420i	1.99304	−	0.163420i	0.995185	−	0.0980171i	$-0.0312500\pi$
$811$	1.99304	−	0.163420i	1.99304	−	0.163420i	0.997859	−	0.0654031i	$-0.0208333\pi$
$812$		0			0
$813$	−0.496952	+	1.46397i	−0.496952	+	1.46397i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.273579	+	0.424308i	0.273579	+	0.424308i
$818$		0			0
$819$	−0.180892	−	0.280556i	−0.180892	−	0.280556i
$820$		0			0
$821$		0			0		−0.989177	−	0.146730i	$-0.953125\pi$
$821$		0			0		0.989177	+	0.146730i	$0.0468750\pi$
$822$		0			0
$823$	−0.212578	+	0.0840046i	−0.212578	+	0.0840046i	−0.471397	−	0.881921i	$-0.656250\pi$
$823$	−0.212578	+	0.0840046i	−0.212578	+	0.0840046i	0.258819	+	0.965926i	$0.416667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.541892	−	0.840448i	$-0.317708\pi$
$827$		0			0		−0.541892	+	0.840448i	$0.682292\pi$
$828$		0			0
$829$	−0.897168	−	0.897168i	−0.897168	−	0.897168i	0.0980171	−	0.995185i	$-0.468750\pi$
$829$	−0.897168	−	0.897168i	−0.897168	−	0.897168i	−0.995185	+	0.0980171i	$0.968750\pi$
$830$		0			0
$831$	−0.128022	−	1.56133i	−0.128022	−	1.56133i
$832$	0.0445147	+	0.451966i	0.0445147	+	0.451966i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.405994	−	1.87978i	0.405994	−	1.87978i
$838$		0			0
$839$		0			0		0.889516	−	0.456904i	$-0.151042\pi$
$839$		0			0		−0.889516	+	0.456904i	$0.848958\pi$
$840$		0			0
$841$	0.442289	+	0.896873i	0.442289	+	0.896873i
$842$		0			0
$843$		0			0
$844$	0.618189	+	0.267995i	0.618189	+	0.267995i
$845$		0			0
$846$		0			0
$847$	0.418307	+	0.604393i	0.418307	+	0.604393i
$848$		0			0
$849$	−1.09802	+	0.995185i	−1.09802	+	0.995185i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.37413	−	1.05441i	−1.37413	−	1.05441i	−0.991445	−	0.130526i	$-0.958333\pi$
$853$	−1.37413	−	1.05441i	−1.37413	−	1.05441i	−0.382683	−	0.923880i	$-0.625000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.427555	−	0.903989i	$-0.359375\pi$
$857$		0			0		−0.427555	+	0.903989i	$0.640625\pi$
$858$		0			0
$859$	0.877010	−	0.115461i	0.877010	−	0.115461i	0.321439	−	0.946930i	$-0.395833\pi$
$859$	0.877010	−	0.115461i	0.877010	−	0.115461i	0.555570	+	0.831470i	$0.312500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.930017	−	0.367516i	$-0.880208\pi$
$863$		0			0		0.930017	+	0.367516i	$0.119792\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$868$	−0.583385	+	1.28756i	−0.583385	+	1.28756i
$869$		0			0
$870$		0			0
$871$	−0.345920	+	0.0340701i	−0.345920	+	0.0340701i
$872$		0			0
$873$	1.94358	−	0.127389i	1.94358	−	0.127389i
$874$		0			0
$875$		0			0
$876$	−1.77308	−	0.910751i	−1.77308	−	0.910751i
$877$	−1.43373	−	1.02740i	−1.43373	−	1.02740i	−0.991445	−	0.130526i	$-0.958333\pi$
$877$	−1.43373	−	1.02740i	−1.43373	−	1.02740i	−0.442289	−	0.896873i	$-0.645833\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.952063	−	0.305903i	$-0.901042\pi$
$881$		0			0		0.952063	+	0.305903i	$0.0989583\pi$
$882$		0			0
$883$	−0.934183	+	0.266770i	−0.934183	+	0.266770i	−0.707107	−	0.707107i	$-0.750000\pi$
$883$	−0.934183	+	0.266770i	−0.934183	+	0.266770i	−0.227076	+	0.973877i	$0.572917\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$887$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$888$		0			0
$889$	−0.561891	−	1.35653i	−0.561891	−	1.35653i
$890$		0			0
$891$		0			0
$892$	1.68387	−	1.04710i	1.68387	−	1.04710i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.0980171	+	0.995185i	−0.0980171	+	0.995185i
$901$		0			0
$902$		0			0
$903$	−0.263133	+	0.114072i	−0.263133	+	0.114072i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.38922	+	1.21831i	1.38922	+	1.21831i	0.946930	+	0.321439i	$0.104167\pi$
$907$	1.38922	+	1.21831i	1.38922	+	1.21831i	0.442289	+	0.896873i	$0.354167\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$911$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$912$	−1.24418	−	0.355294i	−1.24418	−	0.355294i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.475008	+	0.102592i	−0.475008	+	0.102592i
$917$		0			0
$918$		0			0
$919$	1.24441	−	1.51631i	1.24441	−	1.51631i	0.471397	−	0.881921i	$-0.343750\pi$
$919$	1.24441	−	1.51631i	1.24441	−	1.51631i	0.773010	−	0.634393i	$-0.218750\pi$
$920$		0			0
$921$	0.410356	−	1.09029i	0.410356	−	1.09029i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.72772	+	0.748995i	1.72772	+	0.748995i
$926$		0			0
$927$	0.116724	+	0.786887i	0.116724	+	0.786887i
$928$		0			0
$929$		0			0		−0.0817211	−	0.996655i	$-0.526042\pi$
$929$		0			0		0.0817211	+	0.996655i	$0.473958\pi$
$930$		0			0
$931$	0.594768	+	0.00973274i	0.594768	+	0.00973274i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.555570	+	1.83147i	0.555570	+	1.83147i	0.555570	+	0.831470i	$0.312500\pi$
$937$	0.555570	+	1.83147i	0.555570	+	1.83147i			1.00000i	$0.5\pi$
$938$		0			0
$939$	0.0999004	−	0.241181i	0.0999004	−	0.241181i
$940$		0			0
$941$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$941$		0			0		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		−0.621661	−	0.783287i	$-0.713542\pi$
$947$		0			0		0.621661	+	0.783287i	$0.286458\pi$
$948$	0.0308106	−	0.0110242i	0.0308106	−	0.0110242i
$949$	−0.103460	−	0.899335i	−0.103460	−	0.899335i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.412707	−	0.910864i	$-0.635417\pi$
$953$		0			0		0.412707	+	0.910864i	$0.364583\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2.19339	−	1.57176i	−2.19339	−	1.57176i
$962$		0			0
$963$		0			0
$964$	0.417653	+	0.0619530i	0.417653	+	0.0619530i
$965$		0			0
$966$		0			0
$967$	0.314531	+	0.978916i	0.314531	+	0.978916i	0.973877	+	0.227076i	$0.0729167\pi$
$967$	0.314531	+	0.978916i	0.314531	+	0.978916i	−0.659346	+	0.751840i	$0.729167\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.889516	−	0.456904i	$-0.848958\pi$
$971$		0			0		0.889516	+	0.456904i	$0.151042\pi$
$972$	−0.227076	+	0.973877i	−0.227076	+	0.973877i
$973$	−0.0939798	−	0.0202977i	−0.0939798	−	0.0202977i
$974$		0			0
$975$	−0.400527	+	0.214086i	−0.400527	+	0.214086i
$976$	−1.39923	+	0.606588i	−1.39923	+	0.606588i
$977$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$977$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.451966	−	1.93837i	−0.451966	−	1.93837i
$982$		0			0
$983$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$983$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.179759	−	0.559464i	−0.179759	−	0.559464i
$989$		0			0
$990$		0			0
$991$	−0.794045	+	0.696359i	−0.794045	+	0.696359i	−0.956940	−	0.290285i	$-0.906250\pi$
$991$	−0.794045	+	0.696359i	−0.794045	+	0.696359i	0.162895	+	0.986643i	$0.447917\pi$
$992$		0			0
$993$	0.147750	−	0.0698805i	0.147750	−	0.0698805i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.38922	+	0.575433i	−1.38922	+	0.575433i	−0.946930	−	0.321439i	$-0.895833\pi$
$997$	−1.38922	+	0.575433i	−1.38922	+	0.575433i	−0.442289	+	0.896873i	$0.645833\pi$
$998$		0			0
$999$	1.61518	+	0.968101i	1.61518	+	0.968101i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3459.1.bm.a.1406.1	yes	64
3.2	odd	2	CM	3459.1.bm.a.1406.1	yes	64
1153.278	even	192	inner	3459.1.bm.a.278.1	✓	64
3459.278	odd	192	inner	3459.1.bm.a.278.1	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3459.1.bm.a.278.1	✓	64	1153.278	even	192	inner
3459.1.bm.a.278.1	✓	64	3459.278	odd	192	inner
3459.1.bm.a.1406.1	yes	64	1.1	even	1	trivial
3459.1.bm.a.1406.1	yes	64	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 3459 = 3 \cdot 1153 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3459.bm (of order \(192\), degree \(64\), minimal)

\(f(q)\)	\(=\)	\(q+(0.528068 + 0.849202i) q^{3} +(-0.991445 + 0.130526i) q^{4} +(-0.713004 - 0.178598i) q^{7} +(-0.442289 + 0.896873i) q^{9} +O(q^{10})\)	\(q+(0.528068 + 0.849202i) q^{3} +(-0.991445 + 0.130526i) q^{4} +(-0.713004 - 0.178598i) q^{7} +(-0.442289 + 0.896873i) q^{9} +(-0.634393 - 0.773010i) q^{12} +(0.131834 - 0.434597i) q^{13} +(0.965926 - 0.258819i) q^{16} +(-1.15096 + 0.591194i) q^{19} +(-0.224848 - 0.699796i) q^{21} +(-0.973877 + 0.227076i) q^{25} +(-0.995185 + 0.0980171i) q^{27} +(0.730216 + 0.0840046i) q^{28} +(-0.588289 + 1.83093i) q^{31} +(0.321439 - 0.946930i) q^{36} +(-1.51251 - 1.12175i) q^{37} +(0.438678 - 0.117543i) q^{39} +(-0.0382444 - 0.388302i) q^{43} +(0.729864 + 0.683592i) q^{48} +(-0.405444 - 0.216714i) q^{49} +(-0.0739794 + 0.448087i) q^{52} +(-1.10983 - 0.665204i) q^{57} +(-1.50855 + 0.223772i) q^{61} +(0.475533 - 0.560482i) q^{63} +(-0.923880 + 0.382683i) q^{64} +(-0.292893 - 0.707107i) q^{67} +(1.82885 - 0.792836i) q^{73} +(-0.707107 - 0.707107i) q^{75} +(1.06394 - 0.736366i) q^{76} +(-0.0110242 + 0.0308106i) q^{79} +(-0.608761 - 0.793353i) q^{81} +(0.314267 + 0.664461i) q^{84} +(-0.171616 + 0.286324i) q^{91} +(-1.86549 + 0.467281i) q^{93} +(-0.973877 - 1.68680i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 64 * q	\( 64 q - 64 q^{67}+O(q^{100}) \) 64 * q - 64 * q^67

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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