LMFDB - Embedded newform 2240.2.g.o.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(449,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.g (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:	$17.8864900528$
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} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 $ x^10 + 13x^8 + 56x^6 + 97x^4 + 61x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-1.28447i$ of defining polynomial
Character	$\chi$	$=$	2240.449
Dual form			2240.2.g.o.449.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.63460i q^{3} +(2.23081 - 0.153266i) q^{5} -1.00000i q^{7} +0.328072 q^{9} +O(q^{10})$	$q-1.63460i q^{3} +(2.23081 - 0.153266i) q^{5} -1.00000i q^{7} +0.328072 q^{9} +1.24087 q^{11} +4.20355i q^{13} +(-0.250528 - 3.64649i) q^{15} -3.39596i q^{17} -6.46162 q^{19} -1.63460 q^{21} -2.15509i q^{23} +(4.95302 - 0.683813i) q^{25} -5.44008i q^{27} +3.96490 q^{29} +10.0611 q^{31} -2.02833i q^{33} +(-0.153266 - 2.23081i) q^{35} -6.76815i q^{37} +6.87113 q^{39} -0.131318 q^{41} -7.40498i q^{43} +(0.731866 - 0.0502822i) q^{45} -4.82702i q^{47} -1.00000 q^{49} -5.55105 q^{51} +10.0374i q^{53} +(2.76815 - 0.190183i) q^{55} +10.5622i q^{57} -10.9864 q^{59} +6.33030 q^{61} -0.328072i q^{63} +(0.644259 + 9.37731i) q^{65} +2.65614i q^{67} -3.52271 q^{69} -0.754563 q^{71} -6.03736i q^{73} +(-1.11776 - 8.09622i) q^{75} -1.24087i q^{77} +14.6652 q^{79} -7.90815 q^{81} +14.0813i q^{83} +(-0.520484 - 7.57574i) q^{85} -6.48105i q^{87} +12.6742 q^{89} +4.20355 q^{91} -16.4459i q^{93} +(-14.4146 + 0.990344i) q^{95} +0.914215i q^{97} +0.407096 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{5} - 14 q^{9}+O(q^{10}) $ 10 * q + 2 * q^5 - 14 * q^9	$ 10 q + 2 q^{5} - 14 q^{9} + 8 q^{11} - 4 q^{15} - 24 q^{19} + 4 q^{21} + 6 q^{25} - 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} - 10 q^{45} - 10 q^{49} + 24 q^{51} - 16 q^{55} - 32 q^{59} + 20 q^{61} - 8 q^{65} + 8 q^{69} - 8 q^{71} + 64 q^{75} + 64 q^{79} + 2 q^{81} + 12 q^{85} - 4 q^{89} - 60 q^{95} - 80 q^{99}+O(q^{100}) $ 10 * q + 2 * q^5 - 14 * q^9 + 8 * q^11 - 4 * q^15 - 24 * q^19 + 4 * q^21 + 6 * q^25 - 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 - 10 * q^45 - 10 * q^49 + 24 * q^51 - 16 * q^55 - 32 * q^59 + 20 * q^61 - 8 * q^65 + 8 * q^69 - 8 * q^71 + 64 * q^75 + 64 * q^79 + 2 * q^81 + 12 * q^85 - 4 * q^89 - 60 * q^95 - 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	1.63460i		−	0.943739i	−0.881668	−	0.471869i	$-0.843579\pi$
$3$		−	1.63460i		−	0.943739i	0.881668	−	0.471869i	$-0.156421\pi$
$4$		0			0
$5$	2.23081	−	0.153266i	0.997648	−	0.0685425i
$5$	2.23081	−	0.153266i	0.997648	−	0.0685425i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	0.328072			0.109357
$10$		0			0
$11$	1.24087			0.374137			0.187069	−	0.982347i	$-0.440101\pi$
$11$	1.24087			0.374137			0.187069	+	0.982347i	$0.440101\pi$
$12$		0			0
$13$			4.20355i			1.16585i	0.812524	+	0.582927i	$0.198093\pi$
$13$			4.20355i			1.16585i	−0.812524	+	0.582927i	$0.801907\pi$
$14$		0			0
$15$	−0.250528	−	3.64649i	−0.0646862	−	0.941519i
$16$		0			0
$17$		−	3.39596i		−	0.823641i	−0.911265	−	0.411821i	$-0.864893\pi$
$17$		−	3.39596i		−	0.823641i	0.911265	−	0.411821i	$-0.135107\pi$
$18$		0			0
$19$	−6.46162			−1.48240			−0.741198	−	0.671286i	$-0.765742\pi$
$19$	−6.46162			−1.48240			−0.741198	+	0.671286i	$0.765742\pi$
$20$		0			0
$21$	−1.63460			−0.356700
$22$		0			0
$23$		−	2.15509i		−	0.449367i	−0.974432	−	0.224683i	$-0.927865\pi$
$23$		−	2.15509i		−	0.449367i	0.974432	−	0.224683i	$-0.0721348\pi$
$24$		0			0
$25$	4.95302	−	0.683813i	0.990604	−	0.136763i
$26$		0			0
$27$		−	5.44008i		−	1.04694i
$28$		0			0
$29$	3.96490			0.736264			0.368132	−	0.929773i	$-0.379997\pi$
$29$	3.96490			0.736264			0.368132	+	0.929773i	$0.379997\pi$
$30$		0			0
$31$	10.0611			1.80703			0.903516	−	0.428555i	$-0.140977\pi$
$31$	10.0611			1.80703			0.903516	+	0.428555i	$0.140977\pi$
$32$		0			0
$33$		−	2.02833i		−	0.353088i
$34$		0			0
$35$	−0.153266	−	2.23081i	−0.0259066	−	0.377076i
$36$		0			0
$37$		−	6.76815i		−	1.11268i	−0.830956	−	0.556338i	$-0.812206\pi$
$37$		−	6.76815i		−	1.11268i	0.830956	−	0.556338i	$-0.187794\pi$
$38$		0			0
$39$	6.87113			1.10026
$40$		0			0
$41$	−0.131318			−0.0205084			−0.0102542	−	0.999947i	$-0.503264\pi$
$41$	−0.131318			−0.0205084			−0.0102542	+	0.999947i	$0.503264\pi$
$42$		0			0
$43$		−	7.40498i		−	1.12925i	−0.825348	−	0.564625i	$-0.809021\pi$
$43$		−	7.40498i		−	1.12925i	0.825348	−	0.564625i	$-0.190979\pi$
$44$		0			0
$45$	0.731866	−	0.0502822i	0.109100	−	0.00749562i
$46$		0			0
$47$		−	4.82702i		−	0.704093i	−0.935983	−	0.352046i	$-0.885486\pi$
$47$		−	4.82702i		−	0.704093i	0.935983	−	0.352046i	$-0.114514\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−5.55105			−0.777302
$52$		0			0
$53$			10.0374i			1.37874i	0.724411	+	0.689368i	$0.242112\pi$
$53$			10.0374i			1.37874i	−0.724411	+	0.689368i	$0.757888\pi$
$54$		0			0
$55$	2.76815	−	0.190183i	0.373257	−	0.0256443i
$56$		0			0
$57$			10.5622i			1.39900i
$58$		0			0
$59$	−10.9864			−1.43031			−0.715157	−	0.698964i	$-0.753645\pi$
$59$	−10.9864			−1.43031			−0.715157	+	0.698964i	$0.753645\pi$
$60$		0			0
$61$	6.33030			0.810512			0.405256	−	0.914203i	$-0.367182\pi$
$61$	6.33030			0.810512			0.405256	+	0.914203i	$0.367182\pi$
$62$		0			0
$63$		−	0.328072i		−	0.0413332i
$64$		0			0
$65$	0.644259	+	9.37731i	0.0799105	+	1.16311i
$66$		0			0
$67$			2.65614i			0.324500i	0.986750	+	0.162250i	$0.0518750\pi$
$67$			2.65614i			0.324500i	−0.986750	+	0.162250i	$0.948125\pi$
$68$		0			0
$69$	−3.52271			−0.424085
$70$		0			0
$71$	−0.754563			−0.0895501			−0.0447751	−	0.998997i	$-0.514257\pi$
$71$	−0.754563			−0.0895501			−0.0447751	+	0.998997i	$0.514257\pi$
$72$		0			0
$73$		−	6.03736i		−	0.706619i	−0.935506	−	0.353310i	$-0.885056\pi$
$73$		−	6.03736i		−	0.706619i	0.935506	−	0.353310i	$-0.114944\pi$
$74$		0			0
$75$	−1.11776	−	8.09622i	−0.129068	−	0.934871i
$76$		0			0
$77$		−	1.24087i		−	0.141411i
$78$		0			0
$79$	14.6652			1.64996			0.824980	−	0.565162i	$-0.191186\pi$
$79$	14.6652			1.64996			0.824980	+	0.565162i	$0.191186\pi$
$80$		0			0
$81$	−7.90815			−0.878684
$82$		0			0
$83$			14.0813i			1.54562i	0.634639	+	0.772809i	$0.281149\pi$
$83$			14.0813i			1.54562i	−0.634639	+	0.772809i	$0.718851\pi$
$84$		0			0
$85$	−0.520484	−	7.57574i	−0.0564544	−	0.821704i
$86$		0			0
$87$		−	6.48105i		−	0.694841i
$88$		0			0
$89$	12.6742			1.34346			0.671731	−	0.740795i	$-0.265551\pi$
$89$	12.6742			1.34346			0.671731	+	0.740795i	$0.265551\pi$
$90$		0			0
$91$	4.20355			0.440652
$92$		0			0
$93$		−	16.4459i		−	1.70537i
$94$		0			0
$95$	−14.4146	+	0.990344i	−1.47891	+	0.101607i
$96$		0			0
$97$			0.914215i			0.0928244i	0.998922	+	0.0464122i	$0.0147788\pi$
$97$			0.914215i			0.0928244i	−0.998922	+	0.0464122i	$0.985221\pi$
$98$		0			0
$99$	0.407096			0.0409146
$100$		0			0
$101$	−13.4818			−1.34149			−0.670743	−	0.741689i	$-0.734025\pi$
$101$	−13.4818			−1.34149			−0.670743	+	0.741689i	$0.734025\pi$
$102$		0			0
$103$			0.516841i			0.0509258i	0.999676	+	0.0254629i	$0.00810597\pi$
$103$			0.516841i			0.0509258i	−0.999676	+	0.0254629i	$0.991894\pi$
$104$		0			0
$105$	−3.64649	+	0.250528i	−0.355861	+	0.0244491i
$106$		0			0
$107$		−	18.8486i		−	1.82216i	−0.412228	−	0.911081i	$-0.635249\pi$
$107$		−	18.8486i		−	1.82216i	0.412228	−	0.911081i	$-0.364751\pi$
$108$		0			0
$109$	0.345270			0.0330709			0.0165354	−	0.999863i	$-0.494736\pi$
$109$	0.345270			0.0330709			0.0165354	+	0.999863i	$0.494736\pi$
$110$		0			0
$111$	−11.0632			−1.05008
$112$		0			0
$113$		−	3.77026i		−	0.354677i	−0.984150	−	0.177338i	$-0.943251\pi$
$113$		−	3.77026i		−	0.354677i	0.984150	−	0.177338i	$-0.0567487\pi$
$114$		0			0
$115$	−0.330301	−	4.80759i	−0.0308007	−	0.448310i
$116$		0			0
$117$			1.37907i			0.127495i
$118$		0			0
$119$	−3.39596			−0.311307
$120$		0			0
$121$	−9.46024			−0.860021
$122$		0			0
$123$			0.214652i			0.0193545i
$124$		0			0
$125$	10.9444	−	2.28458i	0.978900	−	0.204339i
$126$		0			0
$127$		−	5.63895i		−	0.500376i	−0.968197	−	0.250188i	$-0.919508\pi$
$127$		−	5.63895i		−	0.500376i	0.968197	−	0.250188i	$-0.0804924\pi$
$128$		0			0
$129$	−12.1042			−1.06572
$130$		0			0
$131$	10.5183			0.918987			0.459493	−	0.888181i	$-0.348031\pi$
$131$	10.5183			0.918987			0.459493	+	0.888181i	$0.348031\pi$
$132$		0			0
$133$			6.46162i			0.560293i
$134$		0			0
$135$	−0.833777	−	12.1358i	−0.0717601	−	1.04448i
$136$		0			0
$137$		−	6.96632i		−	0.595173i	−0.954695	−	0.297586i	$-0.903818\pi$
$137$		−	6.96632i		−	0.595173i	0.954695	−	0.297586i	$-0.0961816\pi$
$138$		0			0
$139$	−10.3258			−0.875827			−0.437913	−	0.899017i	$-0.644282\pi$
$139$	−10.3258			−0.875827			−0.437913	+	0.899017i	$0.644282\pi$
$140$		0			0
$141$	−7.89025			−0.664479
$142$		0			0
$143$			5.21607i			0.436189i
$144$		0			0
$145$	8.84494	−	0.607683i	0.734533	−	0.0504654i
$146$		0			0
$147$			1.63460i			0.134820i
$148$		0			0
$149$	−0.773875			−0.0633983			−0.0316992	−	0.999497i	$-0.510092\pi$
$149$	−0.773875			−0.0633983			−0.0316992	+	0.999497i	$0.510092\pi$
$150$		0			0
$151$	−6.14607			−0.500160			−0.250080	−	0.968225i	$-0.580457\pi$
$151$	−6.14607			−0.500160			−0.250080	+	0.968225i	$0.580457\pi$
$152$		0			0
$153$		−	1.11412i		−	0.0900712i
$154$		0			0
$155$	22.4445	−	1.54202i	1.80278	−	0.123858i
$156$		0			0
$157$			18.5421i			1.47982i	0.672707	+	0.739909i	$0.265131\pi$
$157$			18.5421i			1.47982i	−0.672707	+	0.739909i	$0.734869\pi$
$158$		0			0
$159$	16.4071			1.30117
$160$		0			0
$161$	−2.15509			−0.169845
$162$		0			0
$163$			0.993429i			0.0778113i	0.999243	+	0.0389057i	$0.0123872\pi$
$163$			0.993429i			0.0778113i	−0.999243	+	0.0389057i	$0.987613\pi$
$164$		0			0
$165$	−0.310874	−	4.52483i	−0.0242015	−	0.352257i
$166$		0			0
$167$			3.22052i			0.249212i	0.992206	+	0.124606i	$0.0397666\pi$
$167$			3.22052i			0.249212i	−0.992206	+	0.124606i	$0.960233\pi$
$168$		0			0
$169$	−4.66981			−0.359216
$170$		0			0
$171$	−2.11988			−0.162111
$172$		0			0
$173$			15.9187i			1.21028i	0.796120	+	0.605138i	$0.206882\pi$
$173$			15.9187i			1.21028i	−0.796120	+	0.605138i	$0.793118\pi$
$174$		0			0
$175$	−0.683813	−	4.95302i	−0.0516914	−	0.374413i
$176$		0			0
$177$			17.9585i			1.34984i
$178$		0			0
$179$	9.84647			0.735960			0.367980	−	0.929834i	$-0.380049\pi$
$179$	9.84647			0.735960			0.367980	+	0.929834i	$0.380049\pi$
$180$		0			0
$181$	−21.6562			−1.60969			−0.804845	−	0.593484i	$-0.797752\pi$
$181$	−21.6562			−1.60969			−0.804845	+	0.593484i	$0.797752\pi$
$182$		0			0
$183$		−	10.3475i		−	0.764911i
$184$		0			0
$185$	−1.03732	−	15.0985i	−0.0762656	−	1.11006i
$186$		0			0
$187$		−	4.21395i		−	0.308155i
$188$		0			0
$189$	−5.44008			−0.395707
$190$		0			0
$191$	−17.7197			−1.28215			−0.641077	−	0.767477i	$-0.721512\pi$
$191$	−17.7197			−1.28215			−0.641077	+	0.767477i	$0.721512\pi$
$192$		0			0
$193$		−	12.0238i		−	0.865490i	−0.901516	−	0.432745i	$-0.857545\pi$
$193$		−	12.0238i		−	0.865490i	0.901516	−	0.432745i	$-0.142455\pi$
$194$		0			0
$195$	15.3282	−	1.05311i	1.09767	−	0.0754147i
$196$		0			0
$197$			4.12069i			0.293587i	0.989167	+	0.146794i	$0.0468953\pi$
$197$			4.12069i			0.293587i	−0.989167	+	0.146794i	$0.953105\pi$
$198$		0			0
$199$	9.86868			0.699572			0.349786	−	0.936830i	$-0.386254\pi$
$199$	9.86868			0.699572			0.349786	+	0.936830i	$0.386254\pi$
$200$		0			0
$201$	4.34174			0.306243
$202$		0			0
$203$		−	3.96490i		−	0.278282i
$204$		0			0
$205$	−0.292945	+	0.0201265i	−0.0204601	+	0.00140569i
$206$		0			0
$207$		−	0.707024i		−	0.0491416i
$208$		0			0
$209$	−8.01804			−0.554620
$210$		0			0
$211$	−6.93076			−0.477133			−0.238567	−	0.971126i	$-0.576678\pi$
$211$	−6.93076			−0.477133			−0.238567	+	0.971126i	$0.576678\pi$
$212$		0			0
$213$			1.23341i			0.0845119i
$214$		0			0
$215$	−1.13493	−	16.5191i	−0.0774015	−	1.12659i
$216$		0			0
$217$		−	10.0611i		−	0.682994i
$218$		0			0
$219$	−9.86868			−0.666864
$220$		0			0
$221$	14.2751			0.960246
$222$		0			0
$223$			23.0625i			1.54438i	0.635391	+	0.772191i	$0.280839\pi$
$223$			23.0625i			1.54438i	−0.635391	+	0.772191i	$0.719161\pi$
$224$		0			0
$225$	1.62495	−	0.224340i	0.108330	−	0.0149560i
$226$		0			0
$227$			2.86073i			0.189873i	0.995483	+	0.0949366i	$0.0302648\pi$
$227$			2.86073i			0.189873i	−0.995483	+	0.0949366i	$0.969735\pi$
$228$		0			0
$229$	18.5227			1.22402			0.612009	−	0.790851i	$-0.290362\pi$
$229$	18.5227			1.22402			0.612009	+	0.790851i	$0.290362\pi$
$230$		0			0
$231$	−2.02833			−0.133455
$232$		0			0
$233$			23.1112i			1.51407i	0.653376	+	0.757033i	$0.273352\pi$
$233$			23.1112i			1.51407i	−0.653376	+	0.757033i	$0.726648\pi$
$234$		0			0
$235$	−0.739815	−	10.7681i	−0.0482602	−	0.702437i
$236$		0			0
$237$		−	23.9717i		−	1.55713i
$238$		0			0
$239$	−9.01270			−0.582983			−0.291491	−	0.956573i	$-0.594151\pi$
$239$	−9.01270			−0.582983			−0.291491	+	0.956573i	$0.594151\pi$
$240$		0			0
$241$	−12.4415			−0.801427			−0.400713	−	0.916203i	$-0.631238\pi$
$241$	−12.4415			−0.801427			−0.400713	+	0.916203i	$0.631238\pi$
$242$		0			0
$243$		−	3.39354i		−	0.217696i
$244$		0			0
$245$	−2.23081	+	0.153266i	−0.142521	+	0.00979178i
$246$		0			0
$247$		−	27.1617i		−	1.72826i
$248$		0			0
$249$	23.0173			1.45866
$250$		0			0
$251$	21.3984			1.35066			0.675329	−	0.737517i	$-0.264002\pi$
$251$	21.3984			1.35066			0.675329	+	0.737517i	$0.264002\pi$
$252$		0			0
$253$		−	2.67419i		−	0.168125i
$254$		0			0
$255$	−12.3833	+	0.850785i	−0.775474	+	0.0532782i
$256$		0			0
$257$		−	27.5737i		−	1.72000i	−0.510296	−	0.859999i	$-0.670464\pi$
$257$		−	27.5737i		−	1.72000i	0.510296	−	0.859999i	$-0.329536\pi$
$258$		0			0
$259$	−6.76815			−0.420552
$260$		0			0
$261$	1.30077			0.0805159
$262$		0			0
$263$			6.77388i			0.417695i	0.977948	+	0.208848i	$0.0669712\pi$
$263$			6.77388i			0.417695i	−0.977948	+	0.208848i	$0.933029\pi$
$264$		0			0
$265$	1.53838	+	22.3914i	0.0945020	+	1.37549i
$266$		0			0
$267$		−	20.7173i		−	1.26788i
$268$		0			0
$269$	−14.5794			−0.888922			−0.444461	−	0.895798i	$-0.646605\pi$
$269$	−14.5794			−0.888922			−0.444461	+	0.895798i	$0.646605\pi$
$270$		0			0
$271$	−23.3842			−1.42049			−0.710244	−	0.703956i	$-0.751415\pi$
$271$	−23.3842			−1.42049			−0.710244	+	0.703956i	$0.751415\pi$
$272$		0			0
$273$		−	6.87113i		−	0.415860i
$274$		0			0
$275$	6.14607	−	0.848524i	0.370622	−	0.0511679i
$276$		0			0
$277$			8.57577i			0.515268i	0.966243	+	0.257634i	$0.0829429\pi$
$277$			8.57577i			0.515268i	−0.966243	+	0.257634i	$0.917057\pi$
$278$		0			0
$279$	3.30077			0.197612
$280$		0			0
$281$	−3.49675			−0.208598			−0.104299	−	0.994546i	$-0.533260\pi$
$281$	−3.49675			−0.208598			−0.104299	+	0.994546i	$0.533260\pi$
$282$		0			0
$283$			18.2952i			1.08754i	0.839235	+	0.543769i	$0.183003\pi$
$283$			18.2952i			1.08754i	−0.839235	+	0.543769i	$0.816997\pi$
$284$		0			0
$285$	1.61882	+	23.5622i	0.0958906	+	1.39571i
$286$		0			0
$287$			0.131318i			0.00775143i
$288$		0			0
$289$	5.46746			0.321615
$290$		0			0
$291$	1.49438			0.0876020
$292$		0			0
$293$		−	30.4752i		−	1.78038i	−0.455592	−	0.890189i	$-0.650572\pi$
$293$		−	30.4752i		−	1.78038i	0.455592	−	0.890189i	$-0.349428\pi$
$294$		0			0
$295$	−24.5087	+	1.68384i	−1.42695	+	0.0980372i
$296$		0			0
$297$		−	6.75044i		−	0.391700i
$298$		0			0
$299$	9.05901			0.523896
$300$		0			0
$301$	−7.40498			−0.426816
$302$		0			0
$303$			22.0374i			1.26601i
$304$		0			0
$305$	14.1217	−	0.970217i	0.808606	−	0.0555545i
$306$		0			0
$307$			24.8725i			1.41955i	0.704430	+	0.709773i	$0.251203\pi$
$307$			24.8725i			1.41955i	−0.704430	+	0.709773i	$0.748797\pi$
$308$		0			0
$309$	0.844830			0.0480607
$310$		0			0
$311$	−7.42579			−0.421078			−0.210539	−	0.977585i	$-0.567522\pi$
$311$	−7.42579			−0.421078			−0.210539	+	0.977585i	$0.567522\pi$
$312$		0			0
$313$			8.10594i			0.458175i	0.973406	+	0.229088i	$0.0735742\pi$
$313$			8.10594i			0.458175i	−0.973406	+	0.229088i	$0.926426\pi$
$314$		0			0
$315$	−0.0502822	−	0.731866i	−0.00283308	−	0.0412360i
$316$		0			0
$317$		−	14.6010i		−	0.820076i	−0.912068	−	0.410038i	$-0.865515\pi$
$317$		−	14.6010i		−	0.820076i	0.912068	−	0.410038i	$-0.134485\pi$
$318$		0			0
$319$	4.91994			0.275464
$320$		0			0
$321$	−30.8100			−1.71964
$322$		0			0
$323$			21.9434i			1.22096i
$324$		0			0
$325$	2.87444	+	20.8203i	0.159445	+	1.15490i
$326$		0			0
$327$		−	0.564380i		−	0.0312103i
$328$		0			0
$329$	−4.82702			−0.266122
$330$		0			0
$331$	29.7374			1.63452			0.817258	−	0.576271i	$-0.195493\pi$
$331$	29.7374			1.63452			0.817258	+	0.576271i	$0.195493\pi$
$332$		0			0
$333$		−	2.22044i		−	0.121679i
$334$		0			0
$335$	0.407096	+	5.92535i	0.0222420	+	0.323736i
$336$		0			0
$337$			7.88799i			0.429686i	0.976649	+	0.214843i	$0.0689241\pi$
$337$			7.88799i			0.429686i	−0.976649	+	0.214843i	$0.931076\pi$
$338$		0			0
$339$	−6.16289			−0.334722
$340$		0			0
$341$	12.4846			0.676078
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−7.85850	+	0.539911i	−0.423087	+	0.0290678i
$346$		0			0
$347$			3.28725i			0.176469i	0.996100	+	0.0882344i	$0.0281225\pi$
$347$			3.28725i			0.176469i	−0.996100	+	0.0882344i	$0.971878\pi$
$348$		0			0
$349$	6.68073			0.357611			0.178806	−	0.983884i	$-0.442777\pi$
$349$	6.68073			0.357611			0.178806	+	0.983884i	$0.442777\pi$
$350$		0			0
$351$	22.8676			1.22058
$352$		0			0
$353$			26.3234i			1.40105i	0.713626	+	0.700527i	$0.247052\pi$
$353$			26.3234i			1.40105i	−0.713626	+	0.700527i	$0.752948\pi$
$354$		0			0
$355$	−1.68329	+	0.115649i	−0.0893395	+	0.00613799i
$356$		0			0
$357$			5.55105i			0.293793i
$358$		0			0
$359$	−8.27155			−0.436556			−0.218278	−	0.975887i	$-0.570044\pi$
$359$	−8.27155			−0.436556			−0.218278	+	0.975887i	$0.570044\pi$
$360$		0			0
$361$	22.7525			1.19750
$362$		0			0
$363$			15.4637i			0.811635i
$364$		0			0
$365$	−0.925319	−	13.4682i	−0.0484334	−	0.704957i
$366$		0			0
$367$			33.7548i			1.76198i	0.473130	+	0.880992i	$0.343124\pi$
$367$			33.7548i			1.76198i	−0.473130	+	0.880992i	$0.656876\pi$
$368$		0			0
$369$	−0.0430816			−0.00224274
$370$		0			0
$371$	10.0374			0.521114
$372$		0			0
$373$		−	28.0582i		−	1.45280i	−0.687272	−	0.726400i	$-0.741192\pi$
$373$		−	28.0582i		−	1.45280i	0.687272	−	0.726400i	$-0.258808\pi$
$374$		0			0
$375$	−3.73439	−	17.8898i	−0.192843	−	0.923826i
$376$		0			0
$377$			16.6667i			0.858377i
$378$		0			0
$379$	−15.6585			−0.804322			−0.402161	−	0.915569i	$-0.631741\pi$
$379$	−15.6585			−0.804322			−0.402161	+	0.915569i	$0.631741\pi$
$380$		0			0
$381$	−9.21744			−0.472224
$382$		0			0
$383$			18.6950i			0.955270i	0.878558	+	0.477635i	$0.158506\pi$
$383$			18.6950i			0.955270i	−0.878558	+	0.477635i	$0.841494\pi$
$384$		0			0
$385$	−0.190183	−	2.76815i	−0.00969263	−	0.141078i
$386$		0			0
$387$		−	2.42937i		−	0.123492i
$388$		0			0
$389$	33.7798			1.71270			0.856352	−	0.516392i	$-0.172725\pi$
$389$	33.7798			1.71270			0.856352	+	0.516392i	$0.172725\pi$
$390$		0			0
$391$	−7.31859			−0.370117
$392$		0			0
$393$		−	17.1932i		−	0.867284i
$394$		0			0
$395$	32.7152	−	2.24767i	1.64608	−	0.113092i
$396$		0			0
$397$			0.466183i			0.0233971i	0.999932	+	0.0116985i	$0.00372384\pi$
$397$			0.466183i			0.0233971i	−0.999932	+	0.0116985i	$0.996276\pi$
$398$		0			0
$399$	10.5622			0.528771
$400$		0			0
$401$	33.2221			1.65903			0.829516	−	0.558483i	$-0.188616\pi$
$401$	33.2221			1.65903			0.829516	+	0.558483i	$0.188616\pi$
$402$		0			0
$403$			42.2924i			2.10674i
$404$		0			0
$405$	−17.6416	+	1.21205i	−0.876617	+	0.0602271i
$406$		0			0
$407$		−	8.39841i		−	0.416294i
$408$		0			0
$409$	15.4588			0.764390			0.382195	−	0.924082i	$-0.375168\pi$
$409$	15.4588			0.764390			0.382195	+	0.924082i	$0.375168\pi$
$410$		0			0
$411$	−11.3872			−0.561688
$412$		0			0
$413$			10.9864i			0.540608i
$414$		0			0
$415$	2.15817	+	31.4126i	0.105940	+	1.54198i
$416$		0			0
$417$			16.8787i			0.826551i
$418$		0			0
$419$	−29.0268			−1.41805			−0.709025	−	0.705183i	$-0.750865\pi$
$419$	−29.0268			−1.41805			−0.709025	+	0.705183i	$0.750865\pi$
$420$		0			0
$421$	5.31745			0.259156			0.129578	−	0.991569i	$-0.458638\pi$
$421$	5.31745			0.259156			0.129578	+	0.991569i	$0.458638\pi$
$422$		0			0
$423$		−	1.58361i		−	0.0769977i
$424$		0			0
$425$	−2.32220	−	16.8203i	−0.112643	−	0.815902i
$426$		0			0
$427$		−	6.33030i		−	0.306345i
$428$		0			0
$429$	8.52620			0.411649
$430$		0			0
$431$	17.6480			0.850073			0.425036	−	0.905176i	$-0.360261\pi$
$431$	17.6480			0.850073			0.425036	+	0.905176i	$0.360261\pi$
$432$		0			0
$433$			18.5011i			0.889107i	0.895752	+	0.444554i	$0.146638\pi$
$433$			18.5011i			0.889107i	−0.895752	+	0.444554i	$0.853362\pi$
$434$		0			0
$435$	−0.993321	−	14.4580i	−0.0476261	−	0.693207i
$436$		0			0
$437$			13.9254i			0.666140i
$438$		0			0
$439$	14.7444			0.703711			0.351855	−	0.936054i	$-0.385551\pi$
$439$	14.7444			0.703711			0.351855	+	0.936054i	$0.385551\pi$
$440$		0			0
$441$	−0.328072			−0.0156225
$442$		0			0
$443$			18.0223i			0.856264i	0.903716	+	0.428132i	$0.140828\pi$
$443$			18.0223i			0.856264i	−0.903716	+	0.428132i	$0.859172\pi$
$444$		0			0
$445$	28.2737	−	1.94252i	1.34030	−	0.0920842i
$446$		0			0
$447$			1.26498i			0.0598314i
$448$		0			0
$449$	2.46746			0.116447			0.0582233	−	0.998304i	$-0.481456\pi$
$449$	2.46746			0.116447			0.0582233	+	0.998304i	$0.481456\pi$
$450$		0			0
$451$	−0.162948			−0.00767294
$452$		0			0
$453$			10.0464i			0.472020i
$454$		0			0
$455$	9.37731	−	0.644259i	0.439615	−	0.0302033i
$456$		0			0
$457$		−	1.73587i		−	0.0812004i	−0.999175	−	0.0406002i	$-0.987073\pi$
$457$		−	1.73587i		−	0.0812004i	0.999175	−	0.0406002i	$-0.0129270\pi$
$458$		0			0
$459$	−18.4743			−0.862306
$460$		0			0
$461$	−9.00725			−0.419510			−0.209755	−	0.977754i	$-0.567267\pi$
$461$	−9.00725			−0.419510			−0.209755	+	0.977754i	$0.567267\pi$
$462$		0			0
$463$			39.1293i			1.81849i	0.416260	+	0.909246i	$0.363341\pi$
$463$			39.1293i			1.81849i	−0.416260	+	0.909246i	$0.636659\pi$
$464$		0			0
$465$	−2.52060	−	36.6878i	−0.116890	−	1.70135i
$466$		0			0
$467$			6.47873i			0.299800i	0.988701	+	0.149900i	$0.0478952\pi$
$467$			6.47873i			0.299800i	−0.988701	+	0.149900i	$0.952105\pi$
$468$		0			0
$469$	2.65614			0.122649
$470$		0			0
$471$	30.3089			1.39656
$472$		0			0
$473$		−	9.18864i		−	0.422494i
$474$		0			0
$475$	−32.0045	+	4.41854i	−1.46847	+	0.202736i
$476$		0			0
$477$			3.29298i			0.150775i
$478$		0			0
$479$	−22.5651			−1.03103			−0.515514	−	0.856881i	$-0.672399\pi$
$479$	−22.5651			−1.03103			−0.515514	+	0.856881i	$0.672399\pi$
$480$		0			0
$481$	28.4502			1.29722
$482$		0			0
$483$			3.52271i			0.160289i
$484$		0			0
$485$	0.140118	+	2.03944i	0.00636242	+	0.0926061i
$486$		0			0
$487$			39.2500i			1.77859i	0.457339	+	0.889293i	$0.348803\pi$
$487$			39.2500i			1.77859i	−0.457339	+	0.889293i	$0.651197\pi$
$488$		0			0
$489$	1.62386			0.0734336
$490$		0			0
$491$	−33.7793			−1.52444			−0.762220	−	0.647318i	$-0.775891\pi$
$491$	−33.7793			−1.52444			−0.762220	+	0.647318i	$0.775891\pi$
$492$		0			0
$493$		−	13.4647i		−	0.606418i
$494$		0			0
$495$	0.908153	−	0.0623938i	0.0408184	−	0.00280439i
$496$		0			0
$497$			0.754563i			0.0338468i
$498$		0			0
$499$	−11.6529			−0.521654			−0.260827	−	0.965386i	$-0.583995\pi$
$499$	−11.6529			−0.521654			−0.260827	+	0.965386i	$0.583995\pi$
$500$		0			0
$501$	5.26428			0.235191
$502$		0			0
$503$			15.0639i			0.671668i	0.941921	+	0.335834i	$0.109018\pi$
$503$			15.0639i			0.671668i	−0.941921	+	0.335834i	$0.890982\pi$
$504$		0			0
$505$	−30.0753	+	2.06629i	−1.33833	+	0.0919488i
$506$		0			0
$507$			7.63329i			0.339006i
$508$		0			0
$509$	−19.8458			−0.879649			−0.439825	−	0.898084i	$-0.644959\pi$
$509$	−19.8458			−0.879649			−0.439825	+	0.898084i	$0.644959\pi$
$510$		0			0
$511$	−6.03736			−0.267077
$512$		0			0
$513$			35.1517i			1.55199i
$514$		0			0
$515$	0.0792139	+	1.15297i	0.00349058	+	0.0508061i
$516$		0			0
$517$		−	5.98971i		−	0.263427i
$518$		0			0
$519$	26.0208			1.14218
$520$		0			0
$521$	−18.7964			−0.823484			−0.411742	−	0.911300i	$-0.635080\pi$
$521$	−18.7964			−0.823484			−0.411742	+	0.911300i	$0.635080\pi$
$522$		0			0
$523$			31.0907i			1.35950i	0.733444	+	0.679750i	$0.237912\pi$
$523$			31.0907i			1.35950i	−0.733444	+	0.679750i	$0.762088\pi$
$524$		0			0
$525$	−8.09622	+	1.11776i	−0.353348	+	0.0487832i
$526$		0			0
$527$		−	34.1672i		−	1.48835i
$528$		0			0
$529$	18.3556			0.798070
$530$		0			0
$531$	−3.60435			−0.156415
$532$		0			0
$533$		−	0.552000i		−	0.0239098i
$534$		0			0
$535$	−2.88884	−	42.0476i	−0.124895	−	1.81788i
$536$		0			0
$537$		−	16.0951i		−	0.694554i
$538$		0			0
$539$	−1.24087			−0.0534482
$540$		0			0
$541$	35.8816			1.54267			0.771336	−	0.636428i	$-0.219589\pi$
$541$	35.8816			1.54267			0.771336	+	0.636428i	$0.219589\pi$
$542$		0			0
$543$			35.3993i			1.51913i
$544$		0			0
$545$	0.770232	−	0.0529181i	0.0329931	−	0.00226676i
$546$		0			0
$547$			25.4869i			1.08974i	0.838520	+	0.544871i	$0.183421\pi$
$547$			25.4869i			1.08974i	−0.838520	+	0.544871i	$0.816579\pi$
$548$		0			0
$549$	2.07679			0.0886354
$550$		0			0
$551$	−25.6197			−1.09144
$552$		0			0
$553$		−	14.6652i		−	0.623626i
$554$		0			0
$555$	−24.6800	+	1.69561i	−1.04761	+	0.0719748i
$556$		0			0
$557$			23.5917i			0.999612i	0.866137	+	0.499806i	$0.166595\pi$
$557$			23.5917i			0.999612i	−0.866137	+	0.499806i	$0.833405\pi$
$558$		0			0
$559$	31.1272			1.31654
$560$		0			0
$561$	−6.88814			−0.290818
$562$		0			0
$563$			1.25815i			0.0530245i	0.999648	+	0.0265123i	$0.00844011\pi$
$563$			1.25815i			0.0530245i	−0.999648	+	0.0265123i	$0.991560\pi$
$564$		0			0
$565$	−0.577852	−	8.41074i	−0.0243104	−	0.353842i
$566$		0			0
$567$			7.90815i			0.332111i
$568$		0			0
$569$	−23.4899			−0.984747			−0.492373	−	0.870384i	$-0.663871\pi$
$569$	−23.4899			−0.984747			−0.492373	+	0.870384i	$0.663871\pi$
$570$		0			0
$571$	−18.0476			−0.755269			−0.377634	−	0.925955i	$-0.623262\pi$
$571$	−18.0476			−0.755269			−0.377634	+	0.925955i	$0.623262\pi$
$572$		0			0
$573$			28.9647i			1.21002i
$574$		0			0
$575$	−1.47368	−	10.6742i	−0.0614565	−	0.445144i
$576$		0			0
$577$		−	45.4141i		−	1.89061i	−0.326184	−	0.945306i	$-0.605763\pi$
$577$		−	45.4141i		−	1.89061i	0.326184	−	0.945306i	$-0.394237\pi$
$578$		0			0
$579$	−19.6541			−0.816796
$580$		0			0
$581$	14.0813			0.584189
$582$		0			0
$583$			12.4551i			0.515837i
$584$		0			0
$585$	0.211363	+	3.07643i	0.00873880	+	0.127195i
$586$		0			0
$587$		−	38.1946i		−	1.57646i	−0.615381	−	0.788230i	$-0.710998\pi$
$587$		−	38.1946i		−	1.57646i	0.615381	−	0.788230i	$-0.289002\pi$
$588$		0			0
$589$	−65.0112			−2.67874
$590$		0			0
$591$	6.73569			0.277069
$592$		0			0
$593$		−	11.0544i		−	0.453952i	−0.973900	−	0.226976i	$-0.927116\pi$
$593$		−	11.0544i		−	0.453952i	0.973900	−	0.226976i	$-0.0728839\pi$
$594$		0			0
$595$	−7.57574	+	0.520484i	−0.310575	+	0.0213378i
$596$		0			0
$597$		−	16.1314i		−	0.660213i
$598$		0			0
$599$	10.4206			0.425773			0.212887	−	0.977077i	$-0.431714\pi$
$599$	10.4206			0.425773			0.212887	+	0.977077i	$0.431714\pi$
$600$		0			0
$601$	−31.5047			−1.28510			−0.642552	−	0.766242i	$-0.722125\pi$
$601$	−31.5047			−1.28510			−0.642552	+	0.766242i	$0.722125\pi$
$602$		0			0
$603$			0.871407i			0.0354864i
$604$		0			0
$605$	−21.1040	+	1.44993i	−0.857999	+	0.0589480i
$606$		0			0
$607$			34.9493i			1.41855i	0.704932	+	0.709275i	$0.250977\pi$
$607$			34.9493i			1.41855i	−0.704932	+	0.709275i	$0.749023\pi$
$608$		0			0
$609$	−6.48105			−0.262625
$610$		0			0
$611$	20.2906			0.820869
$612$		0			0
$613$			37.9464i			1.53264i	0.642460	+	0.766319i	$0.277914\pi$
$613$			37.9464i			1.53264i	−0.642460	+	0.766319i	$0.722086\pi$
$614$		0			0
$615$	0.0328988	+	0.478848i	0.00132661	+	0.0193090i
$616$		0			0
$617$			27.0081i			1.08731i	0.839310	+	0.543653i	$0.182959\pi$
$617$			27.0081i			1.08731i	−0.839310	+	0.543653i	$0.817041\pi$
$618$		0			0
$619$	29.0656			1.16825			0.584123	−	0.811665i	$-0.301439\pi$
$619$	29.0656			1.16825			0.584123	+	0.811665i	$0.301439\pi$
$620$		0			0
$621$	−11.7238			−0.470462
$622$		0			0
$623$		−	12.6742i		−	0.507781i
$624$		0			0
$625$	24.0648	−	6.77388i	0.962592	−	0.270955i
$626$		0			0
$627$			13.1063i			0.523416i
$628$		0			0
$629$	−22.9844			−0.916447
$630$		0			0
$631$	38.8146			1.54518			0.772592	−	0.634903i	$-0.218960\pi$
$631$	38.8146			1.54518			0.772592	+	0.634903i	$0.218960\pi$
$632$		0			0
$633$			11.3290i			0.450289i
$634$		0			0
$635$	−0.864257	−	12.5794i	−0.0342970	−	0.499199i
$636$		0			0
$637$		−	4.20355i		−	0.166551i
$638$		0			0
$639$	−0.247551			−0.00979297
$640$		0			0
$641$	40.2837			1.59111			0.795555	−	0.605881i	$-0.207179\pi$
$641$	40.2837			1.59111			0.795555	+	0.605881i	$0.207179\pi$
$642$		0			0
$643$			2.02226i			0.0797500i	0.999205	+	0.0398750i	$0.0126960\pi$
$643$			2.02226i			0.0797500i	−0.999205	+	0.0398750i	$0.987304\pi$
$644$		0			0
$645$	−27.0022	+	1.85516i	−1.06321	+	0.0730468i
$646$		0			0
$647$		−	8.08779i		−	0.317964i	−0.987281	−	0.158982i	$-0.949179\pi$
$647$		−	8.08779i		−	0.317964i	0.987281	−	0.158982i	$-0.0508212\pi$
$648$		0			0
$649$	−13.6328			−0.535133
$650$		0			0
$651$	−16.4459			−0.644568
$652$		0			0
$653$		−	25.0259i		−	0.979338i	−0.871908	−	0.489669i	$-0.837118\pi$
$653$		−	25.0259i		−	0.979338i	0.871908	−	0.489669i	$-0.162882\pi$
$654$		0			0
$655$	23.4643	−	1.61209i	0.916826	−	0.0629896i
$656$		0			0
$657$		−	1.98069i		−	0.0772740i
$658$		0			0
$659$	35.1054			1.36751			0.683756	−	0.729711i	$-0.260345\pi$
$659$	35.1054			1.36751			0.683756	+	0.729711i	$0.260345\pi$
$660$		0			0
$661$	7.69197			0.299183			0.149592	−	0.988748i	$-0.452204\pi$
$661$	7.69197			0.299183			0.149592	+	0.988748i	$0.452204\pi$
$662$		0			0
$663$		−	23.3341i		−	0.906221i
$664$		0			0
$665$	0.990344	+	14.4146i	0.0384039	+	0.558976i
$666$		0			0
$667$		−	8.54471i		−	0.330853i
$668$		0			0
$669$	37.6981			1.45749
$670$		0			0
$671$	7.85510			0.303243
$672$		0			0
$673$		−	4.07019i		−	0.156894i	−0.996918	−	0.0784472i	$-0.975004\pi$
$673$		−	4.07019i		−	0.156894i	0.996918	−	0.0784472i	$-0.0249962\pi$
$674$		0			0
$675$	−3.71999	−	26.9448i	−0.143183	−	1.03711i
$676$		0			0
$677$			32.6451i			1.25465i	0.778756	+	0.627327i	$0.215851\pi$
$677$			32.6451i			1.25465i	−0.778756	+	0.627327i	$0.784149\pi$
$678$		0			0
$679$	0.914215			0.0350843
$680$		0			0
$681$	4.67616			0.179191
$682$		0			0
$683$		−	35.7284i		−	1.36711i	−0.729900	−	0.683554i	$-0.760433\pi$
$683$		−	35.7284i		−	1.36711i	0.729900	−	0.683554i	$-0.239567\pi$
$684$		0			0
$685$	−1.06770	−	15.5405i	−0.0407946	−	0.593773i
$686$		0			0
$687$		−	30.2773i		−	1.15515i
$688$		0			0
$689$	−42.1925			−1.60741
$690$		0			0
$691$	27.9663			1.06389			0.531945	−	0.846779i	$-0.321461\pi$
$691$	27.9663			1.06389			0.531945	+	0.846779i	$0.321461\pi$
$692$		0			0
$693$		−	0.407096i		−	0.0154643i
$694$		0			0
$695$	−23.0350	+	1.58260i	−0.873767	+	0.0600313i
$696$		0			0
$697$			0.445949i			0.0168915i
$698$		0			0
$699$	37.7777			1.42888
$700$		0			0
$701$	19.7996			0.747822			0.373911	−	0.927465i	$-0.378017\pi$
$701$	19.7996			0.747822			0.373911	+	0.927465i	$0.378017\pi$
$702$		0			0
$703$			43.7332i			1.64943i
$704$		0			0
$705$	−17.6017	+	1.20930i	−0.662917	+	0.0455451i
$706$		0			0
$707$			13.4818i			0.507034i
$708$		0			0
$709$	8.46892			0.318057			0.159029	−	0.987274i	$-0.449164\pi$
$709$	8.46892			0.318057			0.159029	+	0.987274i	$0.449164\pi$
$710$		0			0
$711$	4.81123			0.180435
$712$		0			0
$713$		−	21.6826i		−	0.812020i
$714$		0			0
$715$	0.799444	+	11.6360i	0.0298975	+	0.435164i
$716$		0			0
$717$			14.7322i			0.550183i
$718$		0			0
$719$	−11.2512			−0.419598			−0.209799	−	0.977745i	$-0.567281\pi$
$719$	−11.2512			−0.419598			−0.209799	+	0.977745i	$0.567281\pi$
$720$		0			0
$721$	0.516841			0.0192482
$722$		0			0
$723$			20.3369i			0.756338i
$724$		0			0
$725$	19.6382	−	2.71125i	0.729346	−	0.100693i
$726$		0			0
$727$		−	26.2716i		−	0.974360i	−0.873302	−	0.487180i	$-0.838026\pi$
$727$		−	26.2716i		−	0.974360i	0.873302	−	0.487180i	$-0.161974\pi$
$728$		0			0
$729$	−29.2716			−1.08413
$730$		0			0
$731$	−25.1470			−0.930096
$732$		0			0
$733$		−	48.7804i		−	1.80175i	−0.434082	−	0.900873i	$-0.642927\pi$
$733$		−	48.7804i		−	1.80175i	0.434082	−	0.900873i	$-0.357073\pi$
$734$		0			0
$735$	0.250528	+	3.64649i	0.00924088	+	0.134503i
$736$		0			0
$737$			3.29594i			0.121407i
$738$		0			0
$739$	−37.5871			−1.38267			−0.691333	−	0.722537i	$-0.742976\pi$
$739$	−37.5871			−1.38267			−0.691333	+	0.722537i	$0.742976\pi$
$740$		0			0
$741$	−44.3986			−1.63102
$742$		0			0
$743$			29.2572i			1.07334i	0.843792	+	0.536671i	$0.180318\pi$
$743$			29.2572i			1.07334i	−0.843792	+	0.536671i	$0.819682\pi$
$744$		0			0
$745$	−1.72637	+	0.118608i	−0.0632492	+	0.00434548i
$746$		0			0
$747$			4.61967i			0.169025i
$748$		0			0
$749$	−18.8486			−0.688712
$750$		0			0
$751$	−41.4125			−1.51116			−0.755582	−	0.655054i	$-0.772646\pi$
$751$	−41.4125			−1.51116			−0.755582	+	0.655054i	$0.772646\pi$
$752$		0			0
$753$		−	34.9780i		−	1.27467i
$754$		0			0
$755$	−13.7107	+	0.941980i	−0.498983	+	0.0342822i
$756$		0			0
$757$		−	33.1411i		−	1.20453i	−0.798295	−	0.602266i	$-0.794264\pi$
$757$		−	33.1411i		−	1.20453i	0.798295	−	0.602266i	$-0.205736\pi$
$758$		0			0
$759$	−4.37124			−0.158666
$760$		0			0
$761$	7.08166			0.256710			0.128355	−	0.991728i	$-0.459030\pi$
$761$	7.08166			0.256710			0.128355	+	0.991728i	$0.459030\pi$
$762$		0			0
$763$		−	0.345270i		−	0.0124996i
$764$		0			0
$765$	−0.170756	−	2.48539i	−0.00617370	−	0.0898594i
$766$		0			0
$767$		−	46.1820i		−	1.66754i
$768$		0			0
$769$	−20.8844			−0.753112			−0.376556	−	0.926394i	$-0.622892\pi$
$769$	−20.8844			−0.753112			−0.376556	+	0.926394i	$0.622892\pi$
$770$		0			0
$771$	−45.0720			−1.62323
$772$		0			0
$773$		−	50.0409i		−	1.79985i	−0.436049	−	0.899923i	$-0.643623\pi$
$773$		−	50.0409i		−	1.79985i	0.436049	−	0.899923i	$-0.356377\pi$
$774$		0			0
$775$	49.8329	−	6.87993i	1.79005	−	0.247134i
$776$		0			0
$777$			11.0632i			0.396892i
$778$		0			0
$779$	0.848524			0.0304015
$780$		0			0
$781$	−0.936316			−0.0335040
$782$		0			0
$783$		−	21.5694i		−	0.770827i
$784$		0			0
$785$	2.84186	+	41.3638i	0.101430	+	1.47634i
$786$		0			0
$787$			23.0831i			0.822824i	0.911449	+	0.411412i	$0.134964\pi$
$787$			23.0831i			0.822824i	−0.911449	+	0.411412i	$0.865036\pi$
$788$		0			0
$789$	11.0726			0.394195
$790$		0			0
$791$	−3.77026			−0.134055
$792$		0			0
$793$			26.6097i			0.944939i
$794$		0			0
$795$	36.6011	−	2.51464i	1.29811	−	0.0891852i
$796$		0			0
$797$			0.553975i			0.0196228i	0.999952	+	0.00981140i	$0.00312312\pi$
$797$			0.553975i			0.0196228i	−0.999952	+	0.00981140i	$0.996877\pi$
$798$		0			0
$799$	−16.3923			−0.579920
$800$		0			0
$801$	4.15805			0.146917
$802$		0			0
$803$		−	7.49159i		−	0.264372i
$804$		0			0
$805$	−4.80759	+	0.330301i	−0.169445	+	0.0116416i
$806$		0			0
$807$			23.8316i			0.838910i
$808$		0			0
$809$	42.8647			1.50704			0.753521	−	0.657424i	$-0.228354\pi$
$809$	42.8647			1.50704			0.753521	+	0.657424i	$0.228354\pi$
$810$		0			0
$811$	3.28042			0.115191			0.0575955	−	0.998340i	$-0.481657\pi$
$811$	3.28042			0.115191			0.0575955	+	0.998340i	$0.481657\pi$
$812$		0			0
$813$			38.2238i			1.34057i
$814$		0			0
$815$	0.152258	+	2.21615i	0.00533338	+	0.0776283i
$816$		0			0
$817$			47.8482i			1.67400i
$818$		0			0
$819$	1.37907			0.0481885
$820$		0			0
$821$	7.73361			0.269905			0.134952	−	0.990852i	$-0.456912\pi$
$821$	7.73361			0.269905			0.134952	+	0.990852i	$0.456912\pi$
$822$		0			0
$823$		−	22.9368i		−	0.799527i	−0.916618	−	0.399764i	$-0.869092\pi$
$823$		−	22.9368i		−	0.799527i	0.916618	−	0.399764i	$-0.130908\pi$
$824$		0			0
$825$	−1.38700	−	10.0464i	−0.0482892	−	0.349770i
$826$		0			0
$827$		−	48.4640i		−	1.68526i	−0.538494	−	0.842629i	$-0.681007\pi$
$827$		−	48.4640i		−	1.68526i	0.538494	−	0.842629i	$-0.318993\pi$
$828$		0			0
$829$	−34.4482			−1.19643			−0.598217	−	0.801334i	$-0.704124\pi$
$829$	−34.4482			−1.19643			−0.598217	+	0.801334i	$0.704124\pi$
$830$		0			0
$831$	14.0180			0.486278
$832$		0			0
$833$			3.39596i			0.117663i
$834$		0			0
$835$	0.493596	+	7.18437i	0.0170816	+	0.248626i
$836$		0			0
$837$		−	54.7333i		−	1.89186i
$838$		0			0
$839$	−9.81974			−0.339015			−0.169508	−	0.985529i	$-0.554218\pi$
$839$	−9.81974			−0.339015			−0.169508	+	0.985529i	$0.554218\pi$
$840$		0			0
$841$	−13.2795			−0.457915
$842$		0			0
$843$			5.71579i			0.196862i
$844$		0			0
$845$	−10.4175	+	0.715722i	−0.358372	+	0.0246216i
$846$		0			0
$847$			9.46024i			0.325058i
$848$		0			0
$849$	29.9054			1.02635
$850$		0			0
$851$	−14.5860			−0.500000
$852$		0			0
$853$			42.3139i			1.44880i	0.689380	+	0.724400i	$0.257883\pi$
$853$			42.3139i			1.44880i	−0.689380	+	0.724400i	$0.742117\pi$
$854$		0			0
$855$	−4.72904	+	0.324904i	−0.161730	+	0.0111115i
$856$		0			0
$857$		−	1.54342i		−	0.0527224i	−0.999652	−	0.0263612i	$-0.991608\pi$
$857$		−	1.54342i		−	0.0527224i	0.999652	−	0.0263612i	$-0.00839200\pi$
$858$		0			0
$859$	−4.41854			−0.150759			−0.0753793	−	0.997155i	$-0.524017\pi$
$859$	−4.41854			−0.150759			−0.0753793	+	0.997155i	$0.524017\pi$
$860$		0			0
$861$	0.214652			0.00731533
$862$		0			0
$863$			4.91388i			0.167270i	0.996496	+	0.0836352i	$0.0266530\pi$
$863$			4.91388i			0.167270i	−0.996496	+	0.0836352i	$0.973347\pi$
$864$		0			0
$865$	2.43979	+	35.5116i	0.0829553	+	1.20743i
$866$		0			0
$867$		−	8.93713i		−	0.303521i
$868$		0			0
$869$	18.1976			0.617311
$870$		0			0
$871$	−11.1652			−0.378319
$872$		0			0
$873$			0.299928i			0.0101510i
$874$		0			0
$875$	−2.28458	−	10.9444i	−0.0772330	−	0.369989i
$876$		0			0
$877$			11.7021i			0.395153i	0.980287	+	0.197577i	$0.0633071\pi$
$877$			11.7021i			0.395153i	−0.980287	+	0.197577i	$0.936693\pi$
$878$		0			0
$879$	−49.8148			−1.68021
$880$		0			0
$881$	−29.8823			−1.00676			−0.503380	−	0.864065i	$-0.667910\pi$
$881$	−29.8823			−1.00676			−0.503380	+	0.864065i	$0.667910\pi$
$882$		0			0
$883$			44.6829i			1.50370i	0.659334	+	0.751850i	$0.270838\pi$
$883$			44.6829i			1.50370i	−0.659334	+	0.751850i	$0.729162\pi$
$884$		0			0
$885$	2.75242	+	40.0619i	0.0925215	+	1.34667i
$886$		0			0
$887$			15.8523i			0.532269i	0.963936	+	0.266135i	$0.0857466\pi$
$887$			15.8523i			0.532269i	−0.963936	+	0.266135i	$0.914253\pi$
$888$		0			0
$889$	−5.63895			−0.189124
$890$		0			0
$891$	−9.81301			−0.328748
$892$		0			0
$893$			31.1903i			1.04374i
$894$		0			0
$895$	21.9656	−	1.50913i	0.734229	−	0.0504445i
$896$		0			0
$897$		−	14.8079i		−	0.494421i
$898$		0			0
$899$	39.8914			1.33045
$900$		0			0
$901$	34.0865			1.13558
$902$		0			0
$903$			12.1042i			0.402803i
$904$		0			0
$905$	−48.3108	+	3.31915i	−1.60591	+	0.110332i
$906$		0			0
$907$		−	26.5453i		−	0.881421i	−0.897649	−	0.440710i	$-0.854727\pi$
$907$		−	26.5453i		−	0.881421i	0.897649	−	0.440710i	$-0.145273\pi$
$908$		0			0
$909$	−4.42299			−0.146701
$910$		0			0
$911$	−0.855810			−0.0283543			−0.0141771	−	0.999899i	$-0.504513\pi$
$911$	−0.855810			−0.0283543			−0.0141771	+	0.999899i	$0.504513\pi$
$912$		0			0
$913$			17.4730i			0.578273i
$914$		0			0
$915$	−1.58592	−	23.0834i	−0.0524289	−	0.763113i
$916$		0			0
$917$		−	10.5183i		−	0.347344i
$918$		0			0
$919$	−13.6925			−0.451675			−0.225838	−	0.974165i	$-0.572512\pi$
$919$	−13.6925			−0.451675			−0.225838	+	0.974165i	$0.572512\pi$
$920$		0			0
$921$	40.6566			1.33968
$922$		0			0
$923$		−	3.17184i		−	0.104402i
$924$		0			0
$925$	−4.62815	−	33.5228i	−0.152173	−	1.10222i
$926$		0			0
$927$			0.169561i			0.00556911i
$928$		0			0
$929$	−25.2648			−0.828912			−0.414456	−	0.910069i	$-0.636028\pi$
$929$	−25.2648			−0.828912			−0.414456	+	0.910069i	$0.636028\pi$
$930$		0			0
$931$	6.46162			0.211771
$932$		0			0
$933$			12.1382i			0.397388i
$934$		0			0
$935$	−0.645854	−	9.40052i	−0.0211217	−	0.307430i
$936$		0			0
$937$		−	28.8534i		−	0.942599i	−0.881973	−	0.471299i	$-0.843785\pi$
$937$		−	28.8534i		−	0.942599i	0.881973	−	0.471299i	$-0.156215\pi$
$938$		0			0
$939$	13.2500			0.432397
$940$		0			0
$941$	−4.28998			−0.139850			−0.0699248	−	0.997552i	$-0.522276\pi$
$941$	−4.28998			−0.139850			−0.0699248	+	0.997552i	$0.522276\pi$
$942$		0			0
$943$			0.283001i			0.00921578i
$944$		0			0
$945$	−12.1358	+	0.833777i	−0.394777	+	0.0271228i
$946$		0			0
$947$		−	48.2836i		−	1.56901i	−0.620126	−	0.784503i	$-0.712918\pi$
$947$		−	48.2836i		−	1.56901i	0.620126	−	0.784503i	$-0.287082\pi$
$948$		0			0
$949$	25.3783			0.823815
$950$		0			0
$951$	−23.8669			−0.773937
$952$		0			0
$953$		−	34.2147i		−	1.10832i	−0.832409	−	0.554162i	$-0.813039\pi$
$953$		−	34.2147i		−	1.10832i	0.832409	−	0.554162i	$-0.186961\pi$
$954$		0			0
$955$	−39.5293	+	2.71582i	−1.27914	+	0.0878820i
$956$		0			0
$957$		−	8.04215i		−	0.259966i
$958$		0			0
$959$	−6.96632			−0.224954
$960$		0			0
$961$	70.2263			2.26536
$962$		0			0
$963$		−	6.18369i		−	0.199267i
$964$		0			0
$965$	−1.84283	−	26.8227i	−0.0593228	−	0.863454i
$966$		0			0
$967$			49.1870i			1.58175i	0.611980	+	0.790873i	$0.290373\pi$
$967$			49.1870i			1.58175i	−0.611980	+	0.790873i	$0.709627\pi$
$968$		0			0
$969$	35.8687			1.15227
$970$		0			0
$971$	25.2107			0.809051			0.404526	−	0.914527i	$-0.367437\pi$
$971$	25.2107			0.809051			0.404526	+	0.914527i	$0.367437\pi$
$972$		0			0
$973$			10.3258i			0.331031i
$974$		0			0
$975$	34.0329	−	4.69857i	1.08992	−	0.150475i
$976$		0			0
$977$		−	18.0389i		−	0.577116i	−0.957462	−	0.288558i	$-0.906824\pi$
$977$		−	18.0389i		−	0.577116i	0.957462	−	0.288558i	$-0.0931758\pi$
$978$		0			0
$979$	15.7271			0.502639
$980$		0			0
$981$	0.113274			0.00361654
$982$		0			0
$983$		−	51.6099i		−	1.64610i	−0.567968	−	0.823051i	$-0.692270\pi$
$983$		−	51.6099i		−	1.64610i	0.567968	−	0.823051i	$-0.307730\pi$
$984$		0			0
$985$	0.631560	+	9.19247i	0.0201232	+	0.292897i
$986$		0			0
$987$			7.89025i			0.251150i
$988$		0			0
$989$	−15.9584			−0.507447
$990$		0			0
$991$	15.4063			0.489398			0.244699	−	0.969599i	$-0.421311\pi$
$991$	15.4063			0.489398			0.244699	+	0.969599i	$0.421311\pi$
$992$		0			0
$993$		−	48.6089i		−	1.54256i
$994$		0			0
$995$	22.0151	−	1.51253i	0.697927	−	0.0479504i
$996$		0			0
$997$			13.6176i			0.431274i	0.976474	+	0.215637i	$0.0691827\pi$
$997$			13.6176i			0.431274i	−0.976474	+	0.215637i	$0.930817\pi$
$998$		0			0
$999$	−36.8193			−1.16491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.g.o.449.4		10
4.3	odd	2		2240.2.g.n.449.7		10
5.4	even	2	inner	2240.2.g.o.449.7		10
8.3	odd	2		1120.2.g.c.449.4	yes	10
8.5	even	2		1120.2.g.b.449.7	yes	10
20.19	odd	2		2240.2.g.n.449.4		10
40.3	even	4		5600.2.a.bv.1.4		5
40.13	odd	4		5600.2.a.bw.1.2		5
40.19	odd	2		1120.2.g.c.449.7	yes	10
40.27	even	4		5600.2.a.bx.1.2		5
40.29	even	2		1120.2.g.b.449.4	✓	10
40.37	odd	4		5600.2.a.bu.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.g.b.449.4	✓	10	40.29	even	2
1120.2.g.b.449.7	yes	10	8.5	even	2
1120.2.g.c.449.4	yes	10	8.3	odd	2
1120.2.g.c.449.7	yes	10	40.19	odd	2
2240.2.g.n.449.4		10	20.19	odd	2
2240.2.g.n.449.7		10	4.3	odd	2
2240.2.g.o.449.4		10	1.1	even	1	trivial
2240.2.g.o.449.7		10	5.4	even	2	inner
5600.2.a.bu.1.4		5	40.37	odd	4
5600.2.a.bv.1.4		5	40.3	even	4
5600.2.a.bw.1.2		5	40.13	odd	4
5600.2.a.bx.1.2		5	40.27	even	4

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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.63460i q^{3} +(2.23081 - 0.153266i) q^{5} -1.00000i q^{7} +0.328072 q^{9} +O(q^{10})\)	\(q-1.63460i q^{3} +(2.23081 - 0.153266i) q^{5} -1.00000i q^{7} +0.328072 q^{9} +1.24087 q^{11} +4.20355i q^{13} +(-0.250528 - 3.64649i) q^{15} -3.39596i q^{17} -6.46162 q^{19} -1.63460 q^{21} -2.15509i q^{23} +(4.95302 - 0.683813i) q^{25} -5.44008i q^{27} +3.96490 q^{29} +10.0611 q^{31} -2.02833i q^{33} +(-0.153266 - 2.23081i) q^{35} -6.76815i q^{37} +6.87113 q^{39} -0.131318 q^{41} -7.40498i q^{43} +(0.731866 - 0.0502822i) q^{45} -4.82702i q^{47} -1.00000 q^{49} -5.55105 q^{51} +10.0374i q^{53} +(2.76815 - 0.190183i) q^{55} +10.5622i q^{57} -10.9864 q^{59} +6.33030 q^{61} -0.328072i q^{63} +(0.644259 + 9.37731i) q^{65} +2.65614i q^{67} -3.52271 q^{69} -0.754563 q^{71} -6.03736i q^{73} +(-1.11776 - 8.09622i) q^{75} -1.24087i q^{77} +14.6652 q^{79} -7.90815 q^{81} +14.0813i q^{83} +(-0.520484 - 7.57574i) q^{85} -6.48105i q^{87} +12.6742 q^{89} +4.20355 q^{91} -16.4459i q^{93} +(-14.4146 + 0.990344i) q^{95} +0.914215i q^{97} +0.407096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{5} - 14 q^{9}+O(q^{10}) \) 10 * q + 2 * q^5 - 14 * q^9	\( 10 q + 2 q^{5} - 14 q^{9} + 8 q^{11} - 4 q^{15} - 24 q^{19} + 4 q^{21} + 6 q^{25} - 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} - 10 q^{45} - 10 q^{49} + 24 q^{51} - 16 q^{55} - 32 q^{59} + 20 q^{61} - 8 q^{65} + 8 q^{69} - 8 q^{71} + 64 q^{75} + 64 q^{79} + 2 q^{81} + 12 q^{85} - 4 q^{89} - 60 q^{95} - 80 q^{99}+O(q^{100}) \) 10 * q + 2 * q^5 - 14 * q^9 + 8 * q^11 - 4 * q^15 - 24 * q^19 + 4 * q^21 + 6 * q^25 - 24 * q^29 - 24 * q^31 + 64 * q^39 - 4 * q^41 - 10 * q^45 - 10 * q^49 + 24 * q^51 - 16 * q^55 - 32 * q^59 + 20 * q^61 - 8 * q^65 + 8 * q^69 - 8 * q^71 + 64 * q^75 + 64 * q^79 + 2 * q^81 + 12 * q^85 - 4 * q^89 - 60 * q^95 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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