LMFDB - Embedded newform 3459.1.bm.a.1415.1

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			1415.1
Root			$-0.973877 + 0.227076i$ of defining polynomial
Character	$\chi$	$=$	3459.1415
Dual form			3459.1.bm.a.3437.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.910864 + 0.412707i) q^{3} +(0.793353 - 0.608761i) q^{4} +(1.29235 + 0.0634893i) q^{7} +(0.659346 - 0.751840i) q^{9} +O(q^{10})$	\(q+(-0.910864 + 0.412707i) q^{3} +(0.793353 - 0.608761i) q^{4} +(1.29235 + 0.0634893i) q^{7} +(0.659346 - 0.751840i) q^{9} +(-0.471397 + 0.881921i) q^{12} +(1.18746 - 1.44693i) q^{13} +(0.258819 - 0.965926i) q^{16} +(0.553222 + 1.27613i) q^{19} +(-1.20336 + 0.475533i) q^{21} +(-0.352250 - 0.935906i) q^{25} +(-0.290285 + 0.956940i) q^{27} +(1.06394 - 0.736366i) q^{28} +(-1.56326 - 0.617756i) q^{31} +(0.0654031 - 0.997859i) q^{36} +(-0.207775 + 0.439303i) q^{37} +(-0.484461 + 1.80803i) q^{39} +(-1.06330 - 0.322547i) q^{43} +(0.162895 + 0.986643i) q^{48} +(0.670963 + 0.0660840i) q^{49} +(0.0612440 - 1.87081i) q^{52} +(-1.03058 - 0.934061i) q^{57} +(0.484166 + 1.35315i) q^{61} +(0.899842 - 0.929782i) q^{63} +(-0.382683 - 0.923880i) q^{64} +(-1.70711 + 0.707107i) q^{67} +(1.99304 - 0.163420i) q^{73} +(0.707107 + 0.707107i) q^{75} +(1.21576 + 0.675641i) q^{76} +(-1.84706 + 0.462665i) q^{79} +(-0.130526 - 0.991445i) q^{81} +(-0.665204 + 1.10983i) q^{84} +(1.62649 - 1.79455i) q^{91} +(1.67887 - 0.0824777i) q^{93} +(-0.352250 + 0.610115i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3459\mathbb{Z}\right)^\times$.

$n$	$1154$	$2311$
$\chi(n)$	$-1$	$e\left(\frac{133}{192}\right)$

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		0.946930	−	0.321439i	$-0.104167\pi$
$2$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$3$	−0.910864	+	0.412707i	−0.910864	+	0.412707i
$3$	−0.910864	+	0.412707i	−0.910864	+	0.412707i
$4$	0.793353	−	0.608761i	0.793353	−	0.608761i
$5$		0			0		0.569100	−	0.822268i	$-0.307292\pi$
$5$		0			0		−0.569100	+	0.822268i	$0.692708\pi$
$6$		0			0
$7$	1.29235	+	0.0634893i	1.29235	+	0.0634893i	0.683592	−	0.729864i	$-0.260417\pi$
$7$	1.29235	+	0.0634893i	1.29235	+	0.0634893i	0.608761	+	0.793353i	$0.291667\pi$
$8$		0			0
$9$	0.659346	−	0.751840i	0.659346	−	0.751840i
$10$		0			0
$11$		0			0		−0.528068	−	0.849202i	$-0.677083\pi$
$11$		0			0		0.528068	+	0.849202i	$0.322917\pi$
$12$	−0.471397	+	0.881921i	−0.471397	+	0.881921i
$13$	1.18746	−	1.44693i	1.18746	−	1.44693i	0.321439	−	0.946930i	$-0.395833\pi$
$13$	1.18746	−	1.44693i	1.18746	−	1.44693i	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		−0.993448	−	0.114287i	$-0.963542\pi$
$17$		0			0		0.993448	+	0.114287i	$0.0364583\pi$
$18$		0			0
$19$	0.553222	+	1.27613i	0.553222	+	1.27613i	0.935906	+	0.352250i	$0.114583\pi$
$19$	0.553222	+	1.27613i	0.553222	+	1.27613i	−0.382683	+	0.923880i	$0.625000\pi$
$20$		0			0
$21$	−1.20336	+	0.475533i	−1.20336	+	0.475533i
$22$		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
$25$	−0.352250	−	0.935906i	−0.352250	−	0.935906i
$26$		0			0
$27$	−0.290285	+	0.956940i	−0.290285	+	0.956940i
$28$	1.06394	−	0.736366i	1.06394	−	0.736366i
$29$		0			0		0.412707	−	0.910864i	$-0.364583\pi$
$29$		0			0		−0.412707	+	0.910864i	$0.635417\pi$
$30$		0			0
$31$	−1.56326	−	0.617756i	−1.56326	−	0.617756i	−0.582478	−	0.812847i	$-0.697917\pi$
$31$	−1.56326	−	0.617756i	−1.56326	−	0.617756i	−0.980785	+	0.195090i	$0.937500\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.0654031	−	0.997859i	0.0654031	−	0.997859i
$37$	−0.207775	+	0.439303i	−0.207775	+	0.439303i	−0.980785	−	0.195090i	$-0.937500\pi$
$37$	−0.207775	+	0.439303i	−0.207775	+	0.439303i	0.773010	+	0.634393i	$0.218750\pi$
$38$		0			0
$39$	−0.484461	+	1.80803i	−0.484461	+	1.80803i
$40$		0			0
$41$		0			0		0.935906	−	0.352250i	$-0.114583\pi$
$41$		0			0		−0.935906	+	0.352250i	$0.885417\pi$
$42$		0			0
$43$	−1.06330	−	0.322547i	−1.06330	−	0.322547i	−0.290285	−	0.956940i	$-0.593750\pi$
$43$	−1.06330	−	0.322547i	−1.06330	−	0.322547i	−0.773010	+	0.634393i	$0.781250\pi$
$44$		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	$0.833333\pi$
$48$	0.162895	+	0.986643i	0.162895	+	0.986643i
$49$	0.670963	+	0.0660840i	0.670963	+	0.0660840i
$50$		0			0
$51$		0			0
$52$	0.0612440	−	1.87081i	0.0612440	−	1.87081i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.03058	−	0.934061i	−1.03058	−	0.934061i
$58$		0			0
$59$		0			0		−0.941544	−	0.336890i	$-0.890625\pi$
$59$		0			0		0.941544	+	0.336890i	$0.109375\pi$
$60$		0			0
$61$	0.484166	+	1.35315i	0.484166	+	1.35315i	0.896873	+	0.442289i	$0.145833\pi$
$61$	0.484166	+	1.35315i	0.484166	+	1.35315i	−0.412707	+	0.910864i	$0.635417\pi$
$62$		0			0
$63$	0.899842	−	0.929782i	0.899842	−	0.929782i
$64$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$65$		0			0
$66$		0			0
$67$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$67$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$71$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$72$		0			0
$73$	1.99304	−	0.163420i	1.99304	−	0.163420i	0.995185	−	0.0980171i	$-0.0312500\pi$
$73$	1.99304	−	0.163420i	1.99304	−	0.163420i	0.997859	−	0.0654031i	$-0.0208333\pi$
$74$		0			0
$75$	0.707107	+	0.707107i	0.707107	+	0.707107i
$76$	1.21576	+	0.675641i	1.21576	+	0.675641i
$77$		0			0
$78$		0			0
$79$	−1.84706	+	0.462665i	−1.84706	+	0.462665i	−0.997859	−	0.0654031i	$-0.979167\pi$
$79$	−1.84706	+	0.462665i	−1.84706	+	0.462665i	−0.849202	+	0.528068i	$0.822917\pi$
$80$		0			0
$81$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$82$		0			0
$83$		0			0		0.840448	−	0.541892i	$-0.182292\pi$
$83$		0			0		−0.840448	+	0.541892i	$0.817708\pi$
$84$	−0.665204	+	1.10983i	−0.665204	+	1.10983i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.973877	−	0.227076i	$-0.927083\pi$
$89$		0			0		0.973877	+	0.227076i	$0.0729167\pi$
$90$		0			0
$91$	1.62649	−	1.79455i	1.62649	−	1.79455i
$92$		0			0
$93$	1.67887	−	0.0824777i	1.67887	−	0.0824777i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.352250	+	0.610115i	−0.352250	+	0.610115i	−0.986643	−	0.162895i	$-0.947917\pi$
$97$	−0.352250	+	0.610115i	−0.352250	+	0.610115i	0.634393	+	0.773010i	$0.281250\pi$
$98$		0			0
$99$		0			0
$100$	−0.849202	−	0.528068i	−0.849202	−	0.528068i
$101$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$101$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$102$		0			0
$103$	−0.142863	−	0.0793942i	−0.142863	−	0.0793942i	0.412707	−	0.910864i	$-0.364583\pi$
$103$	−0.142863	−	0.0793942i	−0.142863	−	0.0793942i	−0.555570	+	0.831470i	$0.687500\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$107$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$108$	0.352250	+	0.935906i	0.352250	+	0.935906i
$109$	0.512016	+	0.273678i	0.512016	+	0.273678i	0.707107	−	0.707107i	$-0.250000\pi$
$109$	0.512016	+	0.273678i	0.512016	+	0.273678i	−0.195090	+	0.980785i	$0.562500\pi$
$110$		0			0
$111$	0.00795115	−	0.485895i	0.00795115	−	0.485895i
$112$	0.395812	−	1.23189i	0.395812	−	1.23189i
$113$		0			0		0.999866	−	0.0163617i	$-0.00520833\pi$
$113$		0			0		−0.999866	+	0.0163617i	$0.994792\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.304910	−	1.84681i	−0.304910	−	1.84681i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.442289	+	0.896873i	−0.442289	+	0.896873i
$122$		0			0
$123$		0			0
$124$	−1.61629	+	0.461555i	−1.61629	+	0.461555i
$125$		0			0
$126$		0			0
$127$	1.45218	−	0.686831i	1.45218	−	0.686831i	0.471397	−	0.881921i	$-0.343750\pi$
$127$	1.45218	−	0.686831i	1.45218	−	0.686831i	0.980785	+	0.195090i	$0.0625000\pi$
$128$		0			0
$129$	1.10163	−	0.145033i	1.10163	−	0.145033i
$130$		0			0
$131$		0			0		0.935906	−	0.352250i	$-0.114583\pi$
$131$		0			0		−0.935906	+	0.352250i	$0.885417\pi$
$132$		0			0
$133$	0.633939	+	1.68433i	0.633939	+	1.68433i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.146730	−	0.989177i	$-0.453125\pi$
$137$		0			0		−0.146730	+	0.989177i	$0.546875\pi$
$138$		0			0
$139$	1.53942	+	1.10313i	1.53942	+	1.10313i	0.956940	+	0.290285i	$0.0937500\pi$
$139$	1.53942	+	1.10313i	1.53942	+	1.10313i	0.582478	+	0.812847i	$0.302083\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$145$		0			0
$146$		0			0
$147$	−0.638429	+	0.216717i	−0.638429	+	0.216717i
$148$	0.102592	+	0.475008i	0.102592	+	0.475008i
$149$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$149$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$150$		0			0
$151$	0.784035	+	1.30808i	0.784035	+	1.30808i	0.946930	+	0.321439i	$0.104167\pi$
$151$	0.784035	+	1.30808i	0.784035	+	1.30808i	−0.162895	+	0.986643i	$0.552083\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.716311	+	1.72933i	0.716311	+	1.72933i
$157$	−0.121103	−	1.47695i	−0.121103	−	1.47695i	−0.729864	−	0.683592i	$-0.760417\pi$
$157$	−0.121103	−	1.47695i	−0.121103	−	1.47695i	0.608761	−	0.793353i	$-0.291667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.390703	−	0.826072i	0.390703	−	0.826072i	−0.608761	−	0.793353i	$-0.708333\pi$
$163$	0.390703	−	0.826072i	0.390703	−	0.826072i	0.999465	−	0.0327191i	$-0.0104167\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.211112	−	0.977462i	$-0.432292\pi$
$167$		0			0		−0.211112	+	0.977462i	$0.567708\pi$
$168$		0			0
$169$	−0.488444	−	2.45557i	−0.488444	−	2.45557i
$170$		0			0
$171$	1.32421	+	0.425476i	1.32421	+	0.425476i
$172$	−1.03992	+	0.391399i	−1.03992	+	0.391399i
$173$		0			0		0.0490677	−	0.998795i	$-0.484375\pi$
$173$		0			0		−0.0490677	+	0.998795i	$0.515625\pi$
$174$		0			0
$175$	−0.395812	−	1.23189i	−0.395812	−	1.23189i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.874090	−	0.485763i	$-0.838542\pi$
$179$		0			0		0.874090	+	0.485763i	$0.161458\pi$
$180$		0			0
$181$	1.83585	+	0.690963i	1.83585	+	0.690963i	0.986643	+	0.162895i	$0.0520833\pi$
$181$	1.83585	+	0.690963i	1.83585	+	0.690963i	0.849202	+	0.528068i	$0.177083\pi$
$182$		0			0
$183$	−0.999465	−	1.03272i	−0.999465	−	1.03272i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.435906	+	1.21828i	−0.435906	+	1.21828i
$190$		0			0
$191$		0			0		−0.695443	−	0.718582i	$-0.744792\pi$
$191$		0			0		0.695443	+	0.718582i	$0.255208\pi$
$192$	0.729864	+	0.683592i	0.729864	+	0.683592i
$193$	−1.94839	+	0.420811i	−1.94839	+	0.420811i	−0.956940	+	0.290285i	$0.906250\pi$
$193$	−1.94839	+	0.420811i	−1.94839	+	0.420811i	−0.991445	+	0.130526i	$0.958333\pi$
$194$		0			0
$195$		0			0
$196$	0.572540	−	0.356028i	0.572540	−	0.356028i
$197$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$197$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$198$		0			0
$199$	−0.478235	−	0.198092i	−0.478235	−	0.198092i	0.130526	−	0.991445i	$-0.458333\pi$
$199$	−0.478235	−	0.198092i	−0.478235	−	0.198092i	−0.608761	+	0.793353i	$0.708333\pi$
$200$		0			0
$201$	1.26311	−	1.34861i	1.26311	−	1.34861i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.09029	−	1.52150i	−1.09029	−	1.52150i
$209$		0			0
$210$		0			0
$211$	1.43749	+	1.30287i	1.43749	+	1.30287i	0.881921	+	0.471397i	$0.156250\pi$
$211$	1.43749	+	1.30287i	1.43749	+	1.30287i	0.555570	+	0.831470i	$0.312500\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.98107	−	0.897610i	−1.98107	−	0.897610i
$218$		0			0
$219$	−1.74795	+	0.971397i	−1.74795	+	0.971397i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.360303	−	1.54526i	0.360303	−	1.54526i	−0.412707	−	0.910864i	$-0.635417\pi$
$223$	0.360303	−	1.54526i	0.360303	−	1.54526i	0.773010	−	0.634393i	$-0.218750\pi$
$224$		0			0
$225$	−0.935906	−	0.352250i	−0.935906	−	0.352250i
$226$		0			0
$227$		0			0		0.977462	−	0.211112i	$-0.0677083\pi$
$227$		0			0		−0.977462	+	0.211112i	$0.932292\pi$
$228$	−1.38623	−	0.113665i	−1.38623	−	0.113665i
$229$	0.0981222	−	0.00160566i	0.0981222	−	0.00160566i	0.0327191	−	0.999465i	$-0.489583\pi$
$229$	0.0981222	−	0.00160566i	0.0981222	−	0.00160566i	0.0654031	+	0.997859i	$0.479167\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.993448	−	0.114287i	$-0.0364583\pi$
$233$		0			0		−0.993448	+	0.114287i	$0.963542\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.49148	−	1.18372i	1.49148	−	1.18372i
$238$		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
$241$	−1.04495	+	0.673745i	−1.04495	+	0.673745i	−0.946930	−	0.321439i	$-0.895833\pi$
$241$	−1.04495	+	0.673745i	−1.04495	+	0.673745i	−0.0980171	+	0.995185i	$0.531250\pi$
$242$		0			0
$243$	0.528068	+	0.849202i	0.528068	+	0.849202i
$244$	1.20786	+	0.778787i	1.20786	+	0.778787i
$245$		0			0
$246$		0			0
$247$	2.50340	+	0.714885i	2.50340	+	0.714885i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$251$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$252$	0.147877	−	1.28543i	0.147877	−	1.28543i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$257$		0			0		−0.621661	−	0.783287i	$-0.713542\pi$
$257$		0			0		0.621661	+	0.783287i	$0.286458\pi$
$258$		0			0
$259$	−0.296410	+	0.554543i	−0.296410	+	0.554543i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.485763	−	0.874090i	$-0.338542\pi$
$263$		0			0		−0.485763	+	0.874090i	$0.661458\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$269$		0			0		0.146730	−	0.989177i	$-0.453125\pi$
$269$		0			0		−0.146730	+	0.989177i	$0.546875\pi$
$270$		0			0
$271$	−0.831470	+	1.55557i	−0.831470	+	1.55557i			1.00000i	$0.5\pi$
$271$	−0.831470	+	1.55557i	−0.831470	+	1.55557i	−0.831470	+	0.555570i	$0.812500\pi$
$272$		0			0
$273$	−0.740885	+	2.30586i	−0.740885	+	2.30586i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.142407	+	0.328494i	−0.142407	+	0.328494i	−0.973877	−	0.227076i	$-0.927083\pi$
$277$	−0.142407	+	0.328494i	−0.142407	+	0.328494i	0.831470	+	0.555570i	$0.187500\pi$
$278$		0			0
$279$	−1.49518	+	0.768008i	−1.49518	+	0.768008i
$280$		0			0
$281$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$281$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$282$		0			0
$283$	1.90231	+	0.543233i	1.90231	+	0.543233i	0.991445	+	0.130526i	$0.0416667\pi$
$283$	1.90231	+	0.543233i	1.90231	+	0.543233i	0.910864	+	0.412707i	$0.135417\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.973877	+	0.227076i	0.973877	+	0.227076i
$290$		0			0
$291$	0.0690531	−	0.701108i	0.0690531	−	0.701108i
$292$	1.48170	−	1.34294i	1.48170	−	1.34294i
$293$		0			0		0.242980	−	0.970031i	$-0.421875\pi$
$293$		0			0		−0.242980	+	0.970031i	$0.578125\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.991445	+	0.130526i	0.991445	+	0.130526i
$301$	−1.35368	−	0.484353i	−1.35368	−	0.484353i
$302$		0			0
$303$		0			0
$304$	1.37583	−	0.204085i	1.37583	−	0.204085i
$305$		0			0
$306$		0			0
$307$	0.178453	+	1.35549i	0.178453	+	1.35549i	0.812847	+	0.582478i	$0.197917\pi$
$307$	0.178453	+	1.35549i	0.178453	+	1.35549i	−0.634393	+	0.773010i	$0.718750\pi$
$308$		0			0
$309$	0.162895	+	0.0133567i	0.162895	+	0.0133567i
$310$		0			0
$311$		0			0		0.983846	−	0.179017i	$-0.0572917\pi$
$311$		0			0		−0.983846	+	0.179017i	$0.942708\pi$
$312$		0			0
$313$	−0.832289	−	0.888626i	−0.832289	−	0.888626i	0.162895	−	0.986643i	$-0.447917\pi$
$313$	−0.832289	−	0.888626i	−0.832289	−	0.888626i	−0.995185	+	0.0980171i	$0.968750\pi$
$314$		0			0
$315$		0			0
$316$	−1.18372	+	1.49148i	−1.18372	+	1.49148i
$317$		0			0		−0.0490677	−	0.998795i	$-0.515625\pi$
$317$		0			0		0.0490677	+	0.998795i	$0.484375\pi$
$318$		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
$325$	−1.77248	−	0.601674i	−1.77248	−	0.601674i
$326$		0			0
$327$	−0.579326	−	0.0379711i	−0.579326	−	0.0379711i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.0186229	+	0.0269075i	0.0186229	+	0.0269075i	0.831470	−	0.555570i	$-0.187500\pi$
$331$	0.0186229	+	0.0269075i	0.0186229	+	0.0269075i	−0.812847	+	0.582478i	$0.802083\pi$
$332$		0			0
$333$	0.193290	+	0.445866i	0.193290	+	0.445866i
$334$		0			0
$335$		0			0
$336$	0.147877	+	1.28543i	0.147877	+	1.28543i
$337$	−0.0143994	−	0.293107i	−0.0143994	−	0.293107i	−0.995185	−	0.0980171i	$-0.968750\pi$
$337$	−0.0143994	−	0.293107i	−0.0143994	−	0.293107i	0.980785	−	0.195090i	$-0.0625000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.416982	−	0.0618535i	−0.416982	−	0.0618535i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$347$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$348$		0			0
$349$	−0.0630132	+	0.639783i	−0.0630132	+	0.639783i	0.910864	+	0.412707i	$0.135417\pi$
$349$	−0.0630132	+	0.639783i	−0.0630132	+	0.639783i	−0.973877	+	0.227076i	$0.927083\pi$
$350$		0			0
$351$	1.03992	+	1.55635i	1.03992	+	1.55635i
$352$		0			0
$353$		0			0		0.910864	−	0.412707i	$-0.135417\pi$
$353$		0			0		−0.910864	+	0.412707i	$0.864583\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.595699	−	0.803208i	$-0.703125\pi$
$359$		0			0		0.595699	+	0.803208i	$0.296875\pi$
$360$		0			0
$361$	−0.638859	+	0.682103i	−0.638859	+	0.682103i
$362$		0			0
$363$	0.0327191	−	0.999465i	0.0327191	−	0.999465i
$364$	0.197925	−	2.41386i	0.197925	−	2.41386i
$365$		0			0
$366$		0			0
$367$	−0.735499	+	1.49144i	−0.735499	+	1.49144i	0.130526	+	0.991445i	$0.458333\pi$
$367$	−0.735499	+	1.49144i	−0.735499	+	1.49144i	−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.28173	−	1.08747i	1.28173	−	1.08747i
$373$	−1.60747	−	0.347181i	−1.60747	−	0.347181i	−0.683592	−	0.729864i	$-0.739583\pi$
$373$	−1.60747	−	0.347181i	−1.60747	−	0.347181i	−0.923880	+	0.382683i	$0.875000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.25793	+	1.53279i	−1.25793	+	1.53279i	−0.528068	+	0.849202i	$0.677083\pi$
$379$	−1.25793	+	1.53279i	−1.25793	+	1.53279i	−0.729864	+	0.683592i	$0.760417\pi$
$380$		0			0
$381$	−1.03928	+	1.22494i	−1.03928	+	1.22494i
$382$		0			0
$383$		0			0		0.973877	−	0.227076i	$-0.0729167\pi$
$383$		0			0		−0.973877	+	0.227076i	$0.927083\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.943583	+	0.586758i	−0.943583	+	0.586758i
$388$	0.0919557	+	0.698473i	0.0919557	+	0.698473i
$389$		0			0		−0.989177	−	0.146730i	$-0.953125\pi$
$389$		0			0		0.989177	+	0.146730i	$0.0468750\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.479220	−	1.49148i	−0.479220	−	1.49148i	−0.831470	−	0.555570i	$-0.812500\pi$
$397$	−0.479220	−	1.49148i	−0.479220	−	1.49148i	0.352250	−	0.935906i	$-0.385417\pi$
$398$		0			0
$399$	−1.27257	−	1.27257i	−1.27257	−	1.27257i
$400$	−0.995185	+	0.0980171i	−0.995185	+	0.0980171i
$401$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$401$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$402$		0			0
$403$	−2.75017	+	1.52837i	−2.75017	+	1.52837i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.217066	+	0.362153i	−0.217066	+	0.362153i	−0.946930	−	0.321439i	$-0.895833\pi$
$409$	−0.217066	+	0.362153i	−0.217066	+	0.362153i	0.729864	+	0.683592i	$0.239583\pi$
$410$		0			0
$411$		0			0
$412$	−0.161673	+	0.0239819i	−0.161673	+	0.0239819i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.85747	−	0.369474i	−1.85747	−	0.369474i
$418$		0			0
$419$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$419$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$420$		0			0
$421$	−0.727076	−	0.107852i	−0.727076	−	0.107852i	−0.227076	−	0.973877i	$-0.572917\pi$
$421$	−0.727076	−	0.107852i	−0.727076	−	0.107852i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.539803	+	1.77949i	0.539803	+	1.77949i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.412707	−	0.910864i	$-0.364583\pi$
$431$		0			0		−0.412707	+	0.910864i	$0.635417\pi$
$432$	0.849202	+	0.528068i	0.849202	+	0.528068i
$433$	1.02807	+	0.0168232i	1.02807	+	0.0168232i	0.528068	−	0.849202i	$-0.322917\pi$
$433$	1.02807	+	0.0168232i	1.02807	+	0.0168232i	0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$	0.572815	−	0.0945721i	0.572815	−	0.0945721i
$437$		0			0
$438$		0			0
$439$	1.64073	−	1.09630i	1.64073	−	1.09630i	0.729864	−	0.683592i	$-0.239583\pi$
$439$	1.64073	−	1.09630i	1.64073	−	1.09630i	0.910864	−	0.412707i	$-0.135417\pi$
$440$		0			0
$441$	0.492081	−	0.460884i	0.492081	−	0.460884i
$442$		0			0
$443$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$443$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$444$	−0.289486	−	0.390327i	−0.289486	−	0.390327i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.435906	−	1.21828i	−0.435906	−	1.21828i
$449$		0			0		0.582478	−	0.812847i	$-0.302083\pi$
$449$		0			0		−0.582478	+	0.812847i	$0.697917\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.25400	−	0.867909i	−1.25400	−	0.867909i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.783904	+	0.523788i	0.783904	+	0.523788i	0.881921	−	0.471397i	$-0.156250\pi$
$457$	0.783904	+	0.523788i	0.783904	+	0.523788i	−0.0980171	+	0.995185i	$0.531250\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.569100	−	0.822268i	$-0.692708\pi$
$461$		0			0		0.569100	+	0.822268i	$0.307292\pi$
$462$		0			0
$463$	−1.07668	−	0.123862i	−1.07668	−	0.123862i	−0.442289	−	0.896873i	$-0.645833\pi$
$463$	−1.07668	−	0.123862i	−1.07668	−	0.123862i	−0.634393	+	0.773010i	$0.718750\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.983846	−	0.179017i	$-0.0572917\pi$
$467$		0			0		−0.983846	+	0.179017i	$0.942708\pi$
$468$	−1.36617	−	1.27956i	−1.36617	−	1.27956i
$469$	−2.25108	+	0.805449i	−2.25108	+	0.805449i
$470$		0			0
$471$	0.719854	+	1.29532i	0.719854	+	1.29532i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.999465	−	0.967281i	0.999465	−	0.967281i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.367516	−	0.930017i	$-0.619792\pi$
$479$		0			0		0.367516	+	0.930017i	$0.380208\pi$
$480$		0			0
$481$	0.388915	+	0.822292i	0.388915	+	0.822292i
$482$		0			0
$483$		0			0
$484$	0.195090	+	0.980785i	0.195090	+	0.980785i
$485$		0			0
$486$		0			0
$487$	−1.34150	−	1.48012i	−1.34150	−	1.48012i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	−1.34150	−	1.48012i	−1.34150	−	1.48012i	−0.634393	−	0.773010i	$-0.718750\pi$
$488$		0			0
$489$	−0.0149515	+	0.913685i	−0.0149515	+	0.913685i
$490$		0			0
$491$		0			0		−0.803208	−	0.595699i	$-0.796875\pi$
$491$		0			0		0.803208	+	0.595699i	$0.203125\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.00131	+	1.35011i	−1.00131	+	1.35011i
$497$		0			0
$498$		0			0
$499$	−0.797289	+	0.213633i	−0.797289	+	0.213633i	−0.634393	−	0.773010i	$-0.718750\pi$
$499$	−0.797289	+	0.213633i	−0.797289	+	0.213633i	−0.162895	+	0.986643i	$0.552083\pi$
$500$		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
$507$	1.45834	+	2.03511i	1.45834	+	2.03511i
$508$	0.733977	−	1.42893i	0.733977	−	1.42893i
$509$		0			0		−0.336890	−	0.941544i	$-0.609375\pi$
$509$		0			0		0.336890	+	0.941544i	$0.390625\pi$
$510$		0			0
$511$	2.58609	−	0.0846599i	2.58609	−	0.0846599i
$512$		0			0
$513$	−1.38177	+	0.158960i	−1.38177	+	0.158960i
$514$		0			0
$515$		0			0
$516$	0.785695	−	0.785695i	0.785695	−	0.785695i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.336890	−	0.941544i	$-0.390625\pi$
$521$		0			0		−0.336890	+	0.941544i	$0.609375\pi$
$522$		0			0
$523$	−0.528068	−	0.914640i	−0.528068	−	0.914640i	−0.999465	−	0.0327191i	$-0.989583\pi$
$523$	−0.528068	−	0.914640i	−0.528068	−	0.914640i	0.471397	−	0.881921i	$-0.343750\pi$
$524$		0			0
$525$	0.868938	+	0.958726i	0.868938	+	0.958726i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$530$		0			0
$531$		0			0
$532$	1.52830	+	0.950355i	1.52830	+	0.950355i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.153079	−	0.841297i	0.153079	−	0.841297i	−0.812847	−	0.582478i	$-0.802083\pi$
$541$	0.153079	−	0.841297i	0.153079	−	0.841297i	0.965926	−	0.258819i	$-0.0833333\pi$
$542$		0			0
$543$	−1.95737	+	0.128293i	−1.95737	+	0.128293i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.45694	−	1.15631i	−1.45694	−	1.15631i	−0.956940	−	0.290285i	$-0.906250\pi$
$547$	−1.45694	−	1.15631i	−1.45694	−	1.15631i	−0.500000	−	0.866025i	$-0.666667\pi$
$548$		0			0
$549$	1.33659	+	0.528180i	1.33659	+	0.528180i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.41643	+	0.480658i	−2.41643	+	0.480658i
$554$		0			0
$555$		0			0
$556$	1.89285	−	0.0619654i	1.89285	−	0.0619654i
$557$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$557$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$558$		0			0
$559$	−1.72933	+	1.15550i	−1.72933	+	1.15550i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.582478	−	0.812847i	$-0.302083\pi$
$563$		0			0		−0.582478	+	0.812847i	$0.697917\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.105740	−	1.28958i	−0.105740	−	1.28958i
$568$		0			0
$569$		0			0		0.485763	−	0.874090i	$-0.338542\pi$
$569$		0			0		−0.485763	+	0.874090i	$0.661458\pi$
$570$		0			0
$571$	0.438678	+	1.88139i	0.438678	+	1.88139i	0.471397	+	0.881921i	$0.343750\pi$
$571$	0.438678	+	1.88139i	0.438678	+	1.88139i	−0.0327191	+	0.999465i	$0.510417\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.946930	−	0.321439i	−0.946930	−	0.321439i
$577$	−1.46397	−	0.496952i	−1.46397	−	0.496952i	−0.528068	−	0.849202i	$-0.677083\pi$
$577$	−1.46397	−	0.496952i	−1.46397	−	0.496952i	−0.935906	+	0.352250i	$0.885417\pi$
$578$		0			0
$579$	1.60104	−	1.18741i	1.60104	−	1.18741i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.998795	−	0.0490677i	$-0.0156250\pi$
$587$		0			0		−0.998795	+	0.0490677i	$0.984375\pi$
$588$	−0.374570	+	0.560584i	−0.374570	+	0.560584i
$589$	−0.0764951	−	2.33668i	−0.0764951	−	2.33668i
$590$		0			0
$591$		0			0
$592$	0.370558	+	0.314395i	0.370558	+	0.314395i
$593$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$593$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.517361	−	0.0169366i	0.517361	−	0.0169366i
$598$		0			0
$599$		0			0		0.0817211	−	0.996655i	$-0.473958\pi$
$599$		0			0		−0.0817211	+	0.996655i	$0.526042\pi$
$600$		0			0
$601$	1.54702	+	0.700943i	1.54702	+	0.700943i	0.991445	−	0.130526i	$-0.0416667\pi$
$601$	1.54702	+	0.700943i	1.54702	+	0.700943i	0.555570	+	0.831470i	$0.312500\pi$
$602$		0			0
$603$	−0.593943	+	1.74970i	−0.593943	+	1.74970i
$604$	1.41833	+	0.560482i	1.41833	+	0.560482i
$605$		0			0
$606$		0			0
$607$	−0.0321948	−	1.96743i	−0.0321948	−	1.96743i	−0.195090	−	0.980785i	$-0.562500\pi$
$607$	−0.0321948	−	1.96743i	−0.0321948	−	1.96743i	0.162895	−	0.986643i	$-0.447917\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.86549	+	0.467281i	1.86549	+	0.467281i	0.999465	−	0.0327191i	$-0.0104167\pi$
$613$	1.86549	+	0.467281i	1.86549	+	0.467281i	0.866025	+	0.500000i	$0.166667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$617$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$618$		0			0
$619$	−0.393308	+	0.227076i	−0.393308	+	0.227076i	−0.683592	−	0.729864i	$-0.739583\pi$
$619$	−0.393308	+	0.227076i	−0.393308	+	0.227076i	0.290285	+	0.956940i	$0.406250\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.62104	+	0.935906i	1.62104	+	0.935906i
$625$	−0.751840	+	0.659346i	−0.751840	+	0.659346i
$626$		0			0
$627$		0			0
$628$	−0.995185	−	1.09802i	−0.995185	−	1.09802i
$629$		0			0
$630$		0			0
$631$	1.24532	+	1.32961i	1.24532	+	1.32961i	0.923880	+	0.382683i	$0.125000\pi$
$631$	1.24532	+	1.32961i	1.24532	+	1.32961i	0.321439	+	0.946930i	$0.395833\pi$
$632$		0			0
$633$	−1.84706	−	0.593471i	−1.84706	−	0.593471i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.892363	−	0.892363i	0.892363	−	0.892363i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$641$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$642$		0			0
$643$	0.0477610	+	1.45895i	0.0477610	+	1.45895i	0.707107	+	0.707107i	$0.250000\pi$
$643$	0.0477610	+	1.45895i	0.0477610	+	1.45895i	−0.659346	+	0.751840i	$0.729167\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.812847	−	0.582478i	$-0.197917\pi$
$647$		0			0		−0.812847	+	0.582478i	$0.802083\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	2.17493			2.17493
$652$	−0.192915	−	0.893212i	−0.192915	−	0.893212i
$653$		0			0		0.917494	−	0.397748i	$-0.130208\pi$
$653$		0			0		−0.917494	+	0.397748i	$0.869792\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.19124	−	1.60620i	1.19124	−	1.60620i
$658$		0			0
$659$		0			0		0.998795	−	0.0490677i	$-0.0156250\pi$
$659$		0			0		−0.998795	+	0.0490677i	$0.984375\pi$
$660$		0			0
$661$	−0.165610	+	1.25793i	−0.165610	+	1.25793i	0.683592	+	0.729864i	$0.260417\pi$
$661$	−0.165610	+	1.25793i	−0.165610	+	1.25793i	−0.849202	+	0.528068i	$0.822917\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.309551	+	1.55622i	0.309551	+	1.55622i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.91052	−	0.219788i	−1.91052	−	0.219788i	−0.923880	−	0.382683i	$-0.875000\pi$
$673$	−1.91052	−	0.219788i	−1.91052	−	0.219788i	−0.986643	+	0.162895i	$0.947917\pi$
$674$		0			0
$675$	0.997859	−	0.0654031i	0.997859	−	0.0654031i
$676$	−1.88237	−	1.65079i	−1.88237	−	1.65079i
$677$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$677$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$678$		0			0
$679$	−0.493967	+	0.766120i	−0.493967	+	0.766120i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.242980	−	0.970031i	$-0.578125\pi$
$683$		0			0		0.242980	+	0.970031i	$0.421875\pi$
$684$	1.30958	−	0.468575i	1.30958	−	0.468575i
$685$		0			0
$686$		0			0
$687$	−0.0887133	+	0.0419583i	−0.0887133	+	0.0419583i
$688$	−0.586758	+	0.943583i	−0.586758	+	0.943583i
$689$		0			0
$690$		0			0
$691$	−0.217617	−	0.0699215i	−0.217617	−	0.0699215i	0.195090	−	0.980785i	$-0.437500\pi$
$691$	−0.217617	−	0.0699215i	−0.217617	−	0.0699215i	−0.412707	+	0.910864i	$0.635417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.06394	−	0.736366i	−1.06394	−	0.736366i
$701$		0			0		0.783287	−	0.621661i	$-0.213542\pi$
$701$		0			0		−0.783287	+	0.621661i	$0.786458\pi$
$702$		0			0
$703$	−0.675553	−	0.0221153i	−0.675553	−	0.0221153i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.05977	−	1.42893i	−1.05977	−	1.42893i	−0.896873	−	0.442289i	$-0.854167\pi$
$709$	−1.05977	−	1.42893i	−1.05977	−	1.42893i	−0.162895	−	0.986643i	$-0.552083\pi$
$710$		0			0
$711$	−0.870002	+	1.69375i	−0.870002	+	1.69375i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$719$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$720$		0			0
$721$	−0.179589	−	0.111676i	−0.179589	−	0.111676i
$722$		0			0
$723$	0.673745	−	1.04495i	0.673745	−	1.04495i
$724$	1.87711	−	0.569414i	1.87711	−	0.569414i
$725$		0			0
$726$		0			0
$727$	−1.05205	−	0.834963i	−1.05205	−	0.834963i	−0.0654031	−	0.997859i	$-0.520833\pi$
$727$	−1.05205	−	0.834963i	−1.05205	−	0.834963i	−0.986643	+	0.162895i	$0.947917\pi$
$728$		0			0
$729$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$730$		0			0
$731$		0			0
$732$	−1.42161	−	0.210876i	−1.42161	−	0.210876i
$733$	0.797289	+	1.75965i	0.797289	+	1.75965i	0.634393	+	0.773010i	$0.281250\pi$
$733$	0.797289	+	1.75965i	0.797289	+	1.75965i	0.162895	+	0.986643i	$0.447917\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.12197	−	0.860919i	−1.12197	−	0.860919i	−0.130526	−	0.991445i	$-0.541667\pi$
$739$	−1.12197	−	0.860919i	−1.12197	−	0.860919i	−0.991445	+	0.130526i	$0.958333\pi$
$740$		0			0
$741$	−2.57530	+	0.382009i	−2.57530	+	0.382009i
$742$		0			0
$743$		0			0		−0.0817211	−	0.996655i	$-0.526042\pi$
$743$		0			0		0.0817211	+	0.996655i	$0.473958\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.824972	−	1.82075i	0.824972	−	1.82075i	0.382683	−	0.923880i	$-0.375000\pi$
$751$	0.824972	−	1.82075i	0.824972	−	1.82075i	0.442289	−	0.896873i	$-0.354167\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.395812	+	1.23189i	0.395812	+	1.23189i
$757$	0.881921	−	0.471397i	0.881921	−	0.471397i	0.0327191	−	0.999465i	$-0.489583\pi$
$757$	0.881921	−	0.471397i	0.881921	−	0.471397i	0.849202	+	0.528068i	$0.177083\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$761$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$762$		0			0
$763$	0.644331	+	0.386197i	0.644331	+	0.386197i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.995185	+	0.0980171i	0.995185	+	0.0980171i
$769$	−0.805730	−	0.365071i	−0.805730	−	0.365071i	−0.0327191	−	0.999465i	$-0.510417\pi$
$769$	−0.805730	−	0.365071i	−0.805730	−	0.365071i	−0.773010	+	0.634393i	$0.781250\pi$
$770$		0			0
$771$		0			0
$772$	−1.28958	+	1.51995i	−1.28958	+	1.51995i
$773$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$773$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$774$		0			0
$775$	−0.0275024	+	1.68067i	−0.0275024	+	1.68067i
$776$		0			0
$777$	0.0411248	−	0.627444i	0.0411248	−	0.627444i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.237490	−	0.630996i	0.237490	−	0.630996i
$785$		0			0
$786$		0			0
$787$	0.864189	+	0.443894i	0.864189	+	0.443894i	0.831470	−	0.555570i	$-0.187500\pi$
$787$	0.864189	+	0.443894i	0.864189	+	0.443894i	0.0327191	+	0.999465i	$0.489583\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.53285	+	0.906267i	2.53285	+	0.906267i
$794$		0			0
$795$		0			0
$796$	−0.500000	+	0.133975i	−0.500000	+	0.133975i
$797$		0			0		0.910864	−	0.412707i	$-0.135417\pi$
$797$		0			0		−0.910864	+	0.412707i	$0.864583\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.181112	−	1.83886i	0.181112	−	1.83886i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.773010	−	0.634393i	$-0.218750\pi$
$809$		0			0		−0.773010	+	0.634393i	$0.781250\pi$
$810$		0			0
$811$	0.611724	−	0.0100102i	0.611724	−	0.0100102i	0.290285	−	0.956940i	$-0.406250\pi$
$811$	0.611724	−	0.0100102i	0.611724	−	0.0100102i	0.321439	+	0.946930i	$0.395833\pi$
$812$		0			0
$813$	0.115361	−	1.76007i	0.115361	−	1.76007i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.176627	−	1.53534i	−0.176627	−	1.53534i
$818$		0			0
$819$	−0.276799	−	2.40609i	−0.276799	−	2.40609i
$820$		0			0
$821$		0			0		0.336890	−	0.941544i	$-0.390625\pi$
$821$		0			0		−0.336890	+	0.941544i	$0.609375\pi$
$822$		0			0
$823$	0.867909	−	0.736366i	0.867909	−	0.736366i	−0.0980171	−	0.995185i	$-0.531250\pi$
$823$	0.867909	−	0.736366i	0.867909	−	0.736366i	0.965926	+	0.258819i	$0.0833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.114287	−	0.993448i	$-0.463542\pi$
$827$		0			0		−0.114287	+	0.993448i	$0.536458\pi$
$828$		0			0
$829$	0.666656	+	0.666656i	0.666656	+	0.666656i	0.956940	−	0.290285i	$-0.0937500\pi$
$829$	0.666656	+	0.666656i	0.666656	+	0.666656i	−0.290285	+	0.956940i	$0.593750\pi$
$830$		0			0
$831$	−0.00585805	−	0.357986i	−0.00585805	−	0.357986i
$832$	−1.79121	−	0.543358i	−1.79121	−	0.543358i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.04495	−	1.31662i	1.04495	−	1.31662i
$838$		0			0
$839$		0			0		−0.397748	−	0.917494i	$-0.630208\pi$
$839$		0			0		0.397748	+	0.917494i	$0.369792\pi$
$840$		0			0
$841$	−0.659346	−	0.751840i	−0.659346	−	0.751840i
$842$		0			0
$843$		0			0
$844$	1.93357	+	0.158544i	1.93357	+	0.158544i
$845$		0			0
$846$		0			0
$847$	−0.628535	+	1.13100i	−0.628535	+	1.13100i
$848$		0			0
$849$	−1.95694	+	0.290285i	−1.95694	+	0.290285i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.71723	+	0.226078i	1.71723	+	0.226078i	0.923880	−	0.382683i	$-0.125000\pi$
$853$	1.71723	+	0.226078i	1.71723	+	0.226078i	0.793353	+	0.608761i	$0.208333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.514103	−	0.857729i	$-0.671875\pi$
$857$		0			0		0.514103	+	0.857729i	$0.328125\pi$
$858$		0			0
$859$	1.04619	−	0.802769i	1.04619	−	0.802769i	0.0654031	−	0.997859i	$-0.479167\pi$
$859$	1.04619	−	0.802769i	1.04619	−	0.802769i	0.980785	+	0.195090i	$0.0625000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.762527	−	0.646956i	$-0.776042\pi$
$863$		0			0		0.762527	+	0.646956i	$0.223958\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$868$	−2.11812	+	0.493876i	−2.11812	+	0.493876i
$869$		0			0
$870$		0			0
$871$	−1.00400	+	3.30973i	−1.00400	+	3.30973i
$872$		0			0
$873$	0.226454	+	0.667112i	0.226454	+	0.667112i
$874$		0			0
$875$		0			0
$876$	−0.795390	+	1.83474i	−0.795390	+	1.83474i
$877$	1.45270	+	1.36060i	1.45270	+	1.36060i	0.793353	+	0.608761i	$0.208333\pi$
$877$	1.45270	+	1.36060i	1.45270	+	1.36060i	0.659346	+	0.751840i	$0.270833\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.367516	−	0.930017i	$-0.380208\pi$
$881$		0			0		−0.367516	+	0.930017i	$0.619792\pi$
$882$		0			0
$883$	1.64301	+	1.05936i	1.64301	+	1.05936i	0.935906	+	0.352250i	$0.114583\pi$
$883$	1.64301	+	1.05936i	1.64301	+	1.05936i	0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$887$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$888$		0			0
$889$	1.92034	−	0.795431i	1.92034	−	0.795431i
$890$		0			0
$891$		0			0
$892$	−0.654845	−	1.44527i	−0.654845	−	1.44527i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.956940	+	0.290285i	−0.956940	+	0.290285i
$901$		0			0
$902$		0			0
$903$	1.43291	−	0.117492i	1.43291	−	0.117492i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.338513	−	0.686437i	0.338513	−	0.686437i	−0.659346	−	0.751840i	$-0.729167\pi$
$907$	0.338513	−	0.686437i	0.338513	−	0.686437i	0.997859	+	0.0654031i	$0.0208333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$911$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$912$	−1.16897	+	0.753709i	−1.16897	+	0.753709i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.0768681	−	0.0610069i	0.0768681	−	0.0610069i
$917$		0			0
$918$		0			0
$919$	−0.783904	−	1.46658i	−0.783904	−	1.46658i	−0.881921	−	0.471397i	$-0.843750\pi$
$919$	−0.783904	−	1.46658i	−0.783904	−	1.46658i	0.0980171	−	0.995185i	$-0.468750\pi$
$920$		0			0
$921$	−0.721966	−	1.16102i	−0.721966	−	1.16102i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.484335	+	0.0397132i	0.484335	+	0.0397132i
$926$		0			0
$927$	−0.153888	+	0.0550620i	−0.153888	+	0.0550620i
$928$		0			0
$929$		0			0		−0.0163617	−	0.999866i	$-0.505208\pi$
$929$		0			0		0.0163617	+	0.999866i	$0.494792\pi$
$930$		0			0
$931$	0.286860	+	0.892794i	0.286860	+	0.892794i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.980785	+	1.19509i	0.980785	+	1.19509i	0.980785	+	0.195090i	$0.0625000\pi$
$937$	0.980785	+	1.19509i	0.980785	+	1.19509i			1.00000i	$0.5\pi$
$938$		0			0
$939$	1.12484	+	0.465926i	1.12484	+	0.465926i
$940$		0			0
$941$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$941$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.983846	−	0.179017i	$-0.942708\pi$
$947$		0			0		0.983846	+	0.179017i	$0.0572917\pi$
$948$	0.462665	−	1.84706i	0.462665	−	1.84706i
$949$	2.13021	−	3.07785i	2.13021	−	3.07785i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.973877	−	0.227076i	$-0.927083\pi$
$953$		0			0		0.973877	+	0.227076i	$0.0729167\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.33230	+	1.24784i	1.33230	+	1.24784i
$962$		0			0
$963$		0			0
$964$	−0.418862	+	1.17064i	−0.418862	+	1.17064i
$965$		0			0
$966$		0			0
$967$	1.24912	−	0.493617i	1.24912	−	0.493617i	0.352250	−	0.935906i	$-0.385417\pi$
$967$	1.24912	−	0.493617i	1.24912	−	0.493617i	0.896873	+	0.442289i	$0.145833\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.397748	−	0.917494i	$-0.369792\pi$
$971$		0			0		−0.397748	+	0.917494i	$0.630208\pi$
$972$	0.935906	+	0.352250i	0.935906	+	0.352250i
$973$	1.91944	+	1.52337i	1.91944	+	1.52337i
$974$		0			0
$975$	1.86280	−	0.183470i	1.86280	−	0.183470i
$976$	1.43236	−	0.117447i	1.43236	−	0.117447i
$977$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$977$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.543358	−	0.204506i	0.543358	−	0.204506i
$982$		0			0
$983$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$983$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	2.42128	−	0.956819i	2.42128	−	0.956819i
$989$		0			0
$990$		0			0
$991$	0.805730	+	1.63386i	0.805730	+	1.63386i	0.773010	+	0.634393i	$0.218750\pi$
$991$	0.805730	+	1.63386i	0.805730	+	1.63386i	0.0327191	+	0.999465i	$0.489583\pi$
$992$		0			0
$993$	−0.0280679	−	0.0168232i	−0.0280679	−	0.0168232i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.338513	−	0.817243i	−0.338513	−	0.817243i	−0.997859	−	0.0654031i	$-0.979167\pi$
$997$	−0.338513	−	0.817243i	−0.338513	−	0.817243i	0.659346	−	0.751840i	$-0.270833\pi$
$998$		0			0
$999$	−0.360073	−	0.326351i	−0.360073	−	0.326351i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3459.1.bm.a.1415.1	✓	64
3.2	odd	2	CM	3459.1.bm.a.1415.1	✓	64
1153.1131	even	192	inner	3459.1.bm.a.3437.1	yes	64
3459.3437	odd	192	inner	3459.1.bm.a.3437.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3459.1.bm.a.1415.1	✓	64	1.1	even	1	trivial
3459.1.bm.a.1415.1	✓	64	3.2	odd	2	CM
3459.1.bm.a.3437.1	yes	64	1153.1131	even	192	inner
3459.1.bm.a.3437.1	yes	64	3459.3437	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)

\(f(q)\)	\(=\)	\(q+(-0.910864 + 0.412707i) q^{3} +(0.793353 - 0.608761i) q^{4} +(1.29235 + 0.0634893i) q^{7} +(0.659346 - 0.751840i) q^{9} +O(q^{10})\)	\(q+(-0.910864 + 0.412707i) q^{3} +(0.793353 - 0.608761i) q^{4} +(1.29235 + 0.0634893i) q^{7} +(0.659346 - 0.751840i) q^{9} +(-0.471397 + 0.881921i) q^{12} +(1.18746 - 1.44693i) q^{13} +(0.258819 - 0.965926i) q^{16} +(0.553222 + 1.27613i) q^{19} +(-1.20336 + 0.475533i) q^{21} +(-0.352250 - 0.935906i) q^{25} +(-0.290285 + 0.956940i) q^{27} +(1.06394 - 0.736366i) q^{28} +(-1.56326 - 0.617756i) q^{31} +(0.0654031 - 0.997859i) q^{36} +(-0.207775 + 0.439303i) q^{37} +(-0.484461 + 1.80803i) q^{39} +(-1.06330 - 0.322547i) q^{43} +(0.162895 + 0.986643i) q^{48} +(0.670963 + 0.0660840i) q^{49} +(0.0612440 - 1.87081i) q^{52} +(-1.03058 - 0.934061i) q^{57} +(0.484166 + 1.35315i) q^{61} +(0.899842 - 0.929782i) q^{63} +(-0.382683 - 0.923880i) q^{64} +(-1.70711 + 0.707107i) q^{67} +(1.99304 - 0.163420i) q^{73} +(0.707107 + 0.707107i) q^{75} +(1.21576 + 0.675641i) q^{76} +(-1.84706 + 0.462665i) q^{79} +(-0.130526 - 0.991445i) q^{81} +(-0.665204 + 1.10983i) q^{84} +(1.62649 - 1.79455i) q^{91} +(1.67887 - 0.0824777i) q^{93} +(-0.352250 + 0.610115i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 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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