LMFDB - Embedded newform 496.3.e.d.433.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [496,3,Mod(433,496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("496.433"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 496 = 2^{4} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	496.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$13.5150211123$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.48128.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 14x^{2} + 47 $ x^4 + 14*x^2 + 47
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 62)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.1
Root			$2.90073i$ of defining polynomial
Character	$\chi$	$=$	496.433
Dual form			496.3.e.d.433.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.10225i q^{3} -5.82843 q^{5} +4.41421 q^{7} -7.82843 q^{9} -19.8074i q^{11} -6.50529i q^{13} +23.9096i q^{15} +13.3021i q^{17} -30.2132 q^{19} -18.1082i q^{21} +19.1036i q^{23} +8.97056 q^{25} -4.80608i q^{27} +6.50529i q^{29} +(-24.4142 - 19.1036i) q^{31} -81.2548 q^{33} -25.7279 q^{35} +59.4222i q^{37} -26.6863 q^{39} +16.6569 q^{41} +65.5152i q^{43} +45.6274 q^{45} +67.7401 q^{47} -29.5147 q^{49} +54.5685 q^{51} -48.4024i q^{53} +115.446i q^{55} +123.942i q^{57} -56.2132 q^{59} -4.51454i q^{61} -34.5563 q^{63} +37.9156i q^{65} -21.0538 q^{67} +78.3675 q^{69} -11.2426 q^{71} +25.1966i q^{73} -36.7995i q^{75} -87.4341i q^{77} -36.0956i q^{79} -90.1716 q^{81} -159.284i q^{83} -77.5304i q^{85} +26.6863 q^{87} -143.166i q^{89} -28.7157i q^{91} +(-78.3675 + 100.153i) q^{93} +176.095 q^{95} +27.7107 q^{97} +155.061i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{5} + 12 q^{7} - 20 q^{9} - 36 q^{19} - 32 q^{25} - 92 q^{31} - 144 q^{33} - 52 q^{35} - 152 q^{39} + 44 q^{41} + 92 q^{45} + 56 q^{47} - 152 q^{49} - 8 q^{51} - 140 q^{59} - 76 q^{63} + 176 q^{67}+ \cdots - 172 q^{97}+O(q^{100}) $ 4 * q - 12 * q^5 + 12 * q^7 - 20 * q^9 - 36 * q^19 - 32 * q^25 - 92 * q^31 - 144 * q^33 - 52 * q^35 - 152 * q^39 + 44 * q^41 + 92 * q^45 + 56 * q^47 - 152 * q^49 - 8 * q^51 - 140 * q^59 - 76 * q^63 + 176 * q^67 + 8 * q^69 - 28 * q^71 - 372 * q^81 + 152 * q^87 - 8 * q^93 + 348 * q^95 - 172 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/496\mathbb{Z}\right)^\times$.

$n$	$63$	$65$	$373$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	4.10225i		−	1.36742i	−0.729756	−	0.683708i	$-0.760366\pi$
$3$		−	4.10225i		−	1.36742i	0.729756	−	0.683708i	$-0.239634\pi$
$4$		0			0
$5$	−5.82843			−1.16569			−0.582843	−	0.812585i	$-0.698060\pi$
$5$	−5.82843			−1.16569			−0.582843	+	0.812585i	$0.698060\pi$
$6$		0			0
$7$	4.41421			0.630602			0.315301	−	0.948992i	$-0.397895\pi$
$7$	4.41421			0.630602			0.315301	+	0.948992i	$0.397895\pi$
$8$		0			0
$9$	−7.82843			−0.869825
$10$		0			0
$11$		−	19.8074i		−	1.80067i	−0.435195	−	0.900336i	$-0.643321\pi$
$11$		−	19.8074i		−	1.80067i	0.435195	−	0.900336i	$-0.356679\pi$
$12$		0			0
$13$		−	6.50529i		−	0.500407i	−0.968193	−	0.250203i	$-0.919503\pi$
$13$		−	6.50529i		−	0.500407i	0.968193	−	0.250203i	$-0.0804975\pi$
$14$		0			0
$15$			23.9096i			1.59398i
$16$		0			0
$17$			13.3021i			0.782477i	0.920289	+	0.391239i	$0.127953\pi$
$17$			13.3021i			0.782477i	−0.920289	+	0.391239i	$0.872047\pi$
$18$		0			0
$19$	−30.2132			−1.59017			−0.795084	−	0.606499i	$-0.792573\pi$
$19$	−30.2132			−1.59017			−0.795084	+	0.606499i	$0.792573\pi$
$20$		0			0
$21$		−	18.1082i		−	0.862295i
$22$		0			0
$23$			19.1036i			0.830590i	0.909687	+	0.415295i	$0.136322\pi$
$23$			19.1036i			0.830590i	−0.909687	+	0.415295i	$0.863678\pi$
$24$		0			0
$25$	8.97056			0.358823
$26$		0			0
$27$		−	4.80608i		−	0.178003i
$28$		0			0
$29$			6.50529i			0.224320i	0.993690	+	0.112160i	$0.0357770\pi$
$29$			6.50529i			0.224320i	−0.993690	+	0.112160i	$0.964223\pi$
$30$		0			0
$31$	−24.4142	−	19.1036i	−0.787555	−	0.616244i
$31$	−24.4142	−	19.1036i	−0.787555	−	0.616244i
$32$		0			0
$33$	−81.2548			−2.46227
$34$		0			0
$35$	−25.7279			−0.735083
$36$		0			0
$37$			59.4222i			1.60601i	0.595975	+	0.803003i	$0.296766\pi$
$37$			59.4222i			1.60601i	−0.595975	+	0.803003i	$0.703234\pi$
$38$		0			0
$39$	−26.6863			−0.684264
$40$		0			0
$41$	16.6569			0.406265			0.203132	−	0.979151i	$-0.434888\pi$
$41$	16.6569			0.406265			0.203132	+	0.979151i	$0.434888\pi$
$42$		0			0
$43$			65.5152i			1.52361i	0.647807	+	0.761804i	$0.275686\pi$
$43$			65.5152i			1.52361i	−0.647807	+	0.761804i	$0.724314\pi$
$44$		0			0
$45$	45.6274			1.01394
$46$		0			0
$47$	67.7401			1.44128			0.720640	−	0.693310i	$-0.243848\pi$
$47$	67.7401			1.44128			0.720640	+	0.693310i	$0.243848\pi$
$48$		0			0
$49$	−29.5147			−0.602341
$50$		0			0
$51$	54.5685			1.06997
$52$		0			0
$53$		−	48.4024i		−	0.913252i	−0.889659	−	0.456626i	$-0.849058\pi$
$53$		−	48.4024i		−	0.913252i	0.889659	−	0.456626i	$-0.150942\pi$
$54$		0			0
$55$			115.446i			2.09902i
$56$		0			0
$57$			123.942i			2.17442i
$58$		0			0
$59$	−56.2132			−0.952766			−0.476383	−	0.879238i	$-0.658052\pi$
$59$	−56.2132			−0.952766			−0.476383	+	0.879238i	$0.658052\pi$
$60$		0			0
$61$		−	4.51454i		−	0.0740089i	−0.999315	−	0.0370045i	$-0.988218\pi$
$61$		−	4.51454i		−	0.0740089i	0.999315	−	0.0370045i	$-0.0117816\pi$
$62$		0			0
$63$	−34.5563			−0.548513
$64$		0			0
$65$			37.9156i			0.583317i
$66$		0			0
$67$	−21.0538			−0.314236			−0.157118	−	0.987580i	$-0.550220\pi$
$67$	−21.0538			−0.314236			−0.157118	+	0.987580i	$0.550220\pi$
$68$		0			0
$69$	78.3675			1.13576
$70$		0			0
$71$	−11.2426			−0.158347			−0.0791735	−	0.996861i	$-0.525228\pi$
$71$	−11.2426			−0.158347			−0.0791735	+	0.996861i	$0.525228\pi$
$72$		0			0
$73$			25.1966i			0.345158i	0.984996	+	0.172579i	$0.0552101\pi$
$73$			25.1966i			0.345158i	−0.984996	+	0.172579i	$0.944790\pi$
$74$		0			0
$75$		−	36.7995i		−	0.490659i
$76$		0			0
$77$		−	87.4341i		−	1.13551i
$78$		0			0
$79$		−	36.0956i		−	0.456907i	−0.973555	−	0.228453i	$-0.926633\pi$
$79$		−	36.0956i		−	0.456907i	0.973555	−	0.228453i	$-0.0733668\pi$
$80$		0			0
$81$	−90.1716			−1.11323
$82$		0			0
$83$		−	159.284i		−	1.91908i	−0.281569	−	0.959541i	$-0.590855\pi$
$83$		−	159.284i		−	1.91908i	0.281569	−	0.959541i	$-0.409145\pi$
$84$		0			0
$85$		−	77.5304i		−	0.912122i
$86$		0			0
$87$	26.6863			0.306739
$88$		0			0
$89$		−	143.166i		−	1.60861i	−0.594216	−	0.804305i	$-0.702538\pi$
$89$		−	143.166i		−	1.60861i	0.594216	−	0.804305i	$-0.297462\pi$
$90$		0			0
$91$		−	28.7157i		−	0.315557i
$92$		0			0
$93$	−78.3675	+	100.153i	−0.842662	+	1.07692i
$94$		0			0
$95$	176.095			1.85364
$96$		0			0
$97$	27.7107			0.285677			0.142839	−	0.989746i	$-0.454377\pi$
$97$	27.7107			0.285677			0.142839	+	0.989746i	$0.454377\pi$
$98$		0			0
$99$			155.061i			1.56627i
$100$		0			0
$101$	81.7452			0.809358			0.404679	−	0.914459i	$-0.367383\pi$
$101$	81.7452			0.809358			0.404679	+	0.914459i	$0.367383\pi$
$102$		0			0
$103$	−185.752			−1.80342			−0.901710	−	0.432341i	$-0.857688\pi$
$103$	−185.752			−1.80342			−0.901710	+	0.432341i	$0.857688\pi$
$104$		0			0
$105$			105.542i			1.00516i
$106$		0			0
$107$	−81.1838			−0.758727			−0.379363	−	0.925248i	$-0.623857\pi$
$107$	−81.1838			−0.758727			−0.379363	+	0.925248i	$0.623857\pi$
$108$		0			0
$109$	−94.6812			−0.868635			−0.434318	−	0.900760i	$-0.643010\pi$
$109$	−94.6812			−0.868635			−0.434318	+	0.900760i	$0.643010\pi$
$110$		0			0
$111$	243.765			2.19608
$112$		0			0
$113$	6.14719			0.0543999			0.0271999	−	0.999630i	$-0.491341\pi$
$113$	6.14719			0.0543999			0.0271999	+	0.999630i	$0.491341\pi$
$114$		0			0
$115$		−	111.344i		−	0.968206i
$116$		0			0
$117$			50.9262i			0.435266i
$118$		0			0
$119$			58.7184i			0.493432i
$120$		0			0
$121$	−271.333			−2.24242
$122$		0			0
$123$		−	68.3305i		−	0.555533i
$124$		0			0
$125$	93.4264			0.747411
$126$		0			0
$127$		−	83.4526i		−	0.657107i	−0.944485	−	0.328554i	$-0.893439\pi$
$127$		−	83.4526i		−	0.657107i	0.944485	−	0.328554i	$-0.106561\pi$
$128$		0			0
$129$	268.759			2.08341
$130$		0			0
$131$	−197.848			−1.51029			−0.755144	−	0.655559i	$-0.772433\pi$
$131$	−197.848			−1.51029			−0.755144	+	0.655559i	$0.772433\pi$
$132$		0			0
$133$	−133.368			−1.00276
$134$		0			0
$135$			28.0119i			0.207495i
$136$		0			0
$137$		−	125.350i		−	0.914961i	−0.889220	−	0.457481i	$-0.848752\pi$
$137$		−	125.350i		−	0.914961i	0.889220	−	0.457481i	$-0.151248\pi$
$138$		0			0
$139$		−	62.8206i		−	0.451947i	−0.974133	−	0.225973i	$-0.927444\pi$
$139$		−	62.8206i		−	0.451947i	0.974133	−	0.225973i	$-0.0725562\pi$
$140$		0			0
$141$		−	277.887i		−	1.97083i
$142$		0			0
$143$	−128.853			−0.901069
$144$		0			0
$145$		−	37.9156i		−	0.261487i
$146$		0			0
$147$			121.077i			0.823651i
$148$		0			0
$149$	102.024			0.684727			0.342364	−	0.939568i	$-0.388773\pi$
$149$	102.024			0.684727			0.342364	+	0.939568i	$0.388773\pi$
$150$		0			0
$151$		−	174.164i		−	1.15341i	−0.816954	−	0.576703i	$-0.804339\pi$
$151$		−	174.164i		−	1.15341i	0.816954	−	0.576703i	$-0.195661\pi$
$152$		0			0
$153$		−	104.135i		−	0.680618i
$154$		0			0
$155$	142.296	+	111.344i	0.918042	+	0.718347i
$156$		0			0
$157$	66.4315			0.423130			0.211565	−	0.977364i	$-0.432144\pi$
$157$	66.4315			0.423130			0.211565	+	0.977364i	$0.432144\pi$
$158$		0			0
$159$	−198.558			−1.24880
$160$		0			0
$161$			84.3272i			0.523772i
$162$		0			0
$163$	157.718			0.967594			0.483797	−	0.875180i	$-0.339257\pi$
$163$	157.718			0.967594			0.483797	+	0.875180i	$0.339257\pi$
$164$		0			0
$165$	473.588			2.87023
$166$		0			0
$167$		−	55.1992i		−	0.330534i	−0.986249	−	0.165267i	$-0.947151\pi$
$167$		−	55.1992i		−	0.330534i	0.986249	−	0.165267i	$-0.0528486\pi$
$168$		0			0
$169$	126.681			0.749593
$170$		0			0
$171$	236.522			1.38317
$172$		0			0
$173$	−6.71068			−0.0387900			−0.0193950	−	0.999812i	$-0.506174\pi$
$173$	−6.71068			−0.0387900			−0.0193950	+	0.999812i	$0.506174\pi$
$174$		0			0
$175$	39.5980			0.226274
$176$		0			0
$177$			230.600i			1.30283i
$178$		0			0
$179$			63.1829i			0.352977i	0.984303	+	0.176489i	$0.0564738\pi$
$179$			63.1829i			0.352977i	−0.984303	+	0.176489i	$0.943526\pi$
$180$		0			0
$181$		−	2.04076i		−	0.0112749i	−0.999984	−	0.00563747i	$-0.998206\pi$
$181$		−	2.04076i		−	0.0112749i	0.999984	−	0.00563747i	$-0.00179447\pi$
$182$		0			0
$183$	−18.5198			−0.101201
$184$		0			0
$185$		−	346.338i		−	1.87210i
$186$		0			0
$187$	263.480			1.40899
$188$		0			0
$189$		−	21.2151i		−	0.112249i
$190$		0			0
$191$	207.787			1.08789			0.543944	−	0.839121i	$-0.316930\pi$
$191$	207.787			1.08789			0.543944	+	0.839121i	$0.316930\pi$
$192$		0			0
$193$	48.3970			0.250762			0.125381	−	0.992109i	$-0.459985\pi$
$193$	48.3970			0.250762			0.125381	+	0.992109i	$0.459985\pi$
$194$		0			0
$195$	155.539			0.797636
$196$		0			0
$197$		−	115.788i		−	0.587754i	−0.955843	−	0.293877i	$-0.905054\pi$
$197$		−	115.788i		−	0.587754i	0.955843	−	0.293877i	$-0.0949456\pi$
$198$		0			0
$199$			262.885i			1.32103i	0.750812	+	0.660516i	$0.229662\pi$
$199$			262.885i			1.32103i	−0.750812	+	0.660516i	$0.770338\pi$
$200$		0			0
$201$			86.3680i			0.429691i
$202$		0			0
$203$			28.7157i			0.141457i
$204$		0			0
$205$	−97.0833			−0.473577
$206$		0			0
$207$		−	149.551i		−	0.722468i
$208$		0			0
$209$			598.445i			2.86337i
$210$		0			0
$211$	170.983			0.810345			0.405172	−	0.914240i	$-0.367211\pi$
$211$	170.983			0.810345			0.405172	+	0.914240i	$0.367211\pi$
$212$		0			0
$213$			46.1201i			0.216526i
$214$		0			0
$215$		−	381.850i		−	1.77605i
$216$		0			0
$217$	−107.770	−	84.3272i	−0.496634	−	0.388605i
$218$		0			0
$219$	103.362			0.471975
$220$		0			0
$221$	86.5341			0.391557
$222$		0			0
$223$		−	349.978i		−	1.56941i	−0.619871	−	0.784704i	$-0.712815\pi$
$223$		−	349.978i		−	1.56941i	0.619871	−	0.784704i	$-0.287185\pi$
$224$		0			0
$225$	−70.2254			−0.312113
$226$		0			0
$227$	190.485			0.839142			0.419571	−	0.907722i	$-0.362180\pi$
$227$	190.485			0.839142			0.419571	+	0.907722i	$0.362180\pi$
$228$		0			0
$229$			116.854i			0.510278i	0.966904	+	0.255139i	$0.0821212\pi$
$229$			116.854i			0.510278i	−0.966904	+	0.255139i	$0.917879\pi$
$230$		0			0
$231$	−358.676			−1.55271
$232$		0			0
$233$	−226.019			−0.970040			−0.485020	−	0.874503i	$-0.661188\pi$
$233$	−226.019			−0.970040			−0.485020	+	0.874503i	$0.661188\pi$
$234$		0			0
$235$	−394.818			−1.68008
$236$		0			0
$237$	−148.073			−0.624781
$238$		0			0
$239$			46.1908i			0.193267i	0.995320	+	0.0966335i	$0.0308075\pi$
$239$			46.1908i			0.193267i	−0.995320	+	0.0966335i	$0.969193\pi$
$240$		0			0
$241$			137.827i			0.571897i	0.958245	+	0.285949i	$0.0923086\pi$
$241$			137.827i			0.571897i	−0.958245	+	0.285949i	$0.907691\pi$
$242$		0			0
$243$			326.651i			1.34424i
$244$		0			0
$245$	172.024			0.702140
$246$		0			0
$247$			196.546i			0.795731i
$248$		0			0
$249$	−653.421			−2.62418
$250$		0			0
$251$			26.4835i			0.105512i	0.998607	+	0.0527559i	$0.0168005\pi$
$251$			26.4835i			0.105512i	−0.998607	+	0.0527559i	$0.983199\pi$
$252$		0			0
$253$	378.392			1.49562
$254$		0			0
$255$	−318.049			−1.24725
$256$		0			0
$257$	17.8183			0.0693320			0.0346660	−	0.999399i	$-0.488963\pi$
$257$	17.8183			0.0693320			0.0346660	+	0.999399i	$0.488963\pi$
$258$		0			0
$259$			262.302i			1.01275i
$260$		0			0
$261$		−	50.9262i		−	0.195119i
$262$		0			0
$263$			132.317i			0.503108i	0.967843	+	0.251554i	$0.0809415\pi$
$263$			132.317i			0.503108i	−0.967843	+	0.251554i	$0.919058\pi$
$264$		0			0
$265$			282.110i			1.06456i
$266$		0			0
$267$	−587.304			−2.19964
$268$		0			0
$269$		−	500.233i		−	1.85960i	−0.368064	−	0.929800i	$-0.619979\pi$
$269$		−	500.233i		−	1.85960i	0.368064	−	0.929800i	$-0.380021\pi$
$270$		0			0
$271$			13.7144i			0.0506067i	0.999680	+	0.0253033i	$0.00805516\pi$
$271$			13.7144i			0.0506067i	−0.999680	+	0.0253033i	$0.991945\pi$
$272$		0			0
$273$	−117.799			−0.431498
$274$		0			0
$275$		−	177.684i		−	0.646122i
$276$		0			0
$277$		−	174.235i		−	0.629008i	−0.949256	−	0.314504i	$-0.898162\pi$
$277$		−	174.235i		−	0.629008i	0.949256	−	0.314504i	$-0.101838\pi$
$278$		0			0
$279$	191.125	+	149.551i	0.685035	+	0.536025i
$280$		0			0
$281$	−181.333			−0.645313			−0.322657	−	0.946516i	$-0.604576\pi$
$281$	−181.333			−0.645313			−0.322657	+	0.946516i	$0.604576\pi$
$282$		0			0
$283$	−87.7889			−0.310208			−0.155104	−	0.987898i	$-0.549571\pi$
$283$	−87.7889			−0.310208			−0.155104	+	0.987898i	$0.549571\pi$
$284$		0			0
$285$		−	722.387i		−	2.53469i
$286$		0			0
$287$	73.5269			0.256191
$288$		0			0
$289$	112.054			0.387729
$290$		0			0
$291$		−	113.676i		−	0.390639i
$292$		0			0
$293$	350.593			1.19656			0.598281	−	0.801286i	$-0.295850\pi$
$293$	350.593			1.19656			0.598281	+	0.801286i	$0.295850\pi$
$294$		0			0
$295$	327.635			1.11063
$296$		0			0
$297$	−95.1960			−0.320525
$298$		0			0
$299$	124.274			0.415633
$300$		0			0
$301$			289.198i			0.960791i
$302$		0			0
$303$		−	335.339i		−	1.10673i
$304$		0			0
$305$			26.3127i			0.0862711i
$306$		0			0
$307$	−473.468			−1.54224			−0.771121	−	0.636689i	$-0.780303\pi$
$307$	−473.468			−1.54224			−0.771121	+	0.636689i	$0.780303\pi$
$308$		0			0
$309$			762.002i			2.46603i
$310$		0			0
$311$	−448.252			−1.44132			−0.720662	−	0.693286i	$-0.756162\pi$
$311$	−448.252			−1.44132			−0.720662	+	0.693286i	$0.756162\pi$
$312$		0			0
$313$		−	362.114i		−	1.15691i	−0.815713	−	0.578457i	$-0.803655\pi$
$313$		−	362.114i		−	1.15691i	0.815713	−	0.578457i	$-0.196345\pi$
$314$		0			0
$315$	201.409			0.639394
$316$		0			0
$317$	−226.470			−0.714417			−0.357208	−	0.934025i	$-0.616271\pi$
$317$	−226.470			−0.714417			−0.357208	+	0.934025i	$0.616271\pi$
$318$		0			0
$319$	128.853			0.403927
$320$		0			0
$321$			333.036i			1.03749i
$322$		0			0
$323$		−	401.899i		−	1.24427i
$324$		0			0
$325$		−	58.3561i		−	0.179557i
$326$		0			0
$327$			388.406i			1.18779i
$328$		0			0
$329$	299.019			0.908873
$330$		0			0
$331$			124.575i			0.376360i	0.982135	+	0.188180i	$0.0602588\pi$
$331$			124.575i			0.376360i	−0.982135	+	0.188180i	$0.939741\pi$
$332$		0			0
$333$		−	465.182i		−	1.39694i
$334$		0			0
$335$	122.711			0.366301
$336$		0			0
$337$		−	197.199i		−	0.585161i	−0.956241	−	0.292581i	$-0.905486\pi$
$337$		−	197.199i		−	0.585161i	0.956241	−	0.292581i	$-0.0945140\pi$
$338$		0			0
$339$		−	25.2173i		−	0.0743872i
$340$		0			0
$341$	−378.392	+	483.582i	−1.10965	+	1.41813i
$342$		0			0
$343$	−346.581			−1.01044
$344$		0			0
$345$	−456.759			−1.32394
$346$		0			0
$347$		−	381.026i		−	1.09806i	−0.835804	−	0.549029i	$-0.814998\pi$
$347$		−	381.026i		−	1.09806i	0.835804	−	0.549029i	$-0.185002\pi$
$348$		0			0
$349$	−212.936			−0.610132			−0.305066	−	0.952331i	$-0.598679\pi$
$349$	−212.936			−0.610132			−0.305066	+	0.952331i	$0.598679\pi$
$350$		0			0
$351$	−31.2649			−0.0890739
$352$		0			0
$353$		−	193.368i		−	0.547785i	−0.961760	−	0.273892i	$-0.911689\pi$
$353$		−	193.368i		−	0.547785i	0.961760	−	0.273892i	$-0.0883112\pi$
$354$		0			0
$355$	65.5269			0.184583
$356$		0			0
$357$	240.877			0.674726
$358$		0			0
$359$	−51.1939			−0.142601			−0.0713007	−	0.997455i	$-0.522715\pi$
$359$	−51.1939			−0.142601			−0.0713007	+	0.997455i	$0.522715\pi$
$360$		0			0
$361$	551.838			1.52864
$362$		0			0
$363$			1113.08i			3.06632i
$364$		0			0
$365$		−	146.856i		−	0.402346i
$366$		0			0
$367$		−	26.1212i		−	0.0711749i	−0.999367	−	0.0355874i	$-0.988670\pi$
$367$		−	26.1212i		−	0.0711749i	0.999367	−	0.0355874i	$-0.0113302\pi$
$368$		0			0
$369$	−130.397			−0.353379
$370$		0			0
$371$		−	213.658i		−	0.575899i
$372$		0			0
$373$	−570.539			−1.52960			−0.764798	−	0.644270i	$-0.777161\pi$
$373$	−570.539			−1.52960			−0.764798	+	0.644270i	$0.777161\pi$
$374$		0			0
$375$		−	383.258i		−	1.02202i
$376$		0			0
$377$	42.3188			0.112251
$378$		0			0
$379$	−67.2893			−0.177544			−0.0887722	−	0.996052i	$-0.528294\pi$
$379$	−67.2893			−0.177544			−0.0887722	+	0.996052i	$0.528294\pi$
$380$		0			0
$381$	−342.343			−0.898538
$382$		0			0
$383$			252.832i			0.660135i	0.943957	+	0.330067i	$0.107071\pi$
$383$			252.832i			0.660135i	−0.943957	+	0.330067i	$0.892929\pi$
$384$		0			0
$385$			509.603i			1.32364i
$386$		0			0
$387$		−	512.881i		−	1.32527i
$388$		0			0
$389$			726.077i			1.86652i	0.359199	+	0.933261i	$0.383050\pi$
$389$			726.077i			1.86652i	−0.359199	+	0.933261i	$0.616950\pi$
$390$		0			0
$391$	−254.118			−0.649918
$392$		0			0
$393$			811.620i			2.06519i
$394$		0			0
$395$			210.381i			0.532609i
$396$		0			0
$397$	250.696			0.631475			0.315737	−	0.948847i	$-0.397748\pi$
$397$	250.696			0.631475			0.315737	+	0.948847i	$0.397748\pi$
$398$		0			0
$399$			547.107i			1.37119i
$400$		0			0
$401$		−	602.618i		−	1.50279i	−0.659854	−	0.751394i	$-0.729382\pi$
$401$		−	602.618i		−	1.50279i	0.659854	−	0.751394i	$-0.270618\pi$
$402$		0			0
$403$	−124.274	+	158.821i	−0.308373	+	0.394098i
$404$		0			0
$405$	525.558			1.29768
$406$		0			0
$407$	1177.00			2.89189
$408$		0			0
$409$		−	152.678i		−	0.373297i	−0.982427	−	0.186648i	$-0.940237\pi$
$409$		−	152.678i		−	0.373297i	0.982427	−	0.186648i	$-0.0597625\pi$
$410$		0			0
$411$	−514.215			−1.25113
$412$		0			0
$413$	−248.137			−0.600816
$414$		0			0
$415$			928.374i			2.23705i
$416$		0			0
$417$	−257.706			−0.617999
$418$		0			0
$419$	275.963			0.658624			0.329312	−	0.944221i	$-0.393183\pi$
$419$	275.963			0.658624			0.329312	+	0.944221i	$0.393183\pi$
$420$		0			0
$421$	38.1270			0.0905629			0.0452815	−	0.998974i	$-0.485582\pi$
$421$	38.1270			0.0905629			0.0452815	+	0.998974i	$0.485582\pi$
$422$		0			0
$423$	−530.299			−1.25366
$424$		0			0
$425$			119.327i			0.280770i
$426$		0			0
$427$		−	19.9282i		−	0.0466702i
$428$		0			0
$429$			528.586i			1.23214i
$430$		0			0
$431$	406.828			0.943917			0.471959	−	0.881621i	$-0.343547\pi$
$431$	406.828			0.943917			0.471959	+	0.881621i	$0.343547\pi$
$432$		0			0
$433$		−	788.606i		−	1.82126i	−0.413222	−	0.910630i	$-0.635597\pi$
$433$		−	788.606i		−	1.82126i	0.413222	−	0.910630i	$-0.364403\pi$
$434$		0			0
$435$	−155.539			−0.357561
$436$		0			0
$437$		−	577.180i		−	1.32078i
$438$		0			0
$439$	175.434			0.399621			0.199810	−	0.979835i	$-0.435967\pi$
$439$	175.434			0.399621			0.199810	+	0.979835i	$0.435967\pi$
$440$		0			0
$441$	231.054			0.523932
$442$		0			0
$443$	−395.115			−0.891907			−0.445953	−	0.895056i	$-0.647135\pi$
$443$	−395.115			−0.891907			−0.445953	+	0.895056i	$0.647135\pi$
$444$		0			0
$445$			834.435i			1.87513i
$446$		0			0
$447$		−	418.529i		−	0.936307i
$448$		0			0
$449$			293.854i			0.654463i	0.944944	+	0.327232i	$0.106116\pi$
$449$			293.854i			0.654463i	−0.944944	+	0.327232i	$0.893884\pi$
$450$		0			0
$451$		−	329.929i		−	0.731550i
$452$		0			0
$453$	−714.465			−1.57719
$454$		0			0
$455$			167.368i			0.367841i
$456$		0			0
$457$			220.014i			0.481430i	0.970596	+	0.240715i	$0.0773819\pi$
$457$			220.014i			0.481430i	−0.970596	+	0.240715i	$0.922618\pi$
$458$		0			0
$459$	63.9310			0.139283
$460$		0			0
$461$			319.492i			0.693042i	0.938042	+	0.346521i	$0.112637\pi$
$461$			319.492i			0.693042i	−0.938042	+	0.346521i	$0.887363\pi$
$462$		0			0
$463$			261.136i			0.564009i	0.959413	+	0.282004i	$0.0909993\pi$
$463$			261.136i			0.564009i	−0.959413	+	0.282004i	$0.909001\pi$
$464$		0			0
$465$	456.759	−	583.735i	0.982278	−	1.25534i
$466$		0			0
$467$	271.856			0.582132			0.291066	−	0.956703i	$-0.405990\pi$
$467$	271.856			0.582132			0.291066	+	0.956703i	$0.405990\pi$
$468$		0			0
$469$	−92.9361			−0.198158
$470$		0			0
$471$		−	272.518i		−	0.578595i
$472$		0			0
$473$	1297.69			2.74352
$474$		0			0
$475$	−271.029			−0.570588
$476$		0			0
$477$			378.914i			0.794370i
$478$		0			0
$479$	192.335			0.401535			0.200767	−	0.979639i	$-0.435656\pi$
$479$	192.335			0.401535			0.200767	+	0.979639i	$0.435656\pi$
$480$		0			0
$481$	386.558			0.803656
$482$		0			0
$483$	345.931			0.716213
$484$		0			0
$485$	−161.510			−0.333010
$486$		0			0
$487$			653.665i			1.34223i	0.741354	+	0.671114i	$0.234184\pi$
$487$			653.665i			1.34223i	−0.741354	+	0.671114i	$0.765816\pi$
$488$		0			0
$489$		−	646.997i		−	1.32310i
$490$		0			0
$491$		−	76.7558i		−	0.156325i	−0.996941	−	0.0781627i	$-0.975095\pi$
$491$		−	76.7558i		−	0.156325i	0.996941	−	0.0781627i	$-0.0249054\pi$
$492$		0			0
$493$	−86.5341			−0.175525
$494$		0			0
$495$		−	903.760i		−	1.82578i
$496$		0			0
$497$	−49.6274			−0.0998540
$498$		0			0
$499$		−	668.304i		−	1.33929i	−0.742683	−	0.669643i	$-0.766447\pi$
$499$		−	668.304i		−	1.33929i	0.742683	−	0.669643i	$-0.233553\pi$
$500$		0			0
$501$	−226.441			−0.451977
$502$		0			0
$503$	−410.899			−0.816898			−0.408449	−	0.912781i	$-0.633930\pi$
$503$	−410.899			−0.816898			−0.408449	+	0.912781i	$0.633930\pi$
$504$		0			0
$505$	−476.446			−0.943457
$506$		0			0
$507$		−	519.678i		−	1.02501i
$508$		0			0
$509$			232.833i			0.457432i	0.973493	+	0.228716i	$0.0734527\pi$
$509$			232.833i			0.457432i	−0.973493	+	0.228716i	$0.926547\pi$
$510$		0			0
$511$			111.223i			0.217657i
$512$		0			0
$513$			145.207i			0.283055i
$514$		0			0
$515$	1082.64			2.10222
$516$		0			0
$517$		−	1341.76i		−	2.59527i
$518$		0			0
$519$			27.5289i			0.0530421i
$520$		0			0
$521$	−472.701			−0.907295			−0.453647	−	0.891181i	$-0.649877\pi$
$521$	−472.701			−0.907295			−0.453647	+	0.891181i	$0.649877\pi$
$522$		0			0
$523$		−	352.814i		−	0.674596i	−0.941398	−	0.337298i	$-0.890487\pi$
$523$		−	352.814i		−	0.674596i	0.941398	−	0.337298i	$-0.109513\pi$
$524$		0			0
$525$		−	162.441i		−	0.309411i
$526$		0			0
$527$	254.118	−	324.761i	0.482197	−	0.616244i
$528$		0			0
$529$	164.054			0.310121
$530$		0			0
$531$	440.061			0.828740
$532$		0			0
$533$		−	108.358i		−	0.203298i
$534$		0			0
$535$	473.174			0.884437
$536$		0			0
$537$	259.192			0.482666
$538$		0			0
$539$			584.610i			1.08462i
$540$		0			0
$541$	−685.337			−1.26680			−0.633399	−	0.773826i	$-0.718341\pi$
$541$	−685.337			−1.26680			−0.633399	+	0.773826i	$0.718341\pi$
$542$		0			0
$543$	−8.37172			−0.0154175
$544$		0			0
$545$	551.843			1.01256
$546$		0			0
$547$	−839.947			−1.53555			−0.767776	−	0.640718i	$-0.778637\pi$
$547$	−839.947			−1.53555			−0.767776	+	0.640718i	$0.778637\pi$
$548$		0			0
$549$			35.3418i			0.0643748i
$550$		0			0
$551$		−	196.546i		−	0.356707i
$552$		0			0
$553$		−	159.334i		−	0.288126i
$554$		0			0
$555$	−1420.76			−2.55993
$556$		0			0
$557$			336.143i			0.603488i	0.953389	+	0.301744i	$0.0975688\pi$
$557$			336.143i			0.603488i	−0.953389	+	0.301744i	$0.902431\pi$
$558$		0			0
$559$	426.195			0.762424
$560$		0			0
$561$		−	1080.86i		−	1.92667i
$562$		0			0
$563$	455.444			0.808959			0.404479	−	0.914547i	$-0.367453\pi$
$563$	455.444			0.808959			0.404479	+	0.914547i	$0.367453\pi$
$564$		0			0
$565$	−35.8284			−0.0634131
$566$		0			0
$567$	−398.037			−0.702005
$568$		0			0
$569$		−	370.468i		−	0.651087i	−0.945527	−	0.325543i	$-0.894453\pi$
$569$		−	370.468i		−	0.651087i	0.945527	−	0.325543i	$-0.105547\pi$
$570$		0			0
$571$			558.126i			0.977454i	0.872437	+	0.488727i	$0.162539\pi$
$571$			558.126i			0.977454i	−0.872437	+	0.488727i	$0.837461\pi$
$572$		0			0
$573$		−	852.393i		−	1.48760i
$574$		0			0
$575$			171.370i			0.298034i
$576$		0			0
$577$	−49.6224			−0.0860006			−0.0430003	−	0.999075i	$-0.513692\pi$
$577$	−49.6224			−0.0860006			−0.0430003	+	0.999075i	$0.513692\pi$
$578$		0			0
$579$		−	198.536i		−	0.342895i
$580$		0			0
$581$		−	703.113i		−	1.21018i
$582$		0			0
$583$	−958.725			−1.64447
$584$		0			0
$585$		−	296.819i		−	0.507384i
$586$		0			0
$587$		−	608.882i		−	1.03728i	−0.854994	−	0.518639i	$-0.826439\pi$
$587$		−	608.882i		−	1.03728i	0.854994	−	0.518639i	$-0.173561\pi$
$588$		0			0
$589$	737.632	+	577.180i	1.25235	+	0.979932i
$590$		0			0
$591$	−474.989			−0.803704
$592$		0			0
$593$	719.093			1.21264			0.606318	−	0.795222i	$-0.292646\pi$
$593$	719.093			1.21264			0.606318	+	0.795222i	$0.292646\pi$
$594$		0			0
$595$		−	342.236i		−	0.575186i
$596$		0			0
$597$	1078.42			1.80640
$598$		0			0
$599$	199.090			0.332371			0.166186	−	0.986094i	$-0.446855\pi$
$599$	199.090			0.332371			0.166186	+	0.986094i	$0.446855\pi$
$600$		0			0
$601$			797.535i			1.32701i	0.748170	+	0.663507i	$0.230933\pi$
$601$			797.535i			1.32701i	−0.748170	+	0.663507i	$0.769067\pi$
$602$		0			0
$603$	164.818			0.273331
$604$		0			0
$605$	1581.44			2.61396
$606$		0			0
$607$	197.417			0.325234			0.162617	−	0.986689i	$-0.448006\pi$
$607$	197.417			0.325234			0.162617	+	0.986689i	$0.448006\pi$
$608$		0			0
$609$	117.799			0.193430
$610$		0			0
$611$		−	440.669i		−	0.721226i
$612$		0			0
$613$			766.958i			1.25115i	0.780162	+	0.625577i	$0.215137\pi$
$613$			766.958i			1.25115i	−0.780162	+	0.625577i	$0.784863\pi$
$614$		0			0
$615$			398.259i			0.647576i
$616$		0			0
$617$	314.975			0.510494			0.255247	−	0.966876i	$-0.417843\pi$
$617$	314.975			0.510494			0.255247	+	0.966876i	$0.417843\pi$
$618$		0			0
$619$			6.67607i			0.0107852i	0.999985	+	0.00539262i	$0.00171653\pi$
$619$			6.67607i			0.0107852i	−0.999985	+	0.00539262i	$0.998283\pi$
$620$		0			0
$621$	91.8133			0.147847
$622$		0			0
$623$		−	631.967i		−	1.01439i
$624$		0			0
$625$	−768.793			−1.23007
$626$		0			0
$627$	2454.97			3.91542
$628$		0			0
$629$	−790.441			−1.25666
$630$		0			0
$631$			1058.10i			1.67686i	0.545012	+	0.838428i	$0.316525\pi$
$631$			1058.10i			1.67686i	−0.545012	+	0.838428i	$0.683475\pi$
$632$		0			0
$633$		−	701.413i		−	1.10808i
$634$		0			0
$635$			486.397i			0.765980i
$636$		0			0
$637$			192.002i			0.301416i
$638$		0			0
$639$	88.0122			0.137734
$640$		0			0
$641$		−	796.861i		−	1.24315i	−0.783354	−	0.621576i	$-0.786493\pi$
$641$		−	796.861i		−	1.24315i	0.783354	−	0.621576i	$-0.213507\pi$
$642$		0			0
$643$		−	875.949i		−	1.36228i	−0.732151	−	0.681142i	$-0.761484\pi$
$643$		−	875.949i		−	1.36228i	0.732151	−	0.681142i	$-0.238516\pi$
$644$		0			0
$645$	−1566.44			−2.42860
$646$		0			0
$647$		−	702.288i		−	1.08545i	−0.839910	−	0.542726i	$-0.817392\pi$
$647$		−	702.288i		−	1.08545i	0.839910	−	0.542726i	$-0.182608\pi$
$648$		0			0
$649$			1113.44i			1.71562i
$650$		0			0
$651$	−345.931	+	442.097i	−0.531384	+	0.679105i
$652$		0			0
$653$	−139.653			−0.213863			−0.106932	−	0.994266i	$-0.534103\pi$
$653$	−139.653			−0.213863			−0.106932	+	0.994266i	$0.534103\pi$
$654$		0			0
$655$	1153.14			1.76052
$656$		0			0
$657$		−	197.249i		−	0.300227i
$658$		0			0
$659$	−1241.56			−1.88400			−0.942000	−	0.335614i	$-0.891056\pi$
$659$	−1241.56			−1.88400			−0.942000	+	0.335614i	$0.891056\pi$
$660$		0			0
$661$	−520.200			−0.786990			−0.393495	−	0.919327i	$-0.628734\pi$
$661$	−520.200			−0.786990			−0.393495	+	0.919327i	$0.628734\pi$
$662$		0			0
$663$		−	354.984i		−	0.535421i
$664$		0			0
$665$	777.323			1.16891
$666$		0			0
$667$	−124.274			−0.186318
$668$		0			0
$669$	−1435.70			−2.14603
$670$		0			0
$671$	−89.4214			−0.133266
$672$		0			0
$673$			295.895i			0.439665i	0.975538	+	0.219833i	$0.0705511\pi$
$673$			295.895i			0.439665i	−0.975538	+	0.219833i	$0.929449\pi$
$674$		0			0
$675$		−	43.1132i		−	0.0638715i
$676$		0			0
$677$		−	130.889i		−	0.193337i	−0.995317	−	0.0966683i	$-0.969181\pi$
$677$		−	130.889i		−	0.193337i	0.995317	−	0.0966683i	$-0.0308186\pi$
$678$		0			0
$679$	122.321			0.180149
$680$		0			0
$681$		−	781.418i		−	1.14746i
$682$		0			0
$683$	−731.066			−1.07037			−0.535187	−	0.844733i	$-0.679759\pi$
$683$	−731.066			−1.07037			−0.535187	+	0.844733i	$0.679759\pi$
$684$		0			0
$685$			730.591i			1.06656i
$686$		0			0
$687$	479.362			0.697762
$688$		0			0
$689$	−314.871			−0.456998
$690$		0			0
$691$	287.247			0.415697			0.207849	−	0.978161i	$-0.433354\pi$
$691$	287.247			0.415697			0.207849	+	0.978161i	$0.433354\pi$
$692$		0			0
$693$			684.471i			0.987693i
$694$		0			0
$695$			366.145i			0.526828i
$696$		0			0
$697$			221.571i			0.317893i
$698$		0			0
$699$			927.187i			1.32645i
$700$		0			0
$701$	379.382			0.541201			0.270600	−	0.962692i	$-0.412778\pi$
$701$	379.382			0.541201			0.270600	+	0.962692i	$0.412778\pi$
$702$		0			0
$703$		−	1795.33i		−	2.55382i
$704$		0			0
$705$			1619.64i			2.29736i
$706$		0			0
$707$	360.841			0.510383
$708$		0			0
$709$		−	427.358i		−	0.602762i	−0.953504	−	0.301381i	$-0.902552\pi$
$709$		−	427.358i		−	0.602762i	0.953504	−	0.301381i	$-0.0974476\pi$
$710$		0			0
$711$			282.572i			0.397429i
$712$		0			0
$713$	364.946	−	466.399i	0.511846	−	0.654135i
$714$		0			0
$715$	751.009			1.05036
$716$		0			0
$717$	189.486			0.264276
$718$		0			0
$719$			165.377i			0.230009i	0.993365	+	0.115005i	$0.0366883\pi$
$719$			165.377i			0.230009i	−0.993365	+	0.115005i	$0.963312\pi$
$720$		0			0
$721$	−819.950			−1.13724
$722$		0			0
$723$	565.401			0.782021
$724$		0			0
$725$			58.3561i			0.0804912i
$726$		0			0
$727$	376.217			0.517493			0.258746	−	0.965945i	$-0.416691\pi$
$727$	376.217			0.517493			0.258746	+	0.965945i	$0.416691\pi$
$728$		0			0
$729$	528.460			0.724911
$730$		0			0
$731$	−871.490			−1.19219
$732$		0			0
$733$	389.965			0.532012			0.266006	−	0.963971i	$-0.414296\pi$
$733$	389.965			0.532012			0.266006	+	0.963971i	$0.414296\pi$
$734$		0			0
$735$		−	705.686i		−	0.960118i
$736$		0			0
$737$			417.021i			0.565836i
$738$		0			0
$739$			1079.09i			1.46020i	0.683338	+	0.730102i	$0.260528\pi$
$739$			1079.09i			1.46020i	−0.683338	+	0.730102i	$0.739472\pi$
$740$		0			0
$741$	806.278			1.08809
$742$		0			0
$743$			1176.58i			1.58355i	0.610812	+	0.791776i	$0.290843\pi$
$743$			1176.58i			1.58355i	−0.610812	+	0.791776i	$0.709157\pi$
$744$		0			0
$745$	−594.642			−0.798177
$746$		0			0
$747$			1246.94i			1.66927i
$748$		0			0
$749$	−358.362			−0.478455
$750$		0			0
$751$	88.5421			0.117899			0.0589494	−	0.998261i	$-0.481225\pi$
$751$	88.5421			0.117899			0.0589494	+	0.998261i	$0.481225\pi$
$752$		0			0
$753$	108.642			0.144278
$754$		0			0
$755$			1015.10i			1.34451i
$756$		0			0
$757$			457.411i			0.604242i	0.953270	+	0.302121i	$0.0976946\pi$
$757$			457.411i			0.604242i	−0.953270	+	0.302121i	$0.902305\pi$
$758$		0			0
$759$		−	1552.26i		−	2.04513i
$760$		0			0
$761$		−	878.422i		−	1.15430i	−0.816638	−	0.577150i	$-0.804165\pi$
$761$		−	878.422i		−	1.15430i	0.816638	−	0.577150i	$-0.195835\pi$
$762$		0			0
$763$	−417.943			−0.547763
$764$		0			0
$765$			606.941i			0.793387i
$766$		0			0
$767$			365.683i			0.476771i
$768$		0			0
$769$	772.734			1.00486			0.502428	−	0.864619i	$-0.332440\pi$
$769$	772.734			1.00486			0.502428	+	0.864619i	$0.332440\pi$
$770$		0			0
$771$		−	73.0952i		−	0.0948057i
$772$		0			0
$773$		−	243.752i		−	0.315333i	−0.987492	−	0.157667i	$-0.949603\pi$
$773$		−	243.752i		−	0.315333i	0.987492	−	0.157667i	$-0.0503971\pi$
$774$		0			0
$775$	−219.009	−	171.370i	−0.282593	−	0.221122i
$776$		0			0
$777$	1076.03			1.38485
$778$		0			0
$779$	−503.257			−0.646029
$780$		0			0
$781$			222.687i			0.285131i
$782$		0			0
$783$	31.2649			0.0399297
$784$		0			0
$785$	−387.191			−0.493237
$786$		0			0
$787$		−	210.039i		−	0.266886i	−0.991056	−	0.133443i	$-0.957397\pi$
$787$		−	210.039i		−	0.266886i	0.991056	−	0.133443i	$-0.0426033\pi$
$788$		0			0
$789$	542.798			0.687957
$790$		0			0
$791$	27.1350			0.0343047
$792$		0			0
$793$	−29.3684			−0.0370346
$794$		0			0
$795$	1157.28			1.45570
$796$		0			0
$797$			451.447i			0.566433i	0.959056	+	0.283217i	$0.0914015\pi$
$797$			451.447i			0.566433i	−0.959056	+	0.283217i	$0.908598\pi$
$798$		0			0
$799$			901.087i			1.12777i
$800$		0			0
$801$			1120.77i			1.39921i
$802$		0			0
$803$	499.078			0.621517
$804$		0			0
$805$		−	491.495i		−	0.610553i
$806$		0			0
$807$	−2052.08			−2.54285
$808$		0			0
$809$			439.303i			0.543019i	0.962436	+	0.271510i	$0.0875229\pi$
$809$			439.303i			0.543019i	−0.962436	+	0.271510i	$0.912477\pi$
$810$		0			0
$811$	1336.32			1.64774			0.823870	−	0.566779i	$-0.191811\pi$
$811$	1336.32			1.64774			0.823870	+	0.566779i	$0.191811\pi$
$812$		0			0
$813$	56.2599			0.0692004
$814$		0			0
$815$	−919.247			−1.12791
$816$		0			0
$817$		−	1979.42i		−	2.42280i
$818$		0			0
$819$			224.799i			0.274480i
$820$		0			0
$821$			437.079i			0.532374i	0.963921	+	0.266187i	$0.0857639\pi$
$821$			437.079i			0.532374i	−0.963921	+	0.266187i	$0.914236\pi$
$822$		0			0
$823$		−	405.739i		−	0.493001i	−0.969143	−	0.246500i	$-0.920719\pi$
$823$		−	405.739i		−	0.493001i	0.969143	−	0.246500i	$-0.0792806\pi$
$824$		0			0
$825$	−728.902			−0.883517
$826$		0			0
$827$		−	729.979i		−	0.882683i	−0.897339	−	0.441342i	$-0.854503\pi$
$827$		−	729.979i		−	0.882683i	0.897339	−	0.441342i	$-0.145497\pi$
$828$		0			0
$829$		−	847.054i		−	1.02178i	−0.859647	−	0.510889i	$-0.829316\pi$
$829$		−	847.054i		−	1.02178i	0.859647	−	0.510889i	$-0.170684\pi$
$830$		0			0
$831$	−714.755			−0.860115
$832$		0			0
$833$		−	392.608i		−	0.471318i
$834$		0			0
$835$			321.724i			0.385299i
$836$		0			0
$837$	−91.8133	+	117.337i	−0.109693	+	0.140187i
$838$		0			0
$839$	−407.720			−0.485959			−0.242980	−	0.970031i	$-0.578125\pi$
$839$	−407.720			−0.485959			−0.242980	+	0.970031i	$0.578125\pi$
$840$		0			0
$841$	798.681			0.949680
$842$		0			0
$843$			743.873i			0.882411i
$844$		0			0
$845$	−738.352			−0.873790
$846$		0			0
$847$	−1197.72			−1.41408
$848$		0			0
$849$			360.132i			0.424183i
$850$		0			0
$851$	−1135.18			−1.33393
$852$		0			0
$853$	−1231.52			−1.44376			−0.721878	−	0.692021i	$-0.756721\pi$
$853$	−1231.52			−1.44376			−0.721878	+	0.692021i	$0.756721\pi$
$854$		0			0
$855$	−1378.55			−1.61234
$856$		0			0
$857$	563.750			0.657818			0.328909	−	0.944362i	$-0.393319\pi$
$857$	563.750			0.657818			0.328909	+	0.944362i	$0.393319\pi$
$858$		0			0
$859$		−	1092.81i		−	1.27218i	−0.771614	−	0.636092i	$-0.780550\pi$
$859$		−	1092.81i		−	1.27218i	0.771614	−	0.636092i	$-0.219450\pi$
$860$		0			0
$861$		−	301.626i		−	0.350320i
$862$		0			0
$863$			512.739i			0.594136i	0.954856	+	0.297068i	$0.0960088\pi$
$863$			512.739i			0.594136i	−0.954856	+	0.297068i	$0.903991\pi$
$864$		0			0
$865$	39.1127			0.0452170
$866$		0			0
$867$		−	459.672i		−	0.530187i
$868$		0			0
$869$	−714.960			−0.822739
$870$		0			0
$871$			136.961i			0.157246i
$872$		0			0
$873$	−216.931			−0.248489
$874$		0			0
$875$	412.404			0.471319
$876$		0			0
$877$	1361.26			1.55218			0.776088	−	0.630625i	$-0.217201\pi$
$877$	1361.26			1.55218			0.776088	+	0.630625i	$0.217201\pi$
$878$		0			0
$879$		−	1438.22i		−	1.63620i
$880$		0			0
$881$		−	980.799i		−	1.11328i	−0.830754	−	0.556640i	$-0.812090\pi$
$881$		−	980.799i		−	1.11328i	0.830754	−	0.556640i	$-0.187910\pi$
$882$		0			0
$883$			879.489i			0.996023i	0.867170	+	0.498012i	$0.165936\pi$
$883$			879.489i			0.996023i	−0.867170	+	0.498012i	$0.834064\pi$
$884$		0			0
$885$		−	1344.04i		−	1.51869i
$886$		0			0
$887$	798.079			0.899751			0.449876	−	0.893091i	$-0.351468\pi$
$887$	798.079			0.899751			0.449876	+	0.893091i	$0.351468\pi$
$888$		0			0
$889$		−	368.378i		−	0.414373i
$890$		0			0
$891$			1786.06i			2.00456i
$892$		0			0
$893$	−2046.65			−2.29188
$894$		0			0
$895$		−	368.257i		−	0.411460i
$896$		0			0
$897$		−	509.803i		−	0.568343i
$898$		0			0
$899$	124.274	−	158.821i	0.138236	−	0.176665i
$900$		0			0
$901$	643.854			0.714599
$902$		0			0
$903$	1186.36			1.31380
$904$		0			0
$905$			11.8944i			0.0131430i
$906$		0			0
$907$	−586.301			−0.646417			−0.323209	−	0.946328i	$-0.604762\pi$
$907$	−586.301			−0.646417			−0.323209	+	0.946328i	$0.604762\pi$
$908$		0			0
$909$	−639.936			−0.704000
$910$		0			0
$911$		−	1086.27i		−	1.19239i	−0.802838	−	0.596197i	$-0.796678\pi$
$911$		−	1086.27i		−	1.19239i	0.802838	−	0.596197i	$-0.203322\pi$
$912$		0			0
$913$	−3155.00			−3.45564
$914$		0			0
$915$	107.941			0.117968
$916$		0			0
$917$	−873.342			−0.952391
$918$		0			0
$919$	−362.396			−0.394337			−0.197169	−	0.980370i	$-0.563175\pi$
$919$	−362.396			−0.394337			−0.197169	+	0.980370i	$0.563175\pi$
$920$		0			0
$921$			1942.28i			2.10888i
$922$		0			0
$923$			73.1366i			0.0792379i
$924$		0			0
$925$			533.051i			0.576271i
$926$		0			0
$927$	1454.15			1.56866
$928$		0			0
$929$		−	66.6020i		−	0.0716922i	−0.999357	−	0.0358461i	$-0.988587\pi$
$929$		−	66.6020i		−	0.0716922i	0.999357	−	0.0358461i	$-0.0114126\pi$
$930$		0			0
$931$	891.734			0.957824
$932$		0			0
$933$			1838.84i			1.97089i
$934$		0			0
$935$	−1535.68			−1.64243
$936$		0			0
$937$	803.024			0.857015			0.428508	−	0.903538i	$-0.359039\pi$
$937$	803.024			0.857015			0.428508	+	0.903538i	$0.359039\pi$
$938$		0			0
$939$	−1485.48			−1.58198
$940$		0			0
$941$			1229.02i			1.30608i	0.757322	+	0.653042i	$0.226507\pi$
$941$			1229.02i			1.30608i	−0.757322	+	0.653042i	$0.773493\pi$
$942$		0			0
$943$			318.205i			0.337439i
$944$		0			0
$945$			123.650i			0.130847i
$946$		0			0
$947$		−	755.838i		−	0.798139i	−0.916921	−	0.399070i	$-0.869333\pi$
$947$		−	755.838i		−	0.798139i	0.916921	−	0.399070i	$-0.130667\pi$
$948$		0			0
$949$	163.911			0.172720
$950$		0			0
$951$			929.036i			0.976905i
$952$		0			0
$953$			831.719i			0.872738i	0.899768	+	0.436369i	$0.143736\pi$
$953$			831.719i			0.872738i	−0.899768	+	0.436369i	$0.856264\pi$
$954$		0			0
$955$	−1211.07			−1.26814
$956$		0			0
$957$		−	528.586i		−	0.552336i
$958$		0			0
$959$		−	553.320i		−	0.576976i
$960$		0			0
$961$	231.108	+	932.797i	0.240487	+	0.970652i
$962$		0			0
$963$	635.541			0.659960
$964$		0			0
$965$	−282.078			−0.292309
$966$		0			0
$967$			1276.50i			1.32006i	0.751237	+	0.660032i	$0.229457\pi$
$967$			1276.50i			1.32006i	−0.751237	+	0.660032i	$0.770543\pi$
$968$		0			0
$969$	−1648.69			−1.70143
$970$		0			0
$971$	1899.15			1.95587			0.977936	−	0.208906i	$-0.0669901\pi$
$971$	1899.15			1.95587			0.977936	+	0.208906i	$0.0669901\pi$
$972$		0			0
$973$		−	277.304i		−	0.284999i
$974$		0			0
$975$	−239.391			−0.245529
$976$		0			0
$977$	902.996			0.924254			0.462127	−	0.886814i	$-0.347086\pi$
$977$	902.996			0.924254			0.462127	+	0.886814i	$0.347086\pi$
$978$		0			0
$979$	−2835.75			−2.89658
$980$		0			0
$981$	741.205			0.755561
$982$		0			0
$983$			1559.25i			1.58622i	0.609079	+	0.793110i	$0.291539\pi$
$983$			1559.25i			1.58622i	−0.609079	+	0.793110i	$0.708461\pi$
$984$		0			0
$985$			674.859i			0.685136i
$986$		0			0
$987$		−	1226.65i		−	1.24281i
$988$		0			0
$989$	−1251.57			−1.26549
$990$		0			0
$991$		−	1385.77i		−	1.39836i	−0.714947	−	0.699179i	$-0.753549\pi$
$991$		−	1385.77i		−	1.39836i	0.714947	−	0.699179i	$-0.246451\pi$
$992$		0			0
$993$	511.038			0.514640
$994$		0			0
$995$		−	1532.21i		−	1.53991i
$996$		0			0
$997$	1199.12			1.20273			0.601363	−	0.798976i	$-0.294625\pi$
$997$	1199.12			1.20273			0.601363	+	0.798976i	$0.294625\pi$
$998$		0			0
$999$	285.588			0.285874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	496.3.e.d.433.1		4
4.3	odd	2		62.3.b.a.61.2	yes	4
12.11	even	2		558.3.d.a.433.3		4
20.3	even	4		1550.3.d.a.1549.5		8
20.7	even	4		1550.3.d.a.1549.4		8
20.19	odd	2		1550.3.c.a.1301.3		4
31.30	odd	2	inner	496.3.e.d.433.4		4
124.123	even	2		62.3.b.a.61.1	✓	4
372.371	odd	2		558.3.d.a.433.4		4
620.123	odd	4		1550.3.d.a.1549.8		8
620.247	odd	4		1550.3.d.a.1549.1		8
620.619	even	2		1550.3.c.a.1301.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.3.b.a.61.1	✓	4	124.123	even	2
62.3.b.a.61.2	yes	4	4.3	odd	2
496.3.e.d.433.1		4	1.1	even	1	trivial
496.3.e.d.433.4		4	31.30	odd	2	inner
558.3.d.a.433.3		4	12.11	even	2
558.3.d.a.433.4		4	372.371	odd	2
1550.3.c.a.1301.3		4	20.19	odd	2
1550.3.c.a.1301.4		4	620.619	even	2
1550.3.d.a.1549.1		8	620.247	odd	4
1550.3.d.a.1549.4		8	20.7	even	4
1550.3.d.a.1549.5		8	20.3	even	4
1550.3.d.a.1549.8		8	620.123	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 496 = 2^{4} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	496.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.10225i q^{3} -5.82843 q^{5} +4.41421 q^{7} -7.82843 q^{9} -19.8074i q^{11} -6.50529i q^{13} +23.9096i q^{15} +13.3021i q^{17} -30.2132 q^{19} -18.1082i q^{21} +19.1036i q^{23} +8.97056 q^{25} -4.80608i q^{27} +6.50529i q^{29} +(-24.4142 - 19.1036i) q^{31} -81.2548 q^{33} -25.7279 q^{35} +59.4222i q^{37} -26.6863 q^{39} +16.6569 q^{41} +65.5152i q^{43} +45.6274 q^{45} +67.7401 q^{47} -29.5147 q^{49} +54.5685 q^{51} -48.4024i q^{53} +115.446i q^{55} +123.942i q^{57} -56.2132 q^{59} -4.51454i q^{61} -34.5563 q^{63} +37.9156i q^{65} -21.0538 q^{67} +78.3675 q^{69} -11.2426 q^{71} +25.1966i q^{73} -36.7995i q^{75} -87.4341i q^{77} -36.0956i q^{79} -90.1716 q^{81} -159.284i q^{83} -77.5304i q^{85} +26.6863 q^{87} -143.166i q^{89} -28.7157i q^{91} +(-78.3675 + 100.153i) q^{93} +176.095 q^{95} +27.7107 q^{97} +155.061i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{5} + 12 q^{7} - 20 q^{9} - 36 q^{19} - 32 q^{25} - 92 q^{31} - 144 q^{33} - 52 q^{35} - 152 q^{39} + 44 q^{41} + 92 q^{45} + 56 q^{47} - 152 q^{49} - 8 q^{51} - 140 q^{59} - 76 q^{63} + 176 q^{67}+ \cdots - 172 q^{97}+O(q^{100}) \) 4 * q - 12 * q^5 + 12 * q^7 - 20 * q^9 - 36 * q^19 - 32 * q^25 - 92 * q^31 - 144 * q^33 - 52 * q^35 - 152 * q^39 + 44 * q^41 + 92 * q^45 + 56 * q^47 - 152 * q^49 - 8 * q^51 - 140 * q^59 - 76 * q^63 + 176 * q^67 + 8 * q^69 - 28 * q^71 - 372 * q^81 + 152 * q^87 - 8 * q^93 + 348 * q^95 - 172 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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