LMFDB - Embedded newform 1640.2.b.e.81.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1640,2,Mod(81,1640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("1640.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1640 = 2^{3} \cdot 5 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1640.b (of order $2$, degree $1$, 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:	$13.0954659315$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 21x^{8} + 168x^{6} + 631x^{4} + 1085x^{2} + 648 $ x^10 + 21x^8 + 168x^6 + 631x^4 + 1085x^2 + 648
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.6
Root			$2.55851i$ of defining polynomial
Character	$\chi$	$=$	1640.81
Dual form			1640.2.b.e.81.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.515459i q^{3} -1.00000 q^{5} -1.31235i q^{7} +2.73430 q^{9} +O(q^{10})$	$q+0.515459i q^{3} -1.00000 q^{5} -1.31235i q^{7} +2.73430 q^{9} +3.62264i q^{11} +1.31235i q^{13} -0.515459i q^{15} +0.515459i q^{17} -6.59846i q^{19} +0.676461 q^{21} +3.73430 q^{23} +1.00000 q^{25} +2.95580i q^{27} +3.75268i q^{29} -5.26855 q^{31} -1.86732 q^{33} +1.31235i q^{35} +7.00285 q^{37} -0.676461 q^{39} +(-4.77315 + 4.26814i) q^{41} +8.14507 q^{43} -2.73430 q^{45} +3.23723i q^{47} +5.27775 q^{49} -0.265698 q^{51} +7.24396i q^{53} -3.62264i q^{55} +3.40123 q^{57} +2.54344 q^{59} +2.59209 q^{61} -3.58835i q^{63} -1.31235i q^{65} +13.3612i q^{67} +1.92488i q^{69} -12.0159i q^{71} +10.7372 q^{73} +0.515459i q^{75} +4.75415 q^{77} -0.130049i q^{79} +6.67931 q^{81} +6.27775 q^{83} -0.515459i q^{85} -1.93435 q^{87} -6.58550i q^{89} +1.72225 q^{91} -2.71572i q^{93} +6.59846i q^{95} -0.118404i q^{97} +9.90538i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{5} - 14 q^{9}+O(q^{10}) $ 10 * q - 10 * q^5 - 14 * q^9	$ 10 q - 10 q^{5} - 14 q^{9} + 6 q^{21} - 4 q^{23} + 10 q^{25} - 22 q^{31} - 8 q^{33} - 2 q^{37} - 6 q^{39} + 4 q^{41} - 2 q^{43} + 14 q^{45} - 20 q^{49} - 44 q^{51} + 14 q^{57} - 6 q^{59} - 4 q^{61} - 6 q^{73} + 26 q^{77} - 6 q^{81} - 10 q^{83} + 74 q^{87} + 90 q^{91}+O(q^{100}) $ 10 * q - 10 * q^5 - 14 * q^9 + 6 * q^21 - 4 * q^23 + 10 * q^25 - 22 * q^31 - 8 * q^33 - 2 * q^37 - 6 * q^39 + 4 * q^41 - 2 * q^43 + 14 * q^45 - 20 * q^49 - 44 * q^51 + 14 * q^57 - 6 * q^59 - 4 * q^61 - 6 * q^73 + 26 * q^77 - 6 * q^81 - 10 * q^83 + 74 * q^87 + 90 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1640\mathbb{Z}\right)^\times$.

$n$	$657$	$821$	$1231$	$1441$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			0.515459i			0.297600i	0.988867	+	0.148800i	$0.0475411\pi$
$3$			0.515459i			0.297600i	−0.988867	+	0.148800i	$0.952459\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	1.31235i		−	0.496020i	−0.968757	−	0.248010i	$-0.920223\pi$
$7$		−	1.31235i		−	0.496020i	0.968757	−	0.248010i	$-0.0797767\pi$
$8$		0			0
$9$	2.73430			0.911434
$10$		0			0
$11$			3.62264i			1.09227i	0.837699	+	0.546133i	$0.183901\pi$
$11$			3.62264i			1.09227i	−0.837699	+	0.546133i	$0.816099\pi$
$12$		0			0
$13$			1.31235i			0.363980i	0.983300	+	0.181990i	$0.0582538\pi$
$13$			1.31235i			0.363980i	−0.983300	+	0.181990i	$0.941746\pi$
$14$		0			0
$15$		−	0.515459i		−	0.133091i
$16$		0			0
$17$			0.515459i			0.125017i	0.998044	+	0.0625086i	$0.0199101\pi$
$17$			0.515459i			0.125017i	−0.998044	+	0.0625086i	$0.980090\pi$
$18$		0			0
$19$		−	6.59846i		−	1.51379i	−0.653537	−	0.756895i	$-0.726715\pi$
$19$		−	6.59846i		−	1.51379i	0.653537	−	0.756895i	$-0.273285\pi$
$20$		0			0
$21$	0.676461			0.147616
$22$		0			0
$23$	3.73430			0.778656			0.389328	−	0.921099i	$-0.372707\pi$
$23$	3.73430			0.778656			0.389328	+	0.921099i	$0.372707\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			2.95580i			0.568843i
$28$		0			0
$29$			3.75268i			0.696856i	0.937336	+	0.348428i	$0.113284\pi$
$29$			3.75268i			0.696856i	−0.937336	+	0.348428i	$0.886716\pi$
$30$		0			0
$31$	−5.26855			−0.946260			−0.473130	−	0.880993i	$-0.656876\pi$
$31$	−5.26855			−0.946260			−0.473130	+	0.880993i	$0.656876\pi$
$32$		0			0
$33$	−1.86732			−0.325059
$34$		0			0
$35$			1.31235i			0.221827i
$36$		0			0
$37$	7.00285			1.15126			0.575631	−	0.817710i	$-0.304756\pi$
$37$	7.00285			1.15126			0.575631	+	0.817710i	$0.304756\pi$
$38$		0			0
$39$	−0.676461			−0.108320
$40$		0			0
$41$	−4.77315	+	4.26814i	−0.745441	+	0.666572i
$41$	−4.77315	+	4.26814i	−0.745441	+	0.666572i
$42$		0			0
$43$	8.14507			1.24211			0.621055	−	0.783767i	$-0.286704\pi$
$43$	8.14507			1.24211			0.621055	+	0.783767i	$0.286704\pi$
$44$		0			0
$45$	−2.73430			−0.407606
$46$		0			0
$47$			3.23723i			0.472198i	0.971729	+	0.236099i	$0.0758690\pi$
$47$			3.23723i			0.472198i	−0.971729	+	0.236099i	$0.924131\pi$
$48$		0			0
$49$	5.27775			0.753964
$50$		0			0
$51$	−0.265698			−0.0372051
$52$		0			0
$53$			7.24396i			0.995035i	0.867454	+	0.497518i	$0.165755\pi$
$53$			7.24396i			0.995035i	−0.867454	+	0.497518i	$0.834245\pi$
$54$		0			0
$55$		−	3.62264i		−	0.488476i
$56$		0			0
$57$	3.40123			0.450504
$58$		0			0
$59$	2.54344			0.331128			0.165564	−	0.986199i	$-0.447056\pi$
$59$	2.54344			0.331128			0.165564	+	0.986199i	$0.447056\pi$
$60$		0			0
$61$	2.59209			0.331883			0.165942	−	0.986136i	$-0.446934\pi$
$61$	2.59209			0.331883			0.165942	+	0.986136i	$0.446934\pi$
$62$		0			0
$63$		−	3.58835i		−	0.452090i
$64$		0			0
$65$		−	1.31235i		−	0.162777i
$66$		0			0
$67$			13.3612i			1.63234i	0.577815	+	0.816168i	$0.303906\pi$
$67$			13.3612i			1.63234i	−0.577815	+	0.816168i	$0.696094\pi$
$68$		0			0
$69$			1.92488i			0.231728i
$70$		0			0
$71$		−	12.0159i		−	1.42603i	−0.701150	−	0.713014i	$-0.747330\pi$
$71$		−	12.0159i		−	1.42603i	0.701150	−	0.713014i	$-0.252670\pi$
$72$		0			0
$73$	10.7372			1.25669			0.628345	−	0.777935i	$-0.283733\pi$
$73$	10.7372			1.25669			0.628345	+	0.777935i	$0.283733\pi$
$74$		0			0
$75$			0.515459i			0.0595201i
$76$		0			0
$77$	4.75415			0.541786
$78$		0			0
$79$		−	0.130049i		−	0.0146317i	−0.999973	−	0.00731585i	$-0.997671\pi$
$79$		−	0.130049i		−	0.0146317i	0.999973	−	0.00731585i	$-0.00232873\pi$
$80$		0			0
$81$	6.67931			0.742146
$82$		0			0
$83$	6.27775			0.689072			0.344536	−	0.938773i	$-0.388036\pi$
$83$	6.27775			0.689072			0.344536	+	0.938773i	$0.388036\pi$
$84$		0			0
$85$		−	0.515459i		−	0.0559094i
$86$		0			0
$87$	−1.93435			−0.207385
$88$		0			0
$89$		−	6.58550i		−	0.698062i	−0.937111	−	0.349031i	$-0.886511\pi$
$89$		−	6.58550i		−	0.698062i	0.937111	−	0.349031i	$-0.113489\pi$
$90$		0			0
$91$	1.72225			0.180541
$92$		0			0
$93$		−	2.71572i		−	0.281607i
$94$		0			0
$95$			6.59846i			0.676987i
$96$		0			0
$97$		−	0.118404i		−	0.0120221i	−0.999982	−	0.00601103i	$-0.998087\pi$
$97$		−	0.118404i		−	0.0120221i	0.999982	−	0.00601103i	$-0.00191338\pi$
$98$		0			0
$99$			9.90538i			0.995528i
$100$		0			0
$101$			15.9897i			1.59103i	0.605932	+	0.795516i	$0.292800\pi$
$101$			15.9897i			1.59103i	−0.605932	+	0.795516i	$0.707200\pi$
$102$		0			0
$103$	−7.81200			−0.769739			−0.384869	−	0.922971i	$-0.625754\pi$
$103$	−7.81200			−0.769739			−0.384869	+	0.922971i	$0.625754\pi$
$104$		0			0
$105$	−0.676461			−0.0660158
$106$		0			0
$107$	18.8272			1.82010			0.910049	−	0.414501i	$-0.136044\pi$
$107$	18.8272			1.82010			0.910049	+	0.414501i	$0.136044\pi$
$108$		0			0
$109$			13.4109i			1.28453i	0.766481	+	0.642266i	$0.222006\pi$
$109$			13.4109i			1.28453i	−0.766481	+	0.642266i	$0.777994\pi$
$110$		0			0
$111$			3.60968i			0.342616i
$112$		0			0
$113$	11.4136			1.07370			0.536851	−	0.843677i	$-0.319614\pi$
$113$	11.4136			1.07370			0.536851	+	0.843677i	$0.319614\pi$
$114$		0			0
$115$	−3.73430			−0.348225
$116$		0			0
$117$			3.58835i			0.331743i
$118$		0			0
$119$	0.676461			0.0620111
$120$		0			0
$121$	−2.12349			−0.193044
$122$		0			0
$123$	−2.20005	−	2.46036i	−0.198372	−	0.221843i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−18.6045			−1.65088			−0.825440	−	0.564490i	$-0.809073\pi$
$127$	−18.6045			−1.65088			−0.825440	+	0.564490i	$0.809073\pi$
$128$		0			0
$129$			4.19845i			0.369653i
$130$		0			0
$131$	−10.0901			−0.881574			−0.440787	−	0.897612i	$-0.645301\pi$
$131$	−10.0901			−0.881574			−0.440787	+	0.897612i	$0.645301\pi$
$132$		0			0
$133$	−8.65946			−0.750871
$134$		0			0
$135$		−	2.95580i		−	0.254394i
$136$		0			0
$137$			21.8516i			1.86691i	0.358697	+	0.933454i	$0.383221\pi$
$137$			21.8516i			1.86691i	−0.358697	+	0.933454i	$0.616779\pi$
$138$		0			0
$139$	−3.21990			−0.273109			−0.136554	−	0.990633i	$-0.543603\pi$
$139$	−3.21990			−0.273109			−0.136554	+	0.990633i	$0.543603\pi$
$140$		0			0
$141$	−1.66866			−0.140526
$142$		0			0
$143$	−4.75415			−0.397562
$144$		0			0
$145$		−	3.75268i		−	0.311644i
$146$		0			0
$147$			2.72046i			0.224380i
$148$		0			0
$149$			15.6845i			1.28492i	0.766318	+	0.642462i	$0.222087\pi$
$149$			15.6845i			1.28492i	−0.766318	+	0.642462i	$0.777913\pi$
$150$		0			0
$151$		−	20.9493i		−	1.70483i	−0.522868	−	0.852413i	$-0.675138\pi$
$151$		−	20.9493i		−	1.70483i	0.522868	−	0.852413i	$-0.324862\pi$
$152$		0			0
$153$			1.40942i			0.113945i
$154$		0			0
$155$	5.26855			0.423180
$156$		0			0
$157$		−	7.31497i		−	0.583798i	−0.956449	−	0.291899i	$-0.905713\pi$
$157$		−	7.31497i		−	0.583798i	0.956449	−	0.291899i	$-0.0942871\pi$
$158$		0			0
$159$	−3.73397			−0.296123
$160$		0			0
$161$		−	4.90070i		−	0.386229i
$162$		0			0
$163$	−12.7372			−0.997651			−0.498826	−	0.866702i	$-0.666235\pi$
$163$	−12.7372			−0.997651			−0.498826	+	0.866702i	$0.666235\pi$
$164$		0			0
$165$	1.86732			0.145371
$166$		0			0
$167$		−	6.98387i		−	0.540428i	−0.962800	−	0.270214i	$-0.912906\pi$
$167$		−	6.98387i		−	0.540428i	0.962800	−	0.270214i	$-0.0870944\pi$
$168$		0			0
$169$	11.2777			0.867519
$170$		0			0
$171$		−	18.0422i		−	1.37972i
$172$		0			0
$173$	−2.92516			−0.222396			−0.111198	−	0.993798i	$-0.535469\pi$
$173$	−2.92516			−0.222396			−0.111198	+	0.993798i	$0.535469\pi$
$174$		0			0
$175$		−	1.31235i		−	0.0992041i
$176$		0			0
$177$			1.31104i			0.0985438i
$178$		0			0
$179$		−	11.5702i		−	0.864794i	−0.901683	−	0.432397i	$-0.857668\pi$
$179$		−	11.5702i		−	0.864794i	0.901683	−	0.432397i	$-0.142332\pi$
$180$		0			0
$181$		−	20.1881i		−	1.50057i	−0.661113	−	0.750286i	$-0.729916\pi$
$181$		−	20.1881i		−	1.50057i	0.661113	−	0.750286i	$-0.270084\pi$
$182$		0			0
$183$			1.33612i			0.0987686i
$184$		0			0
$185$	−7.00285			−0.514860
$186$		0			0
$187$	−1.86732			−0.136552
$188$		0			0
$189$	3.87903			0.282158
$190$		0			0
$191$		−	4.69965i		−	0.340054i	−0.985439	−	0.170027i	$-0.945614\pi$
$191$		−	4.69965i		−	0.340054i	0.985439	−	0.170027i	$-0.0543856\pi$
$192$		0			0
$193$			15.2861i			1.10032i	0.835060	+	0.550160i	$0.185433\pi$
$193$			15.2861i			1.10032i	−0.835060	+	0.550160i	$0.814567\pi$
$194$		0			0
$195$	0.676461			0.0484424
$196$		0			0
$197$	4.04865			0.288454			0.144227	−	0.989545i	$-0.453930\pi$
$197$	4.04865			0.288454			0.144227	+	0.989545i	$0.453930\pi$
$198$		0			0
$199$			0.523837i			0.0371338i	0.999828	+	0.0185669i	$0.00591037\pi$
$199$			0.523837i			0.0371338i	−0.999828	+	0.0185669i	$0.994090\pi$
$200$		0			0
$201$	−6.88717			−0.485784
$202$		0			0
$203$	4.92482			0.345655
$204$		0			0
$205$	4.77315	−	4.26814i	0.333371	−	0.298100i
$206$		0			0
$207$	10.2107			0.709693
$208$		0			0
$209$	23.9038			1.65346
$210$		0			0
$211$		−	10.0223i		−	0.689964i	−0.938609	−	0.344982i	$-0.887885\pi$
$211$		−	10.0223i		−	0.689964i	0.938609	−	0.344982i	$-0.112115\pi$
$212$		0			0
$213$	6.19371			0.424386
$214$		0			0
$215$	−8.14507			−0.555489
$216$		0			0
$217$			6.91417i			0.469364i
$218$		0			0
$219$			5.53456i			0.373991i
$220$		0			0
$221$	−0.676461			−0.0455037
$222$		0			0
$223$	−11.6240			−0.778400			−0.389200	−	0.921153i	$-0.627249\pi$
$223$	−11.6240			−0.778400			−0.389200	+	0.921153i	$0.627249\pi$
$224$		0			0
$225$	2.73430			0.182287
$226$		0			0
$227$			14.3721i			0.953912i	0.878927	+	0.476956i	$0.158260\pi$
$227$			14.3721i			0.953912i	−0.878927	+	0.476956i	$0.841740\pi$
$228$		0			0
$229$		−	20.4683i		−	1.35258i	−0.736635	−	0.676290i	$-0.763587\pi$
$229$		−	20.4683i		−	1.35258i	0.736635	−	0.676290i	$-0.236413\pi$
$230$		0			0
$231$			2.45057i			0.161236i
$232$		0			0
$233$		−	26.5582i		−	1.73988i	−0.493155	−	0.869941i	$-0.664157\pi$
$233$		−	26.5582i		−	1.73988i	0.493155	−	0.869941i	$-0.335843\pi$
$234$		0			0
$235$		−	3.23723i		−	0.211173i
$236$		0			0
$237$	0.0670351			0.00435440
$238$		0			0
$239$			2.21191i			0.143076i	0.997438	+	0.0715382i	$0.0227908\pi$
$239$			2.21191i			0.143076i	−0.997438	+	0.0715382i	$0.977209\pi$
$240$		0			0
$241$	−19.9628			−1.28591			−0.642957	−	0.765902i	$-0.722293\pi$
$241$	−19.9628			−1.28591			−0.642957	+	0.765902i	$0.722293\pi$
$242$		0			0
$243$			12.3103i			0.789706i
$244$		0			0
$245$	−5.27775			−0.337183
$246$		0			0
$247$	8.65946			0.550988
$248$		0			0
$249$			3.23592i			0.205068i
$250$		0			0
$251$	−12.9185			−0.815408			−0.407704	−	0.913114i	$-0.633670\pi$
$251$	−12.9185			−0.815408			−0.407704	+	0.913114i	$0.633670\pi$
$252$		0			0
$253$			13.5280i			0.850499i
$254$		0			0
$255$	0.265698			0.0166386
$256$		0			0
$257$		−	6.11597i		−	0.381504i	−0.981638	−	0.190752i	$-0.938907\pi$
$257$		−	6.11597i		−	0.381504i	0.981638	−	0.190752i	$-0.0610926\pi$
$258$		0			0
$259$		−	9.19017i		−	0.571050i
$260$		0			0
$261$			10.2610i			0.635138i
$262$		0			0
$263$		−	13.1272i		−	0.809459i	−0.914437	−	0.404729i	$-0.867366\pi$
$263$		−	13.1272i		−	0.809459i	0.914437	−	0.404729i	$-0.132634\pi$
$264$		0			0
$265$		−	7.24396i		−	0.444993i
$266$		0			0
$267$	3.39456			0.207743
$268$		0			0
$269$	21.0823			1.28541			0.642704	−	0.766114i	$-0.277812\pi$
$269$	21.0823			1.28541			0.642704	+	0.766114i	$0.277812\pi$
$270$		0			0
$271$	−21.5690			−1.31022			−0.655112	−	0.755532i	$-0.727379\pi$
$271$	−21.5690			−1.31022			−0.655112	+	0.755532i	$0.727379\pi$
$272$		0			0
$273$			0.887751i			0.0537291i
$274$		0			0
$275$			3.62264i			0.218453i
$276$		0			0
$277$	−3.94501			−0.237033			−0.118516	−	0.992952i	$-0.537814\pi$
$277$	−3.94501			−0.237033			−0.118516	+	0.992952i	$0.537814\pi$
$278$		0			0
$279$	−14.4058			−0.862453
$280$		0			0
$281$		−	25.0577i		−	1.49482i	−0.664364	−	0.747409i	$-0.731298\pi$
$281$		−	25.0577i		−	1.49482i	0.664364	−	0.747409i	$-0.268702\pi$
$282$		0			0
$283$	−10.8698			−0.646145			−0.323072	−	0.946374i	$-0.604716\pi$
$283$	−10.8698			−0.646145			−0.323072	+	0.946374i	$0.604716\pi$
$284$		0			0
$285$	−3.40123			−0.201472
$286$		0			0
$287$	5.60128	+	6.26403i	0.330633	+	0.369754i
$288$		0			0
$289$	16.7343			0.984371
$290$		0			0
$291$	0.0610322			0.00357777
$292$		0			0
$293$			7.21562i			0.421541i	0.977536	+	0.210771i	$0.0675973\pi$
$293$			7.21562i			0.421541i	−0.977536	+	0.210771i	$0.932403\pi$
$294$		0			0
$295$	−2.54344			−0.148085
$296$		0			0
$297$	−10.7078			−0.621328
$298$		0			0
$299$			4.90070i			0.283415i
$300$		0			0
$301$		−	10.6892i		−	0.616112i
$302$		0			0
$303$	−8.24202			−0.473492
$304$		0			0
$305$	−2.59209			−0.148423
$306$		0			0
$307$	18.5072			1.05626			0.528130	−	0.849163i	$-0.322893\pi$
$307$	18.5072			1.05626			0.528130	+	0.849163i	$0.322893\pi$
$308$		0			0
$309$		−	4.02676i		−	0.229074i
$310$		0			0
$311$			11.7048i			0.663721i	0.943329	+	0.331860i	$0.107676\pi$
$311$			11.7048i			0.663721i	−0.943329	+	0.331860i	$0.892324\pi$
$312$		0			0
$313$			10.0167i			0.566178i	0.959094	+	0.283089i	$0.0913592\pi$
$313$			10.0167i			0.566178i	−0.959094	+	0.283089i	$0.908641\pi$
$314$		0			0
$315$			3.58835i			0.202181i
$316$		0			0
$317$		−	9.47160i		−	0.531978i	−0.963976	−	0.265989i	$-0.914301\pi$
$317$		−	9.47160i		−	0.531978i	0.963976	−	0.265989i	$-0.0856985\pi$
$318$		0			0
$319$	−13.5946			−0.761152
$320$		0			0
$321$			9.70466i			0.541662i
$322$		0			0
$323$	3.40123			0.189250
$324$		0			0
$325$			1.31235i			0.0727959i
$326$		0			0
$327$	−6.91278			−0.382277
$328$		0			0
$329$	4.24836			0.234220
$330$		0			0
$331$			2.68275i			0.147457i	0.997278	+	0.0737285i	$0.0234898\pi$
$331$			2.68275i			0.147457i	−0.997278	+	0.0737285i	$0.976510\pi$
$332$		0			0
$333$	19.1479			1.04930
$334$		0			0
$335$		−	13.3612i		−	0.730003i
$336$		0			0
$337$	−13.5463			−0.737914			−0.368957	−	0.929446i	$-0.620285\pi$
$337$	−13.5463			−0.737914			−0.368957	+	0.929446i	$0.620285\pi$
$338$		0			0
$339$			5.88325i			0.319534i
$340$		0			0
$341$		−	19.0860i		−	1.03357i
$342$		0			0
$343$		−	16.1127i		−	0.870002i
$344$		0			0
$345$		−	1.92488i		−	0.103632i
$346$		0			0
$347$			0.892490i			0.0479114i	0.999713	+	0.0239557i	$0.00762606\pi$
$347$			0.892490i			0.0479114i	−0.999713	+	0.0239557i	$0.992374\pi$
$348$		0			0
$349$	3.18418			0.170445			0.0852227	−	0.996362i	$-0.472840\pi$
$349$	3.18418			0.170445			0.0852227	+	0.996362i	$0.472840\pi$
$350$		0			0
$351$	−3.87903			−0.207047
$352$		0			0
$353$	−24.5679			−1.30762			−0.653808	−	0.756660i	$-0.726830\pi$
$353$	−24.5679			−1.30762			−0.653808	+	0.756660i	$0.726830\pi$
$354$		0			0
$355$			12.0159i			0.637739i
$356$		0			0
$357$			0.348688i			0.0184545i
$358$		0			0
$359$	3.23917			0.170957			0.0854784	−	0.996340i	$-0.472758\pi$
$359$	3.23917			0.170957			0.0854784	+	0.996340i	$0.472758\pi$
$360$		0			0
$361$	−24.5396			−1.29156
$362$		0			0
$363$		−	1.09457i		−	0.0574500i
$364$		0			0
$365$	−10.7372			−0.562008
$366$		0			0
$367$	37.6024			1.96283			0.981414	−	0.191902i	$-0.0614655\pi$
$367$	37.6024			1.96283			0.981414	+	0.191902i	$0.0614655\pi$
$368$		0			0
$369$	−13.0512	+	11.6704i	−0.679420	+	0.607536i
$370$		0			0
$371$	9.50659			0.493558
$372$		0			0
$373$	−7.60128			−0.393580			−0.196790	−	0.980446i	$-0.563052\pi$
$373$	−7.60128			−0.393580			−0.196790	+	0.980446i	$0.563052\pi$
$374$		0			0
$375$		−	0.515459i		−	0.0266182i
$376$		0			0
$377$	−4.92482			−0.253641
$378$		0			0
$379$	−25.8914			−1.32995			−0.664976	−	0.746864i	$-0.731558\pi$
$379$	−25.8914			−1.32995			−0.664976	+	0.746864i	$0.731558\pi$
$380$		0			0
$381$		−	9.58984i		−	0.491302i
$382$		0			0
$383$		−	35.2575i		−	1.80157i	−0.434263	−	0.900786i	$-0.642991\pi$
$383$		−	35.2575i		−	1.80157i	0.434263	−	0.900786i	$-0.357009\pi$
$384$		0			0
$385$	−4.75415			−0.242294
$386$		0			0
$387$	22.2711			1.13210
$388$		0			0
$389$	−24.0561			−1.21969			−0.609846	−	0.792520i	$-0.708769\pi$
$389$	−24.0561			−1.21969			−0.609846	+	0.792520i	$0.708769\pi$
$390$		0			0
$391$			1.92488i			0.0973453i
$392$		0			0
$393$		−	5.20102i		−	0.262357i
$394$		0			0
$395$			0.130049i			0.00654350i
$396$		0			0
$397$		−	7.98971i		−	0.400992i	−0.979695	−	0.200496i	$-0.935745\pi$
$397$		−	7.98971i		−	0.400992i	0.979695	−	0.200496i	$-0.0642553\pi$
$398$		0			0
$399$		−	4.46360i		−	0.223459i
$400$		0			0
$401$	−4.02271			−0.200884			−0.100442	−	0.994943i	$-0.532026\pi$
$401$	−4.02271			−0.200884			−0.100442	+	0.994943i	$0.532026\pi$
$402$		0			0
$403$		−	6.91417i		−	0.344419i
$404$		0			0
$405$	−6.67931			−0.331898
$406$		0			0
$407$			25.3688i			1.25748i
$408$		0			0
$409$	−31.5587			−1.56048			−0.780238	−	0.625482i	$-0.784902\pi$
$409$	−31.5587			−1.56048			−0.780238	+	0.625482i	$0.784902\pi$
$410$		0			0
$411$	−11.2636			−0.555593
$412$		0			0
$413$		−	3.33788i		−	0.164246i
$414$		0			0
$415$	−6.27775			−0.308162
$416$		0			0
$417$		−	1.65973i		−	0.0812772i
$418$		0			0
$419$	7.21957			0.352699			0.176349	−	0.984328i	$-0.443571\pi$
$419$	7.21957			0.352699			0.176349	+	0.984328i	$0.443571\pi$
$420$		0			0
$421$		−	18.3175i		−	0.892740i	−0.894848	−	0.446370i	$-0.852716\pi$
$421$		−	18.3175i		−	0.892740i	0.894848	−	0.446370i	$-0.147284\pi$
$422$		0			0
$423$			8.85155i			0.430377i
$424$		0			0
$425$			0.515459i			0.0250034i
$426$		0			0
$427$		−	3.40172i		−	0.164621i
$428$		0			0
$429$		−	2.45057i		−	0.118315i
$430$		0			0
$431$	−22.0664			−1.06290			−0.531451	−	0.847089i	$-0.678353\pi$
$431$	−22.0664			−1.06290			−0.531451	+	0.847089i	$0.678353\pi$
$432$		0			0
$433$	9.13587			0.439042			0.219521	−	0.975608i	$-0.429551\pi$
$433$	9.13587			0.439042			0.219521	+	0.975608i	$0.429551\pi$
$434$		0			0
$435$	1.93435			0.0927452
$436$		0			0
$437$		−	24.6406i		−	1.17872i
$438$		0			0
$439$		−	40.0754i		−	1.91269i	−0.292236	−	0.956346i	$-0.594399\pi$
$439$		−	40.0754i		−	1.91269i	0.292236	−	0.956346i	$-0.405601\pi$
$440$		0			0
$441$	14.4310			0.687188
$442$		0			0
$443$	19.8730			0.944196			0.472098	−	0.881546i	$-0.343497\pi$
$443$	19.8730			0.944196			0.472098	+	0.881546i	$0.343497\pi$
$444$		0			0
$445$			6.58550i			0.312183i
$446$		0			0
$447$	−8.08471			−0.382393
$448$		0			0
$449$	2.07135			0.0977532			0.0488766	−	0.998805i	$-0.484436\pi$
$449$	2.07135			0.0977532			0.0488766	+	0.998805i	$0.484436\pi$
$450$		0			0
$451$	−15.4619	−	17.2914i	−0.728074	−	0.814219i
$452$		0			0
$453$	10.7985			0.507357
$454$		0			0
$455$	−1.72225			−0.0807405
$456$		0			0
$457$		−	3.47126i		−	0.162378i	−0.996699	−	0.0811892i	$-0.974128\pi$
$457$		−	3.47126i		−	0.162378i	0.996699	−	0.0811892i	$-0.0258718\pi$
$458$		0			0
$459$	−1.52359			−0.0711152
$460$		0			0
$461$	−22.7849			−1.06120			−0.530600	−	0.847622i	$-0.678033\pi$
$461$	−22.7849			−1.06120			−0.530600	+	0.847622i	$0.678033\pi$
$462$		0			0
$463$		−	10.9989i		−	0.511163i	−0.966787	−	0.255582i	$-0.917733\pi$
$463$		−	10.9989i		−	0.511163i	0.966787	−	0.255582i	$-0.0822670\pi$
$464$		0			0
$465$			2.71572i			0.125939i
$466$		0			0
$467$	30.0365			1.38992			0.694961	−	0.719047i	$-0.255421\pi$
$467$	30.0365			1.38992			0.694961	+	0.719047i	$0.255421\pi$
$468$		0			0
$469$	17.5346			0.809672
$470$		0			0
$471$	3.77057			0.173738
$472$		0			0
$473$			29.5066i			1.35671i
$474$		0			0
$475$		−	6.59846i		−	0.302758i
$476$		0			0
$477$			19.8072i			0.906909i
$478$		0			0
$479$			29.6254i			1.35362i	0.736157	+	0.676811i	$0.236638\pi$
$479$			29.6254i			1.35362i	−0.736157	+	0.676811i	$0.763362\pi$
$480$		0			0
$481$			9.19017i			0.419036i
$482$		0			0
$483$	2.52611			0.114942
$484$		0			0
$485$			0.118404i			0.00537643i
$486$		0			0
$487$	−7.20324			−0.326410			−0.163205	−	0.986592i	$-0.552183\pi$
$487$	−7.20324			−0.326410			−0.163205	+	0.986592i	$0.552183\pi$
$488$		0			0
$489$		−	6.56548i		−	0.296901i
$490$		0			0
$491$	−15.5360			−0.701129			−0.350564	−	0.936539i	$-0.614010\pi$
$491$	−15.5360			−0.701129			−0.350564	+	0.936539i	$0.614010\pi$
$492$		0			0
$493$	−1.93435			−0.0871189
$494$		0			0
$495$		−	9.90538i		−	0.445214i
$496$		0			0
$497$	−15.7691			−0.707339
$498$		0			0
$499$		−	17.4083i		−	0.779303i	−0.920962	−	0.389651i	$-0.872595\pi$
$499$		−	17.4083i		−	0.779303i	0.920962	−	0.389651i	$-0.127405\pi$
$500$		0			0
$501$	3.59990			0.160831
$502$		0			0
$503$			36.1941i			1.61381i	0.590678	+	0.806907i	$0.298860\pi$
$503$			36.1941i			1.61381i	−0.590678	+	0.806907i	$0.701140\pi$
$504$		0			0
$505$		−	15.9897i		−	0.711532i
$506$		0			0
$507$			5.81321i			0.258174i
$508$		0			0
$509$			17.2097i			0.762806i	0.924409	+	0.381403i	$0.124559\pi$
$509$			17.2097i			0.762806i	−0.924409	+	0.381403i	$0.875441\pi$
$510$		0			0
$511$		−	14.0909i		−	0.623344i
$512$		0			0
$513$	19.5037			0.861109
$514$		0			0
$515$	7.81200			0.344238
$516$		0			0
$517$	−11.7273			−0.515766
$518$		0			0
$519$		−	1.50780i		−	0.0661851i
$520$		0			0
$521$		−	7.50537i		−	0.328816i	−0.986392	−	0.164408i	$-0.947429\pi$
$521$		−	7.50537i		−	0.328816i	0.986392	−	0.164408i	$-0.0525714\pi$
$522$		0			0
$523$	11.5293			0.504141			0.252071	−	0.967709i	$-0.418888\pi$
$523$	11.5293			0.504141			0.252071	+	0.967709i	$0.418888\pi$
$524$		0			0
$525$	0.676461			0.0295232
$526$		0			0
$527$		−	2.71572i		−	0.118299i
$528$		0			0
$529$	−9.05499			−0.393695
$530$		0			0
$531$	6.95454			0.301801
$532$		0			0
$533$	−5.60128	−	6.26403i	−0.242619	−	0.271325i
$534$		0			0
$535$	−18.8272			−0.813972
$536$		0			0
$537$	5.96394			0.257363
$538$		0			0
$539$			19.1193i			0.823529i
$540$		0			0
$541$	−26.0607			−1.12044			−0.560218	−	0.828345i	$-0.689283\pi$
$541$	−26.0607			−1.12044			−0.560218	+	0.828345i	$0.689283\pi$
$542$		0			0
$543$	10.4061			0.446571
$544$		0			0
$545$		−	13.4109i		−	0.574461i
$546$		0			0
$547$		−	12.4924i		−	0.534135i	−0.963678	−	0.267067i	$-0.913945\pi$
$547$		−	12.4924i		−	0.534135i	0.963678	−	0.267067i	$-0.0860546\pi$
$548$		0			0
$549$	7.08756			0.302490
$550$		0			0
$551$	24.7619			1.05489
$552$		0			0
$553$	−0.170670			−0.00725763
$554$		0			0
$555$		−	3.60968i		−	0.153223i
$556$		0			0
$557$		−	7.78680i		−	0.329937i	−0.986299	−	0.164969i	$-0.947248\pi$
$557$		−	7.78680i		−	0.329937i	0.986299	−	0.164969i	$-0.0527523\pi$
$558$		0			0
$559$			10.6892i			0.452103i
$560$		0			0
$561$		−	0.962526i		−	0.0406379i
$562$		0			0
$563$		−	13.8122i		−	0.582116i	−0.956705	−	0.291058i	$-0.905993\pi$
$563$		−	13.8122i		−	0.582116i	0.956705	−	0.291058i	$-0.0940072\pi$
$564$		0			0
$565$	−11.4136			−0.480174
$566$		0			0
$567$		−	8.76558i		−	0.368120i
$568$		0			0
$569$	35.8177			1.50156			0.750778	−	0.660554i	$-0.229679\pi$
$569$	35.8177			1.50156			0.750778	+	0.660554i	$0.229679\pi$
$570$		0			0
$571$			17.9215i			0.749990i	0.927027	+	0.374995i	$0.122356\pi$
$571$			17.9215i			0.749990i	−0.927027	+	0.374995i	$0.877644\pi$
$572$		0			0
$573$	2.42247			0.101200
$574$		0			0
$575$	3.73430			0.155731
$576$		0			0
$577$		−	13.1554i		−	0.547666i	−0.961777	−	0.273833i	$-0.911708\pi$
$577$		−	13.1554i		−	0.547666i	0.961777	−	0.273833i	$-0.0882916\pi$
$578$		0			0
$579$	−7.87937			−0.327455
$580$		0			0
$581$		−	8.23858i		−	0.341794i
$582$		0			0
$583$	−26.2422			−1.08684
$584$		0			0
$585$		−	3.58835i		−	0.148360i
$586$		0			0
$587$			40.3548i			1.66562i	0.553559	+	0.832810i	$0.313269\pi$
$587$			40.3548i			1.66562i	−0.553559	+	0.832810i	$0.686731\pi$
$588$		0			0
$589$			34.7643i			1.43244i
$590$		0			0
$591$			2.08691i			0.0858441i
$592$		0			0
$593$		−	16.6231i		−	0.682628i	−0.939949	−	0.341314i	$-0.889128\pi$
$593$		−	16.6231i		−	0.682628i	0.939949	−	0.341314i	$-0.110872\pi$
$594$		0			0
$595$	−0.676461			−0.0277322
$596$		0			0
$597$	−0.270016			−0.0110510
$598$		0			0
$599$	9.14758			0.373760			0.186880	−	0.982383i	$-0.440162\pi$
$599$	9.14758			0.373760			0.186880	+	0.982383i	$0.440162\pi$
$600$		0			0
$601$			29.3982i			1.19918i	0.800309	+	0.599588i	$0.204669\pi$
$601$			29.3982i			1.19918i	−0.800309	+	0.599588i	$0.795331\pi$
$602$		0			0
$603$			36.5337i			1.48777i
$604$		0			0
$605$	2.12349			0.0863320
$606$		0			0
$607$	6.28409			0.255063			0.127532	−	0.991835i	$-0.459295\pi$
$607$	6.28409			0.255063			0.127532	+	0.991835i	$0.459295\pi$
$608$		0			0
$609$			2.53854i			0.102867i
$610$		0			0
$611$	−4.24836			−0.171870
$612$		0			0
$613$	−32.4061			−1.30887			−0.654436	−	0.756117i	$-0.727094\pi$
$613$	−32.4061			−1.30887			−0.654436	+	0.756117i	$0.727094\pi$
$614$		0			0
$615$	2.20005	+	2.46036i	0.0887147	+	0.0992114i
$616$		0			0
$617$	21.3586			0.859866			0.429933	−	0.902861i	$-0.358537\pi$
$617$	21.3586			0.859866			0.429933	+	0.902861i	$0.358537\pi$
$618$		0			0
$619$	−6.21705			−0.249884			−0.124942	−	0.992164i	$-0.539875\pi$
$619$	−6.21705			−0.249884			−0.124942	+	0.992164i	$0.539875\pi$
$620$		0			0
$621$			11.0378i			0.442933i
$622$		0			0
$623$	−8.64247			−0.346253
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			12.3214i			0.492070i
$628$		0			0
$629$			3.60968i			0.143927i
$630$		0			0
$631$	2.39204			0.0952256			0.0476128	−	0.998866i	$-0.484839\pi$
$631$	2.39204			0.0952256			0.0476128	+	0.998866i	$0.484839\pi$
$632$		0			0
$633$	5.16609			0.205334
$634$		0			0
$635$	18.6045			0.738296
$636$		0			0
$637$			6.92623i			0.274427i
$638$		0			0
$639$		−	32.8552i		−	1.29973i
$640$		0			0
$641$		−	45.6940i		−	1.80481i	−0.430893	−	0.902403i	$-0.641801\pi$
$641$		−	45.6940i		−	1.80481i	0.430893	−	0.902403i	$-0.358199\pi$
$642$		0			0
$643$			24.3393i			0.959850i	0.877310	+	0.479925i	$0.159336\pi$
$643$			24.3393i			0.959850i	−0.877310	+	0.479925i	$0.840664\pi$
$644$		0			0
$645$		−	4.19845i		−	0.165314i
$646$		0			0
$647$	4.71789			0.185479			0.0927397	−	0.995690i	$-0.470438\pi$
$647$	4.71789			0.185479			0.0927397	+	0.995690i	$0.470438\pi$
$648$		0			0
$649$			9.21397i			0.361680i
$650$		0			0
$651$	−3.56397			−0.139683
$652$		0			0
$653$			47.0982i			1.84310i	0.388264	+	0.921548i	$0.373075\pi$
$653$			47.0982i			1.84310i	−0.388264	+	0.921548i	$0.626925\pi$
$654$		0			0
$655$	10.0901			0.394252
$656$		0			0
$657$	29.3586			1.14539
$658$		0			0
$659$		−	4.92780i		−	0.191960i	−0.995383	−	0.0959798i	$-0.969402\pi$
$659$		−	4.92780i		−	0.191960i	0.995383	−	0.0959798i	$-0.0305984\pi$
$660$		0			0
$661$	−12.5849			−0.489495			−0.244747	−	0.969587i	$-0.578705\pi$
$661$	−12.5849			−0.489495			−0.244747	+	0.969587i	$0.578705\pi$
$662$		0			0
$663$		−	0.348688i		−	0.0135419i
$664$		0			0
$665$	8.65946			0.335800
$666$		0			0
$667$			14.0137i			0.542611i
$668$		0			0
$669$		−	5.99169i		−	0.231652i
$670$		0			0
$671$			9.39020i			0.362505i
$672$		0			0
$673$		−	8.84052i		−	0.340777i	−0.985377	−	0.170388i	$-0.945498\pi$
$673$		−	8.84052i		−	0.340777i	0.985377	−	0.170388i	$-0.0545023\pi$
$674$		0			0
$675$			2.95580i			0.113769i
$676$		0			0
$677$	−22.2164			−0.853846			−0.426923	−	0.904288i	$-0.640403\pi$
$677$	−22.2164			−0.853846			−0.426923	+	0.904288i	$0.640403\pi$
$678$		0			0
$679$	−0.155387			−0.00596319
$680$		0			0
$681$	−7.40825			−0.283885
$682$		0			0
$683$		−	8.62635i		−	0.330078i	−0.986287	−	0.165039i	$-0.947225\pi$
$683$		−	8.62635i		−	0.330078i	0.986287	−	0.165039i	$-0.0527750\pi$
$684$		0			0
$685$		−	21.8516i		−	0.834907i
$686$		0			0
$687$	10.5505			0.402528
$688$		0			0
$689$	−9.50659			−0.362172
$690$		0			0
$691$			20.1291i			0.765746i	0.923801	+	0.382873i	$0.125065\pi$
$691$			20.1291i			0.765746i	−0.923801	+	0.382873i	$0.874935\pi$
$692$		0			0
$693$	12.9993			0.493802
$694$		0			0
$695$	3.21990			0.122138
$696$		0			0
$697$	−2.20005	−	2.46036i	−0.0833329	−	0.0931928i
$698$		0			0
$699$	13.6896			0.517790
$700$		0			0
$701$	−14.3267			−0.541113			−0.270557	−	0.962704i	$-0.587208\pi$
$701$	−14.3267			−0.541113			−0.270557	+	0.962704i	$0.587208\pi$
$702$		0			0
$703$		−	46.2080i		−	1.74277i
$704$		0			0
$705$	1.66866			0.0628452
$706$		0			0
$707$	20.9840			0.789185
$708$		0			0
$709$		−	4.59567i		−	0.172594i	−0.996269	−	0.0862969i	$-0.972497\pi$
$709$		−	4.59567i		−	0.172594i	0.996269	−	0.0862969i	$-0.0275034\pi$
$710$		0			0
$711$		−	0.355595i		−	0.0133358i
$712$		0			0
$713$	−19.6744			−0.736811
$714$		0			0
$715$	4.75415			0.177795
$716$		0			0
$717$	−1.14015			−0.0425796
$718$		0			0
$719$			16.9601i			0.632505i	0.948675	+	0.316253i	$0.102425\pi$
$719$			16.9601i			0.632505i	−0.948675	+	0.316253i	$0.897575\pi$
$720$		0			0
$721$			10.2520i			0.381806i
$722$		0			0
$723$		−	10.2900i		−	0.382689i
$724$		0			0
$725$			3.75268i			0.139371i
$726$		0			0
$727$			47.0544i			1.74515i	0.488478	+	0.872576i	$0.337552\pi$
$727$			47.0544i			1.74515i	−0.488478	+	0.872576i	$0.662448\pi$
$728$		0			0
$729$	13.6925			0.507129
$730$		0			0
$731$			4.19845i			0.155285i
$732$		0			0
$733$	38.9965			1.44037			0.720185	−	0.693783i	$-0.244057\pi$
$733$	38.9965			1.44037			0.720185	+	0.693783i	$0.244057\pi$
$734$		0			0
$735$		−	2.72046i		−	0.100346i
$736$		0			0
$737$	−48.4029			−1.78294
$738$		0			0
$739$	34.3218			1.26255			0.631274	−	0.775560i	$-0.282533\pi$
$739$	34.3218			1.26255			0.631274	+	0.775560i	$0.282533\pi$
$740$		0			0
$741$			4.46360i			0.163974i
$742$		0			0
$743$	−7.02158			−0.257597			−0.128798	−	0.991671i	$-0.541112\pi$
$743$	−7.02158			−0.257597			−0.128798	+	0.991671i	$0.541112\pi$
$744$		0			0
$745$		−	15.6845i		−	0.574635i
$746$		0			0
$747$	17.1653			0.628044
$748$		0			0
$749$		−	24.7079i		−	0.902806i
$750$		0			0
$751$		−	21.7690i		−	0.794360i	−0.917741	−	0.397180i	$-0.869989\pi$
$751$		−	21.7690i		−	0.794360i	0.917741	−	0.397180i	$-0.130011\pi$
$752$		0			0
$753$		−	6.65895i		−	0.242666i
$754$		0			0
$755$			20.9493i			0.762422i
$756$		0			0
$757$			29.3509i			1.06678i	0.845870	+	0.533389i	$0.179082\pi$
$757$			29.3509i			1.06678i	−0.845870	+	0.533389i	$0.820918\pi$
$758$		0			0
$759$	−6.97314			−0.253109
$760$		0			0
$761$	46.3079			1.67866			0.839330	−	0.543623i	$-0.182948\pi$
$761$	46.3079			1.67866			0.839330	+	0.543623i	$0.182948\pi$
$762$		0			0
$763$	17.5998			0.637155
$764$		0			0
$765$		−	1.40942i		−	0.0509577i
$766$		0			0
$767$			3.33788i			0.120524i
$768$		0			0
$769$	−40.3569			−1.45531			−0.727653	−	0.685945i	$-0.759389\pi$
$769$	−40.3569			−1.45531			−0.727653	+	0.685945i	$0.759389\pi$
$770$		0			0
$771$	3.15253			0.113536
$772$		0			0
$773$		−	42.8050i		−	1.53959i	−0.638292	−	0.769794i	$-0.720359\pi$
$773$		−	42.8050i		−	1.53959i	0.638292	−	0.769794i	$-0.279641\pi$
$774$		0			0
$775$	−5.26855			−0.189252
$776$		0			0
$777$	4.73716			0.169945
$778$		0			0
$779$	28.1632	+	31.4954i	1.00905	+	1.12844i
$780$		0			0
$781$	43.5293			1.55760
$782$		0			0
$783$	−11.0922			−0.396402
$784$		0			0
$785$			7.31497i			0.261082i
$786$		0			0
$787$	34.1508			1.21734			0.608672	−	0.793422i	$-0.291703\pi$
$787$	34.1508			1.21734			0.608672	+	0.793422i	$0.291703\pi$
$788$		0			0
$789$	6.76654			0.240895
$790$		0			0
$791$		−	14.9786i		−	0.532579i
$792$		0			0
$793$			3.40172i			0.120799i
$794$		0			0
$795$	3.73397			0.132430
$796$		0			0
$797$	10.0319			0.355350			0.177675	−	0.984089i	$-0.443142\pi$
$797$	10.0319			0.355350			0.177675	+	0.984089i	$0.443142\pi$
$798$		0			0
$799$	−1.66866			−0.0590328
$800$		0			0
$801$		−	18.0068i		−	0.636237i
$802$		0			0
$803$			38.8968i			1.37264i
$804$		0			0
$805$			4.90070i			0.172727i
$806$		0			0
$807$			10.8670i			0.382538i
$808$		0			0
$809$			50.0745i			1.76053i	0.474485	+	0.880264i	$0.342634\pi$
$809$			50.0745i			1.76053i	−0.474485	+	0.880264i	$0.657366\pi$
$810$		0			0
$811$	16.9792			0.596220			0.298110	−	0.954532i	$-0.403644\pi$
$811$	16.9792			0.596220			0.298110	+	0.954532i	$0.403644\pi$
$812$		0			0
$813$		−	11.1179i		−	0.389923i
$814$		0			0
$815$	12.7372			0.446163
$816$		0			0
$817$		−	53.7449i		−	1.88029i
$818$		0			0
$819$	4.70916			0.164551
$820$		0			0
$821$	−17.7213			−0.618477			−0.309239	−	0.950984i	$-0.600074\pi$
$821$	−17.7213			−0.618477			−0.309239	+	0.950984i	$0.600074\pi$
$822$		0			0
$823$			12.6754i			0.441837i	0.975292	+	0.220918i	$0.0709055\pi$
$823$			12.6754i			0.441837i	−0.975292	+	0.220918i	$0.929095\pi$
$824$		0			0
$825$	−1.86732			−0.0650117
$826$		0			0
$827$			38.8299i			1.35025i	0.737705	+	0.675124i	$0.235910\pi$
$827$			38.8299i			1.35025i	−0.737705	+	0.675124i	$0.764090\pi$
$828$		0			0
$829$	18.4352			0.640281			0.320140	−	0.947370i	$-0.396270\pi$
$829$	18.4352			0.640281			0.320140	+	0.947370i	$0.396270\pi$
$830$		0			0
$831$		−	2.03349i		−	0.0705410i
$832$		0			0
$833$			2.72046i			0.0942584i
$834$		0			0
$835$			6.98387i			0.241687i
$836$		0			0
$837$		−	15.5728i		−	0.538274i
$838$		0			0
$839$		−	14.2636i		−	0.492434i	−0.969215	−	0.246217i	$-0.920812\pi$
$839$		−	14.2636i		−	0.492434i	0.969215	−	0.246217i	$-0.0791875\pi$
$840$		0			0
$841$	14.9174			0.514392
$842$		0			0
$843$	12.9162			0.444858
$844$		0			0
$845$	−11.2777			−0.387966
$846$		0			0
$847$			2.78675i			0.0957539i
$848$		0			0
$849$		−	5.60295i		−	0.192293i
$850$		0			0
$851$	26.1508			0.896437
$852$		0			0
$853$	3.06989			0.105111			0.0525555	−	0.998618i	$-0.483263\pi$
$853$	3.06989			0.105111			0.0525555	+	0.998618i	$0.483263\pi$
$854$		0			0
$855$			18.0422i			0.617029i
$856$		0			0
$857$	−8.83475			−0.301789			−0.150895	−	0.988550i	$-0.548215\pi$
$857$	−8.83475			−0.301789			−0.150895	+	0.988550i	$0.548215\pi$
$858$		0			0
$859$	−41.3303			−1.41017			−0.705087	−	0.709121i	$-0.749092\pi$
$859$	−41.3303			−1.41017			−0.705087	+	0.709121i	$0.749092\pi$
$860$		0			0
$861$	−3.22885	+	2.88723i	−0.110039	+	0.0983966i
$862$		0			0
$863$	−2.42281			−0.0824734			−0.0412367	−	0.999149i	$-0.513130\pi$
$863$	−2.42281			−0.0824734			−0.0412367	+	0.999149i	$0.513130\pi$
$864$		0			0
$865$	2.92516			0.0994585
$866$		0			0
$867$			8.62584i			0.292949i
$868$		0			0
$869$	0.471122			0.0159817
$870$		0			0
$871$	−17.5346			−0.594137
$872$		0			0
$873$		−	0.323751i		−	0.0109573i
$874$		0			0
$875$			1.31235i			0.0443654i
$876$		0			0
$877$	−10.0544			−0.339511			−0.169756	−	0.985486i	$-0.554298\pi$
$877$	−10.0544			−0.339511			−0.169756	+	0.985486i	$0.554298\pi$
$878$		0			0
$879$	−3.71936			−0.125451
$880$		0			0
$881$	1.76658			0.0595177			0.0297589	−	0.999557i	$-0.490526\pi$
$881$	1.76658			0.0595177			0.0297589	+	0.999557i	$0.490526\pi$
$882$		0			0
$883$			15.6089i			0.525282i	0.964894	+	0.262641i	$0.0845934\pi$
$883$			15.6089i			0.525282i	−0.964894	+	0.262641i	$0.915407\pi$
$884$		0			0
$885$		−	1.31104i		−	0.0440701i
$886$		0			0
$887$		−	12.6095i		−	0.423384i	−0.977336	−	0.211692i	$-0.932103\pi$
$887$		−	12.6095i		−	0.423384i	0.977336	−	0.211692i	$-0.0678974\pi$
$888$		0			0
$889$			24.4155i			0.818870i
$890$		0			0
$891$			24.1967i			0.810621i
$892$		0			0
$893$	21.3607			0.714808
$894$		0			0
$895$			11.5702i			0.386748i
$896$		0			0
$897$	−2.52611			−0.0843443
$898$		0			0
$899$		−	19.7712i		−	0.659407i
$900$		0			0
$901$	−3.73397			−0.124396
$902$		0			0
$903$	5.50982			0.183355
$904$		0			0
$905$			20.1881i			0.671076i
$906$		0			0
$907$	22.5782			0.749697			0.374848	−	0.927086i	$-0.377695\pi$
$907$	22.5782			0.749697			0.374848	+	0.927086i	$0.377695\pi$
$908$		0			0
$909$			43.7206i			1.45012i
$910$		0			0
$911$	35.6847			1.18229			0.591143	−	0.806567i	$-0.298677\pi$
$911$	35.6847			1.18229			0.591143	+	0.806567i	$0.298677\pi$
$912$		0			0
$913$			22.7420i			0.752650i
$914$		0			0
$915$		−	1.33612i		−	0.0441706i
$916$		0			0
$917$			13.2417i			0.437279i
$918$		0			0
$919$		−	18.3484i		−	0.605257i	−0.953109	−	0.302628i	$-0.902136\pi$
$919$		−	18.3484i		−	0.605257i	0.953109	−	0.302628i	$-0.0978641\pi$
$920$		0			0
$921$			9.53969i			0.314343i
$922$		0			0
$923$	15.7691			0.519045
$924$		0			0
$925$	7.00285			0.230252
$926$		0			0
$927$	−21.3604			−0.701566
$928$		0			0
$929$			34.1814i			1.12146i	0.828000	+	0.560728i	$0.189479\pi$
$929$			34.1814i			1.12146i	−0.828000	+	0.560728i	$0.810521\pi$
$930$		0			0
$931$		−	34.8250i		−	1.14134i
$932$		0			0
$933$	−6.03336			−0.197523
$934$		0			0
$935$	1.86732			0.0610679
$936$		0			0
$937$		−	60.8370i		−	1.98746i	−0.111816	−	0.993729i	$-0.535667\pi$
$937$		−	60.8370i		−	1.98746i	0.111816	−	0.993729i	$-0.464333\pi$
$938$		0			0
$939$	−5.16320			−0.168495
$940$		0			0
$941$	−37.6744			−1.22815			−0.614075	−	0.789248i	$-0.710471\pi$
$941$	−37.6744			−1.22815			−0.614075	+	0.789248i	$0.710471\pi$
$942$		0			0
$943$	−17.8244	+	15.9385i	−0.580442	+	0.519030i
$944$		0			0
$945$	−3.87903			−0.126185
$946$		0			0
$947$	−36.0437			−1.17126			−0.585631	−	0.810577i	$-0.699153\pi$
$947$	−36.0437			−1.17126			−0.585631	+	0.810577i	$0.699153\pi$
$948$		0			0
$949$			14.0909i			0.457409i
$950$		0			0
$951$	4.88222			0.158317
$952$		0			0
$953$	−17.6673			−0.572299			−0.286149	−	0.958185i	$-0.592375\pi$
$953$	−17.6673			−0.572299			−0.286149	+	0.958185i	$0.592375\pi$
$954$		0			0
$955$			4.69965i			0.152077i
$956$		0			0
$957$		−	7.00746i		−	0.226519i
$958$		0			0
$959$	28.6769			0.926025
$960$		0			0
$961$	−3.24236			−0.104592
$962$		0			0
$963$	51.4793			1.65890
$964$		0			0
$965$		−	15.2861i		−	0.492078i
$966$		0			0
$967$		−	19.5779i		−	0.629582i	−0.949161	−	0.314791i	$-0.898066\pi$
$967$		−	19.5779i		−	0.629582i	0.949161	−	0.314791i	$-0.101934\pi$
$968$		0			0
$969$			1.75320i			0.0563208i
$970$		0			0
$971$		−	56.5841i		−	1.81587i	−0.419112	−	0.907935i	$-0.637658\pi$
$971$		−	56.5841i		−	1.81587i	0.419112	−	0.907935i	$-0.362342\pi$
$972$		0			0
$973$			4.22563i			0.135468i
$974$		0			0
$975$	−0.676461			−0.0216641
$976$		0			0
$977$			9.65759i			0.308974i	0.987995	+	0.154487i	$0.0493724\pi$
$977$			9.65759i			0.308974i	−0.987995	+	0.154487i	$0.950628\pi$
$978$		0			0
$979$	23.8569			0.762469
$980$		0			0
$981$			36.6695i			1.17077i
$982$		0			0
$983$	−26.6275			−0.849285			−0.424642	−	0.905361i	$-0.639600\pi$
$983$	−26.6275			−0.849285			−0.424642	+	0.905361i	$0.639600\pi$
$984$		0			0
$985$	−4.04865			−0.129001
$986$		0			0
$987$			2.18986i			0.0697039i
$988$		0			0
$989$	30.4161			0.967177
$990$		0			0
$991$			35.4772i			1.12697i	0.826126	+	0.563486i	$0.190540\pi$
$991$			35.4772i			1.12697i	−0.826126	+	0.563486i	$0.809460\pi$
$992$		0			0
$993$	−1.38284			−0.0438833
$994$		0			0
$995$		−	0.523837i		−	0.0166067i
$996$		0			0
$997$		−	16.2057i		−	0.513238i	−0.966513	−	0.256619i	$-0.917391\pi$
$997$		−	16.2057i		−	0.513238i	0.966513	−	0.256619i	$-0.0826086\pi$
$998$		0			0
$999$			20.6990i			0.654888i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1640.2.b.e.81.6	yes	10
4.3	odd	2		3280.2.b.o.81.5		10
41.40	even	2	inner	1640.2.b.e.81.5	✓	10
164.163	odd	2		3280.2.b.o.81.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1640.2.b.e.81.5	✓	10	41.40	even	2	inner
1640.2.b.e.81.6	yes	10	1.1	even	1	trivial
3280.2.b.o.81.5		10	4.3	odd	2
3280.2.b.o.81.6		10	164.163	odd	2

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:	\( N \)	\(=\)	\( 1640 = 2^{3} \cdot 5 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1640.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.515459i q^{3} -1.00000 q^{5} -1.31235i q^{7} +2.73430 q^{9} +O(q^{10})\)	\(q+0.515459i q^{3} -1.00000 q^{5} -1.31235i q^{7} +2.73430 q^{9} +3.62264i q^{11} +1.31235i q^{13} -0.515459i q^{15} +0.515459i q^{17} -6.59846i q^{19} +0.676461 q^{21} +3.73430 q^{23} +1.00000 q^{25} +2.95580i q^{27} +3.75268i q^{29} -5.26855 q^{31} -1.86732 q^{33} +1.31235i q^{35} +7.00285 q^{37} -0.676461 q^{39} +(-4.77315 + 4.26814i) q^{41} +8.14507 q^{43} -2.73430 q^{45} +3.23723i q^{47} +5.27775 q^{49} -0.265698 q^{51} +7.24396i q^{53} -3.62264i q^{55} +3.40123 q^{57} +2.54344 q^{59} +2.59209 q^{61} -3.58835i q^{63} -1.31235i q^{65} +13.3612i q^{67} +1.92488i q^{69} -12.0159i q^{71} +10.7372 q^{73} +0.515459i q^{75} +4.75415 q^{77} -0.130049i q^{79} +6.67931 q^{81} +6.27775 q^{83} -0.515459i q^{85} -1.93435 q^{87} -6.58550i q^{89} +1.72225 q^{91} -2.71572i q^{93} +6.59846i q^{95} -0.118404i q^{97} +9.90538i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{5} - 14 q^{9}+O(q^{10}) \) 10 * q - 10 * q^5 - 14 * q^9	\( 10 q - 10 q^{5} - 14 q^{9} + 6 q^{21} - 4 q^{23} + 10 q^{25} - 22 q^{31} - 8 q^{33} - 2 q^{37} - 6 q^{39} + 4 q^{41} - 2 q^{43} + 14 q^{45} - 20 q^{49} - 44 q^{51} + 14 q^{57} - 6 q^{59} - 4 q^{61} - 6 q^{73} + 26 q^{77} - 6 q^{81} - 10 q^{83} + 74 q^{87} + 90 q^{91}+O(q^{100}) \) 10 * q - 10 * q^5 - 14 * q^9 + 6 * q^21 - 4 * q^23 + 10 * q^25 - 22 * q^31 - 8 * q^33 - 2 * q^37 - 6 * q^39 + 4 * q^41 - 2 * q^43 + 14 * q^45 - 20 * q^49 - 44 * q^51 + 14 * q^57 - 6 * q^59 - 4 * q^61 - 6 * q^73 + 26 * q^77 - 6 * q^81 - 10 * q^83 + 74 * q^87 + 90 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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