LMFDB - Embedded newform 1680.3.s.c.1441.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,3,Mod(1441,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1680.1441");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.s (of order $2$, degree $1$, not 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:	$45.7766844125$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} + \cdots + 441 $ x^12 - 2x^11 + 21x^10 - 26x^9 + 295x^8 - 372x^7 + 1704x^6 - 1074x^5 + 5613x^4 - 5472x^3 + 4950x^2 - 1638*x + 441
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1441.11
Root			$-1.01714 - 1.76174i$ of defining polynomial
Character	$\chi$	$=$	1680.1441
Dual form			1680.3.s.c.1441.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +2.23607i q^{5} +(3.33344 - 6.15534i) q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +2.23607i q^{5} +(3.33344 - 6.15534i) q^{7} -3.00000 q^{9} -17.0001 q^{11} +16.3319i q^{13} -3.87298 q^{15} +13.4266i q^{17} -13.7499i q^{19} +(10.6614 + 5.77369i) q^{21} +16.6179 q^{23} -5.00000 q^{25} -5.19615i q^{27} +32.1793 q^{29} +6.74366i q^{31} -29.4450i q^{33} +(13.7637 + 7.45380i) q^{35} -69.2141 q^{37} -28.2878 q^{39} -39.7391i q^{41} -43.2210 q^{43} -6.70820i q^{45} -40.1384i q^{47} +(-26.7763 - 41.0369i) q^{49} -23.2556 q^{51} +22.5002 q^{53} -38.0134i q^{55} +23.8155 q^{57} -81.6005i q^{59} +14.9859i q^{61} +(-10.0003 + 18.4660i) q^{63} -36.5193 q^{65} -72.0872 q^{67} +28.7831i q^{69} +25.7338 q^{71} +75.0647i q^{73} -8.66025i q^{75} +(-56.6689 + 104.641i) q^{77} -80.0480 q^{79} +9.00000 q^{81} +102.112i q^{83} -30.0228 q^{85} +55.7362i q^{87} -128.381i q^{89} +(100.529 + 54.4416i) q^{91} -11.6804 q^{93} +30.7457 q^{95} -159.448i q^{97} +51.0003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{7} - 36 q^{9}+O(q^{10}) $ 12 * q + 8 * q^7 - 36 * q^9	$ 12 q + 8 q^{7} - 36 q^{9} + 16 q^{11} + 36 q^{21} + 64 q^{23} - 60 q^{25} + 104 q^{29} - 60 q^{35} + 32 q^{37} + 24 q^{39} - 152 q^{43} + 60 q^{49} - 24 q^{51} + 176 q^{53} - 240 q^{57} - 24 q^{63} - 240 q^{65} - 168 q^{67} - 32 q^{71} + 8 q^{77} - 120 q^{79} + 108 q^{81} + 120 q^{85} - 24 q^{91} + 48 q^{93} - 48 q^{99}+O(q^{100}) $ 12 * q + 8 * q^7 - 36 * q^9 + 16 * q^11 + 36 * q^21 + 64 * q^23 - 60 * q^25 + 104 * q^29 - 60 * q^35 + 32 * q^37 + 24 * q^39 - 152 * q^43 + 60 * q^49 - 24 * q^51 + 176 * q^53 - 240 * q^57 - 24 * q^63 - 240 * q^65 - 168 * q^67 - 32 * q^71 + 8 * q^77 - 120 * q^79 + 108 * q^81 + 120 * q^85 - 24 * q^91 + 48 * q^93 - 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$			1.73205i			0.577350i
$3$			1.73205i			0.577350i
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$	3.33344	−	6.15534i	0.476206	−	0.879334i
$7$	3.33344	−	6.15534i	0.476206	−	0.879334i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	−17.0001			−1.54546			−0.772732	−	0.634732i	$-0.781110\pi$
$11$	−17.0001			−1.54546			−0.772732	+	0.634732i	$0.781110\pi$
$12$		0			0
$13$			16.3319i			1.25630i	0.778091	+	0.628152i	$0.216188\pi$
$13$			16.3319i			1.25630i	−0.778091	+	0.628152i	$0.783812\pi$
$14$		0			0
$15$	−3.87298			−0.258199
$16$		0			0
$17$			13.4266i			0.789801i	0.918724	+	0.394900i	$0.129221\pi$
$17$			13.4266i			0.789801i	−0.918724	+	0.394900i	$0.870779\pi$
$18$		0			0
$19$		−	13.7499i		−	0.723679i	−0.932240	−	0.361839i	$-0.882149\pi$
$19$		−	13.7499i		−	0.723679i	0.932240	−	0.361839i	$-0.117851\pi$
$20$		0			0
$21$	10.6614	+	5.77369i	0.507684	+	0.274938i
$22$		0			0
$23$	16.6179			0.722518			0.361259	−	0.932466i	$-0.382347\pi$
$23$	16.6179			0.722518			0.361259	+	0.932466i	$0.382347\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		−	5.19615i		−	0.192450i
$28$		0			0
$29$	32.1793			1.10963			0.554816	−	0.831973i	$-0.312789\pi$
$29$	32.1793			1.10963			0.554816	+	0.831973i	$0.312789\pi$
$30$		0			0
$31$			6.74366i			0.217538i	0.994067	+	0.108769i	$0.0346908\pi$
$31$			6.74366i			0.217538i	−0.994067	+	0.108769i	$0.965309\pi$
$32$		0			0
$33$		−	29.4450i		−	0.892274i
$34$		0			0
$35$	13.7637	+	7.45380i	0.393250	+	0.212966i
$36$		0			0
$37$	−69.2141			−1.87065			−0.935325	−	0.353789i	$-0.884893\pi$
$37$	−69.2141			−1.87065			−0.935325	+	0.353789i	$0.884893\pi$
$38$		0			0
$39$	−28.2878			−0.725327
$40$		0			0
$41$		−	39.7391i		−	0.969246i	−0.874723	−	0.484623i	$-0.838957\pi$
$41$		−	39.7391i		−	0.969246i	0.874723	−	0.484623i	$-0.161043\pi$
$42$		0			0
$43$	−43.2210			−1.00514			−0.502570	−	0.864537i	$-0.667612\pi$
$43$	−43.2210			−1.00514			−0.502570	+	0.864537i	$0.667612\pi$
$44$		0			0
$45$		−	6.70820i		−	0.149071i
$46$		0			0
$47$		−	40.1384i		−	0.854009i	−0.904249	−	0.427005i	$-0.859569\pi$
$47$		−	40.1384i		−	0.854009i	0.904249	−	0.427005i	$-0.140431\pi$
$48$		0			0
$49$	−26.7763	−	41.0369i	−0.546456	−	0.837488i
$50$		0			0
$51$	−23.2556			−0.455992
$52$		0			0
$53$	22.5002			0.424533			0.212266	−	0.977212i	$-0.431916\pi$
$53$	22.5002			0.424533			0.212266	+	0.977212i	$0.431916\pi$
$54$		0			0
$55$		−	38.0134i		−	0.691153i
$56$		0			0
$57$	23.8155			0.417816
$58$		0			0
$59$		−	81.6005i		−	1.38306i	−0.722348	−	0.691529i	$-0.756937\pi$
$59$		−	81.6005i		−	1.38306i	0.722348	−	0.691529i	$-0.243063\pi$
$60$		0			0
$61$			14.9859i			0.245671i	0.992427	+	0.122836i	$0.0391988\pi$
$61$			14.9859i			0.245671i	−0.992427	+	0.122836i	$0.960801\pi$
$62$		0			0
$63$	−10.0003	+	18.4660i	−0.158735	+	0.293111i
$64$		0			0
$65$	−36.5193			−0.561836
$66$		0			0
$67$	−72.0872			−1.07593			−0.537964	−	0.842968i	$-0.680806\pi$
$67$	−72.0872			−1.07593			−0.537964	+	0.842968i	$0.680806\pi$
$68$		0			0
$69$			28.7831i			0.417146i
$70$		0			0
$71$	25.7338			0.362448			0.181224	−	0.983442i	$-0.441994\pi$
$71$	25.7338			0.362448			0.181224	+	0.983442i	$0.441994\pi$
$72$		0			0
$73$			75.0647i			1.02828i	0.857705	+	0.514142i	$0.171890\pi$
$73$			75.0647i			1.02828i	−0.857705	+	0.514142i	$0.828110\pi$
$74$		0			0
$75$		−	8.66025i		−	0.115470i
$76$		0			0
$77$	−56.6689	+	104.641i	−0.735959	+	1.35898i
$78$		0			0
$79$	−80.0480			−1.01327			−0.506633	−	0.862162i	$-0.669110\pi$
$79$	−80.0480			−1.01327			−0.506633	+	0.862162i	$0.669110\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			102.112i			1.23027i	0.788421	+	0.615135i	$0.210899\pi$
$83$			102.112i			1.23027i	−0.788421	+	0.615135i	$0.789101\pi$
$84$		0			0
$85$	−30.0228			−0.353210
$86$		0			0
$87$			55.7362i			0.640646i
$88$		0			0
$89$		−	128.381i		−	1.44248i	−0.692683	−	0.721242i	$-0.743571\pi$
$89$		−	128.381i		−	1.44248i	0.692683	−	0.721242i	$-0.256429\pi$
$90$		0			0
$91$	100.529	+	54.4416i	1.10471	+	0.598259i
$92$		0			0
$93$	−11.6804			−0.125595
$94$		0			0
$95$	30.7457			0.323639
$96$		0			0
$97$		−	159.448i		−	1.64379i	−0.569636	−	0.821897i	$-0.692916\pi$
$97$		−	159.448i		−	1.64379i	0.569636	−	0.821897i	$-0.307084\pi$
$98$		0			0
$99$	51.0003			0.515155
$100$		0			0
$101$		−	24.1380i		−	0.238990i	−0.992835	−	0.119495i	$-0.961872\pi$
$101$		−	24.1380i		−	0.238990i	0.992835	−	0.119495i	$-0.0381276\pi$
$102$		0			0
$103$		−	87.3469i		−	0.848028i	−0.905656	−	0.424014i	$-0.860621\pi$
$103$		−	87.3469i		−	0.848028i	0.905656	−	0.424014i	$-0.139379\pi$
$104$		0			0
$105$	−12.9104	+	23.8395i	−0.122956	+	0.227043i
$106$		0			0
$107$	−168.359			−1.57344			−0.786722	−	0.617307i	$-0.788224\pi$
$107$	−168.359			−1.57344			−0.786722	+	0.617307i	$0.788224\pi$
$108$		0			0
$109$	−155.570			−1.42725			−0.713624	−	0.700529i	$-0.752947\pi$
$109$	−155.570			−1.42725			−0.713624	+	0.700529i	$0.752947\pi$
$110$		0			0
$111$		−	119.882i		−	1.08002i
$112$		0			0
$113$	−20.9965			−0.185810			−0.0929050	−	0.995675i	$-0.529615\pi$
$113$	−20.9965			−0.185810			−0.0929050	+	0.995675i	$0.529615\pi$
$114$		0			0
$115$			37.1588i			0.323120i
$116$		0			0
$117$		−	48.9958i		−	0.418768i
$118$		0			0
$119$	82.6453	+	44.7568i	0.694498	+	0.376108i
$120$		0			0
$121$	168.004			1.38846
$122$		0			0
$123$	68.8301			0.559594
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	59.8712			0.471427			0.235713	−	0.971823i	$-0.424257\pi$
$127$	59.8712			0.471427			0.235713	+	0.971823i	$0.424257\pi$
$128$		0			0
$129$		−	74.8609i		−	0.580317i
$130$		0			0
$131$			166.868i			1.27380i	0.770947	+	0.636899i	$0.219783\pi$
$131$			166.868i			1.27380i	−0.770947	+	0.636899i	$0.780217\pi$
$132$		0			0
$133$	−84.6352	−	45.8345i	−0.636355	−	0.344620i
$134$		0			0
$135$	11.6190			0.0860663
$136$		0			0
$137$	126.139			0.920726			0.460363	−	0.887731i	$-0.347719\pi$
$137$	126.139			0.920726			0.460363	+	0.887731i	$0.347719\pi$
$138$		0			0
$139$		−	211.650i		−	1.52266i	−0.648365	−	0.761330i	$-0.724547\pi$
$139$		−	211.650i		−	1.52266i	0.648365	−	0.761330i	$-0.275453\pi$
$140$		0			0
$141$	69.5218			0.493062
$142$		0			0
$143$		−	277.645i		−	1.94157i
$144$		0			0
$145$			71.9552i			0.496243i
$146$		0			0
$147$	71.0780	−	46.3780i	0.483524	−	0.315496i
$148$		0			0
$149$	−64.1825			−0.430755			−0.215377	−	0.976531i	$-0.569098\pi$
$149$	−64.1825			−0.430755			−0.215377	+	0.976531i	$0.569098\pi$
$150$		0			0
$151$	−110.915			−0.734538			−0.367269	−	0.930115i	$-0.619707\pi$
$151$	−110.915			−0.734538			−0.367269	+	0.930115i	$0.619707\pi$
$152$		0			0
$153$		−	40.2798i		−	0.263267i
$154$		0			0
$155$	−15.0793			−0.0972858
$156$		0			0
$157$		−	290.451i		−	1.85001i	−0.379956	−	0.925004i	$-0.624061\pi$
$157$		−	290.451i		−	1.85001i	0.379956	−	0.925004i	$-0.375939\pi$
$158$		0			0
$159$			38.9716i			0.245104i
$160$		0			0
$161$	55.3948	−	102.289i	0.344067	−	0.635334i
$162$		0			0
$163$	53.8559			0.330404			0.165202	−	0.986260i	$-0.447172\pi$
$163$	53.8559			0.330404			0.165202	+	0.986260i	$0.447172\pi$
$164$		0			0
$165$	65.8411			0.399037
$166$		0			0
$167$		−	41.7927i		−	0.250255i	−0.992141	−	0.125128i	$-0.960066\pi$
$167$		−	41.7927i		−	0.250255i	0.992141	−	0.125128i	$-0.0399341\pi$
$168$		0			0
$169$	−97.7324			−0.578298
$170$		0			0
$171$			41.2497i			0.241226i
$172$		0			0
$173$		−	130.344i		−	0.753434i	−0.926328	−	0.376717i	$-0.877053\pi$
$173$		−	130.344i		−	0.753434i	0.926328	−	0.376717i	$-0.122947\pi$
$174$		0			0
$175$	−16.6672	+	30.7767i	−0.0952412	+	0.175867i
$176$		0			0
$177$	141.336			0.798509
$178$		0			0
$179$	−44.9934			−0.251360			−0.125680	−	0.992071i	$-0.540111\pi$
$179$	−44.9934			−0.251360			−0.125680	+	0.992071i	$0.540111\pi$
$180$		0			0
$181$			17.8944i			0.0988640i	0.998777	+	0.0494320i	$0.0157411\pi$
$181$			17.8944i			0.0988640i	−0.998777	+	0.0494320i	$0.984259\pi$
$182$		0			0
$183$	−25.9564			−0.141838
$184$		0			0
$185$		−	154.767i		−	0.836581i
$186$		0			0
$187$		−	228.254i		−	1.22061i
$188$		0			0
$189$	−31.9841	−	17.3211i	−0.169228	−	0.0916459i
$190$		0			0
$191$	−178.314			−0.933583			−0.466791	−	0.884367i	$-0.654590\pi$
$191$	−178.314			−0.933583			−0.466791	+	0.884367i	$0.654590\pi$
$192$		0			0
$193$	−336.283			−1.74240			−0.871200	−	0.490928i	$-0.836658\pi$
$193$	−336.283			−1.74240			−0.871200	+	0.490928i	$0.836658\pi$
$194$		0			0
$195$		−	63.2534i		−	0.324376i
$196$		0			0
$197$	49.2082			0.249788			0.124894	−	0.992170i	$-0.460141\pi$
$197$	49.2082			0.249788			0.124894	+	0.992170i	$0.460141\pi$
$198$		0			0
$199$		−	171.789i		−	0.863262i	−0.902050	−	0.431631i	$-0.857938\pi$
$199$		−	171.789i		−	0.863262i	0.902050	−	0.431631i	$-0.142062\pi$
$200$		0			0
$201$		−	124.859i		−	0.621187i
$202$		0			0
$203$	107.268	−	198.075i	0.528414	−	0.975737i
$204$		0			0
$205$	88.8593			0.433460
$206$		0			0
$207$	−49.8537			−0.240839
$208$		0			0
$209$			233.750i			1.11842i
$210$		0			0
$211$	5.09458			0.0241449			0.0120725	−	0.999927i	$-0.496157\pi$
$211$	5.09458			0.0241449			0.0120725	+	0.999927i	$0.496157\pi$
$212$		0			0
$213$			44.5722i			0.209259i
$214$		0			0
$215$		−	96.6451i		−	0.449512i
$216$		0			0
$217$	41.5095	+	22.4796i	0.191288	+	0.103593i
$218$		0			0
$219$	−130.016			−0.593680
$220$		0			0
$221$	−219.283			−0.992229
$222$		0			0
$223$			310.066i			1.39043i	0.718802	+	0.695215i	$0.244691\pi$
$223$			310.066i			1.39043i	−0.718802	+	0.695215i	$0.755309\pi$
$224$		0			0
$225$	15.0000			0.0666667
$226$		0			0
$227$			108.558i			0.478228i	0.970991	+	0.239114i	$0.0768570\pi$
$227$			108.558i			0.478228i	−0.970991	+	0.239114i	$0.923143\pi$
$228$		0			0
$229$			236.483i			1.03268i	0.856384	+	0.516339i	$0.172706\pi$
$229$			236.483i			1.03268i	−0.856384	+	0.516339i	$0.827294\pi$
$230$		0			0
$231$	−181.244	−	98.1534i	−0.784607	−	0.424906i
$232$		0			0
$233$	151.290			0.649312			0.324656	−	0.945832i	$-0.394751\pi$
$233$	151.290			0.649312			0.324656	+	0.945832i	$0.394751\pi$
$234$		0			0
$235$	89.7523			0.381925
$236$		0			0
$237$		−	138.647i		−	0.585009i
$238$		0			0
$239$	−48.2956			−0.202074			−0.101037	−	0.994883i	$-0.532216\pi$
$239$	−48.2956			−0.202074			−0.101037	+	0.994883i	$0.532216\pi$
$240$		0			0
$241$			230.735i			0.957406i	0.877977	+	0.478703i	$0.158893\pi$
$241$			230.735i			0.957406i	−0.877977	+	0.478703i	$0.841107\pi$
$242$		0			0
$243$			15.5885i			0.0641500i
$244$		0			0
$245$	91.7613	−	59.8737i	0.374536	−	0.244382i
$246$		0			0
$247$	224.563			0.909160
$248$		0			0
$249$	−176.864			−0.710297
$250$		0			0
$251$			86.6812i			0.345343i	0.984979	+	0.172672i	$0.0552399\pi$
$251$			86.6812i			0.345343i	−0.984979	+	0.172672i	$0.944760\pi$
$252$		0			0
$253$	−282.506			−1.11663
$254$		0			0
$255$		−	52.0010i		−	0.203926i
$256$		0			0
$257$		−	141.110i		−	0.549065i	−0.961578	−	0.274532i	$-0.911477\pi$
$257$		−	141.110i		−	0.549065i	0.961578	−	0.274532i	$-0.0885231\pi$
$258$		0			0
$259$	−230.721	+	426.036i	−0.890815	+	1.64493i
$260$		0			0
$261$	−96.5380			−0.369877
$262$		0			0
$263$	−38.0901			−0.144829			−0.0724147	−	0.997375i	$-0.523070\pi$
$263$	−38.0901			−0.144829			−0.0724147	+	0.997375i	$0.523070\pi$
$264$		0			0
$265$			50.3121i			0.189857i
$266$		0			0
$267$	222.363			0.832819
$268$		0			0
$269$		−	454.220i		−	1.68855i	−0.535909	−	0.844275i	$-0.680031\pi$
$269$		−	454.220i		−	1.68855i	0.535909	−	0.844275i	$-0.319969\pi$
$270$		0			0
$271$		−	78.7098i		−	0.290442i	−0.989399	−	0.145221i	$-0.953611\pi$
$271$		−	78.7098i		−	0.290442i	0.989399	−	0.145221i	$-0.0463893\pi$
$272$		0			0
$273$	−94.2956	+	174.121i	−0.345405	+	0.637805i
$274$		0			0
$275$	85.0005			0.309093
$276$		0			0
$277$	−85.3396			−0.308085			−0.154043	−	0.988064i	$-0.549229\pi$
$277$	−85.3396			−0.308085			−0.154043	+	0.988064i	$0.549229\pi$
$278$		0			0
$279$		−	20.2310i		−	0.0725125i
$280$		0			0
$281$	167.376			0.595643			0.297821	−	0.954622i	$-0.403740\pi$
$281$	167.376			0.595643			0.297821	+	0.954622i	$0.403740\pi$
$282$		0			0
$283$			149.591i			0.528588i	0.964442	+	0.264294i	$0.0851390\pi$
$283$			149.591i			0.528588i	−0.964442	+	0.264294i	$0.914861\pi$
$284$		0			0
$285$			53.2531i			0.186853i
$286$		0			0
$287$	−244.607	−	132.468i	−0.852291	−	0.461561i
$288$		0			0
$289$	108.726			0.376215
$290$		0			0
$291$	276.172			0.949045
$292$		0			0
$293$			145.805i			0.497627i	0.968551	+	0.248813i	$0.0800406\pi$
$293$			145.805i			0.497627i	−0.968551	+	0.248813i	$0.919959\pi$
$294$		0			0
$295$	182.464			0.618523
$296$		0			0
$297$			88.3351i			0.297425i
$298$		0			0
$299$			271.403i			0.907701i
$300$		0			0
$301$	−144.075	+	266.040i	−0.478653	+	0.883853i
$302$		0			0
$303$	41.8082			0.137981
$304$		0			0
$305$	−33.5096			−0.109868
$306$		0			0
$307$			205.594i			0.669686i	0.942274	+	0.334843i	$0.108683\pi$
$307$			205.594i			0.669686i	−0.942274	+	0.334843i	$0.891317\pi$
$308$		0			0
$309$	151.289			0.489609
$310$		0			0
$311$		−	424.383i		−	1.36458i	−0.731084	−	0.682288i	$-0.760985\pi$
$311$		−	424.383i		−	1.36458i	0.731084	−	0.682288i	$-0.239015\pi$
$312$		0			0
$313$			363.503i			1.16135i	0.814135	+	0.580676i	$0.197212\pi$
$313$			363.503i			1.16135i	−0.814135	+	0.580676i	$0.802788\pi$
$314$		0			0
$315$	−41.2912	−	22.3614i	−0.131083	−	0.0709886i
$316$		0			0
$317$	441.407			1.39245			0.696226	−	0.717822i	$-0.254861\pi$
$317$	441.407			1.39245			0.696226	+	0.717822i	$0.254861\pi$
$318$		0			0
$319$	−547.052			−1.71490
$320$		0			0
$321$		−	291.606i		−	0.908429i
$322$		0			0
$323$	184.615			0.571562
$324$		0			0
$325$		−	81.6597i		−	0.251261i
$326$		0			0
$327$		−	269.455i		−	0.824022i
$328$		0			0
$329$	−247.066	−	133.799i	−0.750959	−	0.406684i
$330$		0			0
$331$	−509.327			−1.53875			−0.769376	−	0.638796i	$-0.779433\pi$
$331$	−509.327			−1.53875			−0.769376	+	0.638796i	$0.779433\pi$
$332$		0			0
$333$	207.642			0.623550
$334$		0			0
$335$		−	161.192i		−	0.481170i
$336$		0			0
$337$	442.557			1.31323			0.656613	−	0.754228i	$-0.271988\pi$
$337$	442.557			1.31323			0.656613	+	0.754228i	$0.271988\pi$
$338$		0			0
$339$		−	36.3671i		−	0.107277i
$340$		0			0
$341$		−	114.643i		−	0.336197i
$342$		0			0
$343$	−341.853	+	28.0231i	−0.996657	+	0.0816999i
$344$		0			0
$345$	−64.3609			−0.186553
$346$		0			0
$347$	493.763			1.42295			0.711475	−	0.702712i	$-0.248028\pi$
$347$	493.763			1.42295			0.711475	+	0.702712i	$0.248028\pi$
$348$		0			0
$349$			324.460i			0.929686i	0.885393	+	0.464843i	$0.153889\pi$
$349$			324.460i			0.929686i	−0.885393	+	0.464843i	$0.846111\pi$
$350$		0			0
$351$	84.8633			0.241776
$352$		0			0
$353$			529.424i			1.49979i	0.661559	+	0.749893i	$0.269895\pi$
$353$			529.424i			1.49979i	−0.661559	+	0.749893i	$0.730105\pi$
$354$		0			0
$355$			57.5425i			0.162092i
$356$		0			0
$357$	−77.5211	+	143.146i	−0.217146	+	0.400969i
$358$		0			0
$359$	−64.2261			−0.178903			−0.0894514	−	0.995991i	$-0.528511\pi$
$359$	−64.2261			−0.178903			−0.0894514	+	0.995991i	$0.528511\pi$
$360$		0			0
$361$	171.940			0.476289
$362$		0			0
$363$			290.991i			0.801627i
$364$		0			0
$365$	−167.850			−0.459863
$366$		0			0
$367$			10.8172i			0.0294747i	0.999891	+	0.0147373i	$0.00469121\pi$
$367$			10.8172i			0.0294747i	−0.999891	+	0.0147373i	$0.995309\pi$
$368$		0			0
$369$			119.217i			0.323082i
$370$		0			0
$371$	75.0033	−	138.497i	0.202165	−	0.373306i
$372$		0			0
$373$	532.850			1.42855			0.714276	−	0.699864i	$-0.246756\pi$
$373$	532.850			1.42855			0.714276	+	0.699864i	$0.246756\pi$
$374$		0			0
$375$	19.3649			0.0516398
$376$		0			0
$377$			525.551i			1.39403i
$378$		0			0
$379$	−516.003			−1.36149			−0.680743	−	0.732523i	$-0.738343\pi$
$379$	−516.003			−1.36149			−0.680743	+	0.732523i	$0.738343\pi$
$380$		0			0
$381$			103.700i			0.272178i
$382$		0			0
$383$		−	415.069i		−	1.08373i	−0.840465	−	0.541866i	$-0.817718\pi$
$383$		−	415.069i		−	1.08373i	0.840465	−	0.541866i	$-0.182282\pi$
$384$		0			0
$385$	−233.985	−	126.715i	−0.607754	−	0.329131i
$386$		0			0
$387$	129.663			0.335046
$388$		0			0
$389$	54.8032			0.140882			0.0704411	−	0.997516i	$-0.477559\pi$
$389$	54.8032			0.140882			0.0704411	+	0.997516i	$0.477559\pi$
$390$		0			0
$391$			223.122i			0.570645i
$392$		0			0
$393$	−289.023			−0.735428
$394$		0			0
$395$		−	178.993i		−	0.453146i
$396$		0			0
$397$			19.9434i			0.0502352i	0.999685	+	0.0251176i	$0.00799602\pi$
$397$			19.9434i			0.0502352i	−0.999685	+	0.0251176i	$0.992004\pi$
$398$		0			0
$399$	79.3877	−	146.593i	0.198967	−	0.367400i
$400$		0			0
$401$	−239.505			−0.597269			−0.298634	−	0.954368i	$-0.596531\pi$
$401$	−239.505			−0.597269			−0.298634	+	0.954368i	$0.596531\pi$
$402$		0			0
$403$	−110.137			−0.273293
$404$		0			0
$405$			20.1246i			0.0496904i
$406$		0			0
$407$	1176.65			2.89102
$408$		0			0
$409$		−	663.541i		−	1.62235i	−0.584804	−	0.811175i	$-0.698829\pi$
$409$		−	663.541i		−	1.62235i	0.584804	−	0.811175i	$-0.301171\pi$
$410$		0			0
$411$			218.480i			0.531581i
$412$		0			0
$413$	−502.278	−	272.010i	−1.21617	−	0.658621i
$414$		0			0
$415$	−228.330			−0.550194
$416$		0			0
$417$	366.588			0.879108
$418$		0			0
$419$		−	110.648i		−	0.264077i	−0.991245	−	0.132039i	$-0.957848\pi$
$419$		−	110.648i		−	0.264077i	0.991245	−	0.132039i	$-0.0421523\pi$
$420$		0			0
$421$	−521.325			−1.23830			−0.619151	−	0.785272i	$-0.712523\pi$
$421$	−521.325			−1.23830			−0.619151	+	0.785272i	$0.712523\pi$
$422$		0			0
$423$			120.415i			0.284670i
$424$		0			0
$425$		−	67.1330i		−	0.157960i
$426$		0			0
$427$	92.2435	+	49.9548i	0.216027	+	0.116990i
$428$		0			0
$429$	480.895			1.12097
$430$		0			0
$431$	−573.019			−1.32951			−0.664755	−	0.747062i	$-0.731464\pi$
$431$	−573.019			−1.32951			−0.664755	+	0.747062i	$0.731464\pi$
$432$		0			0
$433$		−	429.740i		−	0.992472i	−0.868188	−	0.496236i	$-0.834715\pi$
$433$		−	429.740i		−	0.992472i	0.868188	−	0.496236i	$-0.165285\pi$
$434$		0			0
$435$	−124.630			−0.286506
$436$		0			0
$437$		−	228.495i		−	0.522871i
$438$		0			0
$439$		−	83.0494i		−	0.189179i	−0.995516	−	0.0945893i	$-0.969846\pi$
$439$		−	83.0494i		−	0.189179i	0.995516	−	0.0945893i	$-0.0301538\pi$
$440$		0			0
$441$	80.3290	+	123.111i	0.182152	+	0.279163i
$442$		0			0
$443$	410.010			0.925530			0.462765	−	0.886481i	$-0.346857\pi$
$443$	410.010			0.925530			0.462765	+	0.886481i	$0.346857\pi$
$444$		0			0
$445$	287.069			0.645099
$446$		0			0
$447$		−	111.167i		−	0.248696i
$448$		0			0
$449$	−205.948			−0.458682			−0.229341	−	0.973346i	$-0.573657\pi$
$449$	−205.948			−0.458682			−0.229341	+	0.973346i	$0.573657\pi$
$450$		0			0
$451$			675.569i			1.49793i
$452$		0			0
$453$		−	192.111i		−	0.424086i
$454$		0			0
$455$	−121.735	+	224.789i	−0.267550	+	0.494041i
$456$		0			0
$457$	−640.868			−1.40234			−0.701169	−	0.712995i	$-0.747338\pi$
$457$	−640.868			−1.40234			−0.701169	+	0.712995i	$0.747338\pi$
$458$		0			0
$459$	69.7667			0.151997
$460$		0			0
$461$			166.085i			0.360272i	0.983642	+	0.180136i	$0.0576538\pi$
$461$			166.085i			0.360272i	−0.983642	+	0.180136i	$0.942346\pi$
$462$		0			0
$463$	10.4409			0.0225505			0.0112753	−	0.999936i	$-0.496411\pi$
$463$	10.4409			0.0225505			0.0112753	+	0.999936i	$0.496411\pi$
$464$		0			0
$465$		−	26.1181i		−	0.0561680i
$466$		0			0
$467$			269.736i			0.577592i	0.957391	+	0.288796i	$0.0932550\pi$
$467$			269.736i			0.577592i	−0.957391	+	0.288796i	$0.906745\pi$
$468$		0			0
$469$	−240.298	+	443.721i	−0.512364	+	0.946100i
$470$		0			0
$471$	503.077			1.06810
$472$		0			0
$473$	734.761			1.55341
$474$		0			0
$475$			68.7495i			0.144736i
$476$		0			0
$477$	−67.5007			−0.141511
$478$		0			0
$479$			649.820i			1.35662i	0.734777	+	0.678309i	$0.237287\pi$
$479$			649.820i			1.35662i	−0.734777	+	0.678309i	$0.762713\pi$
$480$		0			0
$481$		−	1130.40i		−	2.35011i
$482$		0			0
$483$	177.169	+	95.9467i	0.366810	+	0.198647i
$484$		0			0
$485$	356.537			0.735127
$486$		0			0
$487$	−597.640			−1.22719			−0.613593	−	0.789622i	$-0.710277\pi$
$487$	−597.640			−1.22719			−0.613593	+	0.789622i	$0.710277\pi$
$488$		0			0
$489$			93.2811i			0.190759i
$490$		0			0
$491$	107.625			0.219195			0.109598	−	0.993976i	$-0.465044\pi$
$491$	107.625			0.219195			0.109598	+	0.993976i	$0.465044\pi$
$492$		0			0
$493$			432.059i			0.876388i
$494$		0			0
$495$			114.040i			0.230384i
$496$		0			0
$497$	85.7821	−	158.400i	0.172600	−	0.318713i
$498$		0			0
$499$	420.611			0.842908			0.421454	−	0.906850i	$-0.361520\pi$
$499$	420.611			0.842908			0.421454	+	0.906850i	$0.361520\pi$
$500$		0			0
$501$	72.3870			0.144485
$502$		0			0
$503$			837.716i			1.66544i	0.553694	+	0.832720i	$0.313218\pi$
$503$			837.716i			1.66544i	−0.553694	+	0.832720i	$0.686782\pi$
$504$		0			0
$505$	53.9742			0.106880
$506$		0			0
$507$		−	169.278i		−	0.333881i
$508$		0			0
$509$			511.304i			1.00453i	0.864715	+	0.502264i	$0.167499\pi$
$509$			511.304i			1.00453i	−0.864715	+	0.502264i	$0.832501\pi$
$510$		0			0
$511$	462.049	+	250.224i	0.904205	+	0.489675i
$512$		0			0
$513$	−71.4466			−0.139272
$514$		0			0
$515$	195.314			0.379250
$516$		0			0
$517$			682.358i			1.31984i
$518$		0			0
$519$	225.762			0.434995
$520$		0			0
$521$			958.401i			1.83954i	0.392455	+	0.919771i	$0.371626\pi$
$521$			958.401i			1.83954i	−0.392455	+	0.919771i	$0.628374\pi$
$522$		0			0
$523$			152.860i			0.292275i	0.989264	+	0.146138i	$0.0466843\pi$
$523$			152.860i			0.292275i	−0.989264	+	0.146138i	$0.953316\pi$
$524$		0			0
$525$	−53.3068	−	28.8685i	−0.101537	−	0.0549875i
$526$		0			0
$527$	−90.5445			−0.171811
$528$		0			0
$529$	−252.845			−0.477968
$530$		0			0
$531$			244.801i			0.461020i
$532$		0			0
$533$	649.017			1.21767
$534$		0			0
$535$		−	376.461i		−	0.703666i
$536$		0			0
$537$		−	77.9309i		−	0.145123i
$538$		0			0
$539$	455.200	+	697.632i	0.844527	+	1.29431i
$540$		0			0
$541$	−285.016			−0.526832			−0.263416	−	0.964682i	$-0.584849\pi$
$541$	−285.016			−0.526832			−0.263416	+	0.964682i	$0.584849\pi$
$542$		0			0
$543$	−30.9940			−0.0570791
$544$		0			0
$545$		−	347.865i		−	0.638285i
$546$		0			0
$547$	195.330			0.357092			0.178546	−	0.983932i	$-0.442861\pi$
$547$	195.330			0.357092			0.178546	+	0.983932i	$0.442861\pi$
$548$		0			0
$549$		−	44.9578i		−	0.0818904i
$550$		0			0
$551$		−	442.463i		−	0.803017i
$552$		0			0
$553$	−266.835	+	492.722i	−0.482523	+	0.890999i
$554$		0			0
$555$	268.065			0.483000
$556$		0			0
$557$	−27.9390			−0.0501598			−0.0250799	−	0.999685i	$-0.507984\pi$
$557$	−27.9390			−0.0501598			−0.0250799	+	0.999685i	$0.507984\pi$
$558$		0			0
$559$		−	705.883i		−	1.26276i
$560$		0			0
$561$	395.347			0.704719
$562$		0			0
$563$			459.585i			0.816314i	0.912912	+	0.408157i	$0.133828\pi$
$563$			459.585i			0.816314i	−0.912912	+	0.408157i	$0.866172\pi$
$564$		0			0
$565$		−	46.9497i		−	0.0830968i
$566$		0			0
$567$	30.0010	−	55.3980i	0.0529118	−	0.0977037i
$568$		0			0
$569$	−635.353			−1.11661			−0.558306	−	0.829635i	$-0.688549\pi$
$569$	−635.353			−1.11661			−0.558306	+	0.829635i	$0.688549\pi$
$570$		0			0
$571$	−205.282			−0.359513			−0.179757	−	0.983711i	$-0.557531\pi$
$571$	−205.282			−0.359513			−0.179757	+	0.983711i	$0.557531\pi$
$572$		0			0
$573$		−	308.849i		−	0.539004i
$574$		0			0
$575$	−83.0895			−0.144504
$576$		0			0
$577$		−	185.464i		−	0.321429i	−0.987001	−	0.160714i	$-0.948620\pi$
$577$		−	185.464i		−	0.321429i	0.987001	−	0.160714i	$-0.0513798\pi$
$578$		0			0
$579$		−	582.460i		−	1.00598i
$580$		0			0
$581$	628.537	+	340.386i	1.08182	+	0.585862i
$582$		0			0
$583$	−382.506			−0.656100
$584$		0			0
$585$	109.558			0.187279
$586$		0			0
$587$			673.958i			1.14814i	0.818806	+	0.574070i	$0.194636\pi$
$587$			673.958i			1.14814i	−0.818806	+	0.574070i	$0.805364\pi$
$588$		0			0
$589$	92.7247			0.157427
$590$		0			0
$591$			85.2311i			0.144215i
$592$		0			0
$593$		−	0.486694i		−	0.000820731i	−1.00000		0.000410366i	$-0.999869\pi$
$593$		−	0.486694i		−	0.000820731i	1.00000		0.000410366i	$-0.000130623\pi$
$594$		0			0
$595$	−100.079	+	184.800i	−0.168201	+	0.310589i
$596$		0			0
$597$	297.547			0.498404
$598$		0			0
$599$	−580.285			−0.968756			−0.484378	−	0.874859i	$-0.660954\pi$
$599$	−580.285			−0.968756			−0.484378	+	0.874859i	$0.660954\pi$
$600$		0			0
$601$			781.851i			1.30092i	0.759542	+	0.650459i	$0.225423\pi$
$601$			781.851i			1.30092i	−0.759542	+	0.650459i	$0.774577\pi$
$602$		0			0
$603$	216.262			0.358643
$604$		0			0
$605$			375.667i			0.620938i
$606$		0			0
$607$			907.211i			1.49458i	0.664498	+	0.747290i	$0.268646\pi$
$607$			907.211i			1.49458i	−0.664498	+	0.747290i	$0.731354\pi$
$608$		0			0
$609$	343.075	+	185.794i	0.563342	+	0.305080i
$610$		0			0
$611$	655.539			1.07289
$612$		0			0
$613$	−911.642			−1.48718			−0.743590	−	0.668636i	$-0.766879\pi$
$613$	−911.642			−1.48718			−0.743590	+	0.668636i	$0.766879\pi$
$614$		0			0
$615$			153.909i			0.250258i
$616$		0			0
$617$	637.918			1.03390			0.516951	−	0.856015i	$-0.327067\pi$
$617$	637.918			1.03390			0.516951	+	0.856015i	$0.327067\pi$
$618$		0			0
$619$			862.604i			1.39354i	0.717292	+	0.696772i	$0.245381\pi$
$619$			862.604i			1.39354i	−0.717292	+	0.696772i	$0.754619\pi$
$620$		0			0
$621$		−	86.3492i		−	0.139049i
$622$		0			0
$623$	−790.229	−	427.951i	−1.26843	−	0.686920i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−404.866			−0.645720
$628$		0			0
$629$		−	929.310i		−	1.47744i
$630$		0			0
$631$	601.057			0.952547			0.476273	−	0.879297i	$-0.341987\pi$
$631$	601.057			0.952547			0.476273	+	0.879297i	$0.341987\pi$
$632$		0			0
$633$			8.82406i			0.0139401i
$634$		0			0
$635$			133.876i			0.210828i
$636$		0			0
$637$	670.213	−	437.309i	1.05214	−	0.686514i
$638$		0			0
$639$	−77.2014			−0.120816
$640$		0			0
$641$	−884.432			−1.37977			−0.689885	−	0.723919i	$-0.742339\pi$
$641$	−884.432			−1.37977			−0.689885	+	0.723919i	$0.742339\pi$
$642$		0			0
$643$		−	385.448i		−	0.599453i	−0.954025	−	0.299726i	$-0.903105\pi$
$643$		−	385.448i		−	0.599453i	0.954025	−	0.299726i	$-0.0968954\pi$
$644$		0			0
$645$	167.394			0.259526
$646$		0			0
$647$		−	1096.93i		−	1.69540i	−0.530473	−	0.847702i	$-0.677986\pi$
$647$		−	1096.93i		−	1.69540i	0.530473	−	0.847702i	$-0.322014\pi$
$648$		0			0
$649$			1387.22i			2.13747i
$650$		0			0
$651$	−38.9358	+	71.8966i	−0.0598093	+	0.110440i
$652$		0			0
$653$	−556.888			−0.852814			−0.426407	−	0.904531i	$-0.640221\pi$
$653$	−556.888			−0.852814			−0.426407	+	0.904531i	$0.640221\pi$
$654$		0			0
$655$	−373.127			−0.569660
$656$		0			0
$657$		−	225.194i		−	0.342761i
$658$		0			0
$659$	715.578			1.08585			0.542927	−	0.839780i	$-0.317316\pi$
$659$	715.578			1.08585			0.542927	+	0.839780i	$0.317316\pi$
$660$		0			0
$661$			280.365i			0.424152i	0.977253	+	0.212076i	$0.0680225\pi$
$661$			280.365i			0.424152i	−0.977253	+	0.212076i	$0.931978\pi$
$662$		0			0
$663$		−	379.809i		−	0.572864i
$664$		0			0
$665$	102.489	−	189.250i	0.154119	−	0.284587i
$666$		0			0
$667$	534.753			0.801729
$668$		0			0
$669$	−537.050			−0.802765
$670$		0			0
$671$		−	254.763i		−	0.379676i
$672$		0			0
$673$	1067.08			1.58556			0.792781	−	0.609506i	$-0.208632\pi$
$673$	1067.08			1.58556			0.792781	+	0.609506i	$0.208632\pi$
$674$		0			0
$675$			25.9808i			0.0384900i
$676$		0			0
$677$		−	313.071i		−	0.462439i	−0.972902	−	0.231219i	$-0.925728\pi$
$677$		−	313.071i		−	0.462439i	0.972902	−	0.231219i	$-0.0742715\pi$
$678$		0			0
$679$	−981.456	−	531.511i	−1.44544	−	0.782785i
$680$		0			0
$681$	−188.028			−0.276105
$682$		0			0
$683$	−505.514			−0.740138			−0.370069	−	0.929004i	$-0.620666\pi$
$683$	−505.514			−0.740138			−0.370069	+	0.929004i	$0.620666\pi$
$684$		0			0
$685$			282.056i			0.411761i
$686$		0			0
$687$	−409.601			−0.596217
$688$		0			0
$689$			367.473i			0.533342i
$690$		0			0
$691$			276.726i			0.400472i	0.979748	+	0.200236i	$0.0641709\pi$
$691$			276.726i			0.400472i	−0.979748	+	0.200236i	$0.935829\pi$
$692$		0			0
$693$	170.007	−	313.924i	0.245320	−	0.452993i
$694$		0			0
$695$	473.263			0.680954
$696$		0			0
$697$	533.561			0.765511
$698$		0			0
$699$			262.041i			0.374880i
$700$		0			0
$701$	854.178			1.21851			0.609256	−	0.792973i	$-0.291468\pi$
$701$	854.178			1.21851			0.609256	+	0.792973i	$0.291468\pi$
$702$		0			0
$703$			951.687i			1.35375i
$704$		0			0
$705$			155.455i			0.220504i
$706$		0			0
$707$	−148.577	−	80.4626i	−0.210152	−	0.113809i
$708$		0			0
$709$	−452.996			−0.638922			−0.319461	−	0.947599i	$-0.603502\pi$
$709$	−452.996			−0.638922			−0.319461	+	0.947599i	$0.603502\pi$
$710$		0			0
$711$	240.144			0.337755
$712$		0			0
$713$			112.066i			0.157175i
$714$		0			0
$715$	620.833			0.868297
$716$		0			0
$717$		−	83.6504i		−	0.116667i
$718$		0			0
$719$			840.685i			1.16924i	0.811306	+	0.584621i	$0.198757\pi$
$719$			840.685i			1.16924i	−0.811306	+	0.584621i	$0.801243\pi$
$720$		0			0
$721$	−537.650	−	291.166i	−0.745700	−	0.403836i
$722$		0			0
$723$	−399.645			−0.552759
$724$		0			0
$725$	−160.897			−0.221926
$726$		0			0
$727$		−	1339.58i		−	1.84262i	−0.388829	−	0.921310i	$-0.627120\pi$
$727$		−	1339.58i		−	1.84262i	0.388829	−	0.921310i	$-0.372880\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	580.311i		−	0.793860i
$732$		0			0
$733$		−	536.275i		−	0.731616i	−0.930690	−	0.365808i	$-0.880793\pi$
$733$		−	536.275i		−	0.731616i	0.930690	−	0.365808i	$-0.119207\pi$
$734$		0			0
$735$	103.704	+	158.935i	0.141094	+	0.216238i
$736$		0			0
$737$	1225.49			1.66281
$738$		0			0
$739$	898.552			1.21590			0.607951	−	0.793974i	$-0.291992\pi$
$739$	898.552			1.21590			0.607951	+	0.793974i	$0.291992\pi$
$740$		0			0
$741$			388.954i			0.524904i
$742$		0			0
$743$	−532.720			−0.716985			−0.358493	−	0.933533i	$-0.616709\pi$
$743$	−532.720			−0.716985			−0.358493	+	0.933533i	$0.616709\pi$
$744$		0			0
$745$		−	143.516i		−	0.192639i
$746$		0			0
$747$		−	306.337i		−	0.410090i
$748$		0			0
$749$	−561.214	+	1036.30i	−0.749284	+	1.38358i
$750$		0			0
$751$	−342.085			−0.455506			−0.227753	−	0.973719i	$-0.573138\pi$
$751$	−342.085			−0.455506			−0.227753	+	0.973719i	$0.573138\pi$
$752$		0			0
$753$	−150.136			−0.199384
$754$		0			0
$755$		−	248.014i		−	0.328495i
$756$		0			0
$757$	405.961			0.536276			0.268138	−	0.963381i	$-0.413592\pi$
$757$	405.961			0.536276			0.268138	+	0.963381i	$0.413592\pi$
$758$		0			0
$759$		−	489.315i		−	0.644684i
$760$		0			0
$761$		−	577.340i		−	0.758660i	−0.925261	−	0.379330i	$-0.876155\pi$
$761$		−	577.340i		−	0.758660i	0.925261	−	0.379330i	$-0.123845\pi$
$762$		0			0
$763$	−518.584	+	957.586i	−0.679664	+	1.25503i
$764$		0			0
$765$	90.0684			0.117737
$766$		0			0
$767$	1332.69			1.73754
$768$		0			0
$769$			828.522i			1.07740i	0.842497	+	0.538701i	$0.181085\pi$
$769$			828.522i			1.07740i	−0.842497	+	0.538701i	$0.818915\pi$
$770$		0			0
$771$	244.409			0.317003
$772$		0			0
$773$			438.991i			0.567906i	0.958838	+	0.283953i	$0.0916460\pi$
$773$			438.991i			0.567906i	−0.958838	+	0.283953i	$0.908354\pi$
$774$		0			0
$775$		−	33.7183i		−	0.0435075i
$776$		0			0
$777$	−737.916	−	399.621i	−0.949699	−	0.514312i
$778$		0			0
$779$	−546.408			−0.701423
$780$		0			0
$781$	−437.477			−0.560150
$782$		0			0
$783$		−	167.209i		−	0.213549i
$784$		0			0
$785$	649.469			0.827349
$786$		0			0
$787$			285.209i			0.362400i	0.983446	+	0.181200i	$0.0579982\pi$
$787$			285.209i			0.362400i	−0.983446	+	0.181200i	$0.942002\pi$
$788$		0			0
$789$		−	65.9740i		−	0.0836172i
$790$		0			0
$791$	−69.9907	+	129.241i	−0.0884838	+	0.163389i
$792$		0			0
$793$	−244.750			−0.308638
$794$		0			0
$795$	−87.1431			−0.109614
$796$		0			0
$797$		−	1243.03i		−	1.55964i	−0.626006	−	0.779818i	$-0.715311\pi$
$797$		−	1243.03i		−	1.55964i	0.626006	−	0.779818i	$-0.284689\pi$
$798$		0			0
$799$	538.923			0.674497
$800$		0			0
$801$			385.143i			0.480828i
$802$		0			0
$803$		−	1276.11i		−	1.58918i
$804$		0			0
$805$	228.725	+	123.867i	0.284130	+	0.153872i
$806$		0			0
$807$	786.732			0.974885
$808$		0			0
$809$	−630.338			−0.779157			−0.389578	−	0.920993i	$-0.627379\pi$
$809$	−630.338			−0.779157			−0.389578	+	0.920993i	$0.627379\pi$
$810$		0			0
$811$		−	1121.08i		−	1.38234i	−0.722692	−	0.691170i	$-0.757095\pi$
$811$		−	1121.08i		−	1.38234i	0.722692	−	0.691170i	$-0.242905\pi$
$812$		0			0
$813$	136.329			0.167687
$814$		0			0
$815$			120.425i			0.147761i
$816$		0			0
$817$			594.284i			0.727398i
$818$		0			0
$819$	−301.586	−	163.325i	−0.368237	−	0.199420i
$820$		0			0
$821$	544.285			0.662954			0.331477	−	0.943463i	$-0.392453\pi$
$821$	544.285			0.662954			0.331477	+	0.943463i	$0.392453\pi$
$822$		0			0
$823$	−83.7611			−0.101775			−0.0508877	−	0.998704i	$-0.516205\pi$
$823$	−83.7611			−0.101775			−0.0508877	+	0.998704i	$0.516205\pi$
$824$		0			0
$825$			147.225i			0.178455i
$826$		0			0
$827$	957.378			1.15765			0.578826	−	0.815451i	$-0.303511\pi$
$827$	957.378			1.15765			0.578826	+	0.815451i	$0.303511\pi$
$828$		0			0
$829$		−	9.08184i		−	0.0109552i	−0.999985	−	0.00547759i	$-0.998256\pi$
$829$		−	9.08184i		−	0.0109552i	0.999985	−	0.00547759i	$-0.00174358\pi$
$830$		0			0
$831$		−	147.812i		−	0.177873i
$832$		0			0
$833$	550.987	−	359.515i	0.661449	−	0.431591i
$834$		0			0
$835$	93.4512			0.111918
$836$		0			0
$837$	35.0411			0.0418651
$838$		0			0
$839$			492.860i			0.587437i	0.955892	+	0.293719i	$0.0948929\pi$
$839$			492.860i			0.587437i	−0.955892	+	0.293719i	$0.905107\pi$
$840$		0			0
$841$	194.509			0.231283
$842$		0			0
$843$			289.903i			0.343894i
$844$		0			0
$845$		−	218.536i		−	0.258623i
$846$		0			0
$847$	560.030	−	1034.12i	0.661193	−	1.22092i
$848$		0			0
$849$	−259.098			−0.305181
$850$		0			0
$851$	−1150.19			−1.35158
$852$		0			0
$853$			519.198i			0.608672i	0.952565	+	0.304336i	$0.0984346\pi$
$853$			519.198i			0.608672i	−0.952565	+	0.304336i	$0.901565\pi$
$854$		0			0
$855$	−92.2371			−0.107880
$856$		0			0
$857$			479.468i			0.559473i	0.960077	+	0.279736i	$0.0902470\pi$
$857$			479.468i			0.559473i	−0.960077	+	0.279736i	$0.909753\pi$
$858$		0			0
$859$			1667.63i			1.94136i	0.240378	+	0.970679i	$0.422729\pi$
$859$			1667.63i			1.94136i	−0.240378	+	0.970679i	$0.577271\pi$
$860$		0			0
$861$	229.441	−	423.672i	0.266482	−	0.492070i
$862$		0			0
$863$	−1217.16			−1.41039			−0.705193	−	0.709016i	$-0.749139\pi$
$863$	−1217.16			−1.41039			−0.705193	+	0.709016i	$0.749139\pi$
$864$		0			0
$865$	291.458			0.336946
$866$		0			0
$867$			188.319i			0.217208i
$868$		0			0
$869$	1360.82			1.56597
$870$		0			0
$871$		−	1177.32i		−	1.35169i
$872$		0			0
$873$			478.344i			0.547931i
$874$		0			0
$875$	−68.8187	−	37.2690i	−0.0786500	−	0.0425932i
$876$		0			0
$877$	−1027.18			−1.17125			−0.585623	−	0.810583i	$-0.699150\pi$
$877$	−1027.18			−1.17125			−0.585623	+	0.810583i	$0.699150\pi$
$878$		0			0
$879$	−252.541			−0.287305
$880$		0			0
$881$		−	149.054i		−	0.169187i	−0.996416	−	0.0845937i	$-0.973041\pi$
$881$		−	149.054i		−	0.169187i	0.996416	−	0.0845937i	$-0.0269592\pi$
$882$		0			0
$883$	−201.243			−0.227908			−0.113954	−	0.993486i	$-0.536352\pi$
$883$	−201.243			−0.227908			−0.113954	+	0.993486i	$0.536352\pi$
$884$		0			0
$885$			316.037i			0.357104i
$886$		0			0
$887$		−	951.252i		−	1.07244i	−0.844079	−	0.536219i	$-0.819852\pi$
$887$		−	951.252i		−	1.07244i	0.844079	−	0.536219i	$-0.180148\pi$
$888$		0			0
$889$	199.577	−	368.527i	0.224496	−	0.414542i
$890$		0			0
$891$	−153.001			−0.171718
$892$		0			0
$893$	−551.899			−0.618028
$894$		0			0
$895$		−	100.608i		−	0.112412i
$896$		0			0
$897$	−470.083			−0.524062
$898$		0			0
$899$			217.007i			0.241387i
$900$		0			0
$901$			302.102i			0.335296i
$902$		0			0
$903$	−460.794	−	249.545i	−0.510293	−	0.276351i
$904$		0			0
$905$	−40.0130			−0.0442133
$906$		0			0
$907$	559.990			0.617409			0.308705	−	0.951158i	$-0.400105\pi$
$907$	559.990			0.617409			0.308705	+	0.951158i	$0.400105\pi$
$908$		0			0
$909$			72.4140i			0.0796633i
$910$		0			0
$911$	674.618			0.740525			0.370262	−	0.928927i	$-0.379268\pi$
$911$	674.618			0.740525			0.370262	+	0.928927i	$0.379268\pi$
$912$		0			0
$913$		−	1735.92i		−	1.90134i
$914$		0			0
$915$		−	58.0403i		−	0.0634320i
$916$		0			0
$917$	1027.13	+	556.244i	1.12009	+	0.606591i
$918$		0			0
$919$	−982.395			−1.06898			−0.534491	−	0.845174i	$-0.679497\pi$
$919$	−982.395			−1.06898			−0.534491	+	0.845174i	$0.679497\pi$
$920$		0			0
$921$	−356.099			−0.386644
$922$		0			0
$923$			420.283i			0.455345i
$924$		0			0
$925$	346.070			0.374130
$926$		0			0
$927$			262.041i			0.282676i
$928$		0			0
$929$			344.774i			0.371124i	0.982633	+	0.185562i	$0.0594105\pi$
$929$			344.774i			0.371124i	−0.982633	+	0.185562i	$0.940589\pi$
$930$		0			0
$931$	−564.253	+	368.172i	−0.606072	+	0.395458i
$932$		0			0
$933$	735.053			0.787838
$934$		0			0
$935$	510.391			0.545873
$936$		0			0
$937$		−	635.256i		−	0.677967i	−0.940792	−	0.338984i	$-0.889917\pi$
$937$		−	635.256i		−	0.677967i	0.940792	−	0.338984i	$-0.110083\pi$
$938$		0			0
$939$	−629.606			−0.670507
$940$		0			0
$941$		−	1207.78i		−	1.28351i	−0.766909	−	0.641756i	$-0.778206\pi$
$941$		−	1207.78i		−	1.28351i	0.766909	−	0.641756i	$-0.221794\pi$
$942$		0			0
$943$		−	660.380i		−	0.700297i
$944$		0			0
$945$	38.7311	−	71.5185i	0.0409853	−	0.0756810i
$946$		0			0
$947$	254.133			0.268356			0.134178	−	0.990957i	$-0.457161\pi$
$947$	254.133			0.268356			0.134178	+	0.990957i	$0.457161\pi$
$948$		0			0
$949$	−1225.95			−1.29184
$950$		0			0
$951$			764.540i			0.803933i
$952$		0			0
$953$	424.523			0.445460			0.222730	−	0.974880i	$-0.428503\pi$
$953$	424.523			0.445460			0.222730	+	0.974880i	$0.428503\pi$
$954$		0			0
$955$		−	398.723i		−	0.417511i
$956$		0			0
$957$		−	947.522i		−	0.990096i
$958$		0			0
$959$	420.479	−	776.431i	0.438455	−	0.809625i
$960$		0			0
$961$	915.523			0.952677
$962$		0			0
$963$	505.076			0.524482
$964$		0			0
$965$		−	751.952i		−	0.779225i
$966$		0			0
$967$	46.7338			0.0483286			0.0241643	−	0.999708i	$-0.492308\pi$
$967$	46.7338			0.0483286			0.0241643	+	0.999708i	$0.492308\pi$
$968$		0			0
$969$			319.762i			0.329991i
$970$		0			0
$971$			1724.03i			1.77552i	0.460303	+	0.887762i	$0.347741\pi$
$971$			1724.03i			1.77552i	−0.460303	+	0.887762i	$0.652259\pi$
$972$		0			0
$973$	−1302.77	−	705.522i	−1.33893	−	0.725100i
$974$		0			0
$975$	141.439			0.145065
$976$		0			0
$977$	−1613.20			−1.65118			−0.825589	−	0.564271i	$-0.809157\pi$
$977$	−1613.20			−1.65118			−0.825589	+	0.564271i	$0.809157\pi$
$978$		0			0
$979$			2182.49i			2.22931i
$980$		0			0
$981$	466.710			0.475749
$982$		0			0
$983$			31.6909i			0.0322389i	0.999870	+	0.0161195i	$0.00513121\pi$
$983$			31.6909i			0.0322389i	−0.999870	+	0.0161195i	$0.994869\pi$
$984$		0			0
$985$			110.033i			0.111708i
$986$		0			0
$987$	231.747	−	427.930i	0.234799	−	0.433566i
$988$		0			0
$989$	−718.242			−0.726231
$990$		0			0
$991$	236.475			0.238623			0.119311	−	0.992857i	$-0.461931\pi$
$991$	236.475			0.238623			0.119311	+	0.992857i	$0.461931\pi$
$992$		0			0
$993$		−	882.180i		−	0.888399i
$994$		0			0
$995$	384.132			0.386062
$996$		0			0
$997$		−	948.441i		−	0.951295i	−0.879636	−	0.475648i	$-0.842214\pi$
$997$		−	948.441i		−	0.951295i	0.879636	−	0.475648i	$-0.157786\pi$
$998$		0			0
$999$			359.647i			0.360007i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.s.c.1441.11		12
4.3	odd	2		105.3.h.a.76.5	✓	12
7.6	odd	2	inner	1680.3.s.c.1441.2		12
12.11	even	2		315.3.h.d.181.7		12
20.3	even	4		525.3.e.c.349.16		24
20.7	even	4		525.3.e.c.349.1		24
20.19	odd	2		525.3.h.d.76.8		12
28.27	even	2		105.3.h.a.76.6	yes	12
84.83	odd	2		315.3.h.d.181.8		12
140.27	odd	4		525.3.e.c.349.15		24
140.83	odd	4		525.3.e.c.349.2		24
140.139	even	2		525.3.h.d.76.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.h.a.76.5	✓	12	4.3	odd	2
105.3.h.a.76.6	yes	12	28.27	even	2
315.3.h.d.181.7		12	12.11	even	2
315.3.h.d.181.8		12	84.83	odd	2
525.3.e.c.349.1		24	20.7	even	4
525.3.e.c.349.2		24	140.83	odd	4
525.3.e.c.349.15		24	140.27	odd	4
525.3.e.c.349.16		24	20.3	even	4
525.3.h.d.76.7		12	140.139	even	2
525.3.h.d.76.8		12	20.19	odd	2
1680.3.s.c.1441.2		12	7.6	odd	2	inner
1680.3.s.c.1441.11		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.s (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +2.23607i q^{5} +(3.33344 - 6.15534i) q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +2.23607i q^{5} +(3.33344 - 6.15534i) q^{7} -3.00000 q^{9} -17.0001 q^{11} +16.3319i q^{13} -3.87298 q^{15} +13.4266i q^{17} -13.7499i q^{19} +(10.6614 + 5.77369i) q^{21} +16.6179 q^{23} -5.00000 q^{25} -5.19615i q^{27} +32.1793 q^{29} +6.74366i q^{31} -29.4450i q^{33} +(13.7637 + 7.45380i) q^{35} -69.2141 q^{37} -28.2878 q^{39} -39.7391i q^{41} -43.2210 q^{43} -6.70820i q^{45} -40.1384i q^{47} +(-26.7763 - 41.0369i) q^{49} -23.2556 q^{51} +22.5002 q^{53} -38.0134i q^{55} +23.8155 q^{57} -81.6005i q^{59} +14.9859i q^{61} +(-10.0003 + 18.4660i) q^{63} -36.5193 q^{65} -72.0872 q^{67} +28.7831i q^{69} +25.7338 q^{71} +75.0647i q^{73} -8.66025i q^{75} +(-56.6689 + 104.641i) q^{77} -80.0480 q^{79} +9.00000 q^{81} +102.112i q^{83} -30.0228 q^{85} +55.7362i q^{87} -128.381i q^{89} +(100.529 + 54.4416i) q^{91} -11.6804 q^{93} +30.7457 q^{95} -159.448i q^{97} +51.0003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{7} - 36 q^{9}+O(q^{10}) \) 12 * q + 8 * q^7 - 36 * q^9	\( 12 q + 8 q^{7} - 36 q^{9} + 16 q^{11} + 36 q^{21} + 64 q^{23} - 60 q^{25} + 104 q^{29} - 60 q^{35} + 32 q^{37} + 24 q^{39} - 152 q^{43} + 60 q^{49} - 24 q^{51} + 176 q^{53} - 240 q^{57} - 24 q^{63} - 240 q^{65} - 168 q^{67} - 32 q^{71} + 8 q^{77} - 120 q^{79} + 108 q^{81} + 120 q^{85} - 24 q^{91} + 48 q^{93} - 48 q^{99}+O(q^{100}) \) 12 * q + 8 * q^7 - 36 * q^9 + 16 * q^11 + 36 * q^21 + 64 * q^23 - 60 * q^25 + 104 * q^29 - 60 * q^35 + 32 * q^37 + 24 * q^39 - 152 * q^43 + 60 * q^49 - 24 * q^51 + 176 * q^53 - 240 * q^57 - 24 * q^63 - 240 * q^65 - 168 * q^67 - 32 * q^71 + 8 * q^77 - 120 * q^79 + 108 * q^81 + 120 * q^85 - 24 * q^91 + 48 * q^93 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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