LMFDB - Embedded newform 1344.3.f.e.769.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,3,Mod(769,1344)] 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("1344.769"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1344.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-8,0,-12,0,-24,0,0,0,24,0,0,0,0,0,-12,0,-24,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$36.6213475300$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.4
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1344.769
Dual form			1344.3.f.e.769.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -1.01461i q^{5} +(2.24264 + 6.63103i) q^{7} -3.00000 q^{9} -10.2426 q^{11} +8.95743i q^{13} +1.75736 q^{15} +30.4085i q^{17} -16.1318i q^{19} +(-11.4853 + 3.88437i) q^{21} +6.72792 q^{23} +23.9706 q^{25} -5.19615i q^{27} -30.0000 q^{29} -50.1785i q^{31} -17.7408i q^{33} +(6.72792 - 2.27541i) q^{35} -30.9117 q^{37} -15.5147 q^{39} -7.10228i q^{41} -74.4264 q^{43} +3.04384i q^{45} +58.2954i q^{47} +(-38.9411 + 29.7420i) q^{49} -52.6690 q^{51} +70.9706 q^{53} +10.3923i q^{55} +27.9411 q^{57} -0.492372i q^{59} -2.86976i q^{61} +(-6.72792 - 19.8931i) q^{63} +9.08831 q^{65} +27.0294 q^{67} +11.6531i q^{69} -50.6102 q^{71} -70.6149i q^{73} +41.5182i q^{75} +(-22.9706 - 67.9193i) q^{77} -133.823 q^{79} +9.00000 q^{81} -104.415i q^{83} +30.8528 q^{85} -51.9615i q^{87} +144.970i q^{89} +(-59.3970 + 20.0883i) q^{91} +86.9117 q^{93} -16.3675 q^{95} -100.705i q^{97} +30.7279 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7} - 12 q^{9} - 24 q^{11} + 24 q^{15} - 12 q^{21} - 24 q^{23} + 28 q^{25} - 120 q^{29} - 24 q^{35} + 80 q^{37} - 96 q^{39} - 128 q^{43} - 20 q^{49} - 24 q^{51} + 216 q^{53} - 24 q^{57} + 24 q^{63}+ \cdots + 72 q^{99}+O(q^{100}) $ 4 * q - 8 * q^7 - 12 * q^9 - 24 * q^11 + 24 * q^15 - 12 * q^21 - 24 * q^23 + 28 * q^25 - 120 * q^29 - 24 * q^35 + 80 * q^37 - 96 * q^39 - 128 * q^43 - 20 * q^49 - 24 * q^51 + 216 * q^53 - 24 * q^57 + 24 * q^63 + 240 * q^65 + 176 * q^67 + 120 * q^71 - 24 * q^77 - 128 * q^79 + 36 * q^81 - 216 * q^85 + 144 * q^93 + 240 * q^95 + 72 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$			1.73205i			0.577350i
$3$			1.73205i			0.577350i
$4$		0			0
$5$		−	1.01461i		−	0.202922i	−0.994839	−	0.101461i	$-0.967648\pi$
$5$		−	1.01461i		−	0.202922i	0.994839	−	0.101461i	$-0.0323518\pi$
$6$		0			0
$7$	2.24264	+	6.63103i	0.320377	+	0.947290i
$7$	2.24264	+	6.63103i	0.320377	+	0.947290i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	−10.2426			−0.931149			−0.465575	−	0.885009i	$-0.654152\pi$
$11$	−10.2426			−0.931149			−0.465575	+	0.885009i	$0.654152\pi$
$12$		0			0
$13$			8.95743i			0.689033i	0.938780	+	0.344516i	$0.111957\pi$
$13$			8.95743i			0.689033i	−0.938780	+	0.344516i	$0.888043\pi$
$14$		0			0
$15$	1.75736			0.117157
$16$		0			0
$17$			30.4085i			1.78873i	0.447333	+	0.894367i	$0.352374\pi$
$17$			30.4085i			1.78873i	−0.447333	+	0.894367i	$0.647626\pi$
$18$		0			0
$19$		−	16.1318i		−	0.849043i	−0.905418	−	0.424521i	$-0.860442\pi$
$19$		−	16.1318i		−	0.849043i	0.905418	−	0.424521i	$-0.139558\pi$
$20$		0			0
$21$	−11.4853	+	3.88437i	−0.546918	+	0.184970i
$22$		0			0
$23$	6.72792			0.292518			0.146259	−	0.989246i	$-0.453277\pi$
$23$	6.72792			0.292518			0.146259	+	0.989246i	$0.453277\pi$
$24$		0			0
$25$	23.9706			0.958823
$26$		0			0
$27$		−	5.19615i		−	0.192450i
$28$		0			0
$29$	−30.0000			−1.03448			−0.517241	−	0.855840i	$-0.673041\pi$
$29$	−30.0000			−1.03448			−0.517241	+	0.855840i	$0.673041\pi$
$30$		0			0
$31$		−	50.1785i		−	1.61866i	−0.587354	−	0.809330i	$-0.699830\pi$
$31$		−	50.1785i		−	1.61866i	0.587354	−	0.809330i	$-0.300170\pi$
$32$		0			0
$33$		−	17.7408i		−	0.537599i
$34$		0			0
$35$	6.72792	−	2.27541i	0.192226	−	0.0650117i
$36$		0			0
$37$	−30.9117			−0.835451			−0.417726	−	0.908573i	$-0.637173\pi$
$37$	−30.9117			−0.835451			−0.417726	+	0.908573i	$0.637173\pi$
$38$		0			0
$39$	−15.5147			−0.397813
$40$		0			0
$41$		−	7.10228i		−	0.173226i	−0.996242	−	0.0866132i	$-0.972396\pi$
$41$		−	7.10228i		−	0.173226i	0.996242	−	0.0866132i	$-0.0276044\pi$
$42$		0			0
$43$	−74.4264			−1.73085			−0.865423	−	0.501041i	$-0.832950\pi$
$43$	−74.4264			−1.73085			−0.865423	+	0.501041i	$0.832950\pi$
$44$		0			0
$45$			3.04384i			0.0676408i
$46$		0			0
$47$			58.2954i			1.24033i	0.784472	+	0.620164i	$0.212934\pi$
$47$			58.2954i			1.24033i	−0.784472	+	0.620164i	$0.787066\pi$
$48$		0			0
$49$	−38.9411	+	29.7420i	−0.794717	+	0.606980i
$50$		0			0
$51$	−52.6690			−1.03273
$52$		0			0
$53$	70.9706			1.33907			0.669534	−	0.742782i	$-0.266494\pi$
$53$	70.9706			1.33907			0.669534	+	0.742782i	$0.266494\pi$
$54$		0			0
$55$			10.3923i			0.188951i
$56$		0			0
$57$	27.9411			0.490195
$58$		0			0
$59$		−	0.492372i		−	0.00834529i	−0.999991	−	0.00417265i	$-0.998672\pi$
$59$		−	0.492372i		−	0.00834529i	0.999991	−	0.00417265i	$-0.00132820\pi$
$60$		0			0
$61$		−	2.86976i		−	0.0470452i	−0.999723	−	0.0235226i	$-0.992512\pi$
$61$		−	2.86976i		−	0.0470452i	0.999723	−	0.0235226i	$-0.00748816\pi$
$62$		0			0
$63$	−6.72792	−	19.8931i	−0.106792	−	0.315763i
$64$		0			0
$65$	9.08831			0.139820
$66$		0			0
$67$	27.0294			0.403424			0.201712	−	0.979445i	$-0.435349\pi$
$67$	27.0294			0.403424			0.201712	+	0.979445i	$0.435349\pi$
$68$		0			0
$69$			11.6531i			0.168886i
$70$		0			0
$71$	−50.6102			−0.712819			−0.356410	−	0.934330i	$-0.615999\pi$
$71$	−50.6102			−0.712819			−0.356410	+	0.934330i	$0.615999\pi$
$72$		0			0
$73$		−	70.6149i		−	0.967328i	−0.875254	−	0.483664i	$-0.839306\pi$
$73$		−	70.6149i		−	0.967328i	0.875254	−	0.483664i	$-0.160694\pi$
$74$		0			0
$75$			41.5182i			0.553576i
$76$		0			0
$77$	−22.9706	−	67.9193i	−0.298319	−	0.882068i
$78$		0			0
$79$	−133.823			−1.69397			−0.846983	−	0.531619i	$-0.821584\pi$
$79$	−133.823			−1.69397			−0.846983	+	0.531619i	$0.821584\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$		−	104.415i		−	1.25802i	−0.777398	−	0.629009i	$-0.783461\pi$
$83$		−	104.415i		−	1.25802i	0.777398	−	0.629009i	$-0.216539\pi$
$84$		0			0
$85$	30.8528			0.362974
$86$		0			0
$87$		−	51.9615i		−	0.597259i
$88$		0			0
$89$			144.970i			1.62888i	0.580250	+	0.814438i	$0.302955\pi$
$89$			144.970i			1.62888i	−0.580250	+	0.814438i	$0.697045\pi$
$90$		0			0
$91$	−59.3970	+	20.0883i	−0.652714	+	0.220750i
$92$		0			0
$93$	86.9117			0.934534
$94$		0			0
$95$	−16.3675			−0.172290
$96$		0			0
$97$		−	100.705i		−	1.03820i	−0.854714	−	0.519099i	$-0.826268\pi$
$97$		−	100.705i		−	1.03820i	0.854714	−	0.519099i	$-0.173732\pi$
$98$		0			0
$99$	30.7279			0.310383
$100$		0			0
$101$			138.882i			1.37507i	0.726150	+	0.687536i	$0.241308\pi$
$101$			138.882i			1.37507i	−0.726150	+	0.687536i	$0.758692\pi$
$102$		0			0
$103$		−	78.8760i		−	0.765787i	−0.923793	−	0.382893i	$-0.874928\pi$
$103$		−	78.8760i		−	0.765787i	0.923793	−	0.382893i	$-0.125072\pi$
$104$		0			0
$105$	3.94113	+	11.6531i	0.0375345	+	0.110982i
$106$		0			0
$107$	−37.7574			−0.352873			−0.176436	−	0.984312i	$-0.556457\pi$
$107$	−37.7574			−0.352873			−0.176436	+	0.984312i	$0.556457\pi$
$108$		0			0
$109$	−31.9411			−0.293038			−0.146519	−	0.989208i	$-0.546807\pi$
$109$	−31.9411			−0.293038			−0.146519	+	0.989208i	$0.546807\pi$
$110$		0			0
$111$		−	53.5406i		−	0.482348i
$112$		0			0
$113$	−106.971			−0.946642			−0.473321	−	0.880890i	$-0.656945\pi$
$113$	−106.971			−0.946642			−0.473321	+	0.880890i	$0.656945\pi$
$114$		0			0
$115$		−	6.82623i		−	0.0593585i
$116$		0			0
$117$		−	26.8723i		−	0.229678i
$118$		0			0
$119$	−201.640	+	68.1953i	−1.69445	+	0.573070i
$120$		0			0
$121$	−16.0883			−0.132961
$122$		0			0
$123$	12.3015			0.100012
$124$		0			0
$125$		−	49.6861i		−	0.397489i
$126$		0			0
$127$	−22.0589			−0.173692			−0.0868460	−	0.996222i	$-0.527679\pi$
$127$	−22.0589			−0.173692			−0.0868460	+	0.996222i	$0.527679\pi$
$128$		0			0
$129$		−	128.910i		−	0.999305i
$130$		0			0
$131$		−	70.9631i		−	0.541703i	−0.962621	−	0.270852i	$-0.912695\pi$
$131$		−	70.9631i		−	0.541703i	0.962621	−	0.270852i	$-0.0873052\pi$
$132$		0			0
$133$	106.971	−	36.1779i	0.804290	−	0.272014i
$134$		0			0
$135$	−5.27208			−0.0390524
$136$		0			0
$137$	−105.765			−0.772004			−0.386002	−	0.922498i	$-0.626144\pi$
$137$	−105.765			−0.772004			−0.386002	+	0.922498i	$0.626144\pi$
$138$		0			0
$139$			181.322i			1.30447i	0.758015	+	0.652237i	$0.226169\pi$
$139$			181.322i			1.30447i	−0.758015	+	0.652237i	$0.773831\pi$
$140$		0			0
$141$	−100.971			−0.716103
$142$		0			0
$143$		−	91.7477i		−	0.641592i
$144$		0			0
$145$			30.4384i			0.209920i
$146$		0			0
$147$	−51.5147	−	67.4480i	−0.350440	−	0.458830i
$148$		0			0
$149$	−41.1472			−0.276156			−0.138078	−	0.990421i	$-0.544092\pi$
$149$	−41.1472			−0.276156			−0.138078	+	0.990421i	$0.544092\pi$
$150$		0			0
$151$	−80.3675			−0.532235			−0.266118	−	0.963941i	$-0.585741\pi$
$151$	−80.3675			−0.532235			−0.266118	+	0.963941i	$0.585741\pi$
$152$		0			0
$153$		−	91.2255i		−	0.596245i
$154$		0			0
$155$	−50.9117			−0.328463
$156$		0			0
$157$			57.3106i			0.365036i	0.983203	+	0.182518i	$0.0584248\pi$
$157$			57.3106i			0.365036i	−0.983203	+	0.182518i	$0.941575\pi$
$158$		0			0
$159$			122.925i			0.773111i
$160$		0			0
$161$	15.0883	+	44.6131i	0.0937162	+	0.277100i
$162$		0			0
$163$	−174.912			−1.07308			−0.536539	−	0.843876i	$-0.680269\pi$
$163$	−174.912			−1.07308			−0.536539	+	0.843876i	$0.680269\pi$
$164$		0			0
$165$	−18.0000			−0.109091
$166$		0			0
$167$		−	196.163i		−	1.17463i	−0.809359	−	0.587315i	$-0.800185\pi$
$167$		−	196.163i		−	1.17463i	0.809359	−	0.587315i	$-0.199815\pi$
$168$		0			0
$169$	88.7645			0.525234
$170$		0			0
$171$			48.3954i			0.283014i
$172$		0			0
$173$			38.5254i			0.222690i	0.993782	+	0.111345i	$0.0355159\pi$
$173$			38.5254i			0.222690i	−0.993782	+	0.111345i	$0.964484\pi$
$174$		0			0
$175$	53.7574	+	158.950i	0.307185	+	0.908283i
$176$		0			0
$177$	0.852814			0.00481816
$178$		0			0
$179$	−191.095			−1.06757			−0.533786	−	0.845619i	$-0.679231\pi$
$179$	−191.095			−1.06757			−0.533786	+	0.845619i	$0.679231\pi$
$180$		0			0
$181$		−	120.793i		−	0.667367i	−0.942685	−	0.333683i	$-0.891708\pi$
$181$		−	120.793i		−	0.667367i	0.942685	−	0.333683i	$-0.108292\pi$
$182$		0			0
$183$	4.97056			0.0271615
$184$		0			0
$185$			31.3634i			0.169532i
$186$		0			0
$187$		−	311.463i		−	1.66558i
$188$		0			0
$189$	34.4558	−	11.6531i	0.182306	−	0.0616566i
$190$		0			0
$191$	−112.066			−0.586733			−0.293367	−	0.956000i	$-0.594776\pi$
$191$	−112.066			−0.586733			−0.293367	+	0.956000i	$0.594776\pi$
$192$		0			0
$193$	−22.9117			−0.118713			−0.0593567	−	0.998237i	$-0.518905\pi$
$193$	−22.9117			−0.118713			−0.0593567	+	0.998237i	$0.518905\pi$
$194$		0			0
$195$			15.7414i			0.0807252i
$196$		0			0
$197$	116.059			0.589131			0.294566	−	0.955631i	$-0.404825\pi$
$197$	116.059			0.589131			0.294566	+	0.955631i	$0.404825\pi$
$198$		0			0
$199$		−	163.101i		−	0.819604i	−0.912174	−	0.409802i	$-0.865598\pi$
$199$		−	163.101i		−	0.819604i	0.912174	−	0.409802i	$-0.134402\pi$
$200$		0			0
$201$			46.8164i			0.232917i
$202$		0			0
$203$	−67.2792	−	198.931i	−0.331425	−	0.979955i
$204$		0			0
$205$	−7.20606			−0.0351515
$206$		0			0
$207$	−20.1838			−0.0975061
$208$		0			0
$209$			165.232i			0.790586i
$210$		0			0
$211$	106.426			0.504391			0.252195	−	0.967676i	$-0.418847\pi$
$211$	106.426			0.504391			0.252195	+	0.967676i	$0.418847\pi$
$212$		0			0
$213$		−	87.6594i		−	0.411546i
$214$		0			0
$215$			75.5139i			0.351228i
$216$		0			0
$217$	332.735	−	112.532i	1.53334	−	0.518582i
$218$		0			0
$219$	122.309			0.558487
$220$		0			0
$221$	−272.382			−1.23250
$222$		0			0
$223$			57.0047i			0.255627i	0.991798	+	0.127813i	$0.0407958\pi$
$223$			57.0047i			0.255627i	−0.991798	+	0.127813i	$0.959204\pi$
$224$		0			0
$225$	−71.9117			−0.319608
$226$		0			0
$227$			287.418i			1.26616i	0.774086	+	0.633080i	$0.218210\pi$
$227$			287.418i			1.26616i	−0.774086	+	0.633080i	$0.781790\pi$
$228$		0			0
$229$		−	139.405i		−	0.608754i	−0.952552	−	0.304377i	$-0.901552\pi$
$229$		−	139.405i		−	0.608754i	0.952552	−	0.304377i	$-0.0984482\pi$
$230$		0			0
$231$	117.640	−	39.7862i	0.509262	−	0.172235i
$232$		0			0
$233$	362.735			1.55680			0.778401	−	0.627767i	$-0.216031\pi$
$233$	362.735			1.55680			0.778401	+	0.627767i	$0.216031\pi$
$234$		0			0
$235$	59.1472			0.251690
$236$		0			0
$237$		−	231.789i		−	0.978012i
$238$		0			0
$239$	−18.4781			−0.0773144			−0.0386572	−	0.999253i	$-0.512308\pi$
$239$	−18.4781			−0.0773144			−0.0386572	+	0.999253i	$0.512308\pi$
$240$		0			0
$241$		−	178.104i		−	0.739021i	−0.929227	−	0.369510i	$-0.879525\pi$
$241$		−	178.104i		−	0.739021i	0.929227	−	0.369510i	$-0.120475\pi$
$242$		0			0
$243$			15.5885i			0.0641500i
$244$		0			0
$245$	30.1766	+	39.5101i	0.123170	+	0.161266i
$246$		0			0
$247$	144.500			0.585018
$248$		0			0
$249$	180.853			0.726317
$250$		0			0
$251$			50.6709i			0.201876i	0.994893	+	0.100938i	$0.0321844\pi$
$251$			50.6709i			0.201876i	−0.994893	+	0.100938i	$0.967816\pi$
$252$		0			0
$253$	−68.9117			−0.272378
$254$		0			0
$255$			53.4386i			0.209563i
$256$		0			0
$257$			187.584i			0.729898i	0.931028	+	0.364949i	$0.118914\pi$
$257$			187.584i			0.729898i	−0.931028	+	0.364949i	$0.881086\pi$
$258$		0			0
$259$	−69.3238	−	204.976i	−0.267659	−	0.791414i
$260$		0			0
$261$	90.0000			0.344828
$262$		0			0
$263$	402.978			1.53223			0.766117	−	0.642701i	$-0.222186\pi$
$263$	402.978			1.53223			0.766117	+	0.642701i	$0.222186\pi$
$264$		0			0
$265$		−	72.0076i		−	0.271727i
$266$		0			0
$267$	−251.095			−0.940432
$268$		0			0
$269$			480.538i			1.78639i	0.449674	+	0.893193i	$0.351540\pi$
$269$			480.538i			1.78639i	−0.449674	+	0.893193i	$0.648460\pi$
$270$		0			0
$271$			37.3068i			0.137664i	0.997628	+	0.0688318i	$0.0219272\pi$
$271$			37.3068i			0.137664i	−0.997628	+	0.0688318i	$0.978073\pi$
$272$		0			0
$273$	−34.7939	−	102.879i	−0.127450	−	0.376845i
$274$		0			0
$275$	−245.522			−0.892807
$276$		0			0
$277$	82.6762			0.298470			0.149235	−	0.988802i	$-0.452319\pi$
$277$	82.6762			0.298470			0.149235	+	0.988802i	$0.452319\pi$
$278$		0			0
$279$			150.535i			0.539554i
$280$		0			0
$281$	−150.853			−0.536843			−0.268421	−	0.963302i	$-0.586502\pi$
$281$	−150.853			−0.536843			−0.268421	+	0.963302i	$0.586502\pi$
$282$		0			0
$283$			284.158i			1.00409i	0.864841	+	0.502046i	$0.167419\pi$
$283$			284.158i			1.00409i	−0.864841	+	0.502046i	$0.832581\pi$
$284$		0			0
$285$		−	28.3494i		−	0.0994716i
$286$		0			0
$287$	47.0955	−	15.9279i	0.164096	−	0.0554978i
$288$		0			0
$289$	−635.676			−2.19957
$290$		0			0
$291$	174.426			0.599403
$292$		0			0
$293$			537.237i			1.83357i	0.399379	+	0.916786i	$0.369226\pi$
$293$			537.237i			1.83357i	−0.399379	+	0.916786i	$0.630774\pi$
$294$		0			0
$295$	−0.499567			−0.00169345
$296$		0			0
$297$			53.2223i			0.179200i
$298$		0			0
$299$			60.2649i			0.201555i
$300$		0			0
$301$	−166.912	−	493.524i	−0.554524	−	1.63961i
$302$		0			0
$303$	−240.551			−0.793899
$304$		0			0
$305$	−2.91169			−0.00954652
$306$		0			0
$307$		−	34.7430i		−	0.113169i	−0.998398	−	0.0565847i	$-0.981979\pi$
$307$		−	34.7430i		−	0.113169i	0.998398	−	0.0565847i	$-0.0180211\pi$
$308$		0			0
$309$	136.617			0.442127
$310$		0			0
$311$			89.1664i			0.286709i	0.989671	+	0.143354i	$0.0457888\pi$
$311$			89.1664i			0.286709i	−0.989671	+	0.143354i	$0.954211\pi$
$312$		0			0
$313$			519.700i			1.66038i	0.557479	+	0.830191i	$0.311769\pi$
$313$			519.700i			1.66038i	−0.557479	+	0.830191i	$0.688231\pi$
$314$		0			0
$315$	−20.1838	+	6.82623i	−0.0640754	+	0.0216706i
$316$		0			0
$317$	−542.029			−1.70987			−0.854935	−	0.518736i	$-0.826403\pi$
$317$	−542.029			−1.70987			−0.854935	+	0.518736i	$0.826403\pi$
$318$		0			0
$319$	307.279			0.963258
$320$		0			0
$321$		−	65.3977i		−	0.203731i
$322$		0			0
$323$	490.544			1.51871
$324$		0			0
$325$			214.715i			0.660660i
$326$		0			0
$327$		−	55.3237i		−	0.169185i
$328$		0			0
$329$	−386.558	+	130.736i	−1.17495	+	0.397373i
$330$		0			0
$331$	179.632			0.542696			0.271348	−	0.962481i	$-0.412531\pi$
$331$	179.632			0.542696			0.271348	+	0.962481i	$0.412531\pi$
$332$		0			0
$333$	92.7351			0.278484
$334$		0			0
$335$		−	27.4244i		−	0.0818638i
$336$		0			0
$337$	291.823			0.865945			0.432972	−	0.901407i	$-0.357465\pi$
$337$	291.823			0.865945			0.432972	+	0.901407i	$0.357465\pi$
$338$		0			0
$339$		−	185.278i		−	0.546544i
$340$		0			0
$341$			513.960i			1.50721i
$342$		0			0
$343$	−284.551	−	191.519i	−0.829596	−	0.558365i
$344$		0			0
$345$	11.8234			0.0342707
$346$		0			0
$347$	452.080			1.30283			0.651413	−	0.758724i	$-0.274177\pi$
$347$	452.080			1.30283			0.651413	+	0.758724i	$0.274177\pi$
$348$		0			0
$349$		−	235.067i		−	0.673543i	−0.941586	−	0.336772i	$-0.890665\pi$
$349$		−	235.067i		−	0.673543i	0.941586	−	0.336772i	$-0.109335\pi$
$350$		0			0
$351$	46.5442			0.132604
$352$		0			0
$353$		−	25.2458i		−	0.0715179i	−0.999360	−	0.0357590i	$-0.988615\pi$
$353$		−	25.2458i		−	0.0715179i	0.999360	−	0.0357590i	$-0.0113849\pi$
$354$		0			0
$355$			51.3497i			0.144647i
$356$		0			0
$357$	−118.118	−	349.250i	−0.330862	−	0.978291i
$358$		0			0
$359$	106.243			0.295941			0.147970	−	0.988992i	$-0.452726\pi$
$359$	106.243			0.295941			0.147970	+	0.988992i	$0.452726\pi$
$360$		0			0
$361$	100.765			0.279126
$362$		0			0
$363$		−	27.8658i		−	0.0767652i
$364$		0			0
$365$	−71.6468			−0.196292
$366$		0			0
$367$			286.416i			0.780426i	0.920725	+	0.390213i	$0.127599\pi$
$367$			286.416i			0.780426i	−0.920725	+	0.390213i	$0.872401\pi$
$368$		0			0
$369$			21.3068i			0.0577421i
$370$		0			0
$371$	159.161	+	470.608i	0.429007	+	1.26849i
$372$		0			0
$373$	423.470			1.13531			0.567654	−	0.823267i	$-0.307851\pi$
$373$	423.470			1.13531			0.567654	+	0.823267i	$0.307851\pi$
$374$		0			0
$375$	86.0589			0.229490
$376$		0			0
$377$		−	268.723i		−	0.712793i
$378$		0			0
$379$	101.103			0.266761			0.133381	−	0.991065i	$-0.457417\pi$
$379$	101.103			0.266761			0.133381	+	0.991065i	$0.457417\pi$
$380$		0			0
$381$		−	38.2071i		−	0.100281i
$382$		0			0
$383$		−	220.394i		−	0.575442i	−0.957714	−	0.287721i	$-0.907102\pi$
$383$		−	220.394i		−	0.575442i	0.957714	−	0.287721i	$-0.0928976\pi$
$384$		0			0
$385$	−68.9117	+	23.3062i	−0.178991	+	0.0605356i
$386$		0			0
$387$	223.279			0.576949
$388$		0			0
$389$	383.470			0.985784			0.492892	−	0.870090i	$-0.335940\pi$
$389$	383.470			0.985784			0.492892	+	0.870090i	$0.335940\pi$
$390$		0			0
$391$			204.586i			0.523238i
$392$		0			0
$393$	122.912			0.312752
$394$		0			0
$395$			135.779i			0.343744i
$396$		0			0
$397$			485.178i			1.22211i	0.791588	+	0.611056i	$0.209255\pi$
$397$			485.178i			1.22211i	−0.791588	+	0.611056i	$0.790745\pi$
$398$		0			0
$399$	62.6619	+	185.278i	0.157047	+	0.464357i
$400$		0			0
$401$	−253.176			−0.631361			−0.315680	−	0.948866i	$-0.602233\pi$
$401$	−253.176			−0.631361			−0.315680	+	0.948866i	$0.602233\pi$
$402$		0			0
$403$	449.470			1.11531
$404$		0			0
$405$		−	9.13151i		−	0.0225469i
$406$		0			0
$407$	316.617			0.777930
$408$		0			0
$409$			5.39135i			0.0131818i	0.999978	+	0.00659089i	$0.00209796\pi$
$409$			5.39135i			0.0131818i	−0.999978	+	0.00659089i	$0.997902\pi$
$410$		0			0
$411$		−	183.189i		−	0.445717i
$412$		0			0
$413$	3.26494	−	1.10421i	0.00790541	−	0.00267364i
$414$		0			0
$415$	−105.941			−0.255280
$416$		0			0
$417$	−314.059			−0.753139
$418$		0			0
$419$			294.431i			0.702700i	0.936244	+	0.351350i	$0.114277\pi$
$419$			294.431i			0.702700i	−0.936244	+	0.351350i	$0.885723\pi$
$420$		0			0
$421$	290.441			0.689883			0.344941	−	0.938624i	$-0.387899\pi$
$421$	290.441			0.689883			0.344941	+	0.938624i	$0.387899\pi$
$422$		0			0
$423$		−	174.886i		−	0.413442i
$424$		0			0
$425$			728.909i			1.71508i
$426$		0			0
$427$	19.0294	−	6.43583i	0.0445654	−	0.0150722i
$428$		0			0
$429$	158.912			0.370424
$430$		0			0
$431$	−50.7136			−0.117665			−0.0588325	−	0.998268i	$-0.518738\pi$
$431$	−50.7136			−0.117665			−0.0588325	+	0.998268i	$0.518738\pi$
$432$		0			0
$433$		−	724.761i		−	1.67381i	−0.547347	−	0.836906i	$-0.684362\pi$
$433$		−	724.761i		−	1.67381i	0.547347	−	0.836906i	$-0.315638\pi$
$434$		0			0
$435$	−52.7208			−0.121197
$436$		0			0
$437$		−	108.534i		−	0.248361i
$438$		0			0
$439$			371.728i			0.846761i	0.905952	+	0.423381i	$0.139157\pi$
$439$			371.728i			0.846761i	−0.905952	+	0.423381i	$0.860843\pi$
$440$		0			0
$441$	116.823	−	89.2261i	0.264906	−	0.202327i
$442$		0			0
$443$	−229.258			−0.517512			−0.258756	−	0.965943i	$-0.583313\pi$
$443$	−229.258			−0.517512			−0.258756	+	0.965943i	$0.583313\pi$
$444$		0			0
$445$	147.088			0.330536
$446$		0			0
$447$		−	71.2690i		−	0.159439i
$448$		0			0
$449$	−600.323			−1.33702			−0.668511	−	0.743702i	$-0.733068\pi$
$449$	−600.323			−1.33702			−0.668511	+	0.743702i	$0.733068\pi$
$450$		0			0
$451$			72.7461i			0.161300i
$452$		0			0
$453$		−	139.201i		−	0.307286i
$454$		0			0
$455$	20.3818	+	60.2649i	0.0447952	+	0.132450i
$456$		0			0
$457$	483.823			1.05869			0.529347	−	0.848405i	$-0.322437\pi$
$457$	483.823			1.05869			0.529347	+	0.848405i	$0.322437\pi$
$458$		0			0
$459$	158.007			0.344242
$460$		0			0
$461$		−	274.661i		−	0.595794i	−0.954598	−	0.297897i	$-0.903715\pi$
$461$		−	274.661i		−	0.595794i	0.954598	−	0.297897i	$-0.0962852\pi$
$462$		0			0
$463$	153.470			0.331469			0.165734	−	0.986170i	$-0.447001\pi$
$463$	153.470			0.331469			0.165734	+	0.986170i	$0.447001\pi$
$464$		0			0
$465$		−	88.1816i		−	0.189638i
$466$		0			0
$467$		−	61.7420i		−	0.132210i	−0.997813	−	0.0661049i	$-0.978943\pi$
$467$		−	61.7420i		−	0.132210i	0.997813	−	0.0661049i	$-0.0210572\pi$
$468$		0			0
$469$	60.6173	+	179.233i	0.129248	+	0.382160i
$470$		0			0
$471$	−99.2649			−0.210754
$472$		0			0
$473$	762.323			1.61168
$474$		0			0
$475$		−	386.689i		−	0.814082i
$476$		0			0
$477$	−212.912			−0.446356
$478$		0			0
$479$		−	556.514i		−	1.16182i	−0.813966	−	0.580912i	$-0.802696\pi$
$479$		−	556.514i		−	1.16182i	0.813966	−	0.580912i	$-0.197304\pi$
$480$		0			0
$481$		−	276.889i		−	0.575653i
$482$		0			0
$483$	−77.2721	+	26.1337i	−0.159984	+	0.0541071i
$484$		0			0
$485$	−102.177			−0.210673
$486$		0			0
$487$	325.220			0.667804			0.333902	−	0.942608i	$-0.391635\pi$
$487$	325.220			0.667804			0.333902	+	0.942608i	$0.391635\pi$
$488$		0			0
$489$		−	302.956i		−	0.619542i
$490$		0			0
$491$	643.477			1.31054			0.655272	−	0.755393i	$-0.272554\pi$
$491$	643.477			1.31054			0.655272	+	0.755393i	$0.272554\pi$
$492$		0			0
$493$		−	912.255i		−	1.85042i
$494$		0			0
$495$		−	31.1769i		−	0.0629837i
$496$		0			0
$497$	−113.500	−	335.598i	−0.228371	−	0.675247i
$498$		0			0
$499$	755.426			1.51388			0.756939	−	0.653485i	$-0.226694\pi$
$499$	755.426			1.51388			0.756939	+	0.653485i	$0.226694\pi$
$500$		0			0
$501$	339.765			0.678173
$502$		0			0
$503$			509.409i			1.01274i	0.862316	+	0.506371i	$0.169013\pi$
$503$			509.409i			1.01274i	−0.862316	+	0.506371i	$0.830987\pi$
$504$		0			0
$505$	140.912			0.279033
$506$		0			0
$507$			153.745i			0.303244i
$508$		0			0
$509$			769.433i			1.51166i	0.654770	+	0.755828i	$0.272766\pi$
$509$			769.433i			1.51166i	−0.654770	+	0.755828i	$0.727234\pi$
$510$		0			0
$511$	468.250	−	158.364i	0.916340	−	0.309910i
$512$		0			0
$513$	−83.8234			−0.163398
$514$		0			0
$515$	−80.0286			−0.155395
$516$		0			0
$517$		−	597.099i		−	1.15493i
$518$		0			0
$519$	−66.7279			−0.128570
$520$		0			0
$521$			535.207i			1.02727i	0.858009	+	0.513635i	$0.171701\pi$
$521$			535.207i			1.02727i	−0.858009	+	0.513635i	$0.828299\pi$
$522$		0			0
$523$			839.466i			1.60510i	0.596586	+	0.802549i	$0.296523\pi$
$523$			839.466i			1.60510i	−0.596586	+	0.802549i	$0.703477\pi$
$524$		0			0
$525$	−275.309	+	93.1105i	−0.524397	+	0.177353i
$526$		0			0
$527$	1525.85			2.89535
$528$		0			0
$529$	−483.735			−0.914433
$530$		0			0
$531$			1.47712i			0.00278176i
$532$		0			0
$533$	63.6182			0.119359
$534$		0			0
$535$			38.3091i			0.0716057i
$536$		0			0
$537$		−	330.987i		−	0.616363i
$538$		0			0
$539$	398.860	−	304.637i	0.740000	−	0.565189i
$540$		0			0
$541$	−285.852			−0.528377			−0.264188	−	0.964471i	$-0.585104\pi$
$541$	−285.852			−0.528377			−0.264188	+	0.964471i	$0.585104\pi$
$542$		0			0
$543$	209.220			0.385305
$544$		0			0
$545$			32.4078i			0.0594639i
$546$		0			0
$547$	−741.470			−1.35552			−0.677761	−	0.735283i	$-0.737049\pi$
$547$	−741.470			−1.35552			−0.677761	+	0.735283i	$0.737049\pi$
$548$		0			0
$549$			8.60927i			0.0156817i
$550$		0			0
$551$			483.954i			0.878320i
$552$		0			0
$553$	−300.118	−	887.387i	−0.542708	−	1.60468i
$554$		0			0
$555$	−54.3229			−0.0978792
$556$		0			0
$557$	−834.853			−1.49884			−0.749419	−	0.662096i	$-0.769667\pi$
$557$	−834.853			−1.49884			−0.749419	+	0.662096i	$0.769667\pi$
$558$		0			0
$559$		−	666.669i		−	1.19261i
$560$		0			0
$561$	539.470			0.961622
$562$		0			0
$563$		−	417.781i		−	0.742062i	−0.928620	−	0.371031i	$-0.879004\pi$
$563$		−	417.781i		−	0.742062i	0.928620	−	0.371031i	$-0.120996\pi$
$564$		0			0
$565$			108.534i			0.192095i
$566$		0			0
$567$	20.1838	+	59.6793i	0.0355975	+	0.105254i
$568$		0			0
$569$	673.029			1.18283			0.591414	−	0.806368i	$-0.298570\pi$
$569$	673.029			1.18283			0.591414	+	0.806368i	$0.298570\pi$
$570$		0			0
$571$	−265.088			−0.464253			−0.232126	−	0.972686i	$-0.574568\pi$
$571$	−265.088			−0.464253			−0.232126	+	0.972686i	$0.574568\pi$
$572$		0			0
$573$		−	194.104i		−	0.338750i
$574$		0			0
$575$	161.272			0.280473
$576$		0			0
$577$			468.740i			0.812375i	0.913790	+	0.406188i	$0.133142\pi$
$577$			468.740i			0.812375i	−0.913790	+	0.406188i	$0.866858\pi$
$578$		0			0
$579$		−	39.6842i		−	0.0685392i
$580$		0			0
$581$	692.382	−	234.166i	1.19171	−	0.403040i
$582$		0			0
$583$	−726.926			−1.24687
$584$		0			0
$585$	−27.2649			−0.0466067
$586$		0			0
$587$		−	120.530i		−	0.205332i	−0.994716	−	0.102666i	$-0.967263\pi$
$587$		−	120.530i		−	0.205332i	0.994716	−	0.102666i	$-0.0327372\pi$
$588$		0			0
$589$	−809.470			−1.37431
$590$		0			0
$591$			201.020i			0.340135i
$592$		0			0
$593$			561.468i			0.946826i	0.880841	+	0.473413i	$0.156978\pi$
$593$			561.468i			0.946826i	−0.880841	+	0.473413i	$0.843022\pi$
$594$		0			0
$595$	69.1918	+	204.586i	0.116289	+	0.343842i
$596$		0			0
$597$	282.500			0.473199
$598$		0			0
$599$	−940.109			−1.56946			−0.784732	−	0.619835i	$-0.787199\pi$
$599$	−940.109			−1.56946			−0.784732	+	0.619835i	$0.787199\pi$
$600$		0			0
$601$			563.527i			0.937649i	0.883291	+	0.468824i	$0.155322\pi$
$601$			563.527i			0.937649i	−0.883291	+	0.468824i	$0.844678\pi$
$602$		0			0
$603$	−81.0883			−0.134475
$604$		0			0
$605$			16.3234i			0.0269808i
$606$		0			0
$607$			306.589i			0.505089i	0.967585	+	0.252544i	$0.0812674\pi$
$607$			306.589i			0.505089i	−0.967585	+	0.252544i	$0.918733\pi$
$608$		0			0
$609$	344.558	−	116.531i	0.565777	−	0.191348i
$610$		0			0
$611$	−522.177			−0.854626
$612$		0			0
$613$	−275.588			−0.449572			−0.224786	−	0.974408i	$-0.572168\pi$
$613$	−275.588			−0.449572			−0.224786	+	0.974408i	$0.572168\pi$
$614$		0			0
$615$		−	12.4813i		−	0.0202947i
$616$		0			0
$617$	461.294			0.747639			0.373820	−	0.927501i	$-0.378048\pi$
$617$	461.294			0.747639			0.373820	+	0.927501i	$0.378048\pi$
$618$		0			0
$619$		−	374.904i		−	0.605660i	−0.953045	−	0.302830i	$-0.902068\pi$
$619$		−	374.904i		−	0.605660i	0.953045	−	0.302830i	$-0.0979315\pi$
$620$		0			0
$621$		−	34.9593i		−	0.0562952i
$622$		0			0
$623$	−961.301	+	325.116i	−1.54302	+	0.521855i
$624$		0			0
$625$	548.852			0.878163
$626$		0			0
$627$	−286.191			−0.456445
$628$		0			0
$629$		−	939.978i		−	1.49440i
$630$		0			0
$631$	1127.32			1.78656			0.893282	−	0.449496i	$-0.148397\pi$
$631$	1127.32			1.78656			0.893282	+	0.449496i	$0.148397\pi$
$632$		0			0
$633$			184.336i			0.291210i
$634$		0			0
$635$			22.3812i			0.0352460i
$636$		0			0
$637$	−266.412	−	348.812i	−0.418229	−	0.547586i
$638$		0			0
$639$	151.831			0.237606
$640$		0			0
$641$	−884.352			−1.37964			−0.689822	−	0.723979i	$-0.742311\pi$
$641$	−884.352			−1.37964			−0.689822	+	0.723979i	$0.742311\pi$
$642$		0			0
$643$		−	300.765i		−	0.467753i	−0.972266	−	0.233876i	$-0.924859\pi$
$643$		−	300.765i		−	0.467753i	0.972266	−	0.233876i	$-0.0751411\pi$
$644$		0			0
$645$	−130.794			−0.202781
$646$		0			0
$647$			940.604i			1.45379i	0.686747	+	0.726896i	$0.259038\pi$
$647$			940.604i			1.45379i	−0.686747	+	0.726896i	$0.740962\pi$
$648$		0			0
$649$			5.04319i			0.00777071i
$650$		0			0
$651$	194.912	+	576.314i	0.299404	+	0.885275i
$652$		0			0
$653$	455.970			0.698269			0.349135	−	0.937073i	$-0.386476\pi$
$653$	455.970			0.698269			0.349135	+	0.937073i	$0.386476\pi$
$654$		0			0
$655$	−72.0000			−0.109924
$656$		0			0
$657$			211.845i			0.322443i
$658$		0			0
$659$	403.684			0.612571			0.306285	−	0.951940i	$-0.400914\pi$
$659$	403.684			0.612571			0.306285	+	0.951940i	$0.400914\pi$
$660$		0			0
$661$			1153.41i			1.74495i	0.488663	+	0.872473i	$0.337485\pi$
$661$			1153.41i			1.74495i	−0.488663	+	0.872473i	$0.662515\pi$
$662$		0			0
$663$		−	471.779i		−	0.711582i
$664$		0			0
$665$	−36.7065	−	108.534i	−0.0551977	−	0.163208i
$666$		0			0
$667$	−201.838			−0.302605
$668$		0			0
$669$	−98.7351			−0.147586
$670$		0			0
$671$			29.3939i			0.0438061i
$672$		0			0
$673$	−607.440			−0.902585			−0.451293	−	0.892376i	$-0.649037\pi$
$673$	−607.440			−0.902585			−0.451293	+	0.892376i	$0.649037\pi$
$674$		0			0
$675$		−	124.555i		−	0.184525i
$676$		0			0
$677$			1137.59i			1.68034i	0.542326	+	0.840168i	$0.317544\pi$
$677$			1137.59i			1.68034i	−0.542326	+	0.840168i	$0.682456\pi$
$678$		0			0
$679$	667.779	−	225.845i	0.983474	−	0.332615i
$680$		0			0
$681$	−497.823			−0.731018
$682$		0			0
$683$	−818.111			−1.19782			−0.598910	−	0.800817i	$-0.704399\pi$
$683$	−818.111			−1.19782			−0.598910	+	0.800817i	$0.704399\pi$
$684$		0			0
$685$			107.310i			0.156657i
$686$		0			0
$687$	241.456			0.351464
$688$		0			0
$689$			635.714i			0.922661i
$690$		0			0
$691$			204.280i			0.295630i	0.989015	+	0.147815i	$0.0472239\pi$
$691$			204.280i			0.295630i	−0.989015	+	0.147815i	$0.952776\pi$
$692$		0			0
$693$	68.9117	+	203.758i	0.0994397	+	0.294023i
$694$		0			0
$695$	183.971			0.264707
$696$		0			0
$697$	215.970			0.309856
$698$		0			0
$699$			628.276i			0.898821i
$700$		0			0
$701$	318.853			0.454854			0.227427	−	0.973795i	$-0.426969\pi$
$701$	318.853			0.454854			0.227427	+	0.973795i	$0.426969\pi$
$702$		0			0
$703$			498.662i			0.709334i
$704$		0			0
$705$			102.446i			0.145313i
$706$		0			0
$707$	−920.933	+	311.463i	−1.30259	+	0.440542i
$708$		0			0
$709$	217.647			0.306977			0.153489	−	0.988150i	$-0.450949\pi$
$709$	217.647			0.306977			0.153489	+	0.988150i	$0.450949\pi$
$710$		0			0
$711$	401.470			0.564656
$712$		0			0
$713$		−	337.597i		−	0.473488i
$714$		0			0
$715$	−93.0883			−0.130193
$716$		0			0
$717$		−	32.0051i		−	0.0446375i
$718$		0			0
$719$		−	1207.36i		−	1.67922i	−0.543192	−	0.839609i	$-0.682784\pi$
$719$		−	1207.36i		−	1.67922i	0.543192	−	0.839609i	$-0.317216\pi$
$720$		0			0
$721$	523.029	−	176.891i	0.725422	−	0.245341i
$722$		0			0
$723$	308.485			0.426674
$724$		0			0
$725$	−719.117			−0.991885
$726$		0			0
$727$		−	123.231i		−	0.169506i	−0.996402	−	0.0847528i	$-0.972990\pi$
$727$		−	123.231i		−	0.169506i	0.996402	−	0.0847528i	$-0.0270100\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	2263.19i		−	3.09603i
$732$		0			0
$733$			375.779i			0.512659i	0.966589	+	0.256330i	$0.0825133\pi$
$733$			375.779i			0.512659i	−0.966589	+	0.256330i	$0.917487\pi$
$734$		0			0
$735$	−68.4335	+	52.2674i	−0.0931069	+	0.0711122i
$736$		0			0
$737$	−276.853			−0.375648
$738$		0			0
$739$	722.530			0.977713			0.488856	−	0.872364i	$-0.337414\pi$
$739$	722.530			0.977713			0.488856	+	0.872364i	$0.337414\pi$
$740$		0			0
$741$			250.281i			0.337761i
$742$		0			0
$743$	−1268.48			−1.70724			−0.853618	−	0.520899i	$-0.825597\pi$
$743$	−1268.48			−1.70724			−0.853618	+	0.520899i	$0.825597\pi$
$744$		0			0
$745$			41.7484i			0.0560382i
$746$		0			0
$747$			313.246i			0.419339i
$748$		0			0
$749$	−84.6762	−	250.370i	−0.113052	−	0.334273i
$750$		0			0
$751$	−439.161			−0.584769			−0.292384	−	0.956301i	$-0.594449\pi$
$751$	−439.161			−0.584769			−0.292384	+	0.956301i	$0.594449\pi$
$752$		0			0
$753$	−87.7645			−0.116553
$754$		0			0
$755$			81.5419i			0.108002i
$756$		0			0
$757$	668.530			0.883131			0.441565	−	0.897229i	$-0.354423\pi$
$757$	668.530			0.883131			0.441565	+	0.897229i	$0.354423\pi$
$758$		0			0
$759$		−	119.359i		−	0.157258i
$760$		0			0
$761$			880.116i			1.15653i	0.815851	+	0.578263i	$0.196269\pi$
$761$			880.116i			1.15653i	−0.815851	+	0.578263i	$0.803731\pi$
$762$		0			0
$763$	−71.6325	−	211.803i	−0.0938827	−	0.277592i
$764$		0			0
$765$	−92.5584			−0.120991
$766$		0			0
$767$	4.41039			0.00575018
$768$		0			0
$769$		−	1163.41i		−	1.51289i	−0.654059	−	0.756444i	$-0.726935\pi$
$769$		−	1163.41i		−	1.51289i	0.654059	−	0.756444i	$-0.273065\pi$
$770$		0			0
$771$	−324.905			−0.421407
$772$		0			0
$773$		−	536.371i		−	0.693883i	−0.937887	−	0.346941i	$-0.887220\pi$
$773$		−	536.371i		−	0.693883i	0.937887	−	0.346941i	$-0.112780\pi$
$774$		0			0
$775$		−	1202.81i		−	1.55201i
$776$		0			0
$777$	355.029	−	120.072i	0.456923	−	0.154533i
$778$		0			0
$779$	−114.573			−0.147077
$780$		0			0
$781$	518.382			0.663741
$782$		0			0
$783$			155.885i			0.199086i
$784$		0			0
$785$	58.1481			0.0740740
$786$		0			0
$787$		−	83.9192i		−	0.106632i	−0.998578	−	0.0533159i	$-0.983021\pi$
$787$		−	83.9192i		−	0.106632i	0.998578	−	0.0533159i	$-0.0169790\pi$
$788$		0			0
$789$			697.978i			0.884636i
$790$		0			0
$791$	−239.897	−	709.325i	−0.303283	−	0.896745i
$792$		0			0
$793$	25.7056			0.0324157
$794$		0			0
$795$	124.721			0.156881
$796$		0			0
$797$		−	1005.57i		−	1.26169i	−0.775908	−	0.630846i	$-0.782708\pi$
$797$		−	1005.57i		−	1.26169i	0.775908	−	0.630846i	$-0.217292\pi$
$798$		0			0
$799$	−1772.67			−2.21862
$800$		0			0
$801$		−	434.910i		−	0.542959i
$802$		0			0
$803$			723.283i			0.900727i
$804$		0			0
$805$	45.2649	−	15.3088i	0.0562297	−	0.0190171i
$806$		0			0
$807$	−832.316			−1.03137
$808$		0			0
$809$	−282.853			−0.349633			−0.174816	−	0.984601i	$-0.555933\pi$
$809$	−282.853			−0.349633			−0.174816	+	0.984601i	$0.555933\pi$
$810$		0			0
$811$			923.997i			1.13933i	0.821877	+	0.569665i	$0.192927\pi$
$811$			923.997i			1.13933i	−0.821877	+	0.569665i	$0.807073\pi$
$812$		0			0
$813$	−64.6173			−0.0794801
$814$		0			0
$815$			177.467i			0.217752i
$816$		0			0
$817$			1200.63i			1.46956i
$818$		0			0
$819$	178.191	−	60.2649i	0.217571	−	0.0735835i
$820$		0			0
$821$	−1272.32			−1.54972			−0.774862	−	0.632131i	$-0.782181\pi$
$821$	−1272.32			−1.54972			−0.774862	+	0.632131i	$0.782181\pi$
$822$		0			0
$823$	−381.852			−0.463976			−0.231988	−	0.972719i	$-0.574523\pi$
$823$	−381.852			−0.463976			−0.231988	+	0.972719i	$0.574523\pi$
$824$		0			0
$825$		−	425.256i		−	0.515462i
$826$		0			0
$827$	653.022			0.789628			0.394814	−	0.918761i	$-0.370809\pi$
$827$	653.022			0.789628			0.394814	+	0.918761i	$0.370809\pi$
$828$		0			0
$829$			215.580i			0.260048i	0.991511	+	0.130024i	$0.0415054\pi$
$829$			215.580i			0.260048i	−0.991511	+	0.130024i	$0.958495\pi$
$830$		0			0
$831$			143.199i			0.172322i
$832$		0			0
$833$	−904.410	−	1184.14i	−1.08573	−	1.42154i
$834$		0			0
$835$	−199.029			−0.238359
$836$		0			0
$837$	−260.735			−0.311511
$838$		0			0
$839$			632.267i			0.753596i	0.926295	+	0.376798i	$0.122975\pi$
$839$			632.267i			0.753596i	−0.926295	+	0.376798i	$0.877025\pi$
$840$		0			0
$841$	59.0000			0.0701546
$842$		0			0
$843$		−	261.285i		−	0.309946i
$844$		0			0
$845$		−	90.0615i		−	0.106582i
$846$		0			0
$847$	−36.0803	−	106.682i	−0.0425978	−	0.125953i
$848$		0			0
$849$	−492.177			−0.579713
$850$		0			0
$851$	−207.971			−0.244385
$852$		0			0
$853$		−	919.650i		−	1.07814i	−0.842262	−	0.539068i	$-0.818776\pi$
$853$		−	919.650i		−	1.07814i	0.842262	−	0.539068i	$-0.181224\pi$
$854$		0			0
$855$	49.1026			0.0574299
$856$		0			0
$857$			527.150i			0.615111i	0.951530	+	0.307556i	$0.0995110\pi$
$857$			527.150i			0.615111i	−0.951530	+	0.307556i	$0.900489\pi$
$858$		0			0
$859$		−	239.804i		−	0.279166i	−0.990210	−	0.139583i	$-0.955424\pi$
$859$		−	239.804i		−	0.279166i	0.990210	−	0.139583i	$-0.0445762\pi$
$860$		0			0
$861$	27.5879	+	81.5717i	0.0320417	+	0.0947407i
$862$		0			0
$863$	−362.507			−0.420054			−0.210027	−	0.977696i	$-0.567355\pi$
$863$	−362.507			−0.420054			−0.210027	+	0.977696i	$0.567355\pi$
$864$		0			0
$865$	39.0883			0.0451888
$866$		0			0
$867$		−	1101.02i		−	1.26992i
$868$		0			0
$869$	1370.70			1.57734
$870$		0			0
$871$			242.114i			0.277973i
$872$		0			0
$873$			302.115i			0.346066i
$874$		0			0
$875$	329.470	−	111.428i	0.376537	−	0.127346i
$876$		0			0
$877$	861.647			0.982493			0.491247	−	0.871020i	$-0.336541\pi$
$877$	861.647			0.982493			0.491247	+	0.871020i	$0.336541\pi$
$878$		0			0
$879$	−930.521			−1.05861
$880$		0			0
$881$		−	334.553i		−	0.379742i	−0.981809	−	0.189871i	$-0.939193\pi$
$881$		−	334.553i		−	0.379742i	0.981809	−	0.189871i	$-0.0608071\pi$
$882$		0			0
$883$	−1488.01			−1.68518			−0.842590	−	0.538555i	$-0.818970\pi$
$883$	−1488.01			−1.68518			−0.842590	+	0.538555i	$0.818970\pi$
$884$		0			0
$885$		−	0.865275i		−	0.000977712i
$886$		0			0
$887$			801.498i			0.903606i	0.892118	+	0.451803i	$0.149219\pi$
$887$			801.498i			0.903606i	−0.892118	+	0.451803i	$0.850781\pi$
$888$		0			0
$889$	−49.4701	−	146.273i	−0.0556469	−	0.164537i
$890$		0			0
$891$	−92.1838			−0.103461
$892$		0			0
$893$	940.410			1.05309
$894$		0			0
$895$			193.888i			0.216634i
$896$		0			0
$897$	−104.382			−0.116368
$898$		0			0
$899$			1505.35i			1.67448i
$900$		0			0
$901$			2158.11i			2.39524i
$902$		0			0
$903$	854.808	−	289.100i	0.946631	−	0.320155i
$904$		0			0
$905$	−122.558			−0.135424
$906$		0			0
$907$	−807.279			−0.890054			−0.445027	−	0.895517i	$-0.646806\pi$
$907$	−807.279			−0.890054			−0.445027	+	0.895517i	$0.646806\pi$
$908$		0			0
$909$		−	416.647i		−	0.458358i
$910$		0			0
$911$	442.742			0.485996			0.242998	−	0.970027i	$-0.421869\pi$
$911$	442.742			0.485996			0.242998	+	0.970027i	$0.421869\pi$
$912$		0			0
$913$			1069.49i			1.17140i
$914$		0			0
$915$		−	5.04319i		−	0.00551169i
$916$		0			0
$917$	470.558	−	159.145i	0.513150	−	0.173549i
$918$		0			0
$919$	118.455			0.128896			0.0644478	−	0.997921i	$-0.479471\pi$
$919$	118.455			0.128896			0.0644478	+	0.997921i	$0.479471\pi$
$920$		0			0
$921$	60.1766			0.0653384
$922$		0			0
$923$		−	453.337i		−	0.491156i
$924$		0			0
$925$	−740.971			−0.801049
$926$		0			0
$927$			236.628i			0.255262i
$928$		0			0
$929$		−	629.297i		−	0.677392i	−0.940896	−	0.338696i	$-0.890014\pi$
$929$		−	629.297i		−	0.677392i	0.940896	−	0.338696i	$-0.109986\pi$
$930$		0			0
$931$	479.793	+	628.191i	0.515352	+	0.674749i
$932$		0			0
$933$	−154.441			−0.165531
$934$		0			0
$935$	−316.014			−0.337983
$936$		0			0
$937$		−	210.631i		−	0.224793i	−0.993663	−	0.112397i	$-0.964147\pi$
$937$		−	210.631i		−	0.224793i	0.993663	−	0.112397i	$-0.0358527\pi$
$938$		0			0
$939$	−900.146			−0.958622
$940$		0			0
$941$			178.422i			0.189609i	0.995496	+	0.0948047i	$0.0302226\pi$
$941$			178.422i			0.189609i	−0.995496	+	0.0948047i	$0.969777\pi$
$942$		0			0
$943$		−	47.7836i		−	0.0506719i
$944$		0			0
$945$	−11.8234	−	34.9593i	−0.0125115	−	0.0369940i
$946$		0			0
$947$	423.463			0.447163			0.223581	−	0.974685i	$-0.428225\pi$
$947$	423.463			0.447163			0.223581	+	0.974685i	$0.428225\pi$
$948$		0			0
$949$	632.528			0.666521
$950$		0			0
$951$		−	938.821i		−	0.987194i
$952$		0			0
$953$	−546.706			−0.573669			−0.286834	−	0.957980i	$-0.592603\pi$
$953$	−546.706			−0.573669			−0.286834	+	0.957980i	$0.592603\pi$
$954$		0			0
$955$			113.704i			0.119061i
$956$		0			0
$957$			532.223i			0.556137i
$958$		0			0
$959$	−237.192	−	701.328i	−0.247332	−	0.731311i
$960$		0			0
$961$	−1556.88			−1.62006
$962$		0			0
$963$	113.272			0.117624
$964$		0			0
$965$			23.2465i			0.0240896i
$966$		0			0
$967$	−1262.32			−1.30540			−0.652701	−	0.757616i	$-0.726364\pi$
$967$	−1262.32			−1.30540			−0.652701	+	0.757616i	$0.726364\pi$
$968$		0			0
$969$			849.647i			0.876829i
$970$		0			0
$971$		−	1080.93i		−	1.11322i	−0.830775	−	0.556608i	$-0.812102\pi$
$971$		−	1080.93i		−	1.11322i	0.830775	−	0.556608i	$-0.187898\pi$
$972$		0			0
$973$	−1202.35	+	406.640i	−1.23572	+	0.417924i
$974$		0			0
$975$	−371.897			−0.381432
$976$		0			0
$977$	1800.26			1.84264			0.921322	−	0.388801i	$-0.127111\pi$
$977$	1800.26			1.84264			0.921322	+	0.388801i	$0.127111\pi$
$978$		0			0
$979$		−	1484.88i		−	1.51673i
$980$		0			0
$981$	95.8234			0.0976793
$982$		0			0
$983$		−	996.810i		−	1.01405i	−0.861932	−	0.507025i	$-0.830745\pi$
$983$		−	996.810i		−	1.01405i	0.861932	−	0.507025i	$-0.169255\pi$
$984$		0			0
$985$		−	117.755i		−	0.119548i
$986$		0			0
$987$	−226.441	−	669.539i	−0.229423	−	0.678358i
$988$		0			0
$989$	−500.735			−0.506304
$990$		0			0
$991$	−788.721			−0.795884			−0.397942	−	0.917411i	$-0.630275\pi$
$991$	−788.721			−0.795884			−0.397942	+	0.917411i	$0.630275\pi$
$992$		0			0
$993$			311.133i			0.313326i
$994$		0			0
$995$	−165.484			−0.166316
$996$		0			0
$997$		−	1942.70i		−	1.94854i	−0.225379	−	0.974271i	$-0.572362\pi$
$997$		−	1942.70i		−	1.94854i	0.225379	−	0.974271i	$-0.427638\pi$
$998$		0			0
$999$			160.622i			0.160783i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.f.e.769.4		4
4.3	odd	2		1344.3.f.f.769.2		4
7.6	odd	2	inner	1344.3.f.e.769.1		4
8.3	odd	2		42.3.c.a.13.4	yes	4
8.5	even	2		336.3.f.c.97.1		4
24.5	odd	2		1008.3.f.g.433.2		4
24.11	even	2		126.3.c.b.55.1		4
28.27	even	2		1344.3.f.f.769.3		4
40.3	even	4		1050.3.h.a.349.1		8
40.19	odd	2		1050.3.f.a.601.1		4
40.27	even	4		1050.3.h.a.349.8		8
56.3	even	6		294.3.g.b.19.1		4
56.11	odd	6		294.3.g.c.19.1		4
56.13	odd	2		336.3.f.c.97.4		4
56.19	even	6		294.3.g.c.31.1		4
56.27	even	2		42.3.c.a.13.3	✓	4
56.51	odd	6		294.3.g.b.31.1		4
168.11	even	6		882.3.n.d.19.2		4
168.59	odd	6		882.3.n.a.19.2		4
168.83	odd	2		126.3.c.b.55.2		4
168.107	even	6		882.3.n.a.325.2		4
168.125	even	2		1008.3.f.g.433.3		4
168.131	odd	6		882.3.n.d.325.2		4
280.27	odd	4		1050.3.h.a.349.5		8
280.83	odd	4		1050.3.h.a.349.4		8
280.139	even	2		1050.3.f.a.601.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.c.a.13.3	✓	4	56.27	even	2
42.3.c.a.13.4	yes	4	8.3	odd	2
126.3.c.b.55.1		4	24.11	even	2
126.3.c.b.55.2		4	168.83	odd	2
294.3.g.b.19.1		4	56.3	even	6
294.3.g.b.31.1		4	56.51	odd	6
294.3.g.c.19.1		4	56.11	odd	6
294.3.g.c.31.1		4	56.19	even	6
336.3.f.c.97.1		4	8.5	even	2
336.3.f.c.97.4		4	56.13	odd	2
882.3.n.a.19.2		4	168.59	odd	6
882.3.n.a.325.2		4	168.107	even	6
882.3.n.d.19.2		4	168.11	even	6
882.3.n.d.325.2		4	168.131	odd	6
1008.3.f.g.433.2		4	24.5	odd	2
1008.3.f.g.433.3		4	168.125	even	2
1050.3.f.a.601.1		4	40.19	odd	2
1050.3.f.a.601.2		4	280.139	even	2
1050.3.h.a.349.1		8	40.3	even	4
1050.3.h.a.349.4		8	280.83	odd	4
1050.3.h.a.349.5		8	280.27	odd	4
1050.3.h.a.349.8		8	40.27	even	4
1344.3.f.e.769.1		4	7.6	odd	2	inner
1344.3.f.e.769.4		4	1.1	even	1	trivial
1344.3.f.f.769.2		4	4.3	odd	2
1344.3.f.f.769.3		4	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -1.01461i q^{5} +(2.24264 + 6.63103i) q^{7} -3.00000 q^{9} -10.2426 q^{11} +8.95743i q^{13} +1.75736 q^{15} +30.4085i q^{17} -16.1318i q^{19} +(-11.4853 + 3.88437i) q^{21} +6.72792 q^{23} +23.9706 q^{25} -5.19615i q^{27} -30.0000 q^{29} -50.1785i q^{31} -17.7408i q^{33} +(6.72792 - 2.27541i) q^{35} -30.9117 q^{37} -15.5147 q^{39} -7.10228i q^{41} -74.4264 q^{43} +3.04384i q^{45} +58.2954i q^{47} +(-38.9411 + 29.7420i) q^{49} -52.6690 q^{51} +70.9706 q^{53} +10.3923i q^{55} +27.9411 q^{57} -0.492372i q^{59} -2.86976i q^{61} +(-6.72792 - 19.8931i) q^{63} +9.08831 q^{65} +27.0294 q^{67} +11.6531i q^{69} -50.6102 q^{71} -70.6149i q^{73} +41.5182i q^{75} +(-22.9706 - 67.9193i) q^{77} -133.823 q^{79} +9.00000 q^{81} -104.415i q^{83} +30.8528 q^{85} -51.9615i q^{87} +144.970i q^{89} +(-59.3970 + 20.0883i) q^{91} +86.9117 q^{93} -16.3675 q^{95} -100.705i q^{97} +30.7279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7} - 12 q^{9} - 24 q^{11} + 24 q^{15} - 12 q^{21} - 24 q^{23} + 28 q^{25} - 120 q^{29} - 24 q^{35} + 80 q^{37} - 96 q^{39} - 128 q^{43} - 20 q^{49} - 24 q^{51} + 216 q^{53} - 24 q^{57} + 24 q^{63}+ \cdots + 72 q^{99}+O(q^{100}) \) 4 * q - 8 * q^7 - 12 * q^9 - 24 * q^11 + 24 * q^15 - 12 * q^21 - 24 * q^23 + 28 * q^25 - 120 * q^29 - 24 * q^35 + 80 * q^37 - 96 * q^39 - 128 * q^43 - 20 * q^49 - 24 * q^51 + 216 * q^53 - 24 * q^57 + 24 * q^63 + 240 * q^65 + 176 * q^67 + 120 * q^71 - 24 * q^77 - 128 * q^79 + 36 * q^81 - 216 * q^85 + 144 * q^93 + 240 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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