LMFDB - Embedded newform 1344.3.f.j.769.8

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)

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")

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);

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 = [16,0,0,0,0,0,0,0,-48,0,0,0,0,0,0,0,0,0,0,0,-24,0,0,0,-64] 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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2 x^{14} - 8 x^{13} - 57 x^{12} + 32 x^{11} + 466 x^{10} + 304 x^{9} + 3000 x^{8} + \cdots + 790321 $ x^16 + 2x^14 - 8x^13 - 57x^12 + 32x^11 + 466x^10 + 304x^9 + 3000x^8 + 1328x^7 - 11560x^6 + 22316x^5 - 23333x^4 - 87520x^3 - 135796x^2 - 88900x + 790321
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.8
Root			$0.0550989 - 2.76316i$ of defining polynomial
Character	$\chi$	$=$	1344.769
Dual form			1344.3.f.j.769.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +8.94195i q^{5} +(-1.52790 - 6.83122i) q^{7} -3.00000 q^{9} +0.217948 q^{11} +17.2063i q^{13} +15.4879 q^{15} +7.22871i q^{17} -28.0850i q^{19} +(-11.8320 + 2.64641i) q^{21} -23.9086 q^{23} -54.9584 q^{25} +5.19615i q^{27} +26.9807 q^{29} +27.0903i q^{31} -0.377496i q^{33} +(61.0844 - 13.6624i) q^{35} +17.9441 q^{37} +29.8022 q^{39} +18.5536i q^{41} -56.3643 q^{43} -26.8258i q^{45} -75.7660i q^{47} +(-44.3310 + 20.8749i) q^{49} +12.5205 q^{51} -1.03364 q^{53} +1.94888i q^{55} -48.6447 q^{57} +2.32111i q^{59} -60.7334i q^{61} +(4.58371 + 20.4936i) q^{63} -153.858 q^{65} -102.011 q^{67} +41.4108i q^{69} -22.0554 q^{71} -107.319i q^{73} +95.1908i q^{75} +(-0.333003 - 1.48885i) q^{77} -141.639 q^{79} +9.00000 q^{81} +81.0927i q^{83} -64.6388 q^{85} -46.7320i q^{87} -167.406i q^{89} +(117.540 - 26.2896i) q^{91} +46.9218 q^{93} +251.135 q^{95} -53.8065i q^{97} -0.653843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} - 24 q^{21} - 64 q^{25} + 128 q^{29} - 48 q^{37} - 256 q^{49} + 160 q^{53} - 144 q^{57} - 288 q^{65} - 128 q^{77} + 144 q^{81} + 16 q^{85} - 96 q^{93}+O(q^{100}) $ 16 * q - 48 * q^9 - 24 * q^21 - 64 * q^25 + 128 * q^29 - 48 * q^37 - 256 * q^49 + 160 * q^53 - 144 * q^57 - 288 * q^65 - 128 * q^77 + 144 * q^81 + 16 * q^85 - 96 * q^93

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
$5$			8.94195i			1.78839i	0.447678	+	0.894195i	$0.352251\pi$
$5$			8.94195i			1.78839i	−0.447678	+	0.894195i	$0.647749\pi$
$6$		0			0
$7$	−1.52790	−	6.83122i	−0.218272	−	0.975888i
$7$	−1.52790	−	6.83122i	−0.218272	−	0.975888i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	0.217948			0.0198134			0.00990671	−	0.999951i	$-0.496847\pi$
$11$	0.217948			0.0198134			0.00990671	+	0.999951i	$0.496847\pi$
$12$		0			0
$13$			17.2063i			1.32356i	0.749697	+	0.661781i	$0.230199\pi$
$13$			17.2063i			1.32356i	−0.749697	+	0.661781i	$0.769801\pi$
$14$		0			0
$15$	15.4879			1.03253
$16$		0			0
$17$			7.22871i			0.425218i	0.977137	+	0.212609i	$0.0681961\pi$
$17$			7.22871i			0.425218i	−0.977137	+	0.212609i	$0.931804\pi$
$18$		0			0
$19$		−	28.0850i		−	1.47816i	−0.673617	−	0.739080i	$-0.735260\pi$
$19$		−	28.0850i		−	1.47816i	0.673617	−	0.739080i	$-0.264740\pi$
$20$		0			0
$21$	−11.8320	+	2.64641i	−0.563429	+	0.126019i
$22$		0			0
$23$	−23.9086			−1.03950			−0.519751	−	0.854318i	$-0.673975\pi$
$23$	−23.9086			−1.03950			−0.519751	+	0.854318i	$0.673975\pi$
$24$		0			0
$25$	−54.9584			−2.19834
$26$		0			0
$27$			5.19615i			0.192450i
$28$		0			0
$29$	26.9807			0.930369			0.465185	−	0.885214i	$-0.345988\pi$
$29$	26.9807			0.930369			0.465185	+	0.885214i	$0.345988\pi$
$30$		0			0
$31$			27.0903i			0.873882i	0.899490	+	0.436941i	$0.143938\pi$
$31$			27.0903i			0.873882i	−0.899490	+	0.436941i	$0.856062\pi$
$32$		0			0
$33$		−	0.377496i		−	0.0114393i
$34$		0			0
$35$	61.0844	−	13.6624i	1.74527	−	0.390355i
$36$		0			0
$37$	17.9441			0.484976			0.242488	−	0.970154i	$-0.422037\pi$
$37$	17.9441			0.484976			0.242488	+	0.970154i	$0.422037\pi$
$38$		0			0
$39$	29.8022			0.764159
$40$		0			0
$41$			18.5536i			0.452527i	0.974066	+	0.226264i	$0.0726511\pi$
$41$			18.5536i			0.452527i	−0.974066	+	0.226264i	$0.927349\pi$
$42$		0			0
$43$	−56.3643			−1.31080			−0.655399	−	0.755283i	$-0.727500\pi$
$43$	−56.3643			−1.31080			−0.655399	+	0.755283i	$0.727500\pi$
$44$		0			0
$45$		−	26.8258i		−	0.596130i
$46$		0			0
$47$		−	75.7660i		−	1.61204i	−0.591886	−	0.806021i	$-0.701617\pi$
$47$		−	75.7660i		−	1.61204i	0.591886	−	0.806021i	$-0.298383\pi$
$48$		0			0
$49$	−44.3310	+	20.8749i	−0.904715	+	0.426018i
$50$		0			0
$51$	12.5205			0.245500
$52$		0			0
$53$	−1.03364			−0.0195026			−0.00975131	−	0.999952i	$-0.503104\pi$
$53$	−1.03364			−0.0195026			−0.00975131	+	0.999952i	$0.503104\pi$
$54$		0			0
$55$			1.94888i			0.0354341i
$56$		0			0
$57$	−48.6447			−0.853416
$58$		0			0
$59$			2.32111i			0.0393409i	0.999807	+	0.0196704i	$0.00626170\pi$
$59$			2.32111i			0.0393409i	−0.999807	+	0.0196704i	$0.993738\pi$
$60$		0			0
$61$		−	60.7334i		−	0.995630i	−0.867283	−	0.497815i	$-0.834136\pi$
$61$		−	60.7334i		−	0.995630i	0.867283	−	0.497815i	$-0.165864\pi$
$62$		0			0
$63$	4.58371	+	20.4936i	0.0727573	+	0.325296i
$64$		0			0
$65$	−153.858			−2.36705
$66$		0			0
$67$	−102.011			−1.52255			−0.761276	−	0.648428i	$-0.775427\pi$
$67$	−102.011			−1.52255			−0.761276	+	0.648428i	$0.775427\pi$
$68$		0			0
$69$			41.4108i			0.600157i
$70$		0			0
$71$	−22.0554			−0.310640			−0.155320	−	0.987864i	$-0.549641\pi$
$71$	−22.0554			−0.310640			−0.155320	+	0.987864i	$0.549641\pi$
$72$		0			0
$73$		−	107.319i		−	1.47012i	−0.678000	−	0.735062i	$-0.737153\pi$
$73$		−	107.319i		−	1.47012i	0.678000	−	0.735062i	$-0.262847\pi$
$74$		0			0
$75$			95.1908i			1.26921i
$76$		0			0
$77$	−0.333003	−	1.48885i	−0.00432471	−	0.0193357i
$78$		0			0
$79$	−141.639			−1.79290			−0.896450	−	0.443146i	$-0.853862\pi$
$79$	−141.639			−1.79290			−0.896450	+	0.443146i	$0.853862\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			81.0927i			0.977021i	0.872558	+	0.488510i	$0.162460\pi$
$83$			81.0927i			0.977021i	−0.872558	+	0.488510i	$0.837540\pi$
$84$		0			0
$85$	−64.6388			−0.760456
$86$		0			0
$87$		−	46.7320i		−	0.537149i
$88$		0			0
$89$		−	167.406i		−	1.88096i	−0.339846	−	0.940481i	$-0.610375\pi$
$89$		−	167.406i		−	1.88096i	0.339846	−	0.940481i	$-0.389625\pi$
$90$		0			0
$91$	117.540	−	26.2896i	1.29165	−	0.288897i
$92$		0			0
$93$	46.9218			0.504536
$94$		0			0
$95$	251.135			2.64353
$96$		0			0
$97$		−	53.8065i		−	0.554706i	−0.960768	−	0.277353i	$-0.910543\pi$
$97$		−	53.8065i		−	0.554706i	0.960768	−	0.277353i	$-0.0894572\pi$
$98$		0			0
$99$	−0.653843			−0.00660448
$100$		0			0
$101$			76.3964i			0.756400i	0.925724	+	0.378200i	$0.123457\pi$
$101$			76.3964i			0.756400i	−0.925724	+	0.378200i	$0.876543\pi$
$102$		0			0
$103$			140.563i			1.36469i	0.731031	+	0.682344i	$0.239040\pi$
$103$			140.563i			1.36469i	−0.731031	+	0.682344i	$0.760960\pi$
$104$		0			0
$105$	−23.6640	−	105.801i	−0.225372	−	1.00763i
$106$		0			0
$107$	−0.0411514			−0.000384592			−0.000192296	−	1.00000i	$-0.500061\pi$
$107$	−0.0411514			−0.000384592			−0.000192296		1.00000i	$0.500061\pi$
$108$		0			0
$109$	35.9268			0.329603			0.164802	−	0.986327i	$-0.447302\pi$
$109$	35.9268			0.329603			0.164802	+	0.986327i	$0.447302\pi$
$110$		0			0
$111$		−	31.0801i		−	0.280001i
$112$		0			0
$113$	19.2405			0.170270			0.0851349	−	0.996369i	$-0.472868\pi$
$113$	19.2405			0.170270			0.0851349	+	0.996369i	$0.472868\pi$
$114$		0			0
$115$		−	213.789i		−	1.85904i
$116$		0			0
$117$		−	51.6190i		−	0.441188i
$118$		0			0
$119$	49.3809	−	11.0448i	0.414966	−	0.0928132i
$120$		0			0
$121$	−120.952			−0.999607
$122$		0			0
$123$	32.1358			0.261267
$124$		0			0
$125$		−	267.887i		−	2.14309i
$126$		0			0
$127$	−223.492			−1.75978			−0.879891	−	0.475176i	$-0.842384\pi$
$127$	−223.492			−1.75978			−0.879891	+	0.475176i	$0.842384\pi$
$128$		0			0
$129$			97.6259i			0.756790i
$130$		0			0
$131$		−	97.0197i		−	0.740608i	−0.928911	−	0.370304i	$-0.879253\pi$
$131$		−	97.0197i		−	0.740608i	0.928911	−	0.370304i	$-0.120747\pi$
$132$		0			0
$133$	−191.855	+	42.9112i	−1.44252	+	0.322641i
$134$		0			0
$135$	−46.4637			−0.344176
$136$		0			0
$137$	−91.9842			−0.671417			−0.335709	−	0.941966i	$-0.608976\pi$
$137$	−91.9842			−0.671417			−0.335709	+	0.941966i	$0.608976\pi$
$138$		0			0
$139$			80.0063i			0.575585i	0.957693	+	0.287792i	$0.0929213\pi$
$139$			80.0063i			0.575585i	−0.957693	+	0.287792i	$0.907079\pi$
$140$		0			0
$141$	−131.231			−0.930713
$142$		0			0
$143$			3.75008i			0.0262243i
$144$		0			0
$145$			241.260i			1.66386i
$146$		0			0
$147$	36.1563	+	76.7836i	0.245962	+	0.522337i
$148$		0			0
$149$	−207.058			−1.38965			−0.694827	−	0.719177i	$-0.744519\pi$
$149$	−207.058			−1.38965			−0.694827	+	0.719177i	$0.744519\pi$
$150$		0			0
$151$	92.5634			0.613002			0.306501	−	0.951870i	$-0.400842\pi$
$151$	92.5634			0.613002			0.306501	+	0.951870i	$0.400842\pi$
$152$		0			0
$153$		−	21.6861i		−	0.141739i
$154$		0			0
$155$	−242.240			−1.56284
$156$		0			0
$157$		−	136.412i		−	0.868864i	−0.900705	−	0.434432i	$-0.856949\pi$
$157$		−	136.412i		−	0.868864i	0.900705	−	0.434432i	$-0.143051\pi$
$158$		0			0
$159$			1.79031i			0.0112598i
$160$		0			0
$161$	36.5300	+	163.324i	0.226894	+	1.01444i
$162$		0			0
$163$	69.9103			0.428897			0.214449	−	0.976735i	$-0.431205\pi$
$163$	69.9103			0.428897			0.214449	+	0.976735i	$0.431205\pi$
$164$		0			0
$165$	3.37555			0.0204579
$166$		0			0
$167$		−	223.641i		−	1.33917i	−0.742738	−	0.669583i	$-0.766473\pi$
$167$		−	223.641i		−	1.33917i	0.742738	−	0.669583i	$-0.233527\pi$
$168$		0			0
$169$	−127.057			−0.751819
$170$		0			0
$171$			84.2551i			0.492720i
$172$		0			0
$173$			259.836i			1.50194i	0.660337	+	0.750970i	$0.270414\pi$
$173$			259.836i			1.50194i	−0.660337	+	0.750970i	$0.729586\pi$
$174$		0			0
$175$	83.9712	+	375.433i	0.479835	+	2.14533i
$176$		0			0
$177$	4.02029			0.0227135
$178$		0			0
$179$	122.999			0.687147			0.343574	−	0.939126i	$-0.388362\pi$
$179$	122.999			0.687147			0.343574	+	0.939126i	$0.388362\pi$
$180$		0			0
$181$			165.001i			0.911608i	0.890080	+	0.455804i	$0.150648\pi$
$181$			165.001i			0.911608i	−0.890080	+	0.455804i	$0.849352\pi$
$182$		0			0
$183$	−105.193			−0.574827
$184$		0			0
$185$			160.455i			0.867325i
$186$		0			0
$187$			1.57548i			0.00842503i
$188$		0			0
$189$	35.4960	−	7.93922i	0.187810	−	0.0420064i
$190$		0			0
$191$	−43.7589			−0.229104			−0.114552	−	0.993417i	$-0.536543\pi$
$191$	−43.7589			−0.229104			−0.114552	+	0.993417i	$0.536543\pi$
$192$		0			0
$193$	179.372			0.929386			0.464693	−	0.885472i	$-0.346165\pi$
$193$	179.372			0.929386			0.464693	+	0.885472i	$0.346165\pi$
$194$		0			0
$195$			266.490i			1.36661i
$196$		0			0
$197$	−315.397			−1.60100			−0.800501	−	0.599332i	$-0.795433\pi$
$197$	−315.397			−1.60100			−0.800501	+	0.599332i	$0.795433\pi$
$198$		0			0
$199$			270.434i			1.35896i	0.733692	+	0.679482i	$0.237796\pi$
$199$			270.434i			1.35896i	−0.733692	+	0.679482i	$0.762204\pi$
$200$		0			0
$201$			176.688i			0.879046i
$202$		0			0
$203$	−41.2239	−	184.311i	−0.203073	−	0.907936i
$204$		0			0
$205$	−165.906			−0.809295
$206$		0			0
$207$	71.7257			0.346501
$208$		0			0
$209$		−	6.12107i		−	0.0292874i
$210$		0			0
$211$	−216.197			−1.02463			−0.512315	−	0.858798i	$-0.671212\pi$
$211$	−216.197			−1.02463			−0.512315	+	0.858798i	$0.671212\pi$
$212$		0			0
$213$			38.2012i			0.179348i
$214$		0			0
$215$		−	504.007i		−	2.34422i
$216$		0			0
$217$	185.060	−	41.3914i	0.852811	−	0.190744i
$218$		0			0
$219$	−185.882			−0.848777
$220$		0			0
$221$	−124.380			−0.562803
$222$		0			0
$223$			196.293i			0.880238i	0.897939	+	0.440119i	$0.145064\pi$
$223$			196.293i			0.880238i	−0.897939	+	0.440119i	$0.854936\pi$
$224$		0			0
$225$	164.875			0.732779
$226$		0			0
$227$		−	117.290i		−	0.516695i	−0.966052	−	0.258348i	$-0.916822\pi$
$227$		−	117.290i		−	0.516695i	0.966052	−	0.258348i	$-0.0831780\pi$
$228$		0			0
$229$			20.9887i			0.0916537i	0.998949	+	0.0458269i	$0.0145923\pi$
$229$			20.9887i			0.0916537i	−0.998949	+	0.0458269i	$0.985408\pi$
$230$		0			0
$231$	−2.57876	+	0.576778i	−0.0111635	+	0.00249687i
$232$		0			0
$233$	−66.0856			−0.283629			−0.141815	−	0.989893i	$-0.545294\pi$
$233$	−66.0856			−0.283629			−0.141815	+	0.989893i	$0.545294\pi$
$234$		0			0
$235$	677.496			2.88296
$236$		0			0
$237$			245.326i			1.03513i
$238$		0			0
$239$	−202.686			−0.848057			−0.424028	−	0.905649i	$-0.639384\pi$
$239$	−202.686			−0.848057			−0.424028	+	0.905649i	$0.639384\pi$
$240$		0			0
$241$		−	196.382i		−	0.814862i	−0.913236	−	0.407431i	$-0.866425\pi$
$241$		−	196.382i		−	0.814862i	0.913236	−	0.407431i	$-0.133575\pi$
$242$		0			0
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$	−186.662	−	396.406i	−0.761886	−	1.61798i
$246$		0			0
$247$	483.240			1.95644
$248$		0			0
$249$	140.457			0.564083
$250$		0			0
$251$			341.565i			1.36082i	0.732834	+	0.680408i	$0.238197\pi$
$251$			341.565i			1.36082i	−0.732834	+	0.680408i	$0.761803\pi$
$252$		0			0
$253$	−5.21081			−0.0205961
$254$		0			0
$255$			111.958i			0.439050i
$256$		0			0
$257$		−	437.240i		−	1.70132i	−0.525715	−	0.850661i	$-0.676202\pi$
$257$		−	437.240i		−	1.70132i	0.525715	−	0.850661i	$-0.323798\pi$
$258$		0			0
$259$	−27.4168	−	122.580i	−0.105857	−	0.473282i
$260$		0			0
$261$	−80.9421			−0.310123
$262$		0			0
$263$	−495.520			−1.88411			−0.942054	−	0.335463i	$-0.891107\pi$
$263$	−495.520			−1.88411			−0.942054	+	0.335463i	$0.891107\pi$
$264$		0			0
$265$		−	9.24274i		−	0.0348783i
$266$		0			0
$267$	−289.955			−1.08597
$268$		0			0
$269$			370.984i			1.37912i	0.724227	+	0.689562i	$0.242197\pi$
$269$			370.984i			1.37912i	−0.724227	+	0.689562i	$0.757803\pi$
$270$		0			0
$271$		−	244.038i		−	0.900509i	−0.892900	−	0.450255i	$-0.851333\pi$
$271$		−	244.038i		−	0.900509i	0.892900	−	0.450255i	$-0.148667\pi$
$272$		0			0
$273$	−45.5349	−	203.585i	−0.166795	−	0.745734i
$274$		0			0
$275$	−11.9781			−0.0435566
$276$		0			0
$277$	143.830			0.519243			0.259621	−	0.965710i	$-0.416402\pi$
$277$	143.830			0.519243			0.259621	+	0.965710i	$0.416402\pi$
$278$		0			0
$279$		−	81.2710i		−	0.291294i
$280$		0			0
$281$	296.565			1.05539			0.527695	−	0.849434i	$-0.323056\pi$
$281$	296.565			1.05539			0.527695	+	0.849434i	$0.323056\pi$
$282$		0			0
$283$		−	128.051i		−	0.452477i	−0.974072	−	0.226239i	$-0.927357\pi$
$283$		−	128.051i		−	0.452477i	0.974072	−	0.226239i	$-0.0726429\pi$
$284$		0			0
$285$		−	434.979i		−	1.52624i
$286$		0			0
$287$	126.744	−	28.3481i	0.441616	−	0.0987740i
$288$		0			0
$289$	236.746			0.819189
$290$		0			0
$291$	−93.1956			−0.320260
$292$		0			0
$293$		−	314.235i		−	1.07247i	−0.844068	−	0.536237i	$-0.819845\pi$
$293$		−	314.235i		−	1.07247i	0.844068	−	0.536237i	$-0.180155\pi$
$294$		0			0
$295$	−20.7553			−0.0703569
$296$		0			0
$297$			1.13249i			0.00381310i
$298$		0			0
$299$		−	411.378i		−	1.37585i
$300$		0			0
$301$	86.1192	+	385.037i	0.286110	+	1.27919i
$302$		0			0
$303$	132.322			0.436707
$304$		0			0
$305$	543.075			1.78057
$306$		0			0
$307$			337.365i			1.09891i	0.835523	+	0.549455i	$0.185165\pi$
$307$			337.365i			1.09891i	−0.835523	+	0.549455i	$0.814835\pi$
$308$		0			0
$309$	243.462			0.787903
$310$		0			0
$311$		−	22.0115i		−	0.0707766i	−0.999374	−	0.0353883i	$-0.988733\pi$
$311$		−	22.0115i		−	0.0707766i	0.999374	−	0.0353883i	$-0.0112668\pi$
$312$		0			0
$313$			342.307i			1.09363i	0.837253	+	0.546816i	$0.184160\pi$
$313$			342.307i			1.09363i	−0.837253	+	0.546816i	$0.815840\pi$
$314$		0			0
$315$	−183.253	+	40.9873i	−0.581756	+	0.130118i
$316$		0			0
$317$	562.714			1.77512			0.887562	−	0.460689i	$-0.152398\pi$
$317$	562.714			1.77512			0.887562	+	0.460689i	$0.152398\pi$
$318$		0			0
$319$	5.88038			0.0184338
$320$		0			0
$321$			0.0712763i			0.000222044i
$322$		0			0
$323$	203.019			0.628541
$324$		0			0
$325$		−	945.632i		−	2.90964i
$326$		0			0
$327$		−	62.2270i		−	0.190297i
$328$		0			0
$329$	−517.574	+	115.763i	−1.57317	+	0.351864i
$330$		0			0
$331$	−268.238			−0.810388			−0.405194	−	0.914231i	$-0.632796\pi$
$331$	−268.238			−0.810388			−0.405194	+	0.914231i	$0.632796\pi$
$332$		0			0
$333$	−53.8323			−0.161659
$334$		0			0
$335$		−	912.177i		−	2.72292i
$336$		0			0
$337$	394.060			1.16932			0.584659	−	0.811279i	$-0.301228\pi$
$337$	394.060			1.16932			0.584659	+	0.811279i	$0.301228\pi$
$338$		0			0
$339$		−	33.3255i		−	0.0983053i
$340$		0			0
$341$			5.90427i			0.0173146i
$342$		0			0
$343$	210.334	+	270.940i	0.613219	+	0.789913i
$344$		0			0
$345$	−370.293			−1.07331
$346$		0			0
$347$	452.174			1.30310			0.651548	−	0.758608i	$-0.274120\pi$
$347$	452.174			1.30310			0.651548	+	0.758608i	$0.274120\pi$
$348$		0			0
$349$		−	153.300i		−	0.439256i	−0.975584	−	0.219628i	$-0.929516\pi$
$349$		−	153.300i		−	0.439256i	0.975584	−	0.219628i	$-0.0704843\pi$
$350$		0			0
$351$	−89.4067			−0.254720
$352$		0			0
$353$			51.4528i			0.145759i	0.997341	+	0.0728794i	$0.0232188\pi$
$353$			51.4528i			0.145759i	−0.997341	+	0.0728794i	$0.976781\pi$
$354$		0			0
$355$		−	197.219i		−	0.555546i
$356$		0			0
$357$	−19.1301	−	85.5302i	−0.0535858	−	0.239581i
$358$		0			0
$359$	485.866			1.35339			0.676694	−	0.736264i	$-0.263412\pi$
$359$	485.866			1.35339			0.676694	+	0.736264i	$0.263412\pi$
$360$		0			0
$361$	−427.770			−1.18496
$362$		0			0
$363$			209.496i			0.577124i
$364$		0			0
$365$	959.642			2.62916
$366$		0			0
$367$			503.758i			1.37264i	0.727301	+	0.686318i	$0.240774\pi$
$367$			503.758i			1.37264i	−0.727301	+	0.686318i	$0.759226\pi$
$368$		0			0
$369$		−	55.6609i		−	0.150842i
$370$		0			0
$371$	1.57930	+	7.06101i	0.00425687	+	0.0190324i
$372$		0			0
$373$	−412.067			−1.10474			−0.552369	−	0.833600i	$-0.686276\pi$
$373$	−412.067			−1.10474			−0.552369	+	0.833600i	$0.686276\pi$
$374$		0			0
$375$	−463.994			−1.23732
$376$		0			0
$377$			464.239i			1.23140i
$378$		0			0
$379$	527.336			1.39139			0.695694	−	0.718338i	$-0.255097\pi$
$379$	527.336			1.39139			0.695694	+	0.718338i	$0.255097\pi$
$380$		0			0
$381$			387.100i			1.01601i
$382$		0			0
$383$		−	128.775i		−	0.336227i	−0.985768	−	0.168113i	$-0.946233\pi$
$383$		−	128.775i		−	0.336227i	0.985768	−	0.168113i	$-0.0537674\pi$
$384$		0			0
$385$	13.3132	−	2.97770i	0.0345797	−	0.00773427i
$386$		0			0
$387$	169.093			0.436933
$388$		0			0
$389$	183.645			0.472094			0.236047	−	0.971742i	$-0.424148\pi$
$389$	183.645			0.472094			0.236047	+	0.971742i	$0.424148\pi$
$390$		0			0
$391$		−	172.828i		−	0.442016i
$392$		0			0
$393$	−168.043			−0.427590
$394$		0			0
$395$		−	1266.53i		−	3.20640i
$396$		0			0
$397$			556.331i			1.40134i	0.713487	+	0.700668i	$0.247115\pi$
$397$			556.331i			1.40134i	−0.713487	+	0.700668i	$0.752885\pi$
$398$		0			0
$399$	74.3244	+	332.303i	0.186277	+	0.832839i
$400$		0			0
$401$	697.265			1.73881			0.869407	−	0.494096i	$-0.164501\pi$
$401$	697.265			1.73881			0.869407	+	0.494096i	$0.164501\pi$
$402$		0			0
$403$	−466.125			−1.15664
$404$		0			0
$405$			80.4775i			0.198710i
$406$		0			0
$407$	3.91087			0.00960903
$408$		0			0
$409$		−	324.662i		−	0.793795i	−0.917863	−	0.396898i	$-0.870087\pi$
$409$		−	324.662i		−	0.793795i	0.917863	−	0.396898i	$-0.129913\pi$
$410$		0			0
$411$			159.321i			0.387643i
$412$		0			0
$413$	15.8560	−	3.54644i	0.0383923	−	0.00858701i
$414$		0			0
$415$	−725.127			−1.74729
$416$		0			0
$417$	138.575			0.332314
$418$		0			0
$419$		−	153.023i		−	0.365211i	−0.983186	−	0.182606i	$-0.941547\pi$
$419$		−	153.023i		−	0.365211i	0.983186	−	0.182606i	$-0.0584531\pi$
$420$		0			0
$421$	289.979			0.688786			0.344393	−	0.938826i	$-0.388085\pi$
$421$	289.979			0.688786			0.344393	+	0.938826i	$0.388085\pi$
$422$		0			0
$423$			227.298i			0.537348i
$424$		0			0
$425$		−	397.279i		−	0.934774i
$426$		0			0
$427$	−414.883	+	92.7948i	−0.971623	+	0.217318i
$428$		0			0
$429$	6.49532			0.0151406
$430$		0			0
$431$	492.468			1.14262			0.571309	−	0.820735i	$-0.306436\pi$
$431$	492.468			1.14262			0.571309	+	0.820735i	$0.306436\pi$
$432$		0			0
$433$		−	264.451i		−	0.610741i	−0.952234	−	0.305370i	$-0.901220\pi$
$433$		−	264.451i		−	0.610741i	0.952234	−	0.305370i	$-0.0987802\pi$
$434$		0			0
$435$	417.875			0.960632
$436$		0			0
$437$			671.473i			1.53655i
$438$		0			0
$439$		−	440.202i		−	1.00274i	−0.865233	−	0.501369i	$-0.832830\pi$
$439$		−	440.202i		−	1.00274i	0.865233	−	0.501369i	$-0.167170\pi$
$440$		0			0
$441$	132.993	−	62.6246i	0.301572	−	0.142006i
$442$		0			0
$443$	−397.990			−0.898398			−0.449199	−	0.893432i	$-0.648291\pi$
$443$	−397.990			−0.898398			−0.449199	+	0.893432i	$0.648291\pi$
$444$		0			0
$445$	1496.93			3.36389
$446$		0			0
$447$			358.636i			0.802317i
$448$		0			0
$449$	−315.334			−0.702302			−0.351151	−	0.936319i	$-0.614210\pi$
$449$	−315.334			−0.702302			−0.351151	+	0.936319i	$0.614210\pi$
$450$		0			0
$451$			4.04372i			0.00896612i
$452$		0			0
$453$		−	160.324i		−	0.353917i
$454$		0			0
$455$	235.080	+	1051.04i	0.516660	+	2.30997i
$456$		0			0
$457$	−26.5749			−0.0581507			−0.0290754	−	0.999577i	$-0.509256\pi$
$457$	−26.5749			−0.0581507			−0.0290754	+	0.999577i	$0.509256\pi$
$458$		0			0
$459$	−37.5615			−0.0818333
$460$		0			0
$461$			666.865i			1.44656i	0.690554	+	0.723281i	$0.257367\pi$
$461$			666.865i			1.44656i	−0.690554	+	0.723281i	$0.742633\pi$
$462$		0			0
$463$	385.038			0.831616			0.415808	−	0.909452i	$-0.363499\pi$
$463$	385.038			0.831616			0.415808	+	0.909452i	$0.363499\pi$
$464$		0			0
$465$			419.573i			0.902307i
$466$		0			0
$467$			392.617i			0.840722i	0.907357	+	0.420361i	$0.138097\pi$
$467$			392.617i			0.840722i	−0.907357	+	0.420361i	$0.861903\pi$
$468$		0			0
$469$	155.863	+	696.859i	0.332330	+	1.48584i
$470$		0			0
$471$	−236.272			−0.501639
$472$		0			0
$473$	−12.2845			−0.0259714
$474$		0			0
$475$			1543.51i			3.24950i
$476$		0			0
$477$	3.10092			0.00650087
$478$		0			0
$479$		−	545.583i		−	1.13900i	−0.821990	−	0.569502i	$-0.807136\pi$
$479$		−	545.583i		−	1.13900i	0.821990	−	0.569502i	$-0.192864\pi$
$480$		0			0
$481$			308.752i			0.641896i
$482$		0			0
$483$	282.886	−	63.2717i	0.585686	−	0.130997i
$484$		0			0
$485$	481.135			0.992031
$486$		0			0
$487$	−721.663			−1.48186			−0.740928	−	0.671585i	$-0.765614\pi$
$487$	−721.663			−1.48186			−0.740928	+	0.671585i	$0.765614\pi$
$488$		0			0
$489$		−	121.088i		−	0.247624i
$490$		0			0
$491$	424.072			0.863691			0.431845	−	0.901948i	$-0.357863\pi$
$491$	424.072			0.863691			0.431845	+	0.901948i	$0.357863\pi$
$492$		0			0
$493$			195.036i			0.395610i
$494$		0			0
$495$		−	5.84663i		−	0.0118114i
$496$		0			0
$497$	33.6986	+	150.666i	0.0678040	+	0.303150i
$498$		0			0
$499$	121.152			0.242789			0.121395	−	0.992604i	$-0.461263\pi$
$499$	121.152			0.242789			0.121395	+	0.992604i	$0.461263\pi$
$500$		0			0
$501$	−387.357			−0.773167
$502$		0			0
$503$			617.407i			1.22745i	0.789520	+	0.613724i	$0.210329\pi$
$503$			617.407i			1.22745i	−0.789520	+	0.613724i	$0.789671\pi$
$504$		0			0
$505$	−683.132			−1.35274
$506$		0			0
$507$			220.070i			0.434063i
$508$		0			0
$509$		−	247.898i		−	0.487029i	−0.969897	−	0.243515i	$-0.921700\pi$
$509$		−	247.898i		−	0.487029i	0.969897	−	0.243515i	$-0.0783004\pi$
$510$		0			0
$511$	−733.120	+	163.973i	−1.43468	+	0.320887i
$512$		0			0
$513$	145.934			0.284472
$514$		0			0
$515$	−1256.91			−2.44060
$516$		0			0
$517$		−	16.5130i		−	0.0319401i
$518$		0			0
$519$	450.048			0.867145
$520$		0			0
$521$			121.749i			0.233684i	0.993151	+	0.116842i	$0.0372771\pi$
$521$			121.749i			0.233684i	−0.993151	+	0.116842i	$0.962723\pi$
$522$		0			0
$523$		−	157.002i		−	0.300195i	−0.988671	−	0.150098i	$-0.952041\pi$
$523$		−	157.002i		−	0.300195i	0.988671	−	0.150098i	$-0.0479588\pi$
$524$		0			0
$525$	650.269	−	145.442i	1.23861	−	0.277033i
$526$		0			0
$527$	−195.828			−0.371591
$528$		0			0
$529$	42.6188			0.0805649
$530$		0			0
$531$		−	6.96334i		−	0.0131136i
$532$		0			0
$533$	−319.239			−0.598948
$534$		0			0
$535$		−	0.367973i		−	0.000687801i
$536$		0			0
$537$		−	213.041i		−	0.396725i
$538$		0			0
$539$	−9.66184	+	4.54963i	−0.0179255	+	0.00844087i
$540$		0			0
$541$	−44.7003			−0.0826253			−0.0413126	−	0.999146i	$-0.513154\pi$
$541$	−44.7003			−0.0826253			−0.0413126	+	0.999146i	$0.513154\pi$
$542$		0			0
$543$	285.790			0.526317
$544$		0			0
$545$			321.255i			0.589459i
$546$		0			0
$547$	−135.263			−0.247282			−0.123641	−	0.992327i	$-0.539457\pi$
$547$	−135.263			−0.247282			−0.123641	+	0.992327i	$0.539457\pi$
$548$		0			0
$549$			182.200i			0.331877i
$550$		0			0
$551$		−	757.754i		−	1.37523i
$552$		0			0
$553$	216.411	+	967.567i	0.391339	+	1.74967i
$554$		0			0
$555$	277.917			0.500750
$556$		0			0
$557$	13.3127			0.0239006			0.0119503	−	0.999929i	$-0.496196\pi$
$557$	13.3127			0.0239006			0.0119503	+	0.999929i	$0.496196\pi$
$558$		0			0
$559$		−	969.822i		−	1.73492i
$560$		0			0
$561$	2.72881			0.00486420
$562$		0			0
$563$		−	323.122i		−	0.573928i	−0.957941	−	0.286964i	$-0.907354\pi$
$563$		−	323.122i		−	0.573928i	0.957941	−	0.286964i	$-0.0926461\pi$
$564$		0			0
$565$			172.047i			0.304509i
$566$		0			0
$567$	−13.7511	−	61.4809i	−0.0242524	−	0.108432i
$568$		0			0
$569$	−673.022			−1.18282			−0.591408	−	0.806373i	$-0.701428\pi$
$569$	−673.022			−1.18282			−0.591408	+	0.806373i	$0.701428\pi$
$570$		0			0
$571$	−731.978			−1.28192			−0.640962	−	0.767573i	$-0.721464\pi$
$571$	−731.978			−1.28192			−0.640962	+	0.767573i	$0.721464\pi$
$572$		0			0
$573$			75.7926i			0.132273i
$574$		0			0
$575$	1313.98			2.28518
$576$		0			0
$577$		−	37.6656i		−	0.0652784i	−0.999467	−	0.0326392i	$-0.989609\pi$
$577$		−	37.6656i		−	0.0652784i	0.999467	−	0.0326392i	$-0.0103912\pi$
$578$		0			0
$579$		−	310.681i		−	0.536582i
$580$		0			0
$581$	553.962	−	123.902i	0.953463	−	0.213256i
$582$		0			0
$583$	−0.225279			−0.000386414
$584$		0			0
$585$	461.574			0.789015
$586$		0			0
$587$			825.132i			1.40568i	0.711350	+	0.702838i	$0.248084\pi$
$587$			825.132i			1.40568i	−0.711350	+	0.702838i	$0.751916\pi$
$588$		0			0
$589$	760.833			1.29174
$590$		0			0
$591$			546.284i			0.924339i
$592$		0			0
$593$			391.122i			0.659566i	0.944057	+	0.329783i	$0.106976\pi$
$593$			391.122i			0.659566i	−0.944057	+	0.329783i	$0.893024\pi$
$594$		0			0
$595$	98.7618	+	441.562i	0.165986	+	0.742120i
$596$		0			0
$597$	468.405			0.784599
$598$		0			0
$599$	−338.153			−0.564530			−0.282265	−	0.959337i	$-0.591086\pi$
$599$	−338.153			−0.564530			−0.282265	+	0.959337i	$0.591086\pi$
$600$		0			0
$601$		−	551.622i		−	0.917840i	−0.888478	−	0.458920i	$-0.848237\pi$
$601$		−	551.622i		−	0.917840i	0.888478	−	0.458920i	$-0.151763\pi$
$602$		0			0
$603$	306.033			0.507517
$604$		0			0
$605$		−	1081.55i		−	1.78769i
$606$		0			0
$607$		−	566.613i		−	0.933465i	−0.884399	−	0.466732i	$-0.845431\pi$
$607$		−	566.613i		−	0.933465i	0.884399	−	0.466732i	$-0.154569\pi$
$608$		0			0
$609$	−319.236	+	71.4019i	−0.524197	+	0.117245i
$610$		0			0
$611$	1303.65			2.13364
$612$		0			0
$613$	−486.369			−0.793424			−0.396712	−	0.917943i	$-0.629849\pi$
$613$	−486.369			−0.793424			−0.396712	+	0.917943i	$0.629849\pi$
$614$		0			0
$615$			287.357i			0.467247i
$616$		0			0
$617$	534.919			0.866968			0.433484	−	0.901161i	$-0.357284\pi$
$617$	534.919			0.866968			0.433484	+	0.901161i	$0.357284\pi$
$618$		0			0
$619$			181.263i			0.292833i	0.989223	+	0.146416i	$0.0467739\pi$
$619$			181.263i			0.292833i	−0.989223	+	0.146416i	$0.953226\pi$
$620$		0			0
$621$		−	124.232i		−	0.200052i
$622$		0			0
$623$	−1143.58	+	255.780i	−1.83561	+	0.410561i
$624$		0			0
$625$	1021.47			1.63435
$626$		0			0
$627$	−10.6020			−0.0169091
$628$		0			0
$629$			129.713i			0.206221i
$630$		0			0
$631$	495.437			0.785162			0.392581	−	0.919717i	$-0.371582\pi$
$631$	495.437			0.785162			0.392581	+	0.919717i	$0.371582\pi$
$632$		0			0
$633$			374.464i			0.591570i
$634$		0			0
$635$		−	1998.46i		−	3.14718i
$636$		0			0
$637$	−359.180	−	762.774i	−0.563861	−	1.19745i
$638$		0			0
$639$	66.1663			0.103547
$640$		0			0
$641$	165.776			0.258622			0.129311	−	0.991604i	$-0.458724\pi$
$641$	165.776			0.258622			0.129311	+	0.991604i	$0.458724\pi$
$642$		0			0
$643$			959.473i			1.49218i	0.665844	+	0.746091i	$0.268071\pi$
$643$			959.473i			1.49218i	−0.665844	+	0.746091i	$0.731929\pi$
$644$		0			0
$645$	−872.965			−1.35343
$646$		0			0
$647$			537.895i			0.831368i	0.909509	+	0.415684i	$0.136458\pi$
$647$			537.895i			0.831368i	−0.909509	+	0.415684i	$0.863542\pi$
$648$		0			0
$649$			0.505881i			0.000779478i
$650$		0			0
$651$	−71.6920	−	320.533i	−0.110126	−	0.492370i
$652$		0			0
$653$	−1045.13			−1.60050			−0.800251	−	0.599665i	$-0.795300\pi$
$653$	−1045.13			−1.60050			−0.800251	+	0.599665i	$0.795300\pi$
$654$		0			0
$655$	867.545			1.32450
$656$		0			0
$657$			321.957i			0.490042i
$658$		0			0
$659$	286.734			0.435105			0.217553	−	0.976049i	$-0.430193\pi$
$659$	286.734			0.435105			0.217553	+	0.976049i	$0.430193\pi$
$660$		0			0
$661$		−	397.426i		−	0.601249i	−0.953743	−	0.300625i	$-0.902805\pi$
$661$		−	397.426i		−	0.601249i	0.953743	−	0.300625i	$-0.0971952\pi$
$662$		0			0
$663$			215.432i			0.324935i
$664$		0			0
$665$	−383.710	−	1715.56i	−0.577008	−	2.57979i
$666$		0			0
$667$	−645.070			−0.967121
$668$		0			0
$669$	339.990			0.508206
$670$		0			0
$671$		−	13.2367i		−	0.0197268i
$672$		0			0
$673$	−506.181			−0.752126			−0.376063	−	0.926594i	$-0.622722\pi$
$673$	−506.181			−0.752126			−0.376063	+	0.926594i	$0.622722\pi$
$674$		0			0
$675$		−	285.572i		−	0.423070i
$676$		0			0
$677$		−	360.038i		−	0.531814i	−0.963999	−	0.265907i	$-0.914329\pi$
$677$		−	360.038i		−	0.531814i	0.963999	−	0.265907i	$-0.0856714\pi$
$678$		0			0
$679$	−367.564	+	82.2112i	−0.541331	+	0.121077i
$680$		0			0
$681$	−203.152			−0.298314
$682$		0			0
$683$	426.969			0.625137			0.312568	−	0.949895i	$-0.398811\pi$
$683$	426.969			0.625137			0.312568	+	0.949895i	$0.398811\pi$
$684$		0			0
$685$		−	822.518i		−	1.20076i
$686$		0			0
$687$	36.3535			0.0529163
$688$		0			0
$689$		−	17.7851i		−	0.0258129i
$690$		0			0
$691$		−	322.616i		−	0.466882i	−0.972371	−	0.233441i	$-0.925001\pi$
$691$		−	322.616i		−	0.466882i	0.972371	−	0.233441i	$-0.0749986\pi$
$692$		0			0
$693$	0.999009	+	4.46654i	0.00144157	+	0.00644523i
$694$		0			0
$695$	−715.412			−1.02937
$696$		0			0
$697$	−134.119			−0.192423
$698$		0			0
$699$			114.464i			0.163753i
$700$		0			0
$701$	311.014			0.443672			0.221836	−	0.975084i	$-0.428795\pi$
$701$	311.014			0.443672			0.221836	+	0.975084i	$0.428795\pi$
$702$		0			0
$703$		−	503.961i		−	0.716872i
$704$		0			0
$705$		−	1173.46i		−	1.66448i
$706$		0			0
$707$	521.880	−	116.726i	0.738161	−	0.165101i
$708$		0			0
$709$	−432.555			−0.610092			−0.305046	−	0.952338i	$-0.598672\pi$
$709$	−432.555			−0.610092			−0.305046	+	0.952338i	$0.598672\pi$
$710$		0			0
$711$	424.917			0.597633
$712$		0			0
$713$		−	647.691i		−	0.908402i
$714$		0			0
$715$	−33.5330			−0.0468993
$716$		0			0
$717$			351.062i			0.489626i
$718$		0			0
$719$		−	880.875i		−	1.22514i	−0.790417	−	0.612569i	$-0.790136\pi$
$719$		−	880.875i		−	1.22514i	0.790417	−	0.612569i	$-0.209864\pi$
$720$		0			0
$721$	960.216	−	214.767i	1.33178	−	0.297873i
$722$		0			0
$723$	−340.143			−0.470461
$724$		0			0
$725$	−1482.82			−2.04527
$726$		0			0
$727$			551.001i			0.757910i	0.925415	+	0.378955i	$0.123717\pi$
$727$			551.001i			0.757910i	−0.925415	+	0.378955i	$0.876283\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	407.442i		−	0.557376i
$732$		0			0
$733$			779.917i			1.06401i	0.846742	+	0.532003i	$0.178561\pi$
$733$			779.917i			1.06401i	−0.846742	+	0.532003i	$0.821439\pi$
$734$		0			0
$735$	−686.595	+	323.308i	−0.934143	+	0.439875i
$736$		0			0
$737$	−22.2331			−0.0301670
$738$		0			0
$739$	−1248.94			−1.69004			−0.845019	−	0.534736i	$-0.820411\pi$
$739$	−1248.94			−1.69004			−0.845019	+	0.534736i	$0.820411\pi$
$740$		0			0
$741$		−	836.997i		−	1.12955i
$742$		0			0
$743$	75.4519			0.101550			0.0507752	−	0.998710i	$-0.483831\pi$
$743$	75.4519			0.101550			0.0507752	+	0.998710i	$0.483831\pi$
$744$		0			0
$745$		−	1851.51i		−	2.48524i
$746$		0			0
$747$		−	243.278i		−	0.325674i
$748$		0			0
$749$	0.0628753	+	0.281114i	8.39457e−5	+	0.000375319i
$750$		0			0
$751$	127.424			0.169673			0.0848365	−	0.996395i	$-0.472963\pi$
$751$	127.424			0.169673			0.0848365	+	0.996395i	$0.472963\pi$
$752$		0			0
$753$	591.607			0.785667
$754$		0			0
$755$			827.697i			1.09629i
$756$		0			0
$757$	−952.912			−1.25880			−0.629401	−	0.777081i	$-0.716700\pi$
$757$	−952.912			−1.25880			−0.629401	+	0.777081i	$0.716700\pi$
$758$		0			0
$759$			9.02539i			0.0118912i
$760$		0			0
$761$			675.155i			0.887194i	0.896226	+	0.443597i	$0.146298\pi$
$761$			675.155i			0.887194i	−0.896226	+	0.443597i	$0.853702\pi$
$762$		0			0
$763$	−54.8926	−	245.424i	−0.0719432	−	0.321656i
$764$		0			0
$765$	193.916			0.253485
$766$		0			0
$767$	−39.9378			−0.0520702
$768$		0			0
$769$			382.917i			0.497941i	0.968511	+	0.248971i	$0.0800922\pi$
$769$			382.917i			0.497941i	−0.968511	+	0.248971i	$0.919908\pi$
$770$		0			0
$771$	−757.321			−0.982258
$772$		0			0
$773$			435.628i			0.563555i	0.959480	+	0.281777i	$0.0909239\pi$
$773$			435.628i			0.563555i	−0.959480	+	0.281777i	$0.909076\pi$
$774$		0			0
$775$		−	1488.84i		−	1.92109i
$776$		0			0
$777$	−212.315	+	47.4874i	−0.273249	+	0.0611163i
$778$		0			0
$779$	521.079			0.668908
$780$		0			0
$781$	−4.80693			−0.00615484
$782$		0			0
$783$			140.196i			0.179050i
$784$		0			0
$785$	1219.79			1.55387
$786$		0			0
$787$		−	271.750i		−	0.345299i	−0.984983	−	0.172649i	$-0.944767\pi$
$787$		−	271.750i		−	0.345299i	0.984983	−	0.172649i	$-0.0552327\pi$
$788$		0			0
$789$			858.266i			1.08779i
$790$		0			0
$791$	−29.3976	−	131.436i	−0.0371651	−	0.166164i
$792$		0			0
$793$	1045.00			1.31778
$794$		0			0
$795$	−16.0089			−0.0201370
$796$		0			0
$797$			663.119i			0.832019i	0.909360	+	0.416009i	$0.136572\pi$
$797$			663.119i			0.832019i	−0.909360	+	0.416009i	$0.863428\pi$
$798$		0			0
$799$	547.691			0.685470
$800$		0			0
$801$			502.217i			0.626988i
$802$		0			0
$803$		−	23.3899i		−	0.0291282i
$804$		0			0
$805$	−1460.44	+	326.649i	−1.81421	+	0.405775i
$806$		0			0
$807$	642.563			0.796237
$808$		0			0
$809$	−572.673			−0.707878			−0.353939	−	0.935269i	$-0.615158\pi$
$809$	−572.673			−0.707878			−0.353939	+	0.935269i	$0.615158\pi$
$810$		0			0
$811$		−	471.069i		−	0.580849i	−0.956898	−	0.290425i	$-0.906203\pi$
$811$		−	471.069i		−	0.580849i	0.956898	−	0.290425i	$-0.0937965\pi$
$812$		0			0
$813$	−422.686			−0.519909
$814$		0			0
$815$			625.134i			0.767035i
$816$		0			0
$817$			1582.99i			1.93757i
$818$		0			0
$819$	−352.620	+	78.8688i	−0.430550	+	0.0962989i
$820$		0			0
$821$	−59.6541			−0.0726603			−0.0363301	−	0.999340i	$-0.511567\pi$
$821$	−59.6541			−0.0726603			−0.0363301	+	0.999340i	$0.511567\pi$
$822$		0			0
$823$	−353.320			−0.429307			−0.214654	−	0.976690i	$-0.568862\pi$
$823$	−353.320			−0.429307			−0.214654	+	0.976690i	$0.568862\pi$
$824$		0			0
$825$			20.7466i			0.0251474i
$826$		0			0
$827$	1045.08			1.26370			0.631850	−	0.775090i	$-0.282296\pi$
$827$	1045.08			1.26370			0.631850	+	0.775090i	$0.282296\pi$
$828$		0			0
$829$		−	95.8922i		−	0.115672i	−0.998326	−	0.0578361i	$-0.981580\pi$
$829$		−	95.8922i		−	0.115672i	0.998326	−	0.0578361i	$-0.0184201\pi$
$830$		0			0
$831$		−	249.121i		−	0.299785i
$832$		0			0
$833$	−150.899	−	320.456i	−0.181151	−	0.384701i
$834$		0			0
$835$	1999.78			2.39495
$836$		0			0
$837$	−140.765			−0.168179
$838$		0			0
$839$			903.440i			1.07681i	0.842687	+	0.538403i	$0.180972\pi$
$839$			903.440i			1.07681i	−0.842687	+	0.538403i	$0.819028\pi$
$840$		0			0
$841$	−113.042			−0.134413
$842$		0			0
$843$		−	513.665i		−	0.609330i
$844$		0			0
$845$		−	1136.14i		−	1.34454i
$846$		0			0
$847$	184.804	+	826.253i	0.218186	+	0.975505i
$848$		0			0
$849$	−221.791			−0.261238
$850$		0			0
$851$	−429.017			−0.504133
$852$		0			0
$853$		−	380.188i		−	0.445707i	−0.974852	−	0.222854i	$-0.928463\pi$
$853$		−	380.188i		−	0.445707i	0.974852	−	0.222854i	$-0.0715372\pi$
$854$		0			0
$855$	−753.405			−0.881176
$856$		0			0
$857$			1312.54i			1.53155i	0.643109	+	0.765775i	$0.277644\pi$
$857$			1312.54i			1.53155i	−0.643109	+	0.765775i	$0.722356\pi$
$858$		0			0
$859$			1253.26i			1.45898i	0.683991	+	0.729491i	$0.260243\pi$
$859$			1253.26i			1.45898i	−0.683991	+	0.729491i	$0.739757\pi$
$860$		0			0
$861$	−49.1004	−	219.527i	−0.0570272	−	0.254967i
$862$		0			0
$863$	188.946			0.218941			0.109471	−	0.993990i	$-0.465084\pi$
$863$	188.946			0.218941			0.109471	+	0.993990i	$0.465084\pi$
$864$		0			0
$865$	−2323.44			−2.68605
$866$		0			0
$867$		−	410.056i		−	0.472959i
$868$		0			0
$869$	−30.8699			−0.0355235
$870$		0			0
$871$		−	1755.23i		−	2.01519i
$872$		0			0
$873$			161.420i			0.184902i
$874$		0			0
$875$	−1829.99	+	409.305i	−2.09142	+	0.467777i
$876$		0			0
$877$	−1103.53			−1.25831			−0.629153	−	0.777282i	$-0.716598\pi$
$877$	−1103.53			−1.25831			−0.629153	+	0.777282i	$0.716598\pi$
$878$		0			0
$879$	−544.271			−0.619193
$880$		0			0
$881$			488.950i			0.554994i	0.960727	+	0.277497i	$0.0895048\pi$
$881$			488.950i			0.554994i	−0.960727	+	0.277497i	$0.910495\pi$
$882$		0			0
$883$	254.920			0.288698			0.144349	−	0.989527i	$-0.453891\pi$
$883$	254.920			0.288698			0.144349	+	0.989527i	$0.453891\pi$
$884$		0			0
$885$			35.9492i			0.0406205i
$886$		0			0
$887$			1195.09i			1.34734i	0.739033	+	0.673669i	$0.235283\pi$
$887$			1195.09i			1.34734i	−0.739033	+	0.673669i	$0.764717\pi$
$888$		0			0
$889$	341.475	+	1526.72i	0.384111	+	1.71735i
$890$		0			0
$891$	1.96153			0.00220149
$892$		0			0
$893$	−2127.89			−2.38286
$894$		0			0
$895$			1099.85i			1.22889i
$896$		0			0
$897$	−712.528			−0.794345
$898$		0			0
$899$			730.916i			0.813033i
$900$		0			0
$901$		−	7.47188i		−	0.00829287i
$902$		0			0
$903$	666.903	−	149.163i	0.738542	−	0.165186i
$904$		0			0
$905$	−1475.43			−1.63031
$906$		0			0
$907$	771.911			0.851059			0.425530	−	0.904944i	$-0.360088\pi$
$907$	771.911			0.851059			0.425530	+	0.904944i	$0.360088\pi$
$908$		0			0
$909$		−	229.189i		−	0.252133i
$910$		0			0
$911$	−674.122			−0.739980			−0.369990	−	0.929036i	$-0.620639\pi$
$911$	−674.122			−0.739980			−0.369990	+	0.929036i	$0.620639\pi$
$912$		0			0
$913$			17.6740i			0.0193581i
$914$		0			0
$915$		−	940.634i		−	1.02802i
$916$		0			0
$917$	−662.762	+	148.237i	−0.722751	+	0.161654i
$918$		0			0
$919$	1654.31			1.80012			0.900058	−	0.435769i	$-0.143524\pi$
$919$	1654.31			1.80012			0.900058	+	0.435769i	$0.143524\pi$
$920$		0			0
$921$	584.334			0.634456
$922$		0			0
$923$		−	379.493i		−	0.411152i
$924$		0			0
$925$	−986.180			−1.06614
$926$		0			0
$927$		−	421.689i		−	0.454896i
$928$		0			0
$929$			973.605i			1.04801i	0.851714	+	0.524007i	$0.175563\pi$
$929$			973.605i			1.04801i	−0.851714	+	0.524007i	$0.824437\pi$
$930$		0			0
$931$	586.272	+	1245.04i	0.629723	+	1.33731i
$932$		0			0
$933$	−38.1251			−0.0408629
$934$		0			0
$935$	−14.0879			−0.0150672
$936$		0			0
$937$			606.298i			0.647063i	0.946217	+	0.323532i	$0.104870\pi$
$937$			606.298i			0.647063i	−0.946217	+	0.323532i	$0.895130\pi$
$938$		0			0
$939$	592.893			0.631409
$940$		0			0
$941$		−	640.116i		−	0.680251i	−0.940380	−	0.340125i	$-0.889530\pi$
$941$		−	640.116i		−	0.680251i	0.940380	−	0.340125i	$-0.110470\pi$
$942$		0			0
$943$		−	443.590i		−	0.470403i
$944$		0			0
$945$	70.9921	+	317.404i	0.0751239	+	0.335877i
$946$		0			0
$947$	−1525.84			−1.61124			−0.805619	−	0.592434i	$-0.798167\pi$
$947$	−1525.84			−1.61124			−0.805619	+	0.592434i	$0.798167\pi$
$948$		0			0
$949$	1846.57			1.94580
$950$		0			0
$951$		−	974.649i		−	1.02487i
$952$		0			0
$953$	478.187			0.501770			0.250885	−	0.968017i	$-0.419278\pi$
$953$	478.187			0.501770			0.250885	+	0.968017i	$0.419278\pi$
$954$		0			0
$955$		−	391.289i		−	0.409727i
$956$		0			0
$957$		−	10.1851i		−	0.0106428i
$958$		0			0
$959$	140.543	+	628.364i	0.146552	+	0.655228i
$960$		0			0
$961$	227.114			0.236331
$962$		0			0
$963$	0.123454			0.000128197
$964$		0			0
$965$			1603.93i			1.66211i
$966$		0			0
$967$	−1654.02			−1.71047			−0.855233	−	0.518244i	$-0.826586\pi$
$967$	−1654.02			−1.71047			−0.855233	+	0.518244i	$0.826586\pi$
$968$		0			0
$969$		−	351.639i		−	0.362888i
$970$		0			0
$971$		−	171.930i		−	0.177065i	−0.996073	−	0.0885323i	$-0.971782\pi$
$971$		−	171.930i		−	0.177065i	0.996073	−	0.0885323i	$-0.0282177\pi$
$972$		0			0
$973$	546.540	−	122.242i	0.561706	−	0.125634i
$974$		0			0
$975$	−1637.88			−1.67988
$976$		0			0
$977$	−499.902			−0.511670			−0.255835	−	0.966720i	$-0.582350\pi$
$977$	−499.902			−0.511670			−0.255835	+	0.966720i	$0.582350\pi$
$978$		0			0
$979$		−	36.4857i		−	0.0372683i
$980$		0			0
$981$	−107.780			−0.109868
$982$		0			0
$983$		−	497.601i		−	0.506206i	−0.967439	−	0.253103i	$-0.918549\pi$
$983$		−	497.601i		−	0.506206i	0.967439	−	0.253103i	$-0.0814512\pi$
$984$		0			0
$985$		−	2820.27i		−	2.86322i
$986$		0			0
$987$	200.508	+	896.464i	0.203149	+	0.908272i
$988$		0			0
$989$	1347.59			1.36258
$990$		0			0
$991$	328.000			0.330979			0.165490	−	0.986212i	$-0.447080\pi$
$991$	328.000			0.330979			0.165490	+	0.986212i	$0.447080\pi$
$992$		0			0
$993$			464.603i			0.467878i
$994$		0			0
$995$	−2418.21			−2.43036
$996$		0			0
$997$			966.951i			0.969861i	0.874553	+	0.484930i	$0.161155\pi$
$997$			966.951i			0.969861i	−0.874553	+	0.484930i	$0.838845\pi$
$998$		0			0
$999$			93.2403i			0.0933336i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.f.j.769.8		16
4.3	odd	2	inner	1344.3.f.j.769.16		16
7.6	odd	2	inner	1344.3.f.j.769.9		16
8.3	odd	2		672.3.f.b.97.1	✓	16
8.5	even	2		672.3.f.b.97.9	yes	16
24.5	odd	2		2016.3.f.g.1441.15		16
24.11	even	2		2016.3.f.g.1441.16		16
28.27	even	2	inner	1344.3.f.j.769.1		16
56.13	odd	2		672.3.f.b.97.8	yes	16
56.27	even	2		672.3.f.b.97.16	yes	16
168.83	odd	2		2016.3.f.g.1441.2		16
168.125	even	2		2016.3.f.g.1441.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.3.f.b.97.1	✓	16	8.3	odd	2
672.3.f.b.97.8	yes	16	56.13	odd	2
672.3.f.b.97.9	yes	16	8.5	even	2
672.3.f.b.97.16	yes	16	56.27	even	2
1344.3.f.j.769.1		16	28.27	even	2	inner
1344.3.f.j.769.8		16	1.1	even	1	trivial
1344.3.f.j.769.9		16	7.6	odd	2	inner
1344.3.f.j.769.16		16	4.3	odd	2	inner
2016.3.f.g.1441.1		16	168.125	even	2
2016.3.f.g.1441.2		16	168.83	odd	2
2016.3.f.g.1441.15		16	24.5	odd	2
2016.3.f.g.1441.16		16	24.11	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)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +8.94195i q^{5} +(-1.52790 - 6.83122i) q^{7} -3.00000 q^{9} +0.217948 q^{11} +17.2063i q^{13} +15.4879 q^{15} +7.22871i q^{17} -28.0850i q^{19} +(-11.8320 + 2.64641i) q^{21} -23.9086 q^{23} -54.9584 q^{25} +5.19615i q^{27} +26.9807 q^{29} +27.0903i q^{31} -0.377496i q^{33} +(61.0844 - 13.6624i) q^{35} +17.9441 q^{37} +29.8022 q^{39} +18.5536i q^{41} -56.3643 q^{43} -26.8258i q^{45} -75.7660i q^{47} +(-44.3310 + 20.8749i) q^{49} +12.5205 q^{51} -1.03364 q^{53} +1.94888i q^{55} -48.6447 q^{57} +2.32111i q^{59} -60.7334i q^{61} +(4.58371 + 20.4936i) q^{63} -153.858 q^{65} -102.011 q^{67} +41.4108i q^{69} -22.0554 q^{71} -107.319i q^{73} +95.1908i q^{75} +(-0.333003 - 1.48885i) q^{77} -141.639 q^{79} +9.00000 q^{81} +81.0927i q^{83} -64.6388 q^{85} -46.7320i q^{87} -167.406i q^{89} +(117.540 - 26.2896i) q^{91} +46.9218 q^{93} +251.135 q^{95} -53.8065i q^{97} -0.653843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 24 q^{21} - 64 q^{25} + 128 q^{29} - 48 q^{37} - 256 q^{49} + 160 q^{53} - 144 q^{57} - 288 q^{65} - 128 q^{77} + 144 q^{81} + 16 q^{85} - 96 q^{93}+O(q^{100}) \) 16 * q - 48 * q^9 - 24 * q^21 - 64 * q^25 + 128 * q^29 - 48 * q^37 - 256 * q^49 + 160 * q^53 - 144 * q^57 - 288 * q^65 - 128 * q^77 + 144 * q^81 + 16 * q^85 - 96 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more