LMFDB - Embedded newform 672.3.f.b.97.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,3,Mod(97,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, 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("672.97"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	672.f (of order $2$, degree $1$, 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:	$18.3106737650$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$0.0550989 + 2.76316i$ of defining polynomial
Character	$\chi$	$=$	672.97
Dual form			672.3.f.b.97.16

$q$-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 $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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	$-0.647749\pi$
$5$		−	8.94195i		−	1.78839i	0.447678	−	0.894195i	$-0.352251\pi$
$6$		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	$-0.769801\pi$
$13$		−	17.2063i		−	1.32356i	0.749697	−	0.661781i	$-0.230199\pi$
$14$		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	$-0.326025\pi$
$23$	23.9086			1.03950			0.519751	+	0.854318i	$0.326025\pi$
$24$		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	$-0.654012\pi$
$29$	−26.9807			−0.930369			−0.465185	+	0.885214i	$0.654012\pi$
$30$		0			0
$31$		−	27.0903i		−	0.873882i	−0.899490	−	0.436941i	$-0.856062\pi$
$31$		−	27.0903i		−	0.873882i	0.899490	−	0.436941i	$-0.143938\pi$
$32$		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	$-0.577963\pi$
$37$	−17.9441			−0.484976			−0.242488	+	0.970154i	$0.577963\pi$
$38$		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	$0.298383\pi$
$47$			75.7660i			1.61204i	−0.591886	+	0.806021i	$0.701617\pi$
$48$		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	$-0.496896\pi$
$53$	1.03364			0.0195026			0.00975131	+	0.999952i	$0.496896\pi$
$54$		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	$0.165864\pi$
$61$			60.7334i			0.995630i	−0.867283	+	0.497815i	$0.834136\pi$
$62$		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	$-0.450359\pi$
$71$	22.0554			0.310640			0.155320	+	0.987864i	$0.450359\pi$
$72$		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	$-0.146138\pi$
$79$	141.639			1.79290			0.896450	+	0.443146i	$0.146138\pi$
$80$		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	$-0.876543\pi$
$101$		−	76.3964i		−	0.756400i	0.925724	−	0.378200i	$-0.123457\pi$
$102$		0			0
$103$		−	140.563i		−	1.36469i	−0.731031	−	0.682344i	$-0.760960\pi$
$103$		−	140.563i		−	1.36469i	0.731031	−	0.682344i	$-0.239040\pi$
$104$		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	$-0.552698\pi$
$109$	−35.9268			−0.329603			−0.164802	+	0.986327i	$0.552698\pi$
$110$		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	$-0.157616\pi$
$127$	223.492			1.75978			0.879891	+	0.475176i	$0.157616\pi$
$128$		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	$-0.255481\pi$
$149$	207.058			1.38965			0.694827	+	0.719177i	$0.255481\pi$
$150$		0			0
$151$	−92.5634			−0.613002			−0.306501	−	0.951870i	$-0.599158\pi$
$151$	−92.5634			−0.613002			−0.306501	+	0.951870i	$0.599158\pi$
$152$		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	$0.143051\pi$
$157$			136.412i			0.868864i	−0.900705	+	0.434432i	$0.856949\pi$
$158$		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	$0.233527\pi$
$167$			223.641i			1.33917i	−0.742738	+	0.669583i	$0.766473\pi$
$168$		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	$-0.729586\pi$
$173$		−	259.836i		−	1.50194i	0.660337	−	0.750970i	$-0.270414\pi$
$174$		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	$-0.849352\pi$
$181$		−	165.001i		−	0.911608i	0.890080	−	0.455804i	$-0.150648\pi$
$182$		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	$-0.463457\pi$
$191$	43.7589			0.229104			0.114552	+	0.993417i	$0.463457\pi$
$192$		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	$-0.204567\pi$
$197$	315.397			1.60100			0.800501	+	0.599332i	$0.204567\pi$
$198$		0			0
$199$		−	270.434i		−	1.35896i	−0.733692	−	0.679482i	$-0.762204\pi$
$199$		−	270.434i		−	1.35896i	0.733692	−	0.679482i	$-0.237796\pi$
$200$		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	$-0.854936\pi$
$223$		−	196.293i		−	0.880238i	0.897939	−	0.440119i	$-0.145064\pi$
$224$		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	$-0.985408\pi$
$229$		−	20.9887i		−	0.0916537i	0.998949	−	0.0458269i	$-0.0145923\pi$
$230$		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	$-0.360616\pi$
$239$	202.686			0.848057			0.424028	+	0.905649i	$0.360616\pi$
$240$		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	$-0.108893\pi$
$263$	495.520			1.88411			0.942054	+	0.335463i	$0.108893\pi$
$264$		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	$-0.757803\pi$
$269$		−	370.984i		−	1.37912i	0.724227	−	0.689562i	$-0.242197\pi$
$270$		0			0
$271$			244.038i			0.900509i	0.892900	+	0.450255i	$0.148667\pi$
$271$			244.038i			0.900509i	−0.892900	+	0.450255i	$0.851333\pi$
$272$		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	$-0.583598\pi$
$277$	−143.830			−0.519243			−0.259621	+	0.965710i	$0.583598\pi$
$278$		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	$0.180155\pi$
$293$			314.235i			1.07247i	−0.844068	+	0.536237i	$0.819845\pi$
$294$		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	$0.0112668\pi$
$311$			22.0115i			0.0707766i	−0.999374	+	0.0353883i	$0.988733\pi$
$312$		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	$-0.847602\pi$
$317$	−562.714			−1.77512			−0.887562	+	0.460689i	$0.847602\pi$
$318$		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	$0.0704843\pi$
$349$			153.300i			0.439256i	−0.975584	+	0.219628i	$0.929516\pi$
$350$		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	$-0.736588\pi$
$359$	−485.866			−1.35339			−0.676694	+	0.736264i	$0.736588\pi$
$360$		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	$-0.759226\pi$
$367$		−	503.758i		−	1.37264i	0.727301	−	0.686318i	$-0.240774\pi$
$368$		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	$-0.313724\pi$
$373$	412.067			1.10474			0.552369	+	0.833600i	$0.313724\pi$
$374$		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	$0.0537674\pi$
$383$			128.775i			0.336227i	−0.985768	+	0.168113i	$0.946233\pi$
$384$		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	$-0.575852\pi$
$389$	−183.645			−0.472094			−0.236047	+	0.971742i	$0.575852\pi$
$390$		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	$-0.752885\pi$
$397$		−	556.331i		−	1.40134i	0.713487	−	0.700668i	$-0.247115\pi$
$398$		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	$-0.611915\pi$
$421$	−289.979			−0.688786			−0.344393	+	0.938826i	$0.611915\pi$
$422$		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	$-0.693564\pi$
$431$	−492.468			−1.14262			−0.571309	+	0.820735i	$0.693564\pi$
$432$		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	$0.167170\pi$
$439$			440.202i			1.00274i	−0.865233	+	0.501369i	$0.832830\pi$
$440$		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	$-0.742633\pi$
$461$		−	666.865i		−	1.44656i	0.690554	−	0.723281i	$-0.257367\pi$
$462$		0			0
$463$	−385.038			−0.831616			−0.415808	−	0.909452i	$-0.636501\pi$
$463$	−385.038			−0.831616			−0.415808	+	0.909452i	$0.636501\pi$
$464$		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	$0.192864\pi$
$479$			545.583i			1.13900i	−0.821990	+	0.569502i	$0.807136\pi$
$480$		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	$-0.234386\pi$
$487$	721.663			1.48186			0.740928	+	0.671585i	$0.234386\pi$
$488$		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	$-0.789671\pi$
$503$		−	617.407i		−	1.22745i	0.789520	−	0.613724i	$-0.210329\pi$
$504$		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	$0.0783004\pi$
$509$			247.898i			0.487029i	−0.969897	+	0.243515i	$0.921700\pi$
$510$		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	$-0.486846\pi$
$541$	44.7003			0.0826253			0.0413126	+	0.999146i	$0.486846\pi$
$542$		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	$-0.503804\pi$
$557$	−13.3127			−0.0239006			−0.0119503	+	0.999929i	$0.503804\pi$
$558$		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	$-0.408914\pi$
$599$	338.153			0.564530			0.282265	+	0.959337i	$0.408914\pi$
$600$		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	$0.154569\pi$
$607$			566.613i			0.933465i	−0.884399	+	0.466732i	$0.845431\pi$
$608$		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	$-0.370151\pi$
$613$	486.369			0.793424			0.396712	+	0.917943i	$0.370151\pi$
$614$		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	$-0.628418\pi$
$631$	−495.437			−0.785162			−0.392581	+	0.919717i	$0.628418\pi$
$632$		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	$-0.863542\pi$
$647$		−	537.895i		−	0.831368i	0.909509	−	0.415684i	$-0.136458\pi$
$648$		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	$-0.204700\pi$
$653$	1045.13			1.60050			0.800251	+	0.599665i	$0.204700\pi$
$654$		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	$0.0971952\pi$
$661$			397.426i			0.601249i	−0.953743	+	0.300625i	$0.902805\pi$
$662$		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	$0.0856714\pi$
$677$			360.038i			0.531814i	−0.963999	+	0.265907i	$0.914329\pi$
$678$		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	$-0.571205\pi$
$701$	−311.014			−0.443672			−0.221836	+	0.975084i	$0.571205\pi$
$702$		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	$-0.401328\pi$
$709$	432.555			0.610092			0.305046	+	0.952338i	$0.401328\pi$
$710$		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	$0.209864\pi$
$719$			880.875i			1.22514i	−0.790417	+	0.612569i	$0.790136\pi$
$720$		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	$-0.876283\pi$
$727$		−	551.001i		−	0.757910i	0.925415	−	0.378955i	$-0.123717\pi$
$728$		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	$-0.821439\pi$
$733$		−	779.917i		−	1.06401i	0.846742	−	0.532003i	$-0.178561\pi$
$734$		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	$-0.516169\pi$
$743$	−75.4519			−0.101550			−0.0507752	+	0.998710i	$0.516169\pi$
$744$		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	$-0.527037\pi$
$751$	−127.424			−0.169673			−0.0848365	+	0.996395i	$0.527037\pi$
$752$		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	$-0.283300\pi$
$757$	952.912			1.25880			0.629401	+	0.777081i	$0.283300\pi$
$758$		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	$-0.909076\pi$
$773$		−	435.628i		−	0.563555i	0.959480	−	0.281777i	$-0.0909239\pi$
$774$		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	$-0.863428\pi$
$797$		−	663.119i		−	0.832019i	0.909360	−	0.416009i	$-0.136572\pi$
$798$		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	$-0.488433\pi$
$821$	59.6541			0.0726603			0.0363301	+	0.999340i	$0.488433\pi$
$822$		0			0
$823$	353.320			0.429307			0.214654	−	0.976690i	$-0.431138\pi$
$823$	353.320			0.429307			0.214654	+	0.976690i	$0.431138\pi$
$824$		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	$0.0184201\pi$
$829$			95.8922i			0.115672i	−0.998326	+	0.0578361i	$0.981580\pi$
$830$		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	$-0.819028\pi$
$839$		−	903.440i		−	1.07681i	0.842687	−	0.538403i	$-0.180972\pi$
$840$		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	$0.0715372\pi$
$853$			380.188i			0.445707i	−0.974852	+	0.222854i	$0.928463\pi$
$854$		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	$-0.534916\pi$
$863$	−188.946			−0.218941			−0.109471	+	0.993990i	$0.534916\pi$
$864$		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	$-0.283402\pi$
$877$	1103.53			1.25831			0.629153	+	0.777282i	$0.283402\pi$
$878$		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	$-0.764717\pi$
$887$		−	1195.09i		−	1.34734i	0.739033	−	0.673669i	$-0.235283\pi$
$888$		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	$-0.379361\pi$
$911$	674.122			0.739980			0.369990	+	0.929036i	$0.379361\pi$
$912$		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	$-0.856476\pi$
$919$	−1654.31			−1.80012			−0.900058	+	0.435769i	$0.856476\pi$
$920$		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	$0.110470\pi$
$941$			640.116i			0.680251i	−0.940380	+	0.340125i	$0.889530\pi$
$942$		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	$-0.173414\pi$
$967$	1654.02			1.71047			0.855233	+	0.518244i	$0.173414\pi$
$968$		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	$0.0814512\pi$
$983$			497.601i			0.506206i	−0.967439	+	0.253103i	$0.918549\pi$
$984$		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	$-0.552920\pi$
$991$	−328.000			−0.330979			−0.165490	+	0.986212i	$0.552920\pi$
$992$		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	$-0.838845\pi$
$997$		−	966.951i		−	0.969861i	0.874553	−	0.484930i	$-0.161155\pi$
$998$		0			0
$999$		−	93.2403i		−	0.0933336i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.3.f.b.97.1	✓	16
3.2	odd	2		2016.3.f.g.1441.16		16
4.3	odd	2	inner	672.3.f.b.97.9	yes	16
7.6	odd	2	inner	672.3.f.b.97.16	yes	16
8.3	odd	2		1344.3.f.j.769.8		16
8.5	even	2		1344.3.f.j.769.16		16
12.11	even	2		2016.3.f.g.1441.15		16
21.20	even	2		2016.3.f.g.1441.2		16
28.27	even	2	inner	672.3.f.b.97.8	yes	16
56.13	odd	2		1344.3.f.j.769.1		16
56.27	even	2		1344.3.f.j.769.9		16
84.83	odd	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	1.1	even	1	trivial
672.3.f.b.97.8	yes	16	28.27	even	2	inner
672.3.f.b.97.9	yes	16	4.3	odd	2	inner
672.3.f.b.97.16	yes	16	7.6	odd	2	inner
1344.3.f.j.769.1		16	56.13	odd	2
1344.3.f.j.769.8		16	8.3	odd	2
1344.3.f.j.769.9		16	56.27	even	2
1344.3.f.j.769.16		16	8.5	even	2
2016.3.f.g.1441.1		16	84.83	odd	2
2016.3.f.g.1441.2		16	21.20	even	2
2016.3.f.g.1441.15		16	12.11	even	2
2016.3.f.g.1441.16		16	3.2	odd	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	672.f (of order \(2\), degree \(1\), 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

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