LMFDB - Embedded newform 3459.1.bm.a.176.1

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			176.1
Root			$-0.582478 - 0.812847i$ of defining polynomial
Character	$\chi$	$=$	3459.176
Dual form			3459.1.bm.a.1985.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.999465 + 0.0327191i) q^{3} +(0.608761 + 0.793353i) q^{4} +(-1.20606 - 0.302102i) q^{7} +(0.997859 - 0.0654031i) q^{9} +O(q^{10})$	\(q+(-0.999465 + 0.0327191i) q^{3} +(0.608761 + 0.793353i) q^{4} +(-1.20606 - 0.302102i) q^{7} +(0.997859 - 0.0654031i) q^{9} +(-0.634393 - 0.773010i) q^{12} +(0.423737 - 1.39687i) q^{13} +(-0.258819 + 0.965926i) q^{16} +(-1.65374 - 1.06628i) q^{19} +(1.21530 + 0.262479i) q^{21} +(0.683592 + 0.729864i) q^{25} +(-0.995185 + 0.0980171i) q^{27} +(-0.494529 - 1.14074i) q^{28} +(-1.40477 + 0.303402i) q^{31} +(0.659346 + 0.751840i) q^{36} +(-1.51251 - 1.12175i) q^{37} +(-0.377805 + 1.40999i) q^{39} +(-0.0382444 - 0.388302i) q^{43} +(0.227076 - 0.973877i) q^{48} +(0.481394 + 0.257311i) q^{49} +(1.36617 - 0.514189i) q^{52} +(1.68775 + 1.01160i) q^{57} +(-0.354159 + 0.0525345i) q^{61} +(-1.22324 - 0.222575i) q^{63} +(-0.923880 + 0.382683i) q^{64} +(-0.292893 - 0.707107i) q^{67} +(0.130081 - 1.13074i) q^{73} +(-0.707107 - 0.707107i) q^{75} +(-0.160802 - 1.96111i) q^{76} +(0.588944 - 1.64599i) q^{79} +(0.991445 - 0.130526i) q^{81} +(0.531588 + 1.12395i) q^{84} +(-0.933051 + 1.55670i) q^{91} +(1.39409 - 0.349202i) q^{93} +(0.683592 - 1.18402i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3459\mathbb{Z}\right)^\times$.

$n$	$1154$	$2311$
$\chi(n)$	$-1$	$e\left(\frac{25}{192}\right)$

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		−0.896873	−	0.442289i	$-0.854167\pi$
$2$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$3$	−0.999465	+	0.0327191i	−0.999465	+	0.0327191i
$3$	−0.999465	+	0.0327191i	−0.999465	+	0.0327191i
$4$	0.608761	+	0.793353i	0.608761	+	0.793353i
$5$		0			0		−0.917494	−	0.397748i	$-0.869792\pi$
$5$		0			0		0.917494	+	0.397748i	$0.130208\pi$
$6$		0			0
$7$	−1.20606	−	0.302102i	−1.20606	−	0.302102i	−0.412707	−	0.910864i	$-0.635417\pi$
$7$	−1.20606	−	0.302102i	−1.20606	−	0.302102i	−0.793353	+	0.608761i	$0.791667\pi$
$8$		0			0
$9$	0.997859	−	0.0654031i	0.997859	−	0.0654031i
$10$		0			0
$11$		0			0		0.986643	−	0.162895i	$-0.0520833\pi$
$11$		0			0		−0.986643	+	0.162895i	$0.947917\pi$
$12$	−0.634393	−	0.773010i	−0.634393	−	0.773010i
$13$	0.423737	−	1.39687i	0.423737	−	1.39687i	−0.442289	−	0.896873i	$-0.645833\pi$
$13$	0.423737	−	1.39687i	0.423737	−	1.39687i	0.866025	−	0.500000i	$-0.166667\pi$
$14$		0			0
$15$		0			0
$16$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$17$		0			0		0.889516	−	0.456904i	$-0.151042\pi$
$17$		0			0		−0.889516	+	0.456904i	$0.848958\pi$
$18$		0			0
$19$	−1.65374	−	1.06628i	−1.65374	−	1.06628i	−0.923880	−	0.382683i	$-0.875000\pi$
$19$	−1.65374	−	1.06628i	−1.65374	−	1.06628i	−0.729864	−	0.683592i	$-0.760417\pi$
$20$		0			0
$21$	1.21530	+	0.262479i	1.21530	+	0.262479i
$22$		0			0
$23$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$23$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$24$		0			0
$25$	0.683592	+	0.729864i	0.683592	+	0.729864i
$26$		0			0
$27$	−0.995185	+	0.0980171i	−0.995185	+	0.0980171i
$28$	−0.494529	−	1.14074i	−0.494529	−	1.14074i
$29$		0			0		0.0327191	−	0.999465i	$-0.489583\pi$
$29$		0			0		−0.0327191	+	0.999465i	$0.510417\pi$
$30$		0			0
$31$	−1.40477	+	0.303402i	−1.40477	+	0.303402i	−0.849202	−	0.528068i	$-0.822917\pi$
$31$	−1.40477	+	0.303402i	−1.40477	+	0.303402i	−0.555570	+	0.831470i	$0.687500\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.659346	+	0.751840i	0.659346	+	0.751840i
$37$	−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
$39$	−0.377805	+	1.40999i	−0.377805	+	1.40999i
$40$		0			0
$41$		0			0		0.729864	−	0.683592i	$-0.239583\pi$
$41$		0			0		−0.729864	+	0.683592i	$0.760417\pi$
$42$		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	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$	0.227076	−	0.973877i	0.227076	−	0.973877i
$49$	0.481394	+	0.257311i	0.481394	+	0.257311i
$50$		0			0
$51$		0			0
$52$	1.36617	−	0.514189i	1.36617	−	0.514189i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.68775	+	1.01160i	1.68775	+	1.01160i
$58$		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
$61$	−0.354159	+	0.0525345i	−0.354159	+	0.0525345i	−0.321439	−	0.946930i	$-0.604167\pi$
$61$	−0.354159	+	0.0525345i	−0.354159	+	0.0525345i	−0.0327191	+	0.999465i	$0.510417\pi$
$62$		0			0
$63$	−1.22324	−	0.222575i	−1.22324	−	0.222575i
$64$	−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
$73$	0.130081	−	1.13074i	0.130081	−	1.13074i	−0.751840	−	0.659346i	$-0.770833\pi$
$73$	0.130081	−	1.13074i	0.130081	−	1.13074i	0.881921	−	0.471397i	$-0.156250\pi$
$74$		0			0
$75$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$76$	−0.160802	−	1.96111i	−0.160802	−	1.96111i
$77$		0			0
$78$		0			0
$79$	0.588944	−	1.64599i	0.588944	−	1.64599i	−0.162895	−	0.986643i	$-0.552083\pi$
$79$	0.588944	−	1.64599i	0.588944	−	1.64599i	0.751840	−	0.659346i	$-0.229167\pi$
$80$		0			0
$81$	0.991445	−	0.130526i	0.991445	−	0.130526i
$82$		0			0
$83$		0			0		0.718582	−	0.695443i	$-0.244792\pi$
$83$		0			0		−0.718582	+	0.695443i	$0.755208\pi$
$84$	0.531588	+	1.12395i	0.531588	+	1.12395i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.582478	−	0.812847i	$-0.302083\pi$
$89$		0			0		−0.582478	+	0.812847i	$0.697917\pi$
$90$		0			0
$91$	−0.933051	+	1.55670i	−0.933051	+	1.55670i
$92$		0			0
$93$	1.39409	−	0.349202i	1.39409	−	0.349202i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.683592	−	1.18402i	0.683592	−	1.18402i	−0.290285	−	0.956940i	$-0.593750\pi$
$97$	0.683592	−	1.18402i	0.683592	−	1.18402i	0.973877	−	0.227076i	$-0.0729167\pi$
$98$		0			0
$99$		0			0
$100$	−0.162895	+	0.986643i	−0.162895	+	0.986643i
$101$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$101$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$102$		0			0
$103$	−0.162371	−	1.98025i	−0.162371	−	1.98025i	−0.195090	−	0.980785i	$-0.562500\pi$
$103$	−0.162371	−	1.98025i	−0.162371	−	1.98025i	0.0327191	−	0.999465i	$-0.489583\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$107$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$108$	−0.683592	−	0.729864i	−0.683592	−	0.729864i
$109$	−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
$111$	1.54840	+	1.07167i	1.54840	+	1.07167i
$112$	0.603960	−	1.08678i	0.603960	−	1.08678i
$113$		0			0		−0.569100	−	0.822268i	$-0.692708\pi$
$113$		0			0		0.569100	+	0.822268i	$0.307292\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.331470	−	1.42160i	0.331470	−	1.42160i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.946930	−	0.321439i	0.946930	−	0.321439i
$122$		0			0
$123$		0			0
$124$	−1.09588	−	0.929782i	−1.09588	−	0.929782i
$125$		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
$129$	0.0509288	+	0.386843i	0.0509288	+	0.386843i
$130$		0			0
$131$		0			0		0.729864	−	0.683592i	$-0.239583\pi$
$131$		0			0		−0.729864	+	0.683592i	$0.760417\pi$
$132$		0			0
$133$	1.67239	+	1.78559i	1.67239	+	1.78559i
$134$		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
$139$	0.947219	+	1.52325i	0.947219	+	1.52325i	0.849202	+	0.528068i	$0.177083\pi$
$139$	0.947219	+	1.52325i	0.947219	+	1.52325i	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
$147$	−0.489556	−	0.241422i	−0.489556	−	0.241422i
$148$	−0.0308106	−	1.88284i	−0.0308106	−	1.88284i
$149$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$149$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$150$		0			0
$151$	0.669796	−	1.41617i	0.669796	−	1.41617i	−0.227076	−	0.973877i	$-0.572917\pi$
$151$	0.669796	−	1.41617i	0.669796	−	1.41617i	0.896873	−	0.442289i	$-0.145833\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.34861	+	0.558614i	−1.34861	+	0.558614i
$157$	−1.70422	−	0.196054i	−1.70422	−	0.196054i	−0.793353	−	0.608761i	$-0.791667\pi$
$157$	−1.70422	−	0.196054i	−1.70422	−	0.196054i	−0.910864	+	0.412707i	$0.864583\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.441103	+	0.327144i	0.441103	+	0.327144i	0.793353	−	0.608761i	$-0.208333\pi$
$163$	0.441103	+	0.327144i	0.441103	+	0.327144i	−0.352250	+	0.935906i	$0.614583\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.0163617	−	0.999866i	$-0.494792\pi$
$167$		0			0		−0.0163617	+	0.999866i	$0.505208\pi$
$168$		0			0
$169$	−0.940231	−	0.628242i	−0.940231	−	0.628242i
$170$		0			0
$171$	−1.71994	−	0.955833i	−1.71994	−	0.955833i
$172$	0.284779	−	0.266724i	0.284779	−	0.266724i
$173$		0			0		0.242980	−	0.970031i	$-0.421875\pi$
$173$		0			0		−0.242980	+	0.970031i	$0.578125\pi$
$174$		0			0
$175$	−0.603960	−	1.08678i	−0.603960	−	1.08678i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.0817211	−	0.996655i	$-0.526042\pi$
$179$		0			0		0.0817211	+	0.996655i	$0.473958\pi$
$180$		0			0
$181$	−0.810982	−	0.759567i	−0.810982	−	0.759567i	0.162895	−	0.986643i	$-0.447917\pi$
$181$	−0.810982	−	0.759567i	−0.810982	−	0.759567i	−0.973877	+	0.227076i	$0.927083\pi$
$182$		0			0
$183$	0.352250	−	0.0640941i	0.352250	−	0.0640941i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.22986	+	0.182433i	1.22986	+	0.182433i
$190$		0			0
$191$		0			0		0.983846	−	0.179017i	$-0.0572917\pi$
$191$		0			0		−0.983846	+	0.179017i	$0.942708\pi$
$192$	0.910864	−	0.412707i	0.910864	−	0.412707i
$193$	−0.228543	+	0.00373987i	−0.228543	+	0.00373987i	−0.130526	−	0.991445i	$-0.541667\pi$
$193$	−0.228543	+	0.00373987i	−0.228543	+	0.00373987i	−0.0980171	+	0.995185i	$0.531250\pi$
$194$		0			0
$195$		0			0
$196$	0.0889161	+	0.538557i	0.0889161	+	0.538557i
$197$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$197$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$198$		0			0
$199$	−0.198092	+	0.478235i	−0.198092	+	0.478235i	−0.991445	−	0.130526i	$-0.958333\pi$
$199$	−0.198092	+	0.478235i	−0.198092	+	0.478235i	0.793353	+	0.608761i	$0.208333\pi$
$200$		0			0
$201$	0.315872	+	0.697145i	0.315872	+	0.697145i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.23960	+	0.770836i	1.23960	+	0.770836i
$209$		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
$217$	1.78590	+	0.0584643i	1.78590	+	0.0584643i
$218$		0			0
$219$	−0.0930150	+	1.13439i	−0.0930150	+	1.13439i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.989659	−	0.709180i	−0.989659	−	0.709180i	−0.0327191	−	0.999465i	$-0.510417\pi$
$223$	−0.989659	−	0.709180i	−0.989659	−	0.709180i	−0.956940	+	0.290285i	$0.906250\pi$
$224$		0			0
$225$	0.729864	+	0.683592i	0.729864	+	0.683592i
$226$		0			0
$227$		0			0		0.999866	−	0.0163617i	$-0.00520833\pi$
$227$		0			0		−0.999866	+	0.0163617i	$0.994792\pi$
$228$	0.224882	+	1.95480i	0.224882	+	1.95480i
$229$	−0.276560	−	0.399590i	−0.276560	−	0.399590i	0.659346	−	0.751840i	$-0.270833\pi$
$229$	−0.276560	−	0.399590i	−0.276560	−	0.399590i	−0.935906	+	0.352250i	$0.885417\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.889516	−	0.456904i	$-0.848958\pi$
$233$		0			0		0.889516	+	0.456904i	$0.151042\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.534774	+	1.66438i	−0.534774	+	1.66438i
$238$		0			0
$239$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$239$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$240$		0			0
$241$	−1.36827	+	1.32421i	−1.36827	+	1.32421i	−0.471397	+	0.881921i	$0.656250\pi$
$241$	−1.36827	+	1.32421i	−1.36827	+	1.32421i	−0.896873	+	0.442289i	$0.854167\pi$
$242$		0			0
$243$	−0.986643	+	0.162895i	−0.986643	+	0.162895i
$244$	−0.257276	−	0.248992i	−0.257276	−	0.248992i
$245$		0			0
$246$		0			0
$247$	−2.19020	+	1.85825i	−2.19020	+	1.85825i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$251$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$252$	−0.568078	−	1.10595i	−0.568078	−	1.10595i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$257$		0			0		−0.952063	−	0.305903i	$-0.901042\pi$
$257$		0			0		0.952063	+	0.305903i	$0.0989583\pi$
$258$		0			0
$259$	1.48529	+	1.80984i	1.48529	+	1.80984i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.996655	−	0.0817211i	$-0.0260417\pi$
$263$		0			0		−0.996655	+	0.0817211i	$0.973958\pi$
$264$		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
$273$	0.881617	−	1.58640i	0.881617	−	1.58640i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.56326	+	1.00794i	−1.56326	+	1.00794i	−0.582478	+	0.812847i	$0.697917\pi$
$277$	−1.56326	+	1.00794i	−1.56326	+	1.00794i	−0.980785	+	0.195090i	$0.937500\pi$
$278$		0			0
$279$	−1.38192	+	0.394629i	−1.38192	+	0.394629i
$280$		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
$283$	1.12999	−	0.958726i	1.12999	−	0.958726i	0.130526	−	0.991445i	$-0.458333\pi$
$283$	1.12999	−	0.958726i	1.12999	−	0.958726i	0.999465	+	0.0327191i	$0.0104167\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.582478	−	0.812847i	0.582478	−	0.812847i
$290$		0			0
$291$	−0.644486	+	1.20575i	−0.644486	+	1.20575i
$292$	0.976267	−	0.585152i	0.976267	−	0.585152i
$293$		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
$300$	0.130526	−	0.991445i	0.130526	−	0.991445i
$301$	−0.0711819	+	0.479869i	−0.0711819	+	0.479869i
$302$		0			0
$303$		0			0
$304$	1.45796	−	1.32142i	1.45796	−	1.32142i
$305$		0			0
$306$		0			0
$307$	0.818353	−	0.107738i	0.818353	−	0.107738i	0.290285	−	0.956940i	$-0.406250\pi$
$307$	0.818353	−	0.107738i	0.818353	−	0.107738i	0.528068	+	0.849202i	$0.322917\pi$
$308$		0			0
$309$	0.227076	+	1.97388i	0.227076	+	1.97388i
$310$		0			0
$311$		0			0		−0.367516	−	0.930017i	$-0.619792\pi$
$311$		0			0		0.367516	+	0.930017i	$0.380208\pi$
$312$		0			0
$313$	−0.654845	+	1.44527i	−0.654845	+	1.44527i	0.227076	+	0.973877i	$0.427083\pi$
$313$	−0.654845	+	1.44527i	−0.654845	+	1.44527i	−0.881921	+	0.471397i	$0.843750\pi$
$314$		0			0
$315$		0			0
$316$	1.66438	−	0.534774i	1.66438	−	0.534774i
$317$		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
$325$	1.30919	−	0.645621i	1.30919	−	0.645621i
$326$		0			0
$327$	1.49644	−	1.31234i	1.49644	−	1.31234i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.50885	+	0.654112i	−1.50885	+	0.654112i	−0.980785	−	0.195090i	$-0.937500\pi$
$331$	−1.50885	+	0.654112i	−1.50885	+	0.654112i	−0.528068	+	0.849202i	$0.677083\pi$
$332$		0			0
$333$	−1.58264	−	1.02043i	−1.58264	−	1.02043i
$334$		0			0
$335$		0			0
$336$	−0.568078	+	1.10595i	−0.568078	+	1.10595i
$337$	−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
$343$	0.418384	+	0.379201i	0.418384	+	0.379201i
$344$		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
$349$	0.416987	−	0.780128i	0.416987	−	0.780128i	−0.582478	−	0.812847i	$-0.697917\pi$
$349$	0.416987	−	0.780128i	0.416987	−	0.780128i	0.999465	+	0.0327191i	$0.0104167\pi$
$350$		0			0
$351$	−0.284779	+	1.43168i	−0.284779	+	1.43168i
$352$		0			0
$353$		0			0		0.999465	−	0.0327191i	$-0.0104167\pi$
$353$		0			0		−0.999465	+	0.0327191i	$0.989583\pi$
$354$		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
$361$	1.18522	+	2.61583i	1.18522	+	2.61583i
$362$		0			0
$363$	−0.935906	+	0.352250i	−0.935906	+	0.352250i
$364$	−1.80302	+	0.207421i	−1.80302	+	0.207421i
$365$		0			0
$366$		0			0
$367$	−1.85747	+	0.630526i	−1.85747	+	0.630526i	−0.866025	+	0.500000i	$0.833333\pi$
$367$	−1.85747	+	0.630526i	−1.85747	+	0.630526i	−0.991445	+	0.130526i	$0.958333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.12571	+	0.893428i	1.12571	+	0.893428i
$373$	0.795390	+	0.0130157i	0.795390	+	0.0130157i	0.412707	−	0.910864i	$-0.364583\pi$
$373$	0.795390	+	0.0130157i	0.795390	+	0.0130157i	0.382683	+	0.923880i	$0.375000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.0757795	−	0.249812i	0.0757795	−	0.249812i	−0.910864	−	0.412707i	$-0.864583\pi$
$379$	0.0757795	−	0.249812i	0.0757795	−	0.249812i	0.986643	+	0.162895i	$0.0520833\pi$
$380$		0			0
$381$	−1.24182	−	1.56469i	−1.24182	−	1.56469i
$382$		0			0
$383$		0			0		−0.582478	−	0.812847i	$-0.697917\pi$
$383$		0			0		0.582478	+	0.812847i	$0.302083\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.0635587	−	0.384969i	−0.0635587	−	0.384969i
$388$	1.35549	−	0.178453i	1.35549	−	0.178453i
$389$		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
$397$	0.297193	+	0.534774i	0.297193	+	0.534774i	0.980785	−	0.195090i	$-0.0625000\pi$
$397$	0.297193	+	0.534774i	0.297193	+	0.534774i	−0.683592	+	0.729864i	$0.739583\pi$
$398$		0			0
$399$	−1.72992	−	1.72992i	−1.72992	−	1.72992i
$400$	−0.881921	+	0.471397i	−0.881921	+	0.471397i
$401$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$401$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$402$		0			0
$403$	−0.171440	+	2.09085i	−0.171440	+	2.09085i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.0139911	+	0.0295817i	0.0139911	+	0.0295817i	0.910864	−	0.412707i	$-0.135417\pi$
$409$	0.0139911	+	0.0295817i	0.0139911	+	0.0295817i	−0.896873	+	0.442289i	$0.854167\pi$
$410$		0			0
$411$		0			0
$412$	1.47219	−	1.33432i	1.47219	−	1.33432i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.996552	−	1.49144i	−0.996552	−	1.49144i
$418$		0			0
$419$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$419$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$420$		0			0
$421$	0.312847	+	0.283548i	0.312847	+	0.283548i	0.812847	−	0.582478i	$-0.197917\pi$
$421$	0.312847	+	0.283548i	0.312847	+	0.283548i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.443007	+	0.0436324i	0.443007	+	0.0436324i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.0327191	−	0.999465i	$-0.489583\pi$
$431$		0			0		−0.0327191	+	0.999465i	$0.510417\pi$
$432$	0.162895	−	0.986643i	0.162895	−	0.986643i
$433$	−0.486643	+	0.703130i	−0.486643	+	0.703130i	−0.986643	−	0.162895i	$-0.947917\pi$
$433$	−0.486643	+	0.703130i	−0.486643	+	0.703130i	0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$	−1.93837	−	0.451966i	−1.93837	−	0.451966i
$437$		0			0
$438$		0			0
$439$	1.91033	+	0.379988i	1.91033	+	0.379988i	0.999465	−	0.0327191i	$-0.0104167\pi$
$439$	1.91033	+	0.379988i	1.91033	+	0.379988i	0.910864	+	0.412707i	$0.135417\pi$
$440$		0			0
$441$	0.497193	+	0.225275i	0.497193	+	0.225275i
$442$		0			0
$443$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$443$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$444$	0.0923988	+	1.88082i	0.0923988	+	1.88082i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.22986	−	0.182433i	1.22986	−	0.182433i
$449$		0			0		0.849202	−	0.528068i	$-0.177083\pi$
$449$		0			0		−0.849202	+	0.528068i	$0.822917\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.623102	+	1.43732i	−0.623102	+	1.43732i
$454$		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		0.917494	−	0.397748i	$-0.130208\pi$
$461$		0			0		−0.917494	+	0.397748i	$0.869792\pi$
$462$		0			0
$463$	1.23721	−	0.635501i	1.23721	−	0.635501i	0.290285	−	0.956940i	$-0.406250\pi$
$463$	1.23721	−	0.635501i	1.23721	−	0.635501i	0.946930	+	0.321439i	$0.104167\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.367516	−	0.930017i	$-0.619792\pi$
$467$		0			0		0.367516	+	0.930017i	$0.380208\pi$
$468$	1.32961	−	0.602440i	1.32961	−	0.602440i
$469$	0.139628	+	0.941297i	0.139628	+	0.941297i
$470$		0			0
$471$	1.70972	+	0.140189i	1.70972	+	0.140189i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.352250	−	1.93591i	−0.352250	−	1.93591i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.211112	−	0.977462i	$-0.432292\pi$
$479$		0			0		−0.211112	+	0.977462i	$0.567708\pi$
$480$		0			0
$481$	−2.20785	+	1.63746i	−2.20785	+	1.63746i
$482$		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
$489$	−0.451571	−	0.312537i	−0.451571	−	0.312537i
$490$		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
$496$	0.0705183	−	1.43543i	0.0705183	−	1.43543i
$497$		0			0
$498$		0			0
$499$	0.0632084	−	0.0169366i	0.0632084	−	0.0169366i	−0.227076	−	0.973877i	$-0.572917\pi$
$499$	0.0632084	−	0.0169366i	0.0632084	−	0.0169366i	0.290285	+	0.956940i	$0.406250\pi$
$500$		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
$507$	0.960283	+	0.597142i	0.960283	+	0.597142i
$508$	−0.548516	+	1.92081i	−0.548516	+	1.92081i
$509$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$509$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$510$		0			0
$511$	−0.498486	+	1.32445i	−0.498486	+	1.32445i
$512$		0			0
$513$	1.75029	+	0.899046i	1.75029	+	0.899046i
$514$		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
$523$	0.986643	+	1.70892i	0.986643	+	1.70892i	0.634393	+	0.773010i	$0.281250\pi$
$523$	0.986643	+	1.70892i	0.986643	+	1.70892i	0.352250	+	0.935906i	$0.385417\pi$
$524$		0			0
$525$	0.639195	+	1.06643i	0.639195	+	1.06643i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$530$		0			0
$531$		0			0
$532$	−0.398519	+	2.41380i	−0.398519	+	2.41380i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.49399	−	0.590383i	−1.49399	−	0.590383i	−0.528068	−	0.849202i	$-0.677083\pi$
$541$	−1.49399	−	0.590383i	−1.49399	−	0.590383i	−0.965926	+	0.258819i	$0.916667\pi$
$542$		0			0
$543$	0.835400	+	0.732626i	0.835400	+	0.732626i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.598017	−	1.86121i	−0.598017	−	1.86121i	−0.500000	−	0.866025i	$-0.666667\pi$
$547$	−0.598017	−	1.86121i	−0.598017	−	1.86121i	−0.0980171	−	0.995185i	$-0.531250\pi$
$548$		0			0
$549$	−0.349964	+	0.0755851i	−0.349964	+	0.0755851i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.20756	+	1.80724i	−1.20756	+	1.80724i
$554$		0			0
$555$		0			0
$556$	−0.631847	+	1.67878i	−0.631847	+	1.67878i
$557$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$557$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$558$		0			0
$559$	−0.558614	−	0.111115i	−0.558614	−	0.111115i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.849202	−	0.528068i	$-0.177083\pi$
$563$		0			0		−0.849202	+	0.528068i	$0.822917\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.23517	−	0.142095i	−1.23517	−	0.142095i
$568$		0			0
$569$		0			0		0.996655	−	0.0817211i	$-0.0260417\pi$
$569$		0			0		−0.996655	+	0.0817211i	$0.973958\pi$
$570$		0			0
$571$	1.57030	−	1.12526i	1.57030	−	1.12526i	0.634393	−	0.773010i	$-0.281250\pi$
$571$	1.57030	−	1.12526i	1.57030	−	1.12526i	0.935906	−	0.352250i	$-0.114583\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.896873	+	0.442289i	−0.896873	+	0.442289i
$577$	1.71651	−	0.846488i	1.71651	−	0.846488i	0.729864	−	0.683592i	$-0.239583\pi$
$577$	1.71651	−	0.846488i	1.71651	−	0.846488i	0.986643	−	0.162895i	$-0.0520833\pi$
$578$		0			0
$579$	0.228299	−	0.0112156i	0.228299	−	0.0112156i
$580$		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	$0.921875\pi$
$588$	−0.106490	−	0.535359i	−0.106490	−	0.535359i
$589$	2.64664	+	0.996126i	2.64664	+	0.996126i
$590$		0			0
$591$		0			0
$592$	1.47500	−	1.17064i	1.47500	−	1.17064i
$593$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$593$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.182338	−	0.484461i	0.182338	−	0.484461i
$598$		0			0
$599$		0			0		0.993448	−	0.114287i	$-0.0364583\pi$
$599$		0			0		−0.993448	+	0.114287i	$0.963542\pi$
$600$		0			0
$601$	0.325617	+	0.0106596i	0.325617	+	0.0106596i	0.195090	−	0.980785i	$-0.437500\pi$
$601$	0.325617	+	0.0106596i	0.325617	+	0.0106596i	0.130526	+	0.991445i	$0.458333\pi$
$602$		0			0
$603$	−0.338513	−	0.686437i	−0.338513	−	0.686437i
$604$	1.53127	−	0.330722i	1.53127	−	0.330722i
$605$		0			0
$606$		0			0
$607$	−0.604393	+	0.418307i	−0.604393	+	0.418307i	−0.831470	−	0.555570i	$-0.812500\pi$
$607$	−0.604393	+	0.418307i	−0.604393	+	0.418307i	0.227076	+	0.973877i	$0.427083\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.513775	+	1.43591i	0.513775	+	1.43591i	0.866025	+	0.500000i	$0.166667\pi$
$613$	0.513775	+	1.43591i	0.513775	+	1.43591i	−0.352250	+	0.935906i	$0.614583\pi$
$614$		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
$619$	1.40789	−	0.812847i	1.40789	−	0.812847i	0.412707	−	0.910864i	$-0.364583\pi$
$619$	1.40789	−	0.812847i	1.40789	−	0.812847i	0.995185	+	0.0980171i	$0.0312500\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.26416	−	0.729864i	−1.26416	−	0.729864i
$625$	−0.0654031	+	0.997859i	−0.0654031	+	0.997859i
$626$		0			0
$627$		0			0
$628$	−0.881921	−	1.47140i	−0.881921	−	1.47140i
$629$		0			0
$630$		0			0
$631$	−0.824972	+	1.82075i	−0.824972	+	1.82075i	−0.382683	+	0.923880i	$0.625000\pi$
$631$	−0.824972	+	1.82075i	−0.824972	+	1.82075i	−0.442289	+	0.896873i	$0.645833\pi$
$632$		0			0
$633$	0.588944	+	0.327298i	0.588944	+	0.327298i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.563415	−	0.563415i	0.563415	−	0.563415i
$638$		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
$643$	−1.70497	−	0.641704i	−1.70497	−	0.641704i	−0.707107	−	0.707107i	$-0.750000\pi$
$643$	−1.70497	−	0.641704i	−1.70497	−	0.641704i	−0.997859	+	0.0654031i	$0.979167\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.528068	−	0.849202i	$-0.322917\pi$
$647$		0			0		−0.528068	+	0.849202i	$0.677083\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.78686			−1.78686
$652$	0.00898549	+	0.549104i	0.00898549	+	0.549104i
$653$		0			0		0.541892	−	0.840448i	$-0.317708\pi$
$653$		0			0		−0.541892	+	0.840448i	$0.682292\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.0558488	−	1.13683i	0.0558488	−	1.13683i
$658$		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
$661$	−0.575603	−	0.0757795i	−0.575603	−	0.0757795i	−0.162895	−	0.986643i	$-0.552083\pi$
$661$	−0.575603	−	0.0757795i	−0.575603	−	0.0757795i	−0.412707	+	0.910864i	$0.635417\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.01233	+	0.676419i	1.01233	+	0.676419i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.35656	−	0.696803i	1.35656	−	0.696803i	0.382683	−	0.923880i	$-0.375000\pi$
$673$	1.35656	−	0.696803i	1.35656	−	0.696803i	0.973877	+	0.227076i	$0.0729167\pi$
$674$		0			0
$675$	−0.751840	−	0.659346i	−0.751840	−	0.659346i
$676$	−0.0739583	−	1.12839i	−0.0739583	−	1.12839i
$677$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$677$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$678$		0			0
$679$	−1.18215	+	1.22148i	−1.18215	+	1.22148i
$680$		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	$0.109375\pi$
$684$	−0.288720	−	1.94639i	−0.288720	−	1.94639i
$685$		0			0
$686$		0			0
$687$	0.289486	+	0.390327i	0.289486	+	0.390327i
$688$	0.384969	+	0.0635587i	0.384969	+	0.0635587i
$689$		0			0
$690$		0			0
$691$	0.798751	+	0.443894i	0.798751	+	0.443894i	0.831470	−	0.555570i	$-0.187500\pi$
$691$	0.798751	+	0.443894i	0.798751	+	0.443894i	−0.0327191	+	0.999465i	$0.510417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.494529	−	1.14074i	0.494529	−	1.14074i
$701$		0			0		0.305903	−	0.952063i	$-0.401042\pi$
$701$		0			0		−0.305903	+	0.952063i	$0.598958\pi$
$702$		0			0
$703$	1.30521	+	3.46785i	1.30521	+	3.46785i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.0943632	+	1.92081i	0.0943632	+	1.92081i	0.321439	+	0.946930i	$0.395833\pi$
$709$	0.0943632	+	1.92081i	0.0943632	+	1.92081i	−0.227076	+	0.973877i	$0.572917\pi$
$710$		0			0
$711$	0.480031	−	1.68098i	0.480031	−	1.68098i
$712$		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
$721$	−0.402409	+	2.43735i	−0.402409	+	2.43735i
$722$		0			0
$723$	1.32421	−	1.36827i	1.32421	−	1.36827i
$724$	0.108911	−	1.10579i	0.108911	−	1.10579i
$725$		0			0
$726$		0			0
$727$	0.314531	+	0.978916i	0.314531	+	0.978916i	0.973877	+	0.227076i	$0.0729167\pi$
$727$	0.314531	+	0.978916i	0.314531	+	0.978916i	−0.659346	+	0.751840i	$0.729167\pi$
$728$		0			0
$729$	0.980785	−	0.195090i	0.980785	−	0.195090i
$730$		0			0
$731$		0			0
$732$	0.265285	+	0.240441i	0.265285	+	0.240441i
$733$	−0.0632084	−	1.93082i	−0.0632084	−	1.93082i	−0.290285	−	0.956940i	$-0.593750\pi$
$733$	−0.0632084	−	1.93082i	−0.0632084	−	1.93082i	0.227076	−	0.973877i	$-0.427083\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.860919	−	1.12197i	0.860919	−	1.12197i	−0.130526	−	0.991445i	$-0.541667\pi$
$739$	0.860919	−	1.12197i	0.860919	−	1.12197i	0.991445	−	0.130526i	$-0.0416667\pi$
$740$		0			0
$741$	2.12823	−	1.92892i	2.12823	−	1.92892i
$742$		0			0
$743$		0			0		−0.993448	−	0.114287i	$-0.963542\pi$
$743$		0			0		0.993448	+	0.114287i	$0.0364583\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.0230506	+	0.704123i	−0.0230506	+	0.704123i	0.923880	+	0.382683i	$0.125000\pi$
$751$	−0.0230506	+	0.704123i	−0.0230506	+	0.704123i	−0.946930	+	0.321439i	$0.895833\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.603960	+	1.08678i	0.603960	+	1.08678i
$757$	−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
$763$	2.23707	−	1.05806i	2.23707	−	1.05806i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.881921	+	0.471397i	0.881921	+	0.471397i
$769$	1.89285	+	0.0619654i	1.89285	+	0.0619654i	0.956940	−	0.290285i	$-0.0937500\pi$
$769$	1.89285	+	0.0619654i	1.89285	+	0.0619654i	0.935906	+	0.352250i	$0.114583\pi$
$770$		0			0
$771$		0			0
$772$	−0.142095	−	0.179039i	−0.142095	−	0.179039i
$773$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$773$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$774$		0			0
$775$	−1.18173	−	0.817890i	−1.18173	−	0.817890i
$776$		0			0
$777$	−1.54372	−	1.76027i	−1.54372	−	1.76027i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.373137	+	0.398394i	−0.373137	+	0.398394i
$785$		0			0
$786$		0			0
$787$	−1.91669	−	0.547340i	−1.91669	−	0.547340i	−0.980785	−	0.195090i	$-0.937500\pi$
$787$	−1.91669	−	0.547340i	−1.91669	−	0.547340i	−0.935906	−	0.352250i	$-0.885417\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.0766860	+	0.516975i	−0.0766860	+	0.516975i
$794$		0			0
$795$		0			0
$796$	−0.500000	+	0.133975i	−0.500000	+	0.133975i
$797$		0			0		0.999465	−	0.0327191i	$-0.0104167\pi$
$797$		0			0		−0.999465	+	0.0327191i	$0.989583\pi$
$798$		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
$811$	0.552896	+	0.798856i	0.552896	+	0.798856i	0.995185	−	0.0980171i	$-0.0312500\pi$
$811$	0.552896	+	0.798856i	0.552896	+	0.798856i	−0.442289	+	0.896873i	$0.645833\pi$
$812$		0			0
$813$	−1.01936	−	1.16236i	−1.01936	−	1.16236i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.350790	+	0.682931i	−0.350790	+	0.682931i
$818$		0			0
$819$	−0.829240	+	1.61439i	−0.829240	+	1.61439i
$820$		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
$823$	−1.43732	−	1.14074i	−1.43732	−	1.14074i	−0.965926	−	0.258819i	$-0.916667\pi$
$823$	−1.43732	−	1.14074i	−1.43732	−	1.14074i	−0.471397	−	0.881921i	$-0.656250\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.456904	−	0.889516i	$-0.651042\pi$
$827$		0			0		0.456904	+	0.889516i	$0.348958\pi$
$828$		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
$831$	1.52945	−	1.05855i	1.52945	−	1.05855i
$832$	0.143078	+	1.45270i	0.143078	+	1.45270i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.36827	−	0.439633i	1.36827	−	0.439633i
$838$		0			0
$839$		0			0		−0.840448	−	0.541892i	$-0.817708\pi$
$839$		0			0		0.840448	+	0.541892i	$0.182292\pi$
$840$		0			0
$841$	−0.997859	−	0.0654031i	−0.997859	−	0.0654031i
$842$		0			0
$843$		0			0
$844$	−0.0770042	−	0.669365i	−0.0770042	−	0.669365i
$845$		0			0
$846$		0			0
$847$	−1.23916	+	0.101606i	−1.23916	+	0.101606i
$848$		0			0
$849$	−1.09802	+	0.995185i	−1.09802	+	0.995185i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.226078	−	1.71723i	0.226078	−	1.71723i	−0.382683	−	0.923880i	$-0.625000\pi$
$853$	0.226078	−	1.71723i	0.226078	−	1.71723i	0.608761	−	0.793353i	$-0.291667\pi$
$854$		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
$859$	1.21492	+	1.58331i	1.21492	+	1.58331i	0.659346	+	0.751840i	$0.270833\pi$
$859$	1.21492	+	1.58331i	1.21492	+	1.58331i	0.555570	+	0.831470i	$0.312500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.783287	−	0.621661i	$-0.213542\pi$
$863$		0			0		−0.783287	+	0.621661i	$0.786458\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$868$	1.04080	+	1.45244i	1.04080	+	1.45244i
$869$		0			0
$870$		0			0
$871$	−1.11185	+	0.109507i	−1.11185	+	0.109507i
$872$		0			0
$873$	0.604690	−	1.22619i	0.604690	−	1.22619i
$874$		0			0
$875$		0			0
$876$	−0.956599	+	0.616781i	−0.956599	+	0.616781i
$877$	1.60662	−	0.727950i	1.60662	−	0.727950i	0.608761	−	0.793353i	$-0.291667\pi$
$877$	1.60662	−	0.727950i	1.60662	−	0.727950i	0.997859	+	0.0654031i	$0.0208333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.211112	−	0.977462i	$-0.567708\pi$
$881$		0			0		0.211112	+	0.977462i	$0.432292\pi$
$882$		0			0
$883$	−1.43697	−	1.39070i	−1.43697	−	1.39070i	−0.729864	−	0.683592i	$-0.760417\pi$
$883$	−1.43697	−	1.39070i	−1.43697	−	1.39070i	−0.707107	−	0.707107i	$-0.750000\pi$
$884$		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
$889$	−0.950451	−	2.29459i	−0.950451	−	2.29459i
$890$		0			0
$891$		0			0
$892$	−0.0398362	−	1.21687i	−0.0398362	−	1.21687i
$893$		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
$903$	0.0554429	−	0.481941i	0.0554429	−	0.481941i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.74970	+	0.593943i	−1.74970	+	0.593943i	−0.997859	−	0.0654031i	$-0.979167\pi$
$907$	−1.74970	+	0.593943i	−1.74970	+	0.593943i	−0.751840	+	0.659346i	$0.770833\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$911$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$912$	−1.41395	+	1.36842i	−1.41395	+	1.36842i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.148657	−	0.462665i	0.148657	−	0.462665i
$917$		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
$921$	−0.814389	+	0.134456i	−0.814389	+	0.134456i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.215212	−	1.87075i	−0.215212	−	1.87075i
$926$		0			0
$927$	−0.291538	−	1.96539i	−0.291538	−	1.96539i
$928$		0			0
$929$		0			0		0.822268	−	0.569100i	$-0.192708\pi$
$929$		0			0		−0.822268	+	0.569100i	$0.807292\pi$
$930$		0			0
$931$	−0.521739	−	0.938825i	−0.521739	−	0.938825i
$932$		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
$939$	0.607206	−	1.46593i	0.607206	−	1.46593i
$940$		0			0
$941$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$941$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.367516	−	0.930017i	$-0.380208\pi$
$947$		0			0		−0.367516	+	0.930017i	$0.619792\pi$
$948$	−1.64599	+	0.588944i	−1.64599	+	0.588944i
$949$	−1.52438	−	0.660844i	−1.52438	−	0.660844i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.582478	−	0.812847i	$-0.302083\pi$
$953$		0			0		−0.582478	+	0.812847i	$0.697917\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.970469	−	0.439714i	0.970469	−	0.439714i
$962$		0			0
$963$		0			0
$964$	−1.88352	−	0.279393i	−1.88352	−	0.279393i
$965$		0			0
$966$		0			0
$967$	−1.00503	−	0.217066i	−1.00503	−	0.217066i	−0.321439	−	0.946930i	$-0.604167\pi$
$967$	−1.00503	−	0.217066i	−1.00503	−	0.217066i	−0.683592	+	0.729864i	$0.739583\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.840448	−	0.541892i	$-0.182292\pi$
$971$		0			0		−0.840448	+	0.541892i	$0.817708\pi$
$972$	−0.729864	−	0.683592i	−0.729864	−	0.683592i
$973$	−0.682225	−	2.12329i	−0.682225	−	2.12329i
$974$		0			0
$975$	−1.28737	+	0.688111i	−1.28737	+	0.688111i
$976$	0.0409186	−	0.355688i	0.0409186	−	0.355688i
$977$		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
$981$	−1.45270	+	1.36060i	−1.45270	+	1.36060i
$982$		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
$988$	−2.80756	−	0.606375i	−2.80756	−	0.606375i
$989$		0			0
$990$		0			0
$991$	−1.89285	−	0.642535i	−1.89285	−	0.642535i	−0.956940	−	0.290285i	$-0.906250\pi$
$991$	−1.89285	−	0.642535i	−1.89285	−	0.642535i	−0.935906	−	0.352250i	$-0.885417\pi$
$992$		0			0
$993$	1.48664	−	0.703130i	1.48664	−	0.703130i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.74970	−	0.724749i	1.74970	−	0.724749i	0.751840	−	0.659346i	$-0.229167\pi$
$997$	1.74970	−	0.724749i	1.74970	−	0.724749i	0.997859	−	0.0654031i	$-0.0208333\pi$
$998$		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.176.1	✓	64
3.2	odd	2	CM	3459.1.bm.a.176.1	✓	64
1153.832	even	192	inner	3459.1.bm.a.1985.1	yes	64
3459.1985	odd	192	inner	3459.1.bm.a.1985.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3459.1.bm.a.176.1	✓	64	1.1	even	1	trivial
3459.1.bm.a.176.1	✓	64	3.2	odd	2	CM
3459.1.bm.a.1985.1	yes	64	1153.832	even	192	inner
3459.1.bm.a.1985.1	yes	64	3459.1985	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-0.999465 + 0.0327191i) q^{3} +(0.608761 + 0.793353i) q^{4} +(-1.20606 - 0.302102i) q^{7} +(0.997859 - 0.0654031i) q^{9} +O(q^{10})\)	\(q+(-0.999465 + 0.0327191i) q^{3} +(0.608761 + 0.793353i) q^{4} +(-1.20606 - 0.302102i) q^{7} +(0.997859 - 0.0654031i) q^{9} +(-0.634393 - 0.773010i) q^{12} +(0.423737 - 1.39687i) q^{13} +(-0.258819 + 0.965926i) q^{16} +(-1.65374 - 1.06628i) q^{19} +(1.21530 + 0.262479i) q^{21} +(0.683592 + 0.729864i) q^{25} +(-0.995185 + 0.0980171i) q^{27} +(-0.494529 - 1.14074i) q^{28} +(-1.40477 + 0.303402i) q^{31} +(0.659346 + 0.751840i) q^{36} +(-1.51251 - 1.12175i) q^{37} +(-0.377805 + 1.40999i) q^{39} +(-0.0382444 - 0.388302i) q^{43} +(0.227076 - 0.973877i) q^{48} +(0.481394 + 0.257311i) q^{49} +(1.36617 - 0.514189i) q^{52} +(1.68775 + 1.01160i) q^{57} +(-0.354159 + 0.0525345i) q^{61} +(-1.22324 - 0.222575i) q^{63} +(-0.923880 + 0.382683i) q^{64} +(-0.292893 - 0.707107i) q^{67} +(0.130081 - 1.13074i) q^{73} +(-0.707107 - 0.707107i) q^{75} +(-0.160802 - 1.96111i) q^{76} +(0.588944 - 1.64599i) q^{79} +(0.991445 - 0.130526i) q^{81} +(0.531588 + 1.12395i) q^{84} +(-0.933051 + 1.55670i) q^{91} +(1.39409 - 0.349202i) q^{93} +(0.683592 - 1.18402i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 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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