LMFDB - Embedded newform 1840.3.k.b.321.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1840,3,Mod(321,1840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1840.321"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,2,0,0,0,0,0,-16,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.1363686423$
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} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} + \cdots + 225444 $ x^10 - 2x^9 - 10x^8 + 34x^7 + 346x^6 - 968x^5 + 165x^4 + 6972x^3 + 19344x^2 - 89080*x + 225444
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.6
Root			$3.44206 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.b.321.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0436725 q^{3} +2.23607i q^{5} -2.33372i q^{7} -8.99809 q^{9} +16.3982i q^{11} +1.02663 q^{13} +0.0976548i q^{15} -17.1482i q^{17} -23.2636i q^{19} -0.101920i q^{21} +(-2.92347 - 22.8134i) q^{23} -5.00000 q^{25} -0.786022 q^{27} -5.00315 q^{29} +26.6303 q^{31} +0.716151i q^{33} +5.21836 q^{35} -2.49670i q^{37} +0.0448355 q^{39} +69.6800 q^{41} +26.6617i q^{43} -20.1203i q^{45} -40.1701 q^{47} +43.5537 q^{49} -0.748905i q^{51} +53.7832i q^{53} -36.6675 q^{55} -1.01598i q^{57} -3.94458 q^{59} +44.9799i q^{61} +20.9991i q^{63} +2.29561i q^{65} +93.6284i q^{67} +(-0.127676 - 0.996321i) q^{69} +97.9038 q^{71} +38.8904 q^{73} -0.218363 q^{75} +38.2689 q^{77} -103.418i q^{79} +80.9485 q^{81} +59.3507i q^{83} +38.3445 q^{85} -0.218500 q^{87} +130.230i q^{89} -2.39587i q^{91} +1.16301 q^{93} +52.0190 q^{95} -142.579i q^{97} -147.553i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 16 q^{9} - 2 q^{13} - 44 q^{23} - 50 q^{25} - 40 q^{27} - 46 q^{29} - 16 q^{31} + 60 q^{35} - 72 q^{39} - 84 q^{41} - 112 q^{47} + 50 q^{49} + 10 q^{55} + 262 q^{59} + 124 q^{69} - 236 q^{71}+ \cdots + 90 q^{95}+O(q^{100}) $ 10 * q + 2 * q^3 - 16 * q^9 - 2 * q^13 - 44 * q^23 - 50 * q^25 - 40 * q^27 - 46 * q^29 - 16 * q^31 + 60 * q^35 - 72 * q^39 - 84 * q^41 - 112 * q^47 + 50 * q^49 + 10 * q^55 + 262 * q^59 + 124 * q^69 - 236 * q^71 + 168 * q^73 - 10 * q^75 + 300 * q^77 - 258 * q^81 - 540 * q^87 + 100 * q^93 + 90 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.0436725			0.0145575			0.00727876	−	0.999974i	$-0.497683\pi$
$3$	0.0436725			0.0145575			0.00727876	+	0.999974i	$0.497683\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	2.33372i		−	0.333389i	−0.986009	−	0.166694i	$-0.946691\pi$
$7$		−	2.33372i		−	0.333389i	0.986009	−	0.166694i	$-0.0533094\pi$
$8$		0			0
$9$	−8.99809			−0.999788
$10$		0			0
$11$			16.3982i			1.49075i	0.666648	+	0.745373i	$0.267729\pi$
$11$			16.3982i			1.49075i	−0.666648	+	0.745373i	$0.732271\pi$
$12$		0			0
$13$	1.02663			0.0789715			0.0394858	−	0.999220i	$-0.487428\pi$
$13$	1.02663			0.0789715			0.0394858	+	0.999220i	$0.487428\pi$
$14$		0			0
$15$			0.0976548i			0.00651032i
$16$		0			0
$17$		−	17.1482i		−	1.00872i	−0.863494	−	0.504359i	$-0.831729\pi$
$17$		−	17.1482i		−	1.00872i	0.863494	−	0.504359i	$-0.168271\pi$
$18$		0			0
$19$		−	23.2636i		−	1.22440i	−0.790703	−	0.612200i	$-0.790285\pi$
$19$		−	23.2636i		−	1.22440i	0.790703	−	0.612200i	$-0.209715\pi$
$20$		0			0
$21$		−	0.101920i		−	0.00485331i
$22$		0			0
$23$	−2.92347	−	22.8134i	−0.127108	−	0.991889i
$23$	−2.92347	−	22.8134i	−0.127108	−	0.991889i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−0.786022			−0.0291119
$28$		0			0
$29$	−5.00315			−0.172523			−0.0862613	−	0.996273i	$-0.527492\pi$
$29$	−5.00315			−0.172523			−0.0862613	+	0.996273i	$0.527492\pi$
$30$		0			0
$31$	26.6303			0.859043			0.429521	−	0.903057i	$-0.358682\pi$
$31$	26.6303			0.859043			0.429521	+	0.903057i	$0.358682\pi$
$32$		0			0
$33$			0.716151i			0.0217016i
$34$		0			0
$35$	5.21836			0.149096
$36$		0			0
$37$		−	2.49670i		−	0.0674784i	−0.999431	−	0.0337392i	$-0.989258\pi$
$37$		−	2.49670i		−	0.0674784i	0.999431	−	0.0337392i	$-0.0107416\pi$
$38$		0			0
$39$	0.0448355			0.00114963
$40$		0			0
$41$	69.6800			1.69951			0.849756	−	0.527176i	$-0.176749\pi$
$41$	69.6800			1.69951			0.849756	+	0.527176i	$0.176749\pi$
$42$		0			0
$43$			26.6617i			0.620040i	0.950730	+	0.310020i	$0.100336\pi$
$43$			26.6617i			0.620040i	−0.950730	+	0.310020i	$0.899664\pi$
$44$		0			0
$45$		−	20.1203i		−	0.447119i
$46$		0			0
$47$	−40.1701			−0.854684			−0.427342	−	0.904090i	$-0.640550\pi$
$47$	−40.1701			−0.854684			−0.427342	+	0.904090i	$0.640550\pi$
$48$		0			0
$49$	43.5537			0.888852
$50$		0			0
$51$		−	0.748905i		−	0.0146844i
$52$		0			0
$53$			53.7832i			1.01478i	0.861717	+	0.507389i	$0.169389\pi$
$53$			53.7832i			1.01478i	−0.861717	+	0.507389i	$0.830611\pi$
$54$		0			0
$55$	−36.6675			−0.666682
$56$		0			0
$57$		−	1.01598i		−	0.0178242i
$58$		0			0
$59$	−3.94458			−0.0668572			−0.0334286	−	0.999441i	$-0.510643\pi$
$59$	−3.94458			−0.0668572			−0.0334286	+	0.999441i	$0.510643\pi$
$60$		0			0
$61$			44.9799i			0.737375i	0.929553	+	0.368688i	$0.120193\pi$
$61$			44.9799i			0.737375i	−0.929553	+	0.368688i	$0.879807\pi$
$62$		0			0
$63$			20.9991i			0.333318i
$64$		0			0
$65$			2.29561i			0.0353171i
$66$		0			0
$67$			93.6284i			1.39744i	0.715396	+	0.698719i	$0.246246\pi$
$67$			93.6284i			1.39744i	−0.715396	+	0.698719i	$0.753754\pi$
$68$		0			0
$69$	−0.127676	−	0.996321i	−0.00185037	−	0.0144394i
$70$		0			0
$71$	97.9038			1.37893			0.689463	−	0.724321i	$-0.257847\pi$
$71$	97.9038			1.37893			0.689463	+	0.724321i	$0.257847\pi$
$72$		0			0
$73$	38.8904			0.532745			0.266372	−	0.963870i	$-0.414175\pi$
$73$	38.8904			0.532745			0.266372	+	0.963870i	$0.414175\pi$
$74$		0			0
$75$	−0.218363			−0.00291150
$76$		0			0
$77$	38.2689			0.496998
$78$		0			0
$79$		−	103.418i		−	1.30908i	−0.756026	−	0.654542i	$-0.772862\pi$
$79$		−	103.418i		−	1.30908i	0.756026	−	0.654542i	$-0.227138\pi$
$80$		0			0
$81$	80.9485			0.999364
$82$		0			0
$83$			59.3507i			0.715068i	0.933900	+	0.357534i	$0.116382\pi$
$83$			59.3507i			0.715068i	−0.933900	+	0.357534i	$0.883618\pi$
$84$		0			0
$85$	38.3445			0.451112
$86$		0			0
$87$	−0.218500			−0.00251150
$88$		0			0
$89$			130.230i			1.46326i	0.681701	+	0.731631i	$0.261240\pi$
$89$			130.230i			1.46326i	−0.681701	+	0.731631i	$0.738760\pi$
$90$		0			0
$91$		−	2.39587i		−	0.0263282i
$92$		0			0
$93$	1.16301			0.0125055
$94$		0			0
$95$	52.0190			0.547569
$96$		0			0
$97$		−	142.579i		−	1.46989i	−0.678126	−	0.734945i	$-0.737208\pi$
$97$		−	142.579i		−	1.46989i	0.678126	−	0.734945i	$-0.262792\pi$
$98$		0			0
$99$		−	147.553i		−	1.49043i
$100$		0			0
$101$	96.4847			0.955294			0.477647	−	0.878552i	$-0.341490\pi$
$101$	96.4847			0.955294			0.477647	+	0.878552i	$0.341490\pi$
$102$		0			0
$103$		−	10.1274i		−	0.0983240i	−0.998791	−	0.0491620i	$-0.984345\pi$
$103$		−	10.1274i		−	0.0983240i	0.998791	−	0.0491620i	$-0.0156551\pi$
$104$		0			0
$105$	0.227899			0.00217047
$106$		0			0
$107$			39.0101i			0.364581i	0.983245	+	0.182290i	$0.0583511\pi$
$107$			39.0101i			0.364581i	−0.983245	+	0.182290i	$0.941649\pi$
$108$		0			0
$109$		−	43.1074i		−	0.395481i	−0.980254	−	0.197741i	$-0.936640\pi$
$109$		−	43.1074i		−	0.395481i	0.980254	−	0.197741i	$-0.0633603\pi$
$110$		0			0
$111$		−	0.109037i		−	0.000982317i
$112$		0			0
$113$		−	162.052i		−	1.43409i	−0.697029	−	0.717043i	$-0.745495\pi$
$113$		−	162.052i		−	1.43409i	0.697029	−	0.717043i	$-0.254505\pi$
$114$		0			0
$115$	51.0124	−	6.53709i	0.443586	−	0.0568442i
$116$		0			0
$117$	−9.23771			−0.0789548
$118$		0			0
$119$	−40.0191			−0.336295
$120$		0			0
$121$	−147.901			−1.22232
$122$		0			0
$123$	3.04310			0.0247407
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	178.661			1.40678			0.703389	−	0.710805i	$-0.251669\pi$
$127$	178.661			1.40678			0.703389	+	0.710805i	$0.251669\pi$
$128$		0			0
$129$			1.16439i			0.00902624i
$130$		0			0
$131$	−49.5358			−0.378136			−0.189068	−	0.981964i	$-0.560547\pi$
$131$	−49.5358			−0.378136			−0.189068	+	0.981964i	$0.560547\pi$
$132$		0			0
$133$	−54.2908			−0.408202
$134$		0			0
$135$		−	1.75760i		−	0.0130193i
$136$		0			0
$137$			145.643i			1.06309i	0.847031	+	0.531543i	$0.178388\pi$
$137$			145.643i			1.06309i	−0.847031	+	0.531543i	$0.821612\pi$
$138$		0			0
$139$	198.891			1.43087			0.715435	−	0.698680i	$-0.246229\pi$
$139$	198.891			1.43087			0.715435	+	0.698680i	$0.246229\pi$
$140$		0			0
$141$	−1.75433			−0.0124421
$142$		0			0
$143$			16.8349i			0.117726i
$144$		0			0
$145$		−	11.1874i		−	0.0771544i
$146$		0			0
$147$	1.90210			0.0129395
$148$		0			0
$149$			103.183i			0.692505i	0.938141	+	0.346253i	$0.112546\pi$
$149$			103.183i			0.692505i	−0.938141	+	0.346253i	$0.887454\pi$
$150$		0			0
$151$	199.563			1.32161			0.660806	−	0.750557i	$-0.270215\pi$
$151$	199.563			1.32161			0.660806	+	0.750557i	$0.270215\pi$
$152$		0			0
$153$			154.301i			1.00850i
$154$		0			0
$155$			59.5472i			0.384176i
$156$		0			0
$157$		−	224.444i		−	1.42958i	−0.699340	−	0.714789i	$-0.746523\pi$
$157$		−	224.444i		−	1.42958i	0.699340	−	0.714789i	$-0.253477\pi$
$158$		0			0
$159$			2.34885i			0.0147726i
$160$		0			0
$161$	−53.2403	+	6.82258i	−0.330685	+	0.0423763i
$162$		0			0
$163$	43.8177			0.268820			0.134410	−	0.990926i	$-0.457086\pi$
$163$	43.8177			0.268820			0.134410	+	0.990926i	$0.457086\pi$
$164$		0			0
$165$	−1.60136			−0.00970523
$166$		0			0
$167$	−168.902			−1.01139			−0.505696	−	0.862712i	$-0.668764\pi$
$167$	−168.902			−1.01139			−0.505696	+	0.862712i	$0.668764\pi$
$168$		0			0
$169$	−167.946			−0.993763
$170$		0			0
$171$			209.328i			1.22414i
$172$		0			0
$173$	56.9816			0.329373			0.164687	−	0.986346i	$-0.447339\pi$
$173$	56.9816			0.329373			0.164687	+	0.986346i	$0.447339\pi$
$174$		0			0
$175$			11.6686i			0.0666778i
$176$		0			0
$177$	−0.172270			−0.000973275
$178$		0			0
$179$	243.497			1.36032			0.680160	−	0.733064i	$-0.261910\pi$
$179$	243.497			1.36032			0.680160	+	0.733064i	$0.261910\pi$
$180$		0			0
$181$			189.480i			1.04685i	0.852072	+	0.523425i	$0.175346\pi$
$181$			189.480i			1.04685i	−0.852072	+	0.523425i	$0.824654\pi$
$182$		0			0
$183$			1.96439i			0.0107344i
$184$		0			0
$185$	5.58279			0.0301772
$186$		0			0
$187$	281.200			1.50374
$188$		0			0
$189$			1.83436i			0.00970560i
$190$		0			0
$191$			303.459i			1.58879i	0.607400	+	0.794396i	$0.292213\pi$
$191$			303.459i			1.58879i	−0.607400	+	0.794396i	$0.707787\pi$
$192$		0			0
$193$	−171.946			−0.890912			−0.445456	−	0.895304i	$-0.646958\pi$
$193$	−171.946			−0.890912			−0.445456	+	0.895304i	$0.646958\pi$
$194$		0			0
$195$			0.100255i			0.000514130i
$196$		0			0
$197$	257.231			1.30574			0.652871	−	0.757470i	$-0.273565\pi$
$197$	257.231			1.30574			0.652871	+	0.757470i	$0.273565\pi$
$198$		0			0
$199$		−	179.782i		−	0.903425i	−0.892163	−	0.451713i	$-0.850813\pi$
$199$		−	179.782i		−	0.903425i	0.892163	−	0.451713i	$-0.149187\pi$
$200$		0			0
$201$			4.08899i			0.0203432i
$202$		0			0
$203$			11.6760i			0.0575171i
$204$		0			0
$205$			155.809i			0.760045i
$206$		0			0
$207$	26.3057	+	205.277i	0.127081	+	0.991679i
$208$		0			0
$209$	381.481			1.82527
$210$		0			0
$211$	−338.952			−1.60641			−0.803204	−	0.595704i	$-0.796873\pi$
$211$	−338.952			−1.60641			−0.803204	+	0.595704i	$0.796873\pi$
$212$		0			0
$213$	4.27571			0.0200737
$214$		0			0
$215$	−59.6174			−0.277290
$216$		0			0
$217$		−	62.1478i		−	0.286395i
$218$		0			0
$219$	1.69844			0.00775544
$220$		0			0
$221$		−	17.6049i		−	0.0796600i
$222$		0			0
$223$	302.953			1.35853			0.679266	−	0.733892i	$-0.262298\pi$
$223$	302.953			1.35853			0.679266	+	0.733892i	$0.262298\pi$
$224$		0			0
$225$	44.9905			0.199958
$226$		0			0
$227$		−	391.761i		−	1.72582i	−0.505358	−	0.862910i	$-0.668640\pi$
$227$		−	391.761i		−	1.72582i	0.505358	−	0.862910i	$-0.331360\pi$
$228$		0			0
$229$			130.299i			0.568993i	0.958677	+	0.284497i	$0.0918265\pi$
$229$			130.299i			0.568993i	−0.958677	+	0.284497i	$0.908174\pi$
$230$		0			0
$231$	1.67130			0.00723506
$232$		0			0
$233$	251.192			1.07808			0.539039	−	0.842281i	$-0.318788\pi$
$233$	251.192			1.07808			0.539039	+	0.842281i	$0.318788\pi$
$234$		0			0
$235$		−	89.8231i		−	0.382226i
$236$		0			0
$237$		−	4.51651i		−	0.0190570i
$238$		0			0
$239$	10.6827			0.0446976			0.0223488	−	0.999750i	$-0.492886\pi$
$239$	10.6827			0.0446976			0.0223488	+	0.999750i	$0.492886\pi$
$240$		0			0
$241$		−	414.415i		−	1.71957i	−0.510659	−	0.859783i	$-0.670599\pi$
$241$		−	414.415i		−	1.71957i	0.510659	−	0.859783i	$-0.329401\pi$
$242$		0			0
$243$	10.6094			0.0436602
$244$		0			0
$245$			97.3891i			0.397507i
$246$		0			0
$247$		−	23.8831i		−	0.0966928i
$248$		0			0
$249$			2.59200i			0.0104096i
$250$		0			0
$251$		−	248.441i		−	0.989807i	−0.868948	−	0.494903i	$-0.835203\pi$
$251$		−	248.441i		−	0.989807i	0.868948	−	0.494903i	$-0.164797\pi$
$252$		0			0
$253$	374.100	−	47.9397i	1.47865	−	0.189485i
$254$		0			0
$255$	1.67460			0.00656707
$256$		0			0
$257$	−390.471			−1.51934			−0.759671	−	0.650308i	$-0.774640\pi$
$257$	−390.471			−1.51934			−0.759671	+	0.650308i	$0.774640\pi$
$258$		0			0
$259$	−5.82660			−0.0224965
$260$		0			0
$261$	45.0188			0.172486
$262$		0			0
$263$			302.380i			1.14973i	0.818247	+	0.574867i	$0.194946\pi$
$263$			302.380i			1.14973i	−0.818247	+	0.574867i	$0.805054\pi$
$264$		0			0
$265$	−120.263			−0.453822
$266$		0			0
$267$			5.68749i			0.0213014i
$268$		0			0
$269$	295.154			1.09723			0.548614	−	0.836076i	$-0.315156\pi$
$269$	295.154			1.09723			0.548614	+	0.836076i	$0.315156\pi$
$270$		0			0
$271$	118.669			0.437892			0.218946	−	0.975737i	$-0.429738\pi$
$271$	118.669			0.437892			0.218946	+	0.975737i	$0.429738\pi$
$272$		0			0
$273$		−	0.104634i		−	0.000383274i
$274$		0			0
$275$		−	81.9910i		−	0.298149i
$276$		0			0
$277$	359.091			1.29636			0.648178	−	0.761489i	$-0.275531\pi$
$277$	359.091			1.29636			0.648178	+	0.761489i	$0.275531\pi$
$278$		0			0
$279$	−239.622			−0.858861
$280$		0			0
$281$			199.129i			0.708645i	0.935123	+	0.354322i	$0.115288\pi$
$281$			199.129i			0.708645i	−0.935123	+	0.354322i	$0.884712\pi$
$282$		0			0
$283$		−	120.934i		−	0.427330i	−0.976907	−	0.213665i	$-0.931460\pi$
$283$		−	120.934i		−	0.427330i	0.976907	−	0.213665i	$-0.0685401\pi$
$284$		0			0
$285$	2.27180			0.00797124
$286$		0			0
$287$		−	162.614i		−	0.566599i
$288$		0			0
$289$	−5.06080			−0.0175114
$290$		0			0
$291$		−	6.22680i		−	0.0213979i
$292$		0			0
$293$		−	431.892i		−	1.47404i	−0.675874	−	0.737018i	$-0.736234\pi$
$293$		−	431.892i		−	1.47404i	0.675874	−	0.737018i	$-0.263766\pi$
$294$		0			0
$295$		−	8.82034i		−	0.0298995i
$296$		0			0
$297$		−	12.8894i		−	0.0433985i
$298$		0			0
$299$	−3.00133	−	23.4210i	−0.0100379	−	0.0783310i
$300$		0			0
$301$	62.2211			0.206715
$302$		0			0
$303$	4.21373			0.0139067
$304$		0			0
$305$	−100.578			−0.329764
$306$		0			0
$307$	−265.975			−0.866368			−0.433184	−	0.901306i	$-0.642610\pi$
$307$	−265.975			−0.866368			−0.433184	+	0.901306i	$0.642610\pi$
$308$		0			0
$309$		−	0.442288i		−	0.00143135i
$310$		0			0
$311$	−110.534			−0.355414			−0.177707	−	0.984083i	$-0.556868\pi$
$311$	−110.534			−0.355414			−0.177707	+	0.984083i	$0.556868\pi$
$312$		0			0
$313$		−	462.215i		−	1.47673i	−0.674404	−	0.738363i	$-0.735599\pi$
$313$		−	462.215i		−	1.47673i	0.674404	−	0.738363i	$-0.264401\pi$
$314$		0			0
$315$	−46.9553			−0.149064
$316$		0			0
$317$	552.548			1.74305			0.871527	−	0.490348i	$-0.163130\pi$
$317$	552.548			1.74305			0.871527	+	0.490348i	$0.163130\pi$
$318$		0			0
$319$		−	82.0427i		−	0.257187i
$320$		0			0
$321$			1.70367i			0.00530739i
$322$		0			0
$323$	−398.929			−1.23507
$324$		0			0
$325$	−5.13315			−0.0157943
$326$		0			0
$327$		−	1.88261i		−	0.00575722i
$328$		0			0
$329$			93.7459i			0.284942i
$330$		0			0
$331$	−33.4106			−0.100938			−0.0504692	−	0.998726i	$-0.516072\pi$
$331$	−33.4106			−0.100938			−0.0504692	+	0.998726i	$0.516072\pi$
$332$		0			0
$333$			22.4655i			0.0674641i
$334$		0			0
$335$	−209.359			−0.624953
$336$		0			0
$337$			232.884i			0.691049i	0.938410	+	0.345525i	$0.112299\pi$
$337$			232.884i			0.691049i	−0.938410	+	0.345525i	$0.887701\pi$
$338$		0			0
$339$		−	7.07721i		−	0.0208767i
$340$		0			0
$341$			436.690i			1.28061i
$342$		0			0
$343$		−	215.995i		−	0.629722i
$344$		0			0
$345$	2.22784	−	0.285491i	0.00645751	−	0.000827511i
$346$		0			0
$347$	409.753			1.18084			0.590422	−	0.807095i	$-0.298961\pi$
$347$	409.753			1.18084			0.590422	+	0.807095i	$0.298961\pi$
$348$		0			0
$349$	−21.3405			−0.0611476			−0.0305738	−	0.999533i	$-0.509733\pi$
$349$	−21.3405			−0.0611476			−0.0305738	+	0.999533i	$0.509733\pi$
$350$		0			0
$351$	−0.806954			−0.00229901
$352$		0			0
$353$	−146.395			−0.414716			−0.207358	−	0.978265i	$-0.566486\pi$
$353$	−146.395			−0.414716			−0.207358	+	0.978265i	$0.566486\pi$
$354$		0			0
$355$			218.919i			0.616675i
$356$		0			0
$357$	−1.74774			−0.00489562
$358$		0			0
$359$			598.255i			1.66645i	0.552936	+	0.833224i	$0.313508\pi$
$359$			598.255i			1.66645i	−0.552936	+	0.833224i	$0.686492\pi$
$360$		0			0
$361$	−180.196			−0.499157
$362$		0			0
$363$	−6.45922			−0.0177940
$364$		0			0
$365$			86.9615i			0.238251i
$366$		0			0
$367$		−	619.595i		−	1.68827i	−0.536131	−	0.844135i	$-0.680115\pi$
$367$		−	619.595i		−	1.68827i	0.536131	−	0.844135i	$-0.319885\pi$
$368$		0			0
$369$	−626.987			−1.69915
$370$		0			0
$371$	125.515			0.338316
$372$		0			0
$373$			68.4568i			0.183530i	0.995781	+	0.0917651i	$0.0292509\pi$
$373$			68.4568i			0.183530i	−0.995781	+	0.0917651i	$0.970749\pi$
$374$		0			0
$375$		−	0.488274i		−	0.00130206i
$376$		0			0
$377$	−5.13639			−0.0136244
$378$		0			0
$379$			516.055i			1.36162i	0.732459	+	0.680811i	$0.238372\pi$
$379$			516.055i			1.36162i	−0.732459	+	0.680811i	$0.761628\pi$
$380$		0			0
$381$	7.80257			0.0204792
$382$		0			0
$383$			209.197i			0.546207i	0.961985	+	0.273103i	$0.0880501\pi$
$383$			209.197i			0.546207i	−0.961985	+	0.273103i	$0.911950\pi$
$384$		0			0
$385$			85.5718i			0.222264i
$386$		0			0
$387$		−	239.905i		−	0.619909i
$388$		0			0
$389$		−	620.747i		−	1.59575i	−0.602823	−	0.797875i	$-0.705958\pi$
$389$		−	620.747i		−	1.59575i	0.602823	−	0.797875i	$-0.294042\pi$
$390$		0			0
$391$	−391.210	+	50.1323i	−1.00054	+	0.128216i
$392$		0			0
$393$	−2.16335			−0.00550472
$394$		0			0
$395$	231.249			0.585440
$396$		0			0
$397$	−465.455			−1.17243			−0.586216	−	0.810155i	$-0.699383\pi$
$397$	−465.455			−1.17243			−0.586216	+	0.810155i	$0.699383\pi$
$398$		0			0
$399$	−2.37102			−0.00594240
$400$		0			0
$401$		−	59.9490i		−	0.149499i	−0.997202	−	0.0747494i	$-0.976184\pi$
$401$		−	59.9490i		−	0.149499i	0.997202	−	0.0747494i	$-0.0238157\pi$
$402$		0			0
$403$	27.3395			0.0678399
$404$		0			0
$405$			181.006i			0.446929i
$406$		0			0
$407$	40.9414			0.100593
$408$		0			0
$409$	−252.327			−0.616936			−0.308468	−	0.951235i	$-0.599816\pi$
$409$	−252.327			−0.616936			−0.308468	+	0.951235i	$0.599816\pi$
$410$		0			0
$411$			6.36059i			0.0154759i
$412$		0			0
$413$			9.20555i			0.0222895i
$414$		0			0
$415$	−132.712			−0.319788
$416$		0			0
$417$	8.68607			0.0208299
$418$		0			0
$419$		−	390.700i		−	0.932458i	−0.884664	−	0.466229i	$-0.845612\pi$
$419$		−	390.700i		−	0.932458i	0.884664	−	0.466229i	$-0.154388\pi$
$420$		0			0
$421$			378.872i			0.899934i	0.893045	+	0.449967i	$0.148564\pi$
$421$			378.872i			0.899934i	−0.893045	+	0.449967i	$0.851436\pi$
$422$		0			0
$423$	361.455			0.854503
$424$		0			0
$425$			85.7410i			0.201744i
$426$		0			0
$427$	104.971			0.245833
$428$		0			0
$429$			0.735222i			0.00171380i
$430$		0			0
$431$			327.359i			0.759534i	0.925082	+	0.379767i	$0.123996\pi$
$431$			327.359i			0.759534i	−0.925082	+	0.379767i	$0.876004\pi$
$432$		0			0
$433$			133.102i			0.307396i	0.988118	+	0.153698i	$0.0491182\pi$
$433$			133.102i			0.307396i	−0.988118	+	0.153698i	$0.950882\pi$
$434$		0			0
$435$		−	0.488582i		−	0.00112318i
$436$		0			0
$437$	−530.723	+	68.0106i	−1.21447	+	0.155631i
$438$		0			0
$439$	−247.458			−0.563686			−0.281843	−	0.959461i	$-0.590946\pi$
$439$	−247.458			−0.563686			−0.281843	+	0.959461i	$0.590946\pi$
$440$		0			0
$441$	−391.901			−0.888663
$442$		0			0
$443$	−626.572			−1.41438			−0.707192	−	0.707021i	$-0.750039\pi$
$443$	−626.572			−1.41438			−0.707192	+	0.707021i	$0.750039\pi$
$444$		0			0
$445$	−291.204			−0.654390
$446$		0			0
$447$			4.50628i			0.0100812i
$448$		0			0
$449$	203.993			0.454327			0.227164	−	0.973857i	$-0.427055\pi$
$449$	203.993			0.454327			0.227164	+	0.973857i	$0.427055\pi$
$450$		0			0
$451$			1142.63i			2.53354i
$452$		0			0
$453$	8.71544			0.0192394
$454$		0			0
$455$	5.35733			0.0117743
$456$		0			0
$457$			341.855i			0.748041i	0.927421	+	0.374020i	$0.122021\pi$
$457$			341.855i			0.748041i	−0.927421	+	0.374020i	$0.877979\pi$
$458$		0			0
$459$			13.4789i			0.0293657i
$460$		0			0
$461$	368.514			0.799380			0.399690	−	0.916650i	$-0.369118\pi$
$461$	368.514			0.799380			0.399690	+	0.916650i	$0.369118\pi$
$462$		0			0
$463$	419.855			0.906814			0.453407	−	0.891304i	$-0.350208\pi$
$463$	419.855			0.906814			0.453407	+	0.891304i	$0.350208\pi$
$464$		0			0
$465$			2.60058i			0.00559264i
$466$		0			0
$467$		−	135.385i		−	0.289904i	−0.989439	−	0.144952i	$-0.953697\pi$
$467$		−	135.385i		−	0.289904i	0.989439	−	0.144952i	$-0.0463028\pi$
$468$		0			0
$469$	218.503			0.465891
$470$		0			0
$471$		−	9.80203i		−	0.0208111i
$472$		0			0
$473$	−437.205			−0.924323
$474$		0			0
$475$			116.318i			0.244880i
$476$		0			0
$477$		−	483.946i		−	1.01456i
$478$		0			0
$479$		−	834.130i		−	1.74140i	−0.491816	−	0.870699i	$-0.663667\pi$
$479$		−	834.130i		−	1.74140i	0.491816	−	0.870699i	$-0.336333\pi$
$480$		0			0
$481$		−	2.56319i		−	0.00532887i
$482$		0			0
$483$	−2.32514	+	0.297959i	−0.00481395	+	0.000616893i
$484$		0			0
$485$	318.817			0.657355
$486$		0			0
$487$	−204.555			−0.420032			−0.210016	−	0.977698i	$-0.567352\pi$
$487$	−204.555			−0.420032			−0.210016	+	0.977698i	$0.567352\pi$
$488$		0			0
$489$	1.91363			0.00391335
$490$		0			0
$491$	275.996			0.562109			0.281054	−	0.959692i	$-0.409316\pi$
$491$	275.996			0.562109			0.281054	+	0.959692i	$0.409316\pi$
$492$		0			0
$493$			85.7951i			0.174027i
$494$		0			0
$495$	329.938			0.666541
$496$		0			0
$497$		−	228.480i		−	0.459719i
$498$		0			0
$499$	−246.943			−0.494875			−0.247437	−	0.968904i	$-0.579588\pi$
$499$	−246.943			−0.494875			−0.247437	+	0.968904i	$0.579588\pi$
$500$		0			0
$501$	−7.37639			−0.0147233
$502$		0			0
$503$		−	365.693i		−	0.727025i	−0.931589	−	0.363512i	$-0.881577\pi$
$503$		−	365.693i		−	0.727025i	0.931589	−	0.363512i	$-0.118423\pi$
$504$		0			0
$505$			215.746i			0.427221i
$506$		0			0
$507$	−7.33463			−0.0144667
$508$		0			0
$509$	186.361			0.366132			0.183066	−	0.983101i	$-0.441398\pi$
$509$	186.361			0.366132			0.183066	+	0.983101i	$0.441398\pi$
$510$		0			0
$511$		−	90.7593i		−	0.177611i
$512$		0			0
$513$			18.2857i			0.0356447i
$514$		0			0
$515$	22.6455			0.0439718
$516$		0			0
$517$		−	658.718i		−	1.27412i
$518$		0			0
$519$	2.48853			0.00479486
$520$		0			0
$521$			197.384i			0.378857i	0.981895	+	0.189428i	$0.0606635\pi$
$521$			197.384i			0.378857i	−0.981895	+	0.189428i	$0.939337\pi$
$522$		0			0
$523$			225.053i			0.430313i	0.976580	+	0.215156i	$0.0690261\pi$
$523$			225.053i			0.430313i	−0.976580	+	0.215156i	$0.930974\pi$
$524$		0			0
$525$			0.509598i			0.000970663i
$526$		0			0
$527$		−	456.662i		−	0.866532i
$528$		0			0
$529$	−511.907	+	133.389i	−0.967687	+	0.252153i
$530$		0			0
$531$	35.4937			0.0668430
$532$		0			0
$533$	71.5356			0.134213
$534$		0			0
$535$	−87.2293			−0.163045
$536$		0			0
$537$	10.6341			0.0198029
$538$		0			0
$539$			714.203i			1.32505i
$540$		0			0
$541$	−568.646			−1.05110			−0.525550	−	0.850762i	$-0.676141\pi$
$541$	−568.646			−1.05110			−0.525550	+	0.850762i	$0.676141\pi$
$542$		0			0
$543$			8.27506i			0.0152395i
$544$		0			0
$545$	96.3911			0.176864
$546$		0			0
$547$	−145.784			−0.266515			−0.133257	−	0.991081i	$-0.542544\pi$
$547$	−145.784			−0.266515			−0.133257	+	0.991081i	$0.542544\pi$
$548$		0			0
$549$		−	404.733i		−	0.737219i
$550$		0			0
$551$			116.391i			0.211237i
$552$		0			0
$553$	−241.348			−0.436434
$554$		0			0
$555$	0.243815			0.000439306
$556$		0			0
$557$			507.563i			0.911245i	0.890173	+	0.455622i	$0.150583\pi$
$557$			507.563i			0.911245i	−0.890173	+	0.455622i	$0.849417\pi$
$558$		0			0
$559$			27.3717i			0.0489655i
$560$		0			0
$561$	12.2807			0.0218907
$562$		0			0
$563$			173.558i			0.308274i	0.988049	+	0.154137i	$0.0492597\pi$
$563$			173.558i			0.308274i	−0.988049	+	0.154137i	$0.950740\pi$
$564$		0			0
$565$	362.358			0.641342
$566$		0			0
$567$		−	188.911i		−	0.333177i
$568$		0			0
$569$		−	456.073i		−	0.801534i	−0.916180	−	0.400767i	$-0.868744\pi$
$569$		−	456.073i		−	0.801534i	0.916180	−	0.400767i	$-0.131256\pi$
$570$		0			0
$571$		−	193.134i		−	0.338238i	−0.985596	−	0.169119i	$-0.945908\pi$
$571$		−	193.134i		−	0.338238i	0.985596	−	0.169119i	$-0.0540922\pi$
$572$		0			0
$573$			13.2528i			0.0231289i
$574$		0			0
$575$	14.6174	+	114.067i	0.0254215	+	0.198378i
$576$		0			0
$577$	−256.538			−0.444606			−0.222303	−	0.974978i	$-0.571357\pi$
$577$	−256.538			−0.444606			−0.222303	+	0.974978i	$0.571357\pi$
$578$		0			0
$579$	−7.50932			−0.0129695
$580$		0			0
$581$	138.508			0.238396
$582$		0			0
$583$	−881.948			−1.51278
$584$		0			0
$585$		−	20.6561i		−	0.0353097i
$586$		0			0
$587$	−1026.07			−1.74799			−0.873993	−	0.485938i	$-0.838478\pi$
$587$	−1026.07			−1.74799			−0.873993	+	0.485938i	$0.838478\pi$
$588$		0			0
$589$		−	619.518i		−	1.05181i
$590$		0			0
$591$	11.2339			0.0190083
$592$		0			0
$593$	−974.347			−1.64308			−0.821540	−	0.570151i	$-0.806885\pi$
$593$	−974.347			−1.64308			−0.821540	+	0.570151i	$0.806885\pi$
$594$		0			0
$595$		−	89.4855i		−	0.150396i
$596$		0			0
$597$		−	7.85152i		−	0.0131516i
$598$		0			0
$599$	593.457			0.990746			0.495373	−	0.868680i	$-0.335031\pi$
$599$	593.457			0.990746			0.495373	+	0.868680i	$0.335031\pi$
$600$		0			0
$601$	−167.443			−0.278608			−0.139304	−	0.990250i	$-0.544486\pi$
$601$	−167.443			−0.278608			−0.139304	+	0.990250i	$0.544486\pi$
$602$		0			0
$603$		−	842.477i		−	1.39714i
$604$		0			0
$605$		−	330.717i		−	0.546640i
$606$		0			0
$607$	−238.549			−0.392996			−0.196498	−	0.980504i	$-0.562957\pi$
$607$	−238.549			−0.392996			−0.196498	+	0.980504i	$0.562957\pi$
$608$		0			0
$609$			0.509919i			0.000837306i
$610$		0			0
$611$	−41.2399			−0.0674957
$612$		0			0
$613$			686.568i			1.12001i	0.828488	+	0.560007i	$0.189201\pi$
$613$			686.568i			1.12001i	−0.828488	+	0.560007i	$0.810799\pi$
$614$		0			0
$615$			6.80458i			0.0110644i
$616$		0			0
$617$		−	387.430i		−	0.627925i	−0.949435	−	0.313962i	$-0.898343\pi$
$617$		−	387.430i		−	0.627925i	0.949435	−	0.313962i	$-0.101657\pi$
$618$		0			0
$619$		−	310.811i		−	0.502118i	−0.967972	−	0.251059i	$-0.919221\pi$
$619$		−	310.811i		−	0.502118i	0.967972	−	0.251059i	$-0.0807789\pi$
$620$		0			0
$621$	2.29792	+	17.9319i	0.00370035	+	0.0288758i
$622$		0			0
$623$	303.921			0.487835
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	16.6603			0.0265714
$628$		0			0
$629$	−42.8139			−0.0680666
$630$		0			0
$631$		−	252.000i		−	0.399366i	−0.979861	−	0.199683i	$-0.936009\pi$
$631$		−	252.000i		−	0.399366i	0.979861	−	0.199683i	$-0.0639913\pi$
$632$		0			0
$633$	−14.8029			−0.0233853
$634$		0			0
$635$			399.497i			0.629130i
$636$		0			0
$637$	44.7136			0.0701940
$638$		0			0
$639$	−880.947			−1.37863
$640$		0			0
$641$			1062.69i			1.65787i	0.559348	+	0.828933i	$0.311052\pi$
$641$			1062.69i			1.65787i	−0.559348	+	0.828933i	$0.688948\pi$
$642$		0			0
$643$		−	456.325i		−	0.709681i	−0.934927	−	0.354841i	$-0.884535\pi$
$643$		−	456.325i		−	0.709681i	0.934927	−	0.354841i	$-0.115465\pi$
$644$		0			0
$645$	−2.60365			−0.00403666
$646$		0			0
$647$	−52.0609			−0.0804651			−0.0402325	−	0.999190i	$-0.512810\pi$
$647$	−52.0609			−0.0804651			−0.0402325	+	0.999190i	$0.512810\pi$
$648$		0			0
$649$		−	64.6840i		−	0.0996671i
$650$		0			0
$651$		−	2.71415i		−	0.00416921i
$652$		0			0
$653$	617.978			0.946368			0.473184	−	0.880964i	$-0.343105\pi$
$653$	617.978			0.946368			0.473184	+	0.880964i	$0.343105\pi$
$654$		0			0
$655$		−	110.765i		−	0.169108i
$656$		0			0
$657$	−349.939			−0.532632
$658$		0			0
$659$		−	558.354i		−	0.847275i	−0.905832	−	0.423637i	$-0.860753\pi$
$659$		−	558.354i		−	0.847275i	0.905832	−	0.423637i	$-0.139247\pi$
$660$		0			0
$661$			371.214i			0.561595i	0.959767	+	0.280798i	$0.0905990\pi$
$661$			371.214i			0.561595i	−0.959767	+	0.280798i	$0.909401\pi$
$662$		0			0
$663$		−	0.768849i		−	0.00115965i
$664$		0			0
$665$		−	121.398i		−	0.182553i
$666$		0			0
$667$	14.6266	+	114.139i	0.0219289	+	0.171123i
$668$		0			0
$669$	13.2307			0.0197769
$670$		0			0
$671$	−737.590			−1.09924
$672$		0			0
$673$	630.324			0.936589			0.468294	−	0.883573i	$-0.344869\pi$
$673$	630.324			0.936589			0.468294	+	0.883573i	$0.344869\pi$
$674$		0			0
$675$	3.93011			0.00582239
$676$		0			0
$677$		−	966.907i		−	1.42822i	−0.700032	−	0.714112i	$-0.746831\pi$
$677$		−	966.907i		−	1.42822i	0.700032	−	0.714112i	$-0.253169\pi$
$678$		0			0
$679$	−332.741			−0.490045
$680$		0			0
$681$		−	17.1092i		−	0.0251236i
$682$		0			0
$683$	265.192			0.388275			0.194137	−	0.980974i	$-0.437809\pi$
$683$	265.192			0.388275			0.194137	+	0.980974i	$0.437809\pi$
$684$		0			0
$685$	−325.667			−0.475426
$686$		0			0
$687$			5.69051i			0.00828313i
$688$		0			0
$689$			55.2155i			0.0801385i
$690$		0			0
$691$	−702.913			−1.01724			−0.508620	−	0.860991i	$-0.669844\pi$
$691$	−702.913			−1.01724			−0.508620	+	0.860991i	$0.669844\pi$
$692$		0			0
$693$	−344.347			−0.496893
$694$		0			0
$695$			444.733i			0.639904i
$696$		0			0
$697$		−	1194.89i		−	1.71433i
$698$		0			0
$699$	10.9702			0.0156941
$700$		0			0
$701$			905.753i			1.29209i	0.763301	+	0.646044i	$0.223578\pi$
$701$			905.753i			1.29209i	−0.763301	+	0.646044i	$0.776422\pi$
$702$		0			0
$703$	−58.0822			−0.0826206
$704$		0			0
$705$		−	3.92280i		−	0.00556426i
$706$		0			0
$707$		−	225.169i		−	0.318485i
$708$		0			0
$709$		−	917.543i		−	1.29414i	−0.762432	−	0.647068i	$-0.775995\pi$
$709$		−	917.543i		−	1.29414i	0.762432	−	0.647068i	$-0.224005\pi$
$710$		0			0
$711$			930.561i			1.30881i
$712$		0			0
$713$	−77.8531	−	607.530i	−0.109191	−	0.852075i
$714$		0			0
$715$	−37.6440			−0.0526489
$716$		0			0
$717$	0.466542			0.000650686
$718$		0			0
$719$	1126.87			1.56728			0.783639	−	0.621216i	$-0.213361\pi$
$719$	1126.87			1.56728			0.783639	+	0.621216i	$0.213361\pi$
$720$		0			0
$721$	−23.6345			−0.0327801
$722$		0			0
$723$		−	18.0986i		−	0.0250326i
$724$		0			0
$725$	25.0158			0.0345045
$726$		0			0
$727$			176.237i			0.242417i	0.992627	+	0.121209i	$0.0386770\pi$
$727$			176.237i			0.242417i	−0.992627	+	0.121209i	$0.961323\pi$
$728$		0			0
$729$	−728.073			−0.998729
$730$		0			0
$731$	457.201			0.625446
$732$		0			0
$733$			1089.28i			1.48605i	0.669262	+	0.743027i	$0.266611\pi$
$733$			1089.28i			1.48605i	−0.669262	+	0.743027i	$0.733389\pi$
$734$		0			0
$735$			4.25323i			0.00578671i
$736$		0			0
$737$	−1535.34			−2.08323
$738$		0			0
$739$	−667.801			−0.903655			−0.451827	−	0.892105i	$-0.649228\pi$
$739$	−667.801			−0.903655			−0.451827	+	0.892105i	$0.649228\pi$
$740$		0			0
$741$		−	1.04304i		−	0.00140761i
$742$		0			0
$743$			111.460i			0.150014i	0.997183	+	0.0750068i	$0.0238978\pi$
$743$			111.460i			0.150014i	−0.997183	+	0.0750068i	$0.976102\pi$
$744$		0			0
$745$	−230.725			−0.309698
$746$		0			0
$747$		−	534.043i		−	0.714917i
$748$		0			0
$749$	91.0388			0.121547
$750$		0			0
$751$			452.088i			0.601981i	0.953627	+	0.300991i	$0.0973173\pi$
$751$			452.088i			0.601981i	−0.953627	+	0.300991i	$0.902683\pi$
$752$		0			0
$753$		−	10.8501i		−	0.0144091i
$754$		0			0
$755$			446.237i			0.591042i
$756$		0			0
$757$		−	216.404i		−	0.285871i	−0.989732	−	0.142935i	$-0.954346\pi$
$757$		−	216.404i		−	0.285871i	0.989732	−	0.142935i	$-0.0456541\pi$
$758$		0			0
$759$	16.3379	−	2.09365i	0.0215255	−	0.00275843i
$760$		0			0
$761$	−653.530			−0.858778			−0.429389	−	0.903120i	$-0.641271\pi$
$761$	−653.530			−0.858778			−0.429389	+	0.903120i	$0.641271\pi$
$762$		0			0
$763$	−100.601			−0.131849
$764$		0			0
$765$	−345.028			−0.451017
$766$		0			0
$767$	−4.04962			−0.00527982
$768$		0			0
$769$			713.959i			0.928426i	0.885724	+	0.464213i	$0.153663\pi$
$769$			713.959i			0.928426i	−0.885724	+	0.464213i	$0.846337\pi$
$770$		0			0
$771$	−17.0529			−0.0221178
$772$		0			0
$773$			31.4153i			0.0406407i	0.999794	+	0.0203204i	$0.00646862\pi$
$773$			31.4153i			0.0406407i	−0.999794	+	0.0203204i	$0.993531\pi$
$774$		0			0
$775$	−133.152			−0.171809
$776$		0			0
$777$	−0.254463			−0.000327494
$778$		0			0
$779$		−	1621.01i		−	2.08088i
$780$		0			0
$781$			1605.45i			2.05563i
$782$		0			0
$783$	3.93259			0.00502247
$784$		0			0
$785$	501.872			0.639327
$786$		0			0
$787$			311.831i			0.396227i	0.980179	+	0.198113i	$0.0634815\pi$
$787$			311.831i			0.396227i	−0.980179	+	0.198113i	$0.936519\pi$
$788$		0			0
$789$			13.2057i			0.0167373i
$790$		0			0
$791$	−378.184			−0.478108
$792$		0			0
$793$			46.1777i			0.0582317i
$794$		0			0
$795$	−5.25219			−0.00660652
$796$		0			0
$797$			94.6765i			0.118791i	0.998235	+	0.0593955i	$0.0189173\pi$
$797$			94.6765i			0.118791i	−0.998235	+	0.0593955i	$0.981083\pi$
$798$		0			0
$799$			688.845i			0.862135i
$800$		0			0
$801$		−	1171.82i		−	1.46295i
$802$		0			0
$803$			637.732i			0.794187i
$804$		0			0
$805$	−15.2557	−	119.049i	−0.0189512	−	0.147887i
$806$		0			0
$807$	12.8901			0.0159729
$808$		0			0
$809$	−645.736			−0.798190			−0.399095	−	0.916910i	$-0.630676\pi$
$809$	−645.736			−0.798190			−0.399095	+	0.916910i	$0.630676\pi$
$810$		0			0
$811$	906.212			1.11740			0.558700	−	0.829370i	$-0.311300\pi$
$811$	906.212			1.11740			0.558700	+	0.829370i	$0.311300\pi$
$812$		0			0
$813$	5.18257			0.00637462
$814$		0			0
$815$			97.9794i			0.120220i
$816$		0			0
$817$	620.248			0.759178
$818$		0			0
$819$			21.5583i			0.0263227i
$820$		0			0
$821$	654.650			0.797381			0.398690	−	0.917086i	$-0.369465\pi$
$821$	654.650			0.797381			0.398690	+	0.917086i	$0.369465\pi$
$822$		0			0
$823$	−995.028			−1.20903			−0.604513	−	0.796595i	$-0.706632\pi$
$823$	−995.028			−1.20903			−0.604513	+	0.796595i	$0.706632\pi$
$824$		0			0
$825$		−	3.58076i		−	0.00434031i
$826$		0			0
$827$		−	326.367i		−	0.394640i	−0.980339	−	0.197320i	$-0.936776\pi$
$827$		−	326.367i		−	0.394640i	0.980339	−	0.197320i	$-0.0632238\pi$
$828$		0			0
$829$	693.960			0.837104			0.418552	−	0.908193i	$-0.362538\pi$
$829$	693.960			0.837104			0.418552	+	0.908193i	$0.362538\pi$
$830$		0			0
$831$	15.6824			0.0188717
$832$		0			0
$833$		−	746.868i		−	0.896601i
$834$		0			0
$835$		−	377.677i		−	0.452308i
$836$		0			0
$837$	−20.9320			−0.0250084
$838$		0			0
$839$		−	343.739i		−	0.409700i	−0.978793	−	0.204850i	$-0.934329\pi$
$839$		−	343.739i		−	0.409700i	0.978793	−	0.204850i	$-0.0656707\pi$
$840$		0			0
$841$	−815.968			−0.970236
$842$		0			0
$843$			8.69648i			0.0103161i
$844$		0			0
$845$		−	375.539i		−	0.444425i
$846$		0			0
$847$			345.160i			0.407509i
$848$		0			0
$849$		−	5.28151i		−	0.00622086i
$850$		0			0
$851$	−56.9583	+	7.29904i	−0.0669310	+	0.00857701i
$852$		0			0
$853$	−24.5062			−0.0287295			−0.0143647	−	0.999897i	$-0.504573\pi$
$853$	−24.5062			−0.0287295			−0.0143647	+	0.999897i	$0.504573\pi$
$854$		0			0
$855$	−468.072			−0.547453
$856$		0			0
$857$	1674.09			1.95343			0.976715	−	0.214542i	$-0.0688260\pi$
$857$	1674.09			1.95343			0.976715	+	0.214542i	$0.0688260\pi$
$858$		0			0
$859$	137.574			0.160156			0.0800778	−	0.996789i	$-0.474483\pi$
$859$	137.574			0.160156			0.0800778	+	0.996789i	$0.474483\pi$
$860$		0			0
$861$		−	7.10176i		−	0.00824827i
$862$		0			0
$863$	312.225			0.361791			0.180895	−	0.983502i	$-0.442100\pi$
$863$	312.225			0.361791			0.180895	+	0.983502i	$0.442100\pi$
$864$		0			0
$865$			127.415i			0.147300i
$866$		0			0
$867$	−0.221018			−0.000254923
$868$		0			0
$869$	1695.86			1.95151
$870$		0			0
$871$			96.1217i			0.110358i
$872$		0			0
$873$			1282.94i			1.46958i
$874$		0			0
$875$	−26.0918			−0.0298192
$876$		0			0
$877$	984.015			1.12202			0.561012	−	0.827808i	$-0.310412\pi$
$877$	984.015			1.12202			0.561012	+	0.827808i	$0.310412\pi$
$878$		0			0
$879$		−	18.8618i		−	0.0214583i
$880$		0			0
$881$			1250.48i			1.41939i	0.704509	+	0.709695i	$0.251167\pi$
$881$			1250.48i			1.41939i	−0.704509	+	0.709695i	$0.748833\pi$
$882$		0			0
$883$	788.969			0.893510			0.446755	−	0.894656i	$-0.352580\pi$
$883$	788.969			0.893510			0.446755	+	0.894656i	$0.352580\pi$
$884$		0			0
$885$		−	0.385207i		−	0.000435262i
$886$		0			0
$887$	298.078			0.336051			0.168026	−	0.985783i	$-0.446261\pi$
$887$	298.078			0.336051			0.168026	+	0.985783i	$0.446261\pi$
$888$		0			0
$889$		−	416.945i		−	0.469004i
$890$		0			0
$891$			1327.41i			1.48980i
$892$		0			0
$893$			934.502i			1.04648i
$894$		0			0
$895$			544.476i			0.608354i
$896$		0			0
$897$	−0.131076	−	1.02285i	−0.000146127	−	0.00114030i
$898$		0			0
$899$	−133.236			−0.148204
$900$		0			0
$901$	922.285			1.02362
$902$		0			0
$903$	2.71735			0.00300925
$904$		0			0
$905$	−423.689			−0.468165
$906$		0			0
$907$		−	371.856i		−	0.409984i	−0.978764	−	0.204992i	$-0.934283\pi$
$907$		−	371.856i		−	0.409984i	0.978764	−	0.204992i	$-0.0657169\pi$
$908$		0			0
$909$	−868.178			−0.955092
$910$		0			0
$911$		−	564.051i		−	0.619156i	−0.950874	−	0.309578i	$-0.899812\pi$
$911$		−	564.051i		−	0.619156i	0.950874	−	0.309578i	$-0.100188\pi$
$912$		0			0
$913$	−973.245			−1.06599
$914$		0			0
$915$	−4.39250			−0.00480055
$916$		0			0
$917$			115.603i			0.126066i
$918$		0			0
$919$		−	31.1434i		−	0.0338883i	−0.999856	−	0.0169442i	$-0.994606\pi$
$919$		−	31.1434i		−	0.0338883i	0.999856	−	0.0169442i	$-0.00539376\pi$
$920$		0			0
$921$	−11.6158			−0.0126122
$922$		0			0
$923$	100.511			0.108896
$924$		0			0
$925$			12.4835i			0.0134957i
$926$		0			0
$927$			91.1270i			0.0983032i
$928$		0			0
$929$	−348.046			−0.374646			−0.187323	−	0.982298i	$-0.559981\pi$
$929$	−348.046			−0.374646			−0.187323	+	0.982298i	$0.559981\pi$
$930$		0			0
$931$		−	1013.22i		−	1.08831i
$932$		0			0
$933$	−4.82729			−0.00517395
$934$		0			0
$935$			628.782i			0.672494i
$936$		0			0
$937$		−	1626.37i		−	1.73572i	−0.496808	−	0.867860i	$-0.665495\pi$
$937$		−	1626.37i		−	1.73572i	0.496808	−	0.867860i	$-0.334505\pi$
$938$		0			0
$939$		−	20.1861i		−	0.0214975i
$940$		0			0
$941$		−	604.582i		−	0.642489i	−0.946996	−	0.321244i	$-0.895899\pi$
$941$		−	604.582i		−	0.642489i	0.946996	−	0.321244i	$-0.104101\pi$
$942$		0			0
$943$	−203.708	−	1589.64i	−0.216021	−	1.68573i
$944$		0			0
$945$	−4.10175			−0.00434048
$946$		0			0
$947$	370.557			0.391296			0.195648	−	0.980674i	$-0.437319\pi$
$947$	370.557			0.391296			0.195648	+	0.980674i	$0.437319\pi$
$948$		0			0
$949$	39.9260			0.0420717
$950$		0			0
$951$	24.1312			0.0253745
$952$		0			0
$953$			477.703i			0.501263i	0.968083	+	0.250631i	$0.0806382\pi$
$953$			477.703i			0.501263i	−0.968083	+	0.250631i	$0.919362\pi$
$954$		0			0
$955$	−678.556			−0.710530
$956$		0			0
$957$		−	3.58301i		−	0.00374401i
$958$		0			0
$959$	339.890			0.354421
$960$		0			0
$961$	−251.825			−0.262045
$962$		0			0
$963$		−	351.017i		−	0.364503i
$964$		0			0
$965$		−	384.483i		−	0.398428i
$966$		0			0
$967$	668.967			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$967$	668.967			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$968$		0			0
$969$	−17.4222			−0.0179796
$970$		0			0
$971$			1839.11i			1.89404i	0.321178	+	0.947019i	$0.395921\pi$
$971$			1839.11i			1.89404i	−0.321178	+	0.947019i	$0.604079\pi$
$972$		0			0
$973$		−	464.156i		−	0.477036i
$974$		0			0
$975$	−0.224178			−0.000229926
$976$		0			0
$977$		−	1078.15i		−	1.10353i	−0.834000	−	0.551764i	$-0.813955\pi$
$977$		−	1078.15i		−	1.10353i	0.834000	−	0.551764i	$-0.186045\pi$
$978$		0			0
$979$	−2135.54			−2.18135
$980$		0			0
$981$			387.885i			0.395397i
$982$		0			0
$983$		−	1248.18i		−	1.26977i	−0.772608	−	0.634884i	$-0.781048\pi$
$983$		−	1248.18i		−	1.26977i	0.772608	−	0.634884i	$-0.218952\pi$
$984$		0			0
$985$			575.186i			0.583945i
$986$		0			0
$987$			4.09412i			0.00414805i
$988$		0			0
$989$	608.246	−	77.9449i	0.615011	−	0.0788118i
$990$		0			0
$991$	−800.574			−0.807845			−0.403922	−	0.914793i	$-0.632353\pi$
$991$	−800.574			−0.807845			−0.403922	+	0.914793i	$0.632353\pi$
$992$		0			0
$993$	−1.45912			−0.00146941
$994$		0			0
$995$	402.004			0.404024
$996$		0			0
$997$	372.888			0.374010			0.187005	−	0.982359i	$-0.440122\pi$
$997$	372.888			0.374010			0.187005	+	0.982359i	$0.440122\pi$
$998$		0			0
$999$			1.96246i			0.00196443i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.b.321.6		10
4.3	odd	2		115.3.d.b.91.10	yes	10
12.11	even	2		1035.3.g.b.91.1		10
20.3	even	4		575.3.c.d.574.2		20
20.7	even	4		575.3.c.d.574.19		20
20.19	odd	2		575.3.d.g.551.1		10
23.22	odd	2	inner	1840.3.k.b.321.5		10
92.91	even	2		115.3.d.b.91.9	✓	10
276.275	odd	2		1035.3.g.b.91.2		10
460.183	odd	4		575.3.c.d.574.1		20
460.367	odd	4		575.3.c.d.574.20		20
460.459	even	2		575.3.d.g.551.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.d.b.91.9	✓	10	92.91	even	2
115.3.d.b.91.10	yes	10	4.3	odd	2
575.3.c.d.574.1		20	460.183	odd	4
575.3.c.d.574.2		20	20.3	even	4
575.3.c.d.574.19		20	20.7	even	4
575.3.c.d.574.20		20	460.367	odd	4
575.3.d.g.551.1		10	20.19	odd	2
575.3.d.g.551.2		10	460.459	even	2
1035.3.g.b.91.1		10	12.11	even	2
1035.3.g.b.91.2		10	276.275	odd	2
1840.3.k.b.321.5		10	23.22	odd	2	inner
1840.3.k.b.321.6		10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.0436725 q^{3} +2.23607i q^{5} -2.33372i q^{7} -8.99809 q^{9} +16.3982i q^{11} +1.02663 q^{13} +0.0976548i q^{15} -17.1482i q^{17} -23.2636i q^{19} -0.101920i q^{21} +(-2.92347 - 22.8134i) q^{23} -5.00000 q^{25} -0.786022 q^{27} -5.00315 q^{29} +26.6303 q^{31} +0.716151i q^{33} +5.21836 q^{35} -2.49670i q^{37} +0.0448355 q^{39} +69.6800 q^{41} +26.6617i q^{43} -20.1203i q^{45} -40.1701 q^{47} +43.5537 q^{49} -0.748905i q^{51} +53.7832i q^{53} -36.6675 q^{55} -1.01598i q^{57} -3.94458 q^{59} +44.9799i q^{61} +20.9991i q^{63} +2.29561i q^{65} +93.6284i q^{67} +(-0.127676 - 0.996321i) q^{69} +97.9038 q^{71} +38.8904 q^{73} -0.218363 q^{75} +38.2689 q^{77} -103.418i q^{79} +80.9485 q^{81} +59.3507i q^{83} +38.3445 q^{85} -0.218500 q^{87} +130.230i q^{89} -2.39587i q^{91} +1.16301 q^{93} +52.0190 q^{95} -142.579i q^{97} -147.553i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 16 q^{9} - 2 q^{13} - 44 q^{23} - 50 q^{25} - 40 q^{27} - 46 q^{29} - 16 q^{31} + 60 q^{35} - 72 q^{39} - 84 q^{41} - 112 q^{47} + 50 q^{49} + 10 q^{55} + 262 q^{59} + 124 q^{69} - 236 q^{71}+ \cdots + 90 q^{95}+O(q^{100}) \) 10 * q + 2 * q^3 - 16 * q^9 - 2 * q^13 - 44 * q^23 - 50 * q^25 - 40 * q^27 - 46 * q^29 - 16 * q^31 + 60 * q^35 - 72 * q^39 - 84 * q^41 - 112 * q^47 + 50 * q^49 + 10 * q^55 + 262 * q^59 + 124 * q^69 - 236 * q^71 + 168 * q^73 - 10 * q^75 + 300 * q^77 - 258 * q^81 - 540 * q^87 + 100 * q^93 + 90 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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