LMFDB - Embedded newform 672.3.f.b.97.12

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			97.12
Root			$-2.42052 - 1.33386i$ of defining polynomial
Character	$\chi$	$=$	672.97
Dual form			672.3.f.b.97.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -0.817910i q^{5} +(6.47914 + 2.64968i) q^{7} -3.00000 q^{9} +7.22005 q^{11} +5.77193i q^{13} +1.41666 q^{15} +15.3528i q^{17} -17.6927i q^{19} +(-4.58937 + 11.2222i) q^{21} -8.69922 q^{23} +24.3310 q^{25} -5.19615i q^{27} +19.4660 q^{29} +0.274321i q^{31} +12.5055i q^{33} +(2.16720 - 5.29935i) q^{35} -4.34017 q^{37} -9.99728 q^{39} +72.6342i q^{41} +29.3599 q^{43} +2.45373i q^{45} +15.7896i q^{47} +(34.9584 + 34.3352i) q^{49} -26.5917 q^{51} -18.2052 q^{53} -5.90535i q^{55} +30.6447 q^{57} +51.3109i q^{59} +21.5071i q^{61} +(-19.4374 - 7.94903i) q^{63} +4.72092 q^{65} +45.5308 q^{67} -15.0675i q^{69} -100.378 q^{71} +117.366i q^{73} +42.1426i q^{75} +(46.7797 + 19.1308i) q^{77} +61.7155 q^{79} +9.00000 q^{81} -65.3843i q^{83} +12.5572 q^{85} +33.7161i q^{87} +105.785i q^{89} +(-15.2937 + 37.3971i) q^{91} -0.475139 q^{93} -14.4711 q^{95} +59.9300i q^{97} -21.6602 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	0.817910i		−	0.163582i	−0.996650	−	0.0817910i	$-0.973936\pi$
$5$		−	0.817910i		−	0.163582i	0.996650	−	0.0817910i	$-0.0260640\pi$
$6$		0			0
$7$	6.47914	+	2.64968i	0.925591	+	0.378525i
$7$	6.47914	+	2.64968i	0.925591	+	0.378525i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	7.22005			0.656368			0.328184	−	0.944614i	$-0.393563\pi$
$11$	7.22005			0.656368			0.328184	+	0.944614i	$0.393563\pi$
$12$		0			0
$13$			5.77193i			0.443995i	0.975047	+	0.221997i	$0.0712577\pi$
$13$			5.77193i			0.443995i	−0.975047	+	0.221997i	$0.928742\pi$
$14$		0			0
$15$	1.41666			0.0944441
$16$		0			0
$17$			15.3528i			0.903103i	0.892245	+	0.451552i	$0.149129\pi$
$17$			15.3528i			0.903103i	−0.892245	+	0.451552i	$0.850871\pi$
$18$		0			0
$19$		−	17.6927i		−	0.931197i	−0.884996	−	0.465599i	$-0.845839\pi$
$19$		−	17.6927i		−	0.931197i	0.884996	−	0.465599i	$-0.154161\pi$
$20$		0			0
$21$	−4.58937	+	11.2222i	−0.218542	+	0.534390i
$22$		0			0
$23$	−8.69922			−0.378227			−0.189113	−	0.981955i	$-0.560561\pi$
$23$	−8.69922			−0.378227			−0.189113	+	0.981955i	$0.560561\pi$
$24$		0			0
$25$	24.3310			0.973241
$26$		0			0
$27$		−	5.19615i		−	0.192450i
$28$		0			0
$29$	19.4660			0.671241			0.335620	−	0.941997i	$-0.391054\pi$
$29$	19.4660			0.671241			0.335620	+	0.941997i	$0.391054\pi$
$30$		0			0
$31$			0.274321i			0.00884908i	0.999990	+	0.00442454i	$0.00140838\pi$
$31$			0.274321i			0.00884908i	−0.999990	+	0.00442454i	$0.998592\pi$
$32$		0			0
$33$			12.5055i			0.378954i
$34$		0			0
$35$	2.16720	−	5.29935i	0.0619199	−	0.151410i
$36$		0			0
$37$	−4.34017			−0.117302			−0.0586510	−	0.998279i	$-0.518680\pi$
$37$	−4.34017			−0.117302			−0.0586510	+	0.998279i	$0.518680\pi$
$38$		0			0
$39$	−9.99728			−0.256341
$40$		0			0
$41$			72.6342i			1.77156i	0.464101	+	0.885782i	$0.346378\pi$
$41$			72.6342i			1.77156i	−0.464101	+	0.885782i	$0.653622\pi$
$42$		0			0
$43$	29.3599			0.682789			0.341394	−	0.939920i	$-0.389101\pi$
$43$	29.3599			0.682789			0.341394	+	0.939920i	$0.389101\pi$
$44$		0			0
$45$			2.45373i			0.0545273i
$46$		0			0
$47$			15.7896i			0.335948i	0.985791	+	0.167974i	$0.0537225\pi$
$47$			15.7896i			0.335948i	−0.985791	+	0.167974i	$0.946277\pi$
$48$		0			0
$49$	34.9584	+	34.3352i	0.713438	+	0.700719i
$50$		0			0
$51$	−26.5917			−0.521407
$52$		0			0
$53$	−18.2052			−0.343495			−0.171747	−	0.985141i	$-0.554941\pi$
$53$	−18.2052			−0.343495			−0.171747	+	0.985141i	$0.554941\pi$
$54$		0			0
$55$		−	5.90535i		−	0.107370i
$56$		0			0
$57$	30.6447			0.537627
$58$		0			0
$59$			51.3109i			0.869676i	0.900509	+	0.434838i	$0.143194\pi$
$59$			51.3109i			0.869676i	−0.900509	+	0.434838i	$0.856806\pi$
$60$		0			0
$61$			21.5071i			0.352575i	0.984339	+	0.176288i	$0.0564089\pi$
$61$			21.5071i			0.352575i	−0.984339	+	0.176288i	$0.943591\pi$
$62$		0			0
$63$	−19.4374	−	7.94903i	−0.308530	−	0.126175i
$64$		0			0
$65$	4.72092			0.0726296
$66$		0			0
$67$	45.5308			0.679564			0.339782	−	0.940504i	$-0.389647\pi$
$67$	45.5308			0.679564			0.339782	+	0.940504i	$0.389647\pi$
$68$		0			0
$69$		−	15.0675i		−	0.218369i
$70$		0			0
$71$	−100.378			−1.41378			−0.706889	−	0.707324i	$-0.749902\pi$
$71$	−100.378			−1.41378			−0.706889	+	0.707324i	$0.749902\pi$
$72$		0			0
$73$			117.366i			1.60776i	0.594794	+	0.803878i	$0.297234\pi$
$73$			117.366i			1.60776i	−0.594794	+	0.803878i	$0.702766\pi$
$74$		0			0
$75$			42.1426i			0.561901i
$76$		0			0
$77$	46.7797	+	19.1308i	0.607529	+	0.248452i
$78$		0			0
$79$	61.7155			0.781209			0.390605	−	0.920559i	$-0.372266\pi$
$79$	61.7155			0.781209			0.390605	+	0.920559i	$0.372266\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$		−	65.3843i		−	0.787762i	−0.919161	−	0.393881i	$-0.871132\pi$
$83$		−	65.3843i		−	0.787762i	0.919161	−	0.393881i	$-0.128868\pi$
$84$		0			0
$85$	12.5572			0.147731
$86$		0			0
$87$			33.7161i			0.387541i
$88$		0			0
$89$			105.785i			1.18859i	0.804246	+	0.594296i	$0.202569\pi$
$89$			105.785i			1.18859i	−0.804246	+	0.594296i	$0.797431\pi$
$90$		0			0
$91$	−15.2937	+	37.3971i	−0.168063	+	0.410958i
$92$		0			0
$93$	−0.475139			−0.00510902
$94$		0			0
$95$	−14.4711			−0.152327
$96$		0			0
$97$			59.9300i			0.617835i	0.951089	+	0.308918i	$0.0999667\pi$
$97$			59.9300i			0.617835i	−0.951089	+	0.308918i	$0.900033\pi$
$98$		0			0
$99$	−21.6602			−0.218789
$100$		0			0
$101$		−	90.0958i		−	0.892038i	−0.895023	−	0.446019i	$-0.852841\pi$
$101$		−	90.0958i		−	0.892038i	0.895023	−	0.446019i	$-0.147159\pi$
$102$		0			0
$103$		−	122.996i		−	1.19414i	−0.802190	−	0.597069i	$-0.796332\pi$
$103$		−	122.996i		−	1.19414i	0.802190	−	0.597069i	$-0.203668\pi$
$104$		0			0
$105$	9.17874	+	3.75369i	0.0874166	+	0.0357495i
$106$		0			0
$107$	−31.6537			−0.295829			−0.147914	−	0.989000i	$-0.547256\pi$
$107$	−31.6537			−0.295829			−0.147914	+	0.989000i	$0.547256\pi$
$108$		0			0
$109$	−101.612			−0.932223			−0.466112	−	0.884726i	$-0.654345\pi$
$109$	−101.612			−0.932223			−0.466112	+	0.884726i	$0.654345\pi$
$110$		0			0
$111$		−	7.51740i		−	0.0677244i
$112$		0			0
$113$	84.9260			0.751558			0.375779	−	0.926709i	$-0.377375\pi$
$113$	84.9260			0.751558			0.375779	+	0.926709i	$0.377375\pi$
$114$		0			0
$115$			7.11518i			0.0618711i
$116$		0			0
$117$		−	17.3158i		−	0.147998i
$118$		0			0
$119$	−40.6798	+	99.4726i	−0.341847	+	0.835904i
$120$		0			0
$121$	−68.8709			−0.569181
$122$		0			0
$123$	−125.806			−1.02281
$124$		0			0
$125$		−	40.3483i		−	0.322787i
$126$		0			0
$127$	215.993			1.70073			0.850364	−	0.526194i	$-0.176382\pi$
$127$	215.993			1.70073			0.850364	+	0.526194i	$0.176382\pi$
$128$		0			0
$129$			50.8529i			0.394208i
$130$		0			0
$131$		−	16.7510i		−	0.127870i	−0.997954	−	0.0639351i	$-0.979635\pi$
$131$		−	16.7510i		−	0.127870i	0.997954	−	0.0639351i	$-0.0203651\pi$
$132$		0			0
$133$	46.8800	−	114.634i	0.352481	−	0.861908i
$134$		0			0
$135$	−4.24998			−0.0314814
$136$		0			0
$137$	105.072			0.766952			0.383476	−	0.923551i	$-0.374727\pi$
$137$	105.072			0.766952			0.383476	+	0.923551i	$0.374727\pi$
$138$		0			0
$139$		−	218.687i		−	1.57329i	−0.617406	−	0.786645i	$-0.711816\pi$
$139$		−	218.687i		−	1.57329i	0.617406	−	0.786645i	$-0.288184\pi$
$140$		0			0
$141$	−27.3483			−0.193960
$142$		0			0
$143$			41.6736i			0.291424i
$144$		0			0
$145$		−	15.9214i		−	0.109803i
$146$		0			0
$147$	−59.4703	+	60.5498i	−0.404560	+	0.411903i
$148$		0			0
$149$	−195.024			−1.30889			−0.654443	−	0.756112i	$-0.727097\pi$
$149$	−195.024			−1.30889			−0.654443	+	0.756112i	$0.727097\pi$
$150$		0			0
$151$	169.415			1.12195			0.560976	−	0.827832i	$-0.310426\pi$
$151$	169.415			1.12195			0.560976	+	0.827832i	$0.310426\pi$
$152$		0			0
$153$		−	46.0583i		−	0.301034i
$154$		0			0
$155$	0.224370			0.00144755
$156$		0			0
$157$		−	59.7200i		−	0.380382i	−0.981747	−	0.190191i	$-0.939089\pi$
$157$		−	59.7200i		−	0.380382i	0.981747	−	0.190191i	$-0.0609108\pi$
$158$		0			0
$159$		−	31.5324i		−	0.198317i
$160$		0			0
$161$	−56.3634	−	23.0501i	−0.350083	−	0.143168i
$162$		0			0
$163$	187.453			1.15002			0.575010	−	0.818147i	$-0.304998\pi$
$163$	187.453			1.15002			0.575010	+	0.818147i	$0.304998\pi$
$164$		0			0
$165$	10.2284			0.0619901
$166$		0			0
$167$		−	56.2378i		−	0.336754i	−0.985723	−	0.168377i	$-0.946147\pi$
$167$		−	56.2378i		−	0.336754i	0.985723	−	0.168377i	$-0.0538525\pi$
$168$		0			0
$169$	135.685			0.802869
$170$		0			0
$171$			53.0782i			0.310399i
$172$		0			0
$173$		−	168.162i		−	0.972033i	−0.873950	−	0.486016i	$-0.838449\pi$
$173$		−	168.162i		−	0.972033i	0.873950	−	0.486016i	$-0.161551\pi$
$174$		0			0
$175$	157.644	+	64.4693i	0.900823	+	0.368396i
$176$		0			0
$177$	−88.8731			−0.502108
$178$		0			0
$179$	1.56092			0.00872022			0.00436011	−	0.999990i	$-0.498612\pi$
$179$	1.56092			0.00872022			0.00436011	+	0.999990i	$0.498612\pi$
$180$		0			0
$181$		−	258.547i		−	1.42844i	−0.699923	−	0.714219i	$-0.746782\pi$
$181$		−	258.547i		−	1.42844i	0.699923	−	0.714219i	$-0.253218\pi$
$182$		0			0
$183$	−37.2514			−0.203560
$184$		0			0
$185$			3.54987i			0.0191885i
$186$		0			0
$187$			110.848i			0.592768i
$188$		0			0
$189$	13.7681	−	33.6666i	0.0728472	−	0.178130i
$190$		0			0
$191$	−264.128			−1.38287			−0.691433	−	0.722440i	$-0.743020\pi$
$191$	−264.128			−1.38287			−0.691433	+	0.722440i	$0.743020\pi$
$192$		0			0
$193$	−85.7046			−0.444065			−0.222033	−	0.975039i	$-0.571269\pi$
$193$	−85.7046			−0.444065			−0.222033	+	0.975039i	$0.571269\pi$
$194$		0			0
$195$			8.17688i			0.0419327i
$196$		0			0
$197$	−113.893			−0.578136			−0.289068	−	0.957309i	$-0.593345\pi$
$197$	−113.893			−0.578136			−0.289068	+	0.957309i	$0.593345\pi$
$198$		0			0
$199$			52.6525i			0.264586i	0.991211	+	0.132293i	$0.0422339\pi$
$199$			52.6525i			0.264586i	−0.991211	+	0.132293i	$0.957766\pi$
$200$		0			0
$201$			78.8616i			0.392346i
$202$		0			0
$203$	126.123	+	51.5785i	0.621295	+	0.254082i
$204$		0			0
$205$	59.4082			0.289796
$206$		0			0
$207$	26.0977			0.126076
$208$		0			0
$209$		−	127.742i		−	0.611208i
$210$		0			0
$211$	55.9752			0.265285			0.132643	−	0.991164i	$-0.457654\pi$
$211$	55.9752			0.265285			0.132643	+	0.991164i	$0.457654\pi$
$212$		0			0
$213$		−	173.860i		−	0.816245i
$214$		0			0
$215$		−	24.0138i		−	0.111692i
$216$		0			0
$217$	−0.726863	+	1.77737i	−0.00334960	+	0.00819063i
$218$		0			0
$219$	−203.284			−0.928239
$220$		0			0
$221$	−88.6151			−0.400973
$222$		0			0
$223$		−	372.002i		−	1.66817i	−0.551635	−	0.834086i	$-0.685996\pi$
$223$		−	372.002i		−	1.66817i	0.551635	−	0.834086i	$-0.314004\pi$
$224$		0			0
$225$	−72.9931			−0.324414
$226$		0			0
$227$			233.756i			1.02976i	0.857262	+	0.514880i	$0.172164\pi$
$227$			233.756i			1.02976i	−0.857262	+	0.514880i	$0.827836\pi$
$228$		0			0
$229$		−	157.225i		−	0.686573i	−0.939231	−	0.343286i	$-0.888460\pi$
$229$		−	157.225i		−	0.686573i	0.939231	−	0.343286i	$-0.111540\pi$
$230$		0			0
$231$	−33.1355	+	81.0248i	−0.143444	+	0.350757i
$232$		0			0
$233$	−213.395			−0.915857			−0.457928	−	0.888989i	$-0.651408\pi$
$233$	−213.395			−0.915857			−0.457928	+	0.888989i	$0.651408\pi$
$234$		0			0
$235$	12.9144			0.0549551
$236$		0			0
$237$			106.894i			0.451031i
$238$		0			0
$239$	−242.428			−1.01434			−0.507171	−	0.861845i	$-0.669309\pi$
$239$	−242.428			−1.01434			−0.507171	+	0.861845i	$0.669309\pi$
$240$		0			0
$241$		−	402.031i		−	1.66818i	−0.551629	−	0.834090i	$-0.685993\pi$
$241$		−	402.031i		−	1.66818i	0.551629	−	0.834090i	$-0.314007\pi$
$242$		0			0
$243$			15.5885i			0.0641500i
$244$		0			0
$245$	28.0831	−	28.5929i	0.114625	−	0.116706i
$246$		0			0
$247$	102.121			0.413447
$248$		0			0
$249$	113.249			0.454815
$250$		0			0
$251$		−	392.502i		−	1.56375i	−0.623435	−	0.781875i	$-0.714263\pi$
$251$		−	392.502i		−	1.56375i	0.623435	−	0.781875i	$-0.285737\pi$
$252$		0			0
$253$	−62.8088			−0.248256
$254$		0			0
$255$			21.7497i			0.0852928i
$256$		0			0
$257$		−	323.742i		−	1.25970i	−0.776718	−	0.629848i	$-0.783117\pi$
$257$		−	323.742i		−	1.25970i	0.776718	−	0.629848i	$-0.216883\pi$
$258$		0			0
$259$	−28.1206	−	11.5001i	−0.108574	−	0.0444018i
$260$		0			0
$261$	−58.3980			−0.223747
$262$		0			0
$263$	−319.503			−1.21484			−0.607421	−	0.794380i	$-0.707796\pi$
$263$	−319.503			−1.21484			−0.607421	+	0.794380i	$0.707796\pi$
$264$		0			0
$265$			14.8902i			0.0561895i
$266$		0			0
$267$	−183.225			−0.686234
$268$		0			0
$269$			167.751i			0.623609i	0.950146	+	0.311805i	$0.100933\pi$
$269$			167.751i			0.623609i	−0.950146	+	0.311805i	$0.899067\pi$
$270$		0			0
$271$		−	217.222i		−	0.801557i	−0.916175	−	0.400779i	$-0.868740\pi$
$271$		−	217.222i		−	0.801557i	0.916175	−	0.400779i	$-0.131260\pi$
$272$		0			0
$273$	−64.7738	−	26.4895i	−0.237266	−	0.0970313i
$274$		0			0
$275$	175.671			0.638804
$276$		0			0
$277$	−381.699			−1.37797			−0.688987	−	0.724774i	$-0.741944\pi$
$277$	−381.699			−1.37797			−0.688987	+	0.724774i	$0.741944\pi$
$278$		0			0
$279$		−	0.822964i		−	0.00294969i
$280$		0			0
$281$	45.0923			0.160471			0.0802354	−	0.996776i	$-0.474433\pi$
$281$	45.0923			0.160471			0.0802354	+	0.996776i	$0.474433\pi$
$282$		0			0
$283$			316.321i			1.11774i	0.829254	+	0.558871i	$0.188765\pi$
$283$			316.321i			1.11774i	−0.829254	+	0.558871i	$0.811235\pi$
$284$		0			0
$285$		−	25.0646i		−	0.0879461i
$286$		0			0
$287$	−192.457	+	470.607i	−0.670582	+	1.63974i
$288$		0			0
$289$	53.2930			0.184405
$290$		0			0
$291$	−103.802			−0.356707
$292$		0			0
$293$		−	139.994i		−	0.477795i	−0.971045	−	0.238897i	$-0.923214\pi$
$293$		−	139.994i		−	0.477795i	0.971045	−	0.238897i	$-0.0767860\pi$
$294$		0			0
$295$	41.9677			0.142263
$296$		0			0
$297$		−	37.5165i		−	0.126318i
$298$		0			0
$299$		−	50.2113i		−	0.167931i
$300$		0			0
$301$	190.227	+	77.7942i	0.631983	+	0.258453i
$302$		0			0
$303$	156.051			0.515018
$304$		0			0
$305$	17.5909			0.0576750
$306$		0			0
$307$		−	436.775i		−	1.42272i	−0.702827	−	0.711360i	$-0.748079\pi$
$307$		−	436.775i		−	1.42272i	0.702827	−	0.711360i	$-0.251921\pi$
$308$		0			0
$309$	213.036			0.689436
$310$		0			0
$311$			410.856i			1.32108i	0.750790	+	0.660540i	$0.229673\pi$
$311$			410.856i			1.32108i	−0.750790	+	0.660540i	$0.770327\pi$
$312$		0			0
$313$			64.2583i			0.205298i	0.994718	+	0.102649i	$0.0327318\pi$
$313$			64.2583i			0.205298i	−0.994718	+	0.102649i	$0.967268\pi$
$314$		0			0
$315$	−6.50159	+	15.8981i	−0.0206400	+	0.0504700i
$316$		0			0
$317$	419.268			1.32261			0.661306	−	0.750116i	$-0.270002\pi$
$317$	419.268			1.32261			0.661306	+	0.750116i	$0.270002\pi$
$318$		0			0
$319$	140.545			0.440581
$320$		0			0
$321$		−	54.8258i		−	0.170797i
$322$		0			0
$323$	271.632			0.840967
$324$		0			0
$325$			140.437i			0.432114i
$326$		0			0
$327$		−	175.998i		−	0.538219i
$328$		0			0
$329$	−41.8372	+	102.303i	−0.127165	+	0.310951i
$330$		0			0
$331$	61.9410			0.187133			0.0935665	−	0.995613i	$-0.470173\pi$
$331$	61.9410			0.187133			0.0935665	+	0.995613i	$0.470173\pi$
$332$		0			0
$333$	13.0205			0.0391007
$334$		0			0
$335$		−	37.2401i		−	0.111164i
$336$		0			0
$337$	−421.374			−1.25037			−0.625184	−	0.780477i	$-0.714976\pi$
$337$	−421.374			−1.25037			−0.625184	+	0.780477i	$0.714976\pi$
$338$		0			0
$339$			147.096i			0.433912i
$340$		0			0
$341$			1.98061i			0.00580825i
$342$		0			0
$343$	135.523	+	315.091i	0.395112	+	0.918633i
$344$		0			0
$345$	−12.3238			−0.0357213
$346$		0			0
$347$	323.940			0.933544			0.466772	−	0.884378i	$-0.345417\pi$
$347$	323.940			0.933544			0.466772	+	0.884378i	$0.345417\pi$
$348$		0			0
$349$			579.214i			1.65964i	0.558031	+	0.829820i	$0.311557\pi$
$349$			579.214i			1.65964i	−0.558031	+	0.829820i	$0.688443\pi$
$350$		0			0
$351$	29.9918			0.0854468
$352$		0			0
$353$		−	149.339i		−	0.423056i	−0.977372	−	0.211528i	$-0.932156\pi$
$353$		−	149.339i		−	0.423056i	0.977372	−	0.211528i	$-0.0678439\pi$
$354$		0			0
$355$			82.1004i			0.231269i
$356$		0			0
$357$	−172.292	−	70.4595i	−0.482609	−	0.197366i
$358$		0			0
$359$	54.2908			0.151228			0.0756140	−	0.997137i	$-0.475908\pi$
$359$	54.2908			0.151228			0.0756140	+	0.997137i	$0.475908\pi$
$360$		0			0
$361$	47.9668			0.132872
$362$		0			0
$363$		−	119.288i		−	0.328617i
$364$		0			0
$365$	95.9950			0.263000
$366$		0			0
$367$			203.623i			0.554830i	0.960750	+	0.277415i	$0.0894777\pi$
$367$			203.623i			0.554830i	−0.960750	+	0.277415i	$0.910522\pi$
$368$		0			0
$369$		−	217.902i		−	0.590522i
$370$		0			0
$371$	−117.954	−	48.2379i	−0.317935	−	0.130021i
$372$		0			0
$373$	280.696			0.752537			0.376268	−	0.926511i	$-0.377207\pi$
$373$	280.696			0.752537			0.376268	+	0.926511i	$0.377207\pi$
$374$		0			0
$375$	69.8854			0.186361
$376$		0			0
$377$			112.356i			0.298027i
$378$		0			0
$379$	−300.782			−0.793620			−0.396810	−	0.917901i	$-0.629883\pi$
$379$	−300.782			−0.793620			−0.396810	+	0.917901i	$0.629883\pi$
$380$		0			0
$381$			374.110i			0.981916i
$382$		0			0
$383$		−	582.741i		−	1.52152i	−0.649035	−	0.760759i	$-0.724827\pi$
$383$		−	582.741i		−	1.52152i	0.649035	−	0.760759i	$-0.275173\pi$
$384$		0			0
$385$	15.6473	−	38.2616i	0.0406422	−	0.0993807i
$386$		0			0
$387$	−88.0797			−0.227596
$388$		0			0
$389$	300.061			0.771365			0.385683	−	0.922632i	$-0.373966\pi$
$389$	300.061			0.771365			0.385683	+	0.922632i	$0.373966\pi$
$390$		0			0
$391$		−	133.557i		−	0.341578i
$392$		0			0
$393$	29.0136			0.0738259
$394$		0			0
$395$		−	50.4778i		−	0.127792i
$396$		0			0
$397$			487.010i			1.22673i	0.789801	+	0.613363i	$0.210184\pi$
$397$			487.010i			1.22673i	−0.789801	+	0.613363i	$0.789816\pi$
$398$		0			0
$399$	198.551	+	81.1986i	0.497623	+	0.203505i
$400$		0			0
$401$	−156.647			−0.390642			−0.195321	−	0.980739i	$-0.562575\pi$
$401$	−156.647			−0.390642			−0.195321	+	0.980739i	$0.562575\pi$
$402$		0			0
$403$	−1.58336			−0.00392894
$404$		0			0
$405$		−	7.36119i		−	0.0181758i
$406$		0			0
$407$	−31.3363			−0.0769933
$408$		0			0
$409$			278.826i			0.681726i	0.940113	+	0.340863i	$0.110719\pi$
$409$			278.826i			0.681726i	−0.940113	+	0.340863i	$0.889281\pi$
$410$		0			0
$411$			181.991i			0.442800i
$412$		0			0
$413$	−135.957	+	332.450i	−0.329194	+	0.804965i
$414$		0			0
$415$	−53.4785			−0.128864
$416$		0			0
$417$	378.777			0.908339
$418$		0			0
$419$		−	793.949i		−	1.89487i	−0.319953	−	0.947434i	$-0.603667\pi$
$419$		−	793.949i		−	1.89487i	0.319953	−	0.947434i	$-0.396333\pi$
$420$		0			0
$421$	−24.9026			−0.0591510			−0.0295755	−	0.999563i	$-0.509416\pi$
$421$	−24.9026			−0.0591510			−0.0295755	+	0.999563i	$0.509416\pi$
$422$		0			0
$423$		−	47.3687i		−	0.111983i
$424$		0			0
$425$			373.548i			0.878937i
$426$		0			0
$427$	−56.9868	+	139.347i	−0.133459	+	0.326341i
$428$		0			0
$429$	−72.1809			−0.168254
$430$		0			0
$431$	−426.634			−0.989871			−0.494935	−	0.868930i	$-0.664808\pi$
$431$	−426.634			−0.989871			−0.494935	+	0.868930i	$0.664808\pi$
$432$		0			0
$433$		−	475.197i		−	1.09745i	−0.836002	−	0.548727i	$-0.815113\pi$
$433$		−	475.197i		−	1.09745i	0.836002	−	0.548727i	$-0.184887\pi$
$434$		0			0
$435$	27.5767			0.0633947
$436$		0			0
$437$			153.913i			0.352204i
$438$		0			0
$439$		−	563.733i		−	1.28413i	−0.766650	−	0.642065i	$-0.778078\pi$
$439$		−	563.733i		−	1.28413i	0.766650	−	0.642065i	$-0.221922\pi$
$440$		0			0
$441$	−104.875	−	103.006i	−0.237813	−	0.233573i
$442$		0			0
$443$	295.801			0.667722			0.333861	−	0.942622i	$-0.391648\pi$
$443$	295.801			0.667722			0.333861	+	0.942622i	$0.391648\pi$
$444$		0			0
$445$	86.5224			0.194432
$446$		0			0
$447$		−	337.791i		−	0.755685i
$448$		0			0
$449$	137.863			0.307046			0.153523	−	0.988145i	$-0.450938\pi$
$449$	137.863			0.307046			0.153523	+	0.988145i	$0.450938\pi$
$450$		0			0
$451$			524.422i			1.16280i
$452$		0			0
$453$			293.435i			0.647759i
$454$		0			0
$455$	30.5875	+	12.5089i	0.0672253	+	0.0274921i
$456$		0			0
$457$	449.162			0.982849			0.491424	−	0.870920i	$-0.336476\pi$
$457$	449.162			0.982849			0.491424	+	0.870920i	$0.336476\pi$
$458$		0			0
$459$	79.7752			0.173802
$460$		0			0
$461$			605.343i			1.31311i	0.754279	+	0.656554i	$0.227987\pi$
$461$			605.343i			1.31311i	−0.754279	+	0.656554i	$0.772013\pi$
$462$		0			0
$463$	248.016			0.535672			0.267836	−	0.963465i	$-0.413691\pi$
$463$	248.016			0.535672			0.267836	+	0.963465i	$0.413691\pi$
$464$		0			0
$465$			0.388621i			0.000835743i
$466$		0			0
$467$			681.140i			1.45854i	0.684224	+	0.729272i	$0.260141\pi$
$467$			681.140i			1.45854i	−0.684224	+	0.729272i	$0.739859\pi$
$468$		0			0
$469$	295.000	+	120.642i	0.628998	+	0.257232i
$470$		0			0
$471$	103.438			0.219614
$472$		0			0
$473$	211.980			0.448161
$474$		0			0
$475$		−	430.483i		−	0.906279i
$476$		0			0
$477$	54.6156			0.114498
$478$		0			0
$479$			534.681i			1.11624i	0.829759	+	0.558122i	$0.188478\pi$
$479$			534.681i			1.11624i	−0.829759	+	0.558122i	$0.811522\pi$
$480$		0			0
$481$		−	25.0512i		−	0.0520815i
$482$		0			0
$483$	39.9240	−	97.6243i	0.0826583	−	0.202121i
$484$		0			0
$485$	49.0174			0.101067
$486$		0			0
$487$	−206.304			−0.423621			−0.211811	−	0.977311i	$-0.567936\pi$
$487$	−206.304			−0.423621			−0.211811	+	0.977311i	$0.567936\pi$
$488$		0			0
$489$			324.678i			0.663964i
$490$		0			0
$491$	800.160			1.62965			0.814827	−	0.579705i	$-0.196832\pi$
$491$	800.160			1.62965			0.814827	+	0.579705i	$0.196832\pi$
$492$		0			0
$493$			298.856i			0.606200i
$494$		0			0
$495$			17.7161i			0.0357900i
$496$		0			0
$497$	−650.365	−	265.970i	−1.30858	−	0.535151i
$498$		0			0
$499$	−823.805			−1.65091			−0.825456	−	0.564467i	$-0.809082\pi$
$499$	−823.805			−1.65091			−0.825456	+	0.564467i	$0.809082\pi$
$500$		0			0
$501$	97.4068			0.194425
$502$		0			0
$503$		−	282.441i		−	0.561512i	−0.959779	−	0.280756i	$-0.909415\pi$
$503$		−	282.441i		−	0.561512i	0.959779	−	0.280756i	$-0.0905852\pi$
$504$		0			0
$505$	−73.6903			−0.145921
$506$		0			0
$507$			235.013i			0.463536i
$508$		0			0
$509$			929.121i			1.82539i	0.408646	+	0.912693i	$0.366001\pi$
$509$			929.121i			1.82539i	−0.408646	+	0.912693i	$0.633999\pi$
$510$		0			0
$511$	−310.982	+	760.432i	−0.608576	+	1.48813i
$512$		0			0
$513$	−91.9342			−0.179209
$514$		0			0
$515$	−100.600			−0.195340
$516$		0			0
$517$			114.001i			0.220506i
$518$		0			0
$519$	291.265			0.561203
$520$		0			0
$521$			176.308i			0.338403i	0.985581	+	0.169202i	$0.0541189\pi$
$521$			176.308i			0.338403i	−0.985581	+	0.169202i	$0.945881\pi$
$522$		0			0
$523$			380.732i			0.727977i	0.931403	+	0.363988i	$0.118585\pi$
$523$			380.732i			0.727977i	−0.931403	+	0.363988i	$0.881415\pi$
$524$		0			0
$525$	−111.664	+	273.047i	−0.212694	+	0.520090i
$526$		0			0
$527$	−4.21159			−0.00799163
$528$		0			0
$529$	−453.324			−0.856944
$530$		0			0
$531$		−	153.933i		−	0.289892i
$532$		0			0
$533$	−419.239			−0.786566
$534$		0			0
$535$			25.8899i			0.0483923i
$536$		0			0
$537$			2.70359i			0.00503462i
$538$		0			0
$539$	252.402	+	247.902i	0.468278	+	0.459930i
$540$		0			0
$541$	966.632			1.78675			0.893375	−	0.449312i	$-0.148331\pi$
$541$	966.632			1.78675			0.893375	+	0.449312i	$0.148331\pi$
$542$		0			0
$543$	447.817			0.824709
$544$		0			0
$545$			83.1097i			0.152495i
$546$		0			0
$547$	−556.676			−1.01769			−0.508844	−	0.860858i	$-0.669927\pi$
$547$	−556.676			−1.01769			−0.508844	+	0.860858i	$0.669927\pi$
$548$		0			0
$549$		−	64.5213i		−	0.117525i
$550$		0			0
$551$		−	344.407i		−	0.625058i
$552$		0			0
$553$	399.863	+	163.526i	0.723080	+	0.295707i
$554$		0			0
$555$	−6.14856			−0.0110785
$556$		0			0
$557$	744.405			1.33645			0.668227	−	0.743957i	$-0.267053\pi$
$557$	744.405			1.33645			0.668227	+	0.743957i	$0.267053\pi$
$558$		0			0
$559$			169.463i			0.303155i
$560$		0			0
$561$	−191.994			−0.342235
$562$		0			0
$563$			226.176i			0.401733i	0.979619	+	0.200867i	$0.0643758\pi$
$563$			226.176i			0.401733i	−0.979619	+	0.200867i	$0.935624\pi$
$564$		0			0
$565$		−	69.4618i		−	0.122941i
$566$		0			0
$567$	58.3122	+	23.8471i	0.102843	+	0.0420583i
$568$		0			0
$569$	147.081			0.258490			0.129245	−	0.991613i	$-0.458745\pi$
$569$	147.081			0.258490			0.129245	+	0.991613i	$0.458745\pi$
$570$		0			0
$571$	−864.521			−1.51405			−0.757023	−	0.653388i	$-0.773347\pi$
$571$	−864.521			−1.51405			−0.757023	+	0.653388i	$0.773347\pi$
$572$		0			0
$573$		−	457.482i		−	0.798398i
$574$		0			0
$575$	−211.661			−0.368106
$576$		0			0
$577$		−	90.5504i		−	0.156933i	−0.996917	−	0.0784666i	$-0.974998\pi$
$577$		−	90.5504i		−	0.156933i	0.996917	−	0.0784666i	$-0.0250024\pi$
$578$		0			0
$579$		−	148.445i		−	0.256381i
$580$		0			0
$581$	173.247	−	423.634i	0.298188	−	0.729146i
$582$		0			0
$583$	−131.443			−0.225459
$584$		0			0
$585$	−14.1628			−0.0242099
$586$		0			0
$587$		−	942.714i		−	1.60599i	−0.595988	−	0.802993i	$-0.703240\pi$
$587$		−	942.714i		−	1.60599i	0.595988	−	0.802993i	$-0.296760\pi$
$588$		0			0
$589$	4.85350			0.00824023
$590$		0			0
$591$		−	197.268i		−	0.333787i
$592$		0			0
$593$		−	706.976i		−	1.19220i	−0.802910	−	0.596101i	$-0.796716\pi$
$593$		−	706.976i		−	1.19220i	0.802910	−	0.596101i	$-0.203284\pi$
$594$		0			0
$595$	81.3596	+	33.2724i	0.136739	+	0.0559200i
$596$		0			0
$597$	−91.1968			−0.152759
$598$		0			0
$599$	787.489			1.31467			0.657337	−	0.753597i	$-0.271683\pi$
$599$	787.489			1.31467			0.657337	+	0.753597i	$0.271683\pi$
$600$		0			0
$601$		−	170.509i		−	0.283709i	−0.989887	−	0.141855i	$-0.954693\pi$
$601$		−	170.509i		−	0.283709i	0.989887	−	0.141855i	$-0.0453066\pi$
$602$		0			0
$603$	−136.592			−0.226521
$604$		0			0
$605$			56.3302i			0.0931077i
$606$		0			0
$607$		−	459.907i		−	0.757673i	−0.925464	−	0.378836i	$-0.876324\pi$
$607$		−	459.907i		−	0.757673i	0.925464	−	0.378836i	$-0.123676\pi$
$608$		0			0
$609$	−89.3367	+	218.451i	−0.146694	+	0.358705i
$610$		0			0
$611$	−91.1363			−0.149159
$612$		0			0
$613$	−1203.58			−1.96343			−0.981716	−	0.190353i	$-0.939037\pi$
$613$	−1203.58			−1.96343			−0.981716	+	0.190353i	$0.939037\pi$
$614$		0			0
$615$			102.898i			0.167314i
$616$		0			0
$617$	453.296			0.734677			0.367339	−	0.930087i	$-0.380269\pi$
$617$	453.296			0.734677			0.367339	+	0.930087i	$0.380269\pi$
$618$		0			0
$619$			538.168i			0.869415i	0.900572	+	0.434708i	$0.143148\pi$
$619$			538.168i			0.869415i	−0.900572	+	0.434708i	$0.856852\pi$
$620$		0			0
$621$			45.2025i			0.0727898i
$622$		0			0
$623$	−280.295	+	685.394i	−0.449912	+	1.10015i
$624$		0			0
$625$	575.274			0.920439
$626$		0			0
$627$	221.256			0.352881
$628$		0			0
$629$		−	66.6336i		−	0.105936i
$630$		0			0
$631$	−962.673			−1.52563			−0.762815	−	0.646617i	$-0.776183\pi$
$631$	−962.673			−1.52563			−0.762815	+	0.646617i	$0.776183\pi$
$632$		0			0
$633$			96.9519i			0.153163i
$634$		0			0
$635$		−	176.662i		−	0.278209i
$636$		0			0
$637$	−198.181	+	201.778i	−0.311116	+	0.316763i
$638$		0			0
$639$	301.135			0.471260
$640$		0			0
$641$	18.4674			0.0288103			0.0144051	−	0.999896i	$-0.495415\pi$
$641$	18.4674			0.0288103			0.0144051	+	0.999896i	$0.495415\pi$
$642$		0			0
$643$		−	301.527i		−	0.468938i	−0.972124	−	0.234469i	$-0.924665\pi$
$643$		−	301.527i		−	0.468938i	0.972124	−	0.234469i	$-0.0753352\pi$
$644$		0			0
$645$	41.5931			0.0644854
$646$		0			0
$647$		−	488.736i		−	0.755388i	−0.925930	−	0.377694i	$-0.876717\pi$
$647$		−	488.736i		−	0.755388i	0.925930	−	0.377694i	$-0.123283\pi$
$648$		0			0
$649$			370.467i			0.570828i
$650$		0			0
$651$	−3.07849	−	1.25896i	−0.00472886	−	0.00193389i
$652$		0			0
$653$	−575.838			−0.881835			−0.440918	−	0.897548i	$-0.645347\pi$
$653$	−575.838			−0.881835			−0.440918	+	0.897548i	$0.645347\pi$
$654$		0			0
$655$	−13.7008			−0.0209173
$656$		0			0
$657$		−	352.099i		−	0.535919i
$658$		0			0
$659$	−664.692			−1.00864			−0.504318	−	0.863518i	$-0.668256\pi$
$659$	−664.692			−1.00864			−0.504318	+	0.863518i	$0.668256\pi$
$660$		0			0
$661$		−	374.795i		−	0.567012i	−0.958970	−	0.283506i	$-0.908502\pi$
$661$		−	374.795i		−	0.567012i	0.958970	−	0.283506i	$-0.0914976\pi$
$662$		0			0
$663$		−	153.486i		−	0.231502i
$664$		0			0
$665$	−93.7601	−	38.3436i	−0.140993	−	0.0576596i
$666$		0			0
$667$	−169.339			−0.253881
$668$		0			0
$669$	644.327			0.963119
$670$		0			0
$671$			155.282i			0.231419i
$672$		0			0
$673$	−485.975			−0.722103			−0.361052	−	0.932546i	$-0.617582\pi$
$673$	−485.975			−0.722103			−0.361052	+	0.932546i	$0.617582\pi$
$674$		0			0
$675$		−	126.428i		−	0.187300i
$676$		0			0
$677$		−	542.126i		−	0.800778i	−0.916345	−	0.400389i	$-0.868875\pi$
$677$		−	542.126i		−	0.800778i	0.916345	−	0.400389i	$-0.131125\pi$
$678$		0			0
$679$	−158.795	+	388.295i	−0.233866	+	0.571863i
$680$		0			0
$681$	−404.877			−0.594533
$682$		0			0
$683$	749.854			1.09788			0.548942	−	0.835861i	$-0.315031\pi$
$683$	749.854			1.09788			0.548942	+	0.835861i	$0.315031\pi$
$684$		0			0
$685$		−	85.9398i		−	0.125460i
$686$		0			0
$687$	272.322			0.396393
$688$		0			0
$689$		−	105.079i		−	0.152510i
$690$		0			0
$691$			577.762i			0.836125i	0.908418	+	0.418063i	$0.137291\pi$
$691$			577.762i			0.836125i	−0.908418	+	0.418063i	$0.862709\pi$
$692$		0			0
$693$	−140.339	−	57.3924i	−0.202510	−	0.0828173i
$694$		0			0
$695$	−178.866			−0.257362
$696$		0			0
$697$	−1115.13			−1.59991
$698$		0			0
$699$		−	369.610i		−	0.528770i
$700$		0			0
$701$	−652.079			−0.930212			−0.465106	−	0.885255i	$-0.653984\pi$
$701$	−652.079			−0.930212			−0.465106	+	0.885255i	$0.653984\pi$
$702$		0			0
$703$			76.7896i			0.109231i
$704$		0			0
$705$			22.3685i			0.0317283i
$706$		0			0
$707$	238.725	−	583.743i	0.337659	−	0.825662i
$708$		0			0
$709$	131.335			0.185240			0.0926202	−	0.995702i	$-0.470476\pi$
$709$	131.335			0.185240			0.0926202	+	0.995702i	$0.470476\pi$
$710$		0			0
$711$	−185.147			−0.260403
$712$		0			0
$713$		−	2.38638i		−	0.00334696i
$714$		0			0
$715$	34.0853			0.0476717
$716$		0			0
$717$		−	419.897i		−	0.585631i
$718$		0			0
$719$		−	533.056i		−	0.741385i	−0.928756	−	0.370692i	$-0.879120\pi$
$719$		−	533.056i		−	0.741385i	0.928756	−	0.370692i	$-0.120880\pi$
$720$		0			0
$721$	325.900	−	796.910i	0.452011	−	1.10528i
$722$		0			0
$723$	696.339			0.963124
$724$		0			0
$725$	473.627			0.653279
$726$		0			0
$727$			555.602i			0.764239i	0.924113	+	0.382120i	$0.124806\pi$
$727$			555.602i			0.764239i	−0.924113	+	0.382120i	$0.875194\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$			450.755i			0.616628i
$732$		0			0
$733$		−	1274.96i		−	1.73938i	−0.493600	−	0.869689i	$-0.664319\pi$
$733$		−	1274.96i		−	1.73938i	0.493600	−	0.869689i	$-0.335681\pi$
$734$		0			0
$735$	49.5243	+	48.6414i	0.0673800	+	0.0661788i
$736$		0			0
$737$	328.734			0.446044
$738$		0			0
$739$	168.110			0.227482			0.113741	−	0.993510i	$-0.463717\pi$
$739$	168.110			0.227482			0.113741	+	0.993510i	$0.463717\pi$
$740$		0			0
$741$			176.879i			0.238704i
$742$		0			0
$743$	1386.00			1.86541			0.932703	−	0.360645i	$-0.117443\pi$
$743$	1386.00			1.86541			0.932703	+	0.360645i	$0.117443\pi$
$744$		0			0
$745$			159.512i			0.214110i
$746$		0			0
$747$			196.153i			0.262587i
$748$		0			0
$749$	−205.088	−	83.8720i	−0.273816	−	0.111979i
$750$		0			0
$751$	−107.911			−0.143690			−0.0718448	−	0.997416i	$-0.522889\pi$
$751$	−107.911			−0.143690			−0.0718448	+	0.997416i	$0.522889\pi$
$752$		0			0
$753$	679.833			0.902832
$754$		0			0
$755$		−	138.566i		−	0.183531i
$756$		0			0
$757$	94.3323			0.124613			0.0623067	−	0.998057i	$-0.480154\pi$
$757$	94.3323			0.124613			0.0623067	+	0.998057i	$0.480154\pi$
$758$		0			0
$759$		−	108.788i		−	0.143331i
$760$		0			0
$761$		−	238.161i		−	0.312958i	−0.987681	−	0.156479i	$-0.949986\pi$
$761$		−	238.161i		−	0.312958i	0.987681	−	0.156479i	$-0.0500143\pi$
$762$		0			0
$763$	−658.360	−	269.240i	−0.862857	−	0.352870i
$764$		0			0
$765$	−37.6715			−0.0492438
$766$		0			0
$767$	−296.163			−0.386132
$768$		0			0
$769$			751.525i			0.977276i	0.872487	+	0.488638i	$0.162506\pi$
$769$			751.525i			0.977276i	−0.872487	+	0.488638i	$0.837494\pi$
$770$		0			0
$771$	560.738			0.727286
$772$		0			0
$773$		−	1198.57i		−	1.55054i	−0.631628	−	0.775272i	$-0.717613\pi$
$773$		−	1198.57i		−	1.55054i	0.631628	−	0.775272i	$-0.282387\pi$
$774$		0			0
$775$			6.67452i			0.00861228i
$776$		0			0
$777$	19.9187	−	48.7063i	0.0256354	−	0.0626851i
$778$		0			0
$779$	1285.10			1.64968
$780$		0			0
$781$	−724.736			−0.927959
$782$		0			0
$783$		−	101.148i		−	0.129180i
$784$		0			0
$785$	−48.8456			−0.0622237
$786$		0			0
$787$			66.4238i			0.0844012i	0.999109	+	0.0422006i	$0.0134369\pi$
$787$			66.4238i			0.0844012i	−0.999109	+	0.0422006i	$0.986563\pi$
$788$		0			0
$789$		−	553.396i		−	0.701389i
$790$		0			0
$791$	550.247	+	225.026i	0.695635	+	0.284483i
$792$		0			0
$793$	−124.138			−0.156542
$794$		0			0
$795$	−25.7906			−0.0324410
$796$		0			0
$797$		−	705.174i		−	0.884785i	−0.896821	−	0.442393i	$-0.854130\pi$
$797$		−	705.174i		−	0.884785i	0.896821	−	0.442393i	$-0.145870\pi$
$798$		0			0
$799$	−242.413			−0.303396
$800$		0			0
$801$		−	317.354i		−	0.396198i
$802$		0			0
$803$			847.390i			1.05528i
$804$		0			0
$805$	−18.8529	+	46.1002i	−0.0234198	+	0.0572673i
$806$		0			0
$807$	−290.553			−0.360041
$808$		0			0
$809$	84.1821			0.104057			0.0520285	−	0.998646i	$-0.483431\pi$
$809$	84.1821			0.104057			0.0520285	+	0.998646i	$0.483431\pi$
$810$		0			0
$811$		−	1225.17i		−	1.51069i	−0.655324	−	0.755347i	$-0.727468\pi$
$811$		−	1225.17i		−	1.51069i	0.655324	−	0.755347i	$-0.272532\pi$
$812$		0			0
$813$	376.240			0.462779
$814$		0			0
$815$		−	153.320i		−	0.188122i
$816$		0			0
$817$		−	519.457i		−	0.635811i
$818$		0			0
$819$	45.8812	−	112.191i	0.0560211	−	0.136986i
$820$		0			0
$821$	−1545.37			−1.88231			−0.941153	−	0.337979i	$-0.890257\pi$
$821$	−1545.37			−1.88231			−0.941153	+	0.337979i	$0.890257\pi$
$822$		0			0
$823$	−537.199			−0.652732			−0.326366	−	0.945243i	$-0.605824\pi$
$823$	−537.199			−0.652732			−0.326366	+	0.945243i	$0.605824\pi$
$824$		0			0
$825$			304.271i			0.368814i
$826$		0			0
$827$	−382.318			−0.462295			−0.231147	−	0.972919i	$-0.574248\pi$
$827$	−382.318			−0.462295			−0.231147	+	0.972919i	$0.574248\pi$
$828$		0			0
$829$			200.589i			0.241965i	0.992655	+	0.120983i	$0.0386045\pi$
$829$			200.589i			0.241965i	−0.992655	+	0.120983i	$0.961395\pi$
$830$		0			0
$831$		−	661.122i		−	0.795573i
$832$		0			0
$833$	−527.140	+	536.708i	−0.632821	+	0.644308i
$834$		0			0
$835$	−45.9975			−0.0550868
$836$		0			0
$837$	1.42542			0.00170301
$838$		0			0
$839$			902.324i			1.07548i	0.843112	+	0.537738i	$0.180721\pi$
$839$			902.324i			1.07548i	−0.843112	+	0.537738i	$0.819279\pi$
$840$		0			0
$841$	−462.075			−0.549436
$842$		0			0
$843$			78.1021i			0.0926478i
$844$		0			0
$845$		−	110.978i		−	0.131335i
$846$		0			0
$847$	−446.224	−	182.485i	−0.526829	−	0.215449i
$848$		0			0
$849$	−547.884			−0.645329
$850$		0			0
$851$	37.7561			0.0443668
$852$		0			0
$853$		−	1549.79i		−	1.81687i	−0.418030	−	0.908433i	$-0.637279\pi$
$853$		−	1549.79i		−	1.81687i	0.418030	−	0.908433i	$-0.362721\pi$
$854$		0			0
$855$	43.4132			0.0507757
$856$		0			0
$857$		−	295.304i		−	0.344579i	−0.985046	−	0.172290i	$-0.944884\pi$
$857$		−	295.304i		−	0.344579i	0.985046	−	0.172290i	$-0.0551165\pi$
$858$		0			0
$859$			1170.26i			1.36236i	0.732118	+	0.681178i	$0.238532\pi$
$859$			1170.26i			1.36236i	−0.732118	+	0.681178i	$0.761468\pi$
$860$		0			0
$861$	−815.115	−	333.345i	−0.946707	−	0.387161i
$862$		0			0
$863$	−386.789			−0.448192			−0.224096	−	0.974567i	$-0.571943\pi$
$863$	−386.789			−0.448192			−0.224096	+	0.974567i	$0.571943\pi$
$864$		0			0
$865$	−137.541			−0.159007
$866$		0			0
$867$			92.3062i			0.106466i
$868$		0			0
$869$	445.589			0.512761
$870$		0			0
$871$			262.800i			0.301723i
$872$		0			0
$873$		−	179.790i		−	0.205945i
$874$		0			0
$875$	106.910	−	261.422i	0.122183	−	0.298768i
$876$		0			0
$877$	179.268			0.204411			0.102205	−	0.994763i	$-0.467410\pi$
$877$	179.268			0.204411			0.102205	+	0.994763i	$0.467410\pi$
$878$		0			0
$879$	242.477			0.275855
$880$		0			0
$881$		−	1049.76i		−	1.19155i	−0.803150	−	0.595777i	$-0.796844\pi$
$881$		−	1049.76i		−	1.19155i	0.803150	−	0.595777i	$-0.203156\pi$
$882$		0			0
$883$	711.891			0.806219			0.403109	−	0.915152i	$-0.367929\pi$
$883$	711.891			0.806219			0.403109	+	0.915152i	$0.367929\pi$
$884$		0			0
$885$			72.6902i			0.0821358i
$886$		0			0
$887$		−	1182.66i		−	1.33333i	−0.745359	−	0.666663i	$-0.767722\pi$
$887$		−	1182.66i		−	1.33333i	0.745359	−	0.666663i	$-0.232278\pi$
$888$		0			0
$889$	1399.45	+	572.310i	1.57418	+	0.643768i
$890$		0			0
$891$	64.9805			0.0729298
$892$		0			0
$893$	279.361			0.312834
$894$		0			0
$895$		−	1.27669i		−	0.00142647i
$896$		0			0
$897$	86.9685			0.0969549
$898$		0			0
$899$			5.33994i			0.00593986i
$900$		0			0
$901$		−	279.500i		−	0.310211i
$902$		0			0
$903$	−134.744	+	329.483i	−0.149218	+	0.364876i
$904$		0			0
$905$	−211.468			−0.233667
$906$		0			0
$907$	−77.4195			−0.0853578			−0.0426789	−	0.999089i	$-0.513589\pi$
$907$	−77.4195			−0.0853578			−0.0426789	+	0.999089i	$0.513589\pi$
$908$		0			0
$909$			270.288i			0.297346i
$910$		0			0
$911$	−1122.02			−1.23164			−0.615818	−	0.787888i	$-0.711174\pi$
$911$	−1122.02			−1.23164			−0.615818	+	0.787888i	$0.711174\pi$
$912$		0			0
$913$		−	472.078i		−	0.517062i
$914$		0			0
$915$			30.4683i			0.0332987i
$916$		0			0
$917$	44.3847	−	108.532i	0.0484021	−	0.118356i
$918$		0			0
$919$	−531.689			−0.578551			−0.289276	−	0.957246i	$-0.593414\pi$
$919$	−531.689			−0.578551			−0.289276	+	0.957246i	$0.593414\pi$
$920$		0			0
$921$	756.517			0.821408
$922$		0			0
$923$		−	579.377i		−	0.627710i
$924$		0			0
$925$	−105.601			−0.114163
$926$		0			0
$927$			368.989i			0.398046i
$928$		0			0
$929$			1563.00i			1.68245i	0.540684	+	0.841226i	$0.318166\pi$
$929$			1563.00i			1.68245i	−0.540684	+	0.841226i	$0.681834\pi$
$930$		0			0
$931$	607.484	−	618.511i	0.652507	−	0.664351i
$932$		0			0
$933$	−711.624			−0.762726
$934$		0			0
$935$	90.6634			0.0969662
$936$		0			0
$937$			690.326i			0.736741i	0.929679	+	0.368371i	$0.120084\pi$
$937$			690.326i			0.736741i	−0.929679	+	0.368371i	$0.879916\pi$
$938$		0			0
$939$	−111.299			−0.118529
$940$		0			0
$941$		−	405.493i		−	0.430918i	−0.976513	−	0.215459i	$-0.930875\pi$
$941$		−	405.493i		−	0.430918i	0.976513	−	0.215459i	$-0.0691247\pi$
$942$		0			0
$943$		−	631.860i		−	0.670053i
$944$		0			0
$945$	−27.5362	−	11.2611i	−0.0291389	−	0.0119165i
$946$		0			0
$947$	−1844.51			−1.94774			−0.973868	−	0.227116i	$-0.927070\pi$
$947$	−1844.51			−1.94774			−0.973868	+	0.227116i	$0.927070\pi$
$948$		0			0
$949$	−677.430			−0.713836
$950$		0			0
$951$			726.194i			0.763611i
$952$		0			0
$953$	1146.31			1.20285			0.601423	−	0.798931i	$-0.294601\pi$
$953$	1146.31			1.20285			0.601423	+	0.798931i	$0.294601\pi$
$954$		0			0
$955$			216.033i			0.226212i
$956$		0			0
$957$			243.432i			0.254370i
$958$		0			0
$959$	680.779	+	278.408i	0.709884	+	0.290311i
$960$		0			0
$961$	960.925			0.999922
$962$		0			0
$963$	94.9610			0.0986096
$964$		0			0
$965$			70.0987i			0.0726411i
$966$		0			0
$967$	368.569			0.381147			0.190573	−	0.981673i	$-0.438965\pi$
$967$	368.569			0.381147			0.190573	+	0.981673i	$0.438965\pi$
$968$		0			0
$969$			470.481i			0.485532i
$970$		0			0
$971$			364.243i			0.375121i	0.982253	+	0.187561i	$0.0600581\pi$
$971$			364.243i			0.375121i	−0.982253	+	0.187561i	$0.939942\pi$
$972$		0			0
$973$	579.450	−	1416.90i	0.595530	−	1.45622i
$974$		0			0
$975$	−243.244			−0.249481
$976$		0			0
$977$	−681.020			−0.697052			−0.348526	−	0.937299i	$-0.613318\pi$
$977$	−681.020			−0.697052			−0.348526	+	0.937299i	$0.613318\pi$
$978$		0			0
$979$			763.771i			0.780155i
$980$		0			0
$981$	304.837			0.310741
$982$		0			0
$983$		−	837.334i		−	0.851815i	−0.904767	−	0.425907i	$-0.859955\pi$
$983$		−	837.334i		−	0.851815i	0.904767	−	0.425907i	$-0.140045\pi$
$984$		0			0
$985$			93.1540i			0.0945726i
$986$		0			0
$987$	−177.194	−	72.4642i	−0.179527	−	0.0734187i
$988$		0			0
$989$	−255.408			−0.258249
$990$		0			0
$991$	−775.019			−0.782057			−0.391029	−	0.920379i	$-0.627881\pi$
$991$	−775.019			−0.782057			−0.391029	+	0.920379i	$0.627881\pi$
$992$		0			0
$993$			107.285i			0.108041i
$994$		0			0
$995$	43.0650			0.0432814
$996$		0			0
$997$		−	333.556i		−	0.334560i	−0.985909	−	0.167280i	$-0.946502\pi$
$997$		−	333.556i		−	0.334560i	0.985909	−	0.167280i	$-0.0534984\pi$
$998$		0			0
$999$			22.5522i			0.0225748i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.3.f.b.97.12	yes	16
3.2	odd	2		2016.3.f.g.1441.10		16
4.3	odd	2	inner	672.3.f.b.97.4	✓	16
7.6	odd	2	inner	672.3.f.b.97.5	yes	16
8.3	odd	2		1344.3.f.j.769.13		16
8.5	even	2		1344.3.f.j.769.5		16
12.11	even	2		2016.3.f.g.1441.9		16
21.20	even	2		2016.3.f.g.1441.8		16
28.27	even	2	inner	672.3.f.b.97.13	yes	16
56.13	odd	2		1344.3.f.j.769.12		16
56.27	even	2		1344.3.f.j.769.4		16
84.83	odd	2		2016.3.f.g.1441.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.3.f.b.97.4	✓	16	4.3	odd	2	inner
672.3.f.b.97.5	yes	16	7.6	odd	2	inner
672.3.f.b.97.12	yes	16	1.1	even	1	trivial
672.3.f.b.97.13	yes	16	28.27	even	2	inner
1344.3.f.j.769.4		16	56.27	even	2
1344.3.f.j.769.5		16	8.5	even	2
1344.3.f.j.769.12		16	56.13	odd	2
1344.3.f.j.769.13		16	8.3	odd	2
2016.3.f.g.1441.7		16	84.83	odd	2
2016.3.f.g.1441.8		16	21.20	even	2
2016.3.f.g.1441.9		16	12.11	even	2
2016.3.f.g.1441.10		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)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -0.817910i q^{5} +(6.47914 + 2.64968i) q^{7} -3.00000 q^{9} +7.22005 q^{11} +5.77193i q^{13} +1.41666 q^{15} +15.3528i q^{17} -17.6927i q^{19} +(-4.58937 + 11.2222i) q^{21} -8.69922 q^{23} +24.3310 q^{25} -5.19615i q^{27} +19.4660 q^{29} +0.274321i q^{31} +12.5055i q^{33} +(2.16720 - 5.29935i) q^{35} -4.34017 q^{37} -9.99728 q^{39} +72.6342i q^{41} +29.3599 q^{43} +2.45373i q^{45} +15.7896i q^{47} +(34.9584 + 34.3352i) q^{49} -26.5917 q^{51} -18.2052 q^{53} -5.90535i q^{55} +30.6447 q^{57} +51.3109i q^{59} +21.5071i q^{61} +(-19.4374 - 7.94903i) q^{63} +4.72092 q^{65} +45.5308 q^{67} -15.0675i q^{69} -100.378 q^{71} +117.366i q^{73} +42.1426i q^{75} +(46.7797 + 19.1308i) q^{77} +61.7155 q^{79} +9.00000 q^{81} -65.3843i q^{83} +12.5572 q^{85} +33.7161i q^{87} +105.785i q^{89} +(-15.2937 + 37.3971i) q^{91} -0.475139 q^{93} -14.4711 q^{95} +59.9300i q^{97} -21.6602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} + 24 q^{21} - 64 q^{25} - 128 q^{29} + 48 q^{37} - 256 q^{49} - 160 q^{53} - 144 q^{57} - 288 q^{65} + 128 q^{77} + 144 q^{81} - 16 q^{85} + 96 q^{93}+O(q^{100}) \) 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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