LMFDB - Embedded newform 672.2.k.c.545.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(545,672)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("672.545");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			545.8
Root			$1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	672.545
Dual form			672.2.k.c.545.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.58114 + 0.707107i) q^{3} +1.41421 q^{5} +(2.23607 + 1.41421i) q^{7} +(2.00000 + 2.23607i) q^{9} +O(q^{10})$	$q+(1.58114 + 0.707107i) q^{3} +1.41421 q^{5} +(2.23607 + 1.41421i) q^{7} +(2.00000 + 2.23607i) q^{9} -3.16228i q^{13} +(2.23607 + 1.00000i) q^{15} -2.82843 q^{17} +4.24264i q^{19} +(2.53553 + 3.81721i) q^{21} -6.00000i q^{23} -3.00000 q^{25} +(1.58114 + 4.94975i) q^{27} -4.47214i q^{29} +5.65685i q^{31} +(3.16228 + 2.00000i) q^{35} -2.00000 q^{37} +(2.23607 - 5.00000i) q^{39} -8.48528 q^{41} +8.94427 q^{43} +(2.82843 + 3.16228i) q^{45} +6.32456 q^{47} +(3.00000 + 6.32456i) q^{49} +(-4.47214 - 2.00000i) q^{51} -13.4164i q^{53} +(-3.00000 + 6.70820i) q^{57} -3.16228 q^{59} -3.16228i q^{61} +(1.30986 + 7.82843i) q^{63} -4.47214i q^{65} +(4.24264 - 9.48683i) q^{69} +8.00000i q^{71} +12.6491i q^{73} +(-4.74342 - 2.12132i) q^{75} -13.4164 q^{79} +(-1.00000 + 8.94427i) q^{81} +3.16228 q^{83} -4.00000 q^{85} +(3.16228 - 7.07107i) q^{87} -11.3137 q^{89} +(4.47214 - 7.07107i) q^{91} +(-4.00000 + 8.94427i) q^{93} +6.00000i q^{95} -18.9737i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} - 8 q^{21} - 24 q^{25} - 16 q^{37} + 24 q^{49} - 24 q^{57} - 8 q^{81} - 32 q^{85} - 32 q^{93}+O(q^{100}) $ 8 * q + 16 * q^9 - 8 * q^21 - 24 * q^25 - 16 * q^37 + 24 * q^49 - 24 * q^57 - 8 * q^81 - 32 * q^85 - 32 * q^93

Display 100 coefficients 10 coefficients

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
$3$	1.58114	+	0.707107i	0.912871	+	0.408248i
$3$	1.58114	+	0.707107i	0.912871	+	0.408248i
$4$		0			0
$5$	1.41421			0.632456			0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421			0.632456			0.316228	+	0.948683i	$0.397584\pi$
$6$		0			0
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$7$	2.23607	+	1.41421i	0.845154	+	0.534522i
$8$		0			0
$9$	2.00000	+	2.23607i	0.666667	+	0.745356i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		−	3.16228i		−	0.877058i	−0.898717	−	0.438529i	$-0.855500\pi$
$13$		−	3.16228i		−	0.877058i	0.898717	−	0.438529i	$-0.144500\pi$
$14$		0			0
$15$	2.23607	+	1.00000i	0.577350	+	0.258199i
$16$		0			0
$17$	−2.82843			−0.685994			−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843			−0.685994			−0.342997	+	0.939336i	$0.611442\pi$
$18$		0			0
$19$			4.24264i			0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$			4.24264i			0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$		0			0
$21$	2.53553	+	3.81721i	0.553299	+	0.832983i
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$		0			0
$27$	1.58114	+	4.94975i	0.304290	+	0.952579i
$28$		0			0
$29$		−	4.47214i		−	0.830455i	−0.909718	−	0.415227i	$-0.863702\pi$
$29$		−	4.47214i		−	0.830455i	0.909718	−	0.415227i	$-0.136298\pi$
$30$		0			0
$31$			5.65685i			1.01600i	0.861357	+	0.508001i	$0.169615\pi$
$31$			5.65685i			1.01600i	−0.861357	+	0.508001i	$0.830385\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.16228	+	2.00000i	0.534522	+	0.338062i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	2.23607	−	5.00000i	0.358057	−	0.800641i
$40$		0			0
$41$	−8.48528			−1.32518			−0.662589	−	0.748983i	$-0.730542\pi$
$41$	−8.48528			−1.32518			−0.662589	+	0.748983i	$0.730542\pi$
$42$		0			0
$43$	8.94427			1.36399			0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427			1.36399			0.681994	+	0.731357i	$0.261113\pi$
$44$		0			0
$45$	2.82843	+	3.16228i	0.421637	+	0.471405i
$46$		0			0
$47$	6.32456			0.922531			0.461266	−	0.887262i	$-0.347396\pi$
$47$	6.32456			0.922531			0.461266	+	0.887262i	$0.347396\pi$
$48$		0			0
$49$	3.00000	+	6.32456i	0.428571	+	0.903508i
$50$		0			0
$51$	−4.47214	−	2.00000i	−0.626224	−	0.280056i
$52$		0			0
$53$		−	13.4164i		−	1.84289i	−0.388514	−	0.921443i	$-0.627012\pi$
$53$		−	13.4164i		−	1.84289i	0.388514	−	0.921443i	$-0.372988\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.00000	+	6.70820i	−0.397360	+	0.888523i
$58$		0			0
$59$	−3.16228			−0.411693			−0.205847	−	0.978584i	$-0.565995\pi$
$59$	−3.16228			−0.411693			−0.205847	+	0.978584i	$0.565995\pi$
$60$		0			0
$61$		−	3.16228i		−	0.404888i	−0.979294	−	0.202444i	$-0.935112\pi$
$61$		−	3.16228i		−	0.404888i	0.979294	−	0.202444i	$-0.0648884\pi$
$62$		0			0
$63$	1.30986	+	7.82843i	0.165027	+	0.986289i
$64$		0			0
$65$		−	4.47214i		−	0.554700i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	4.24264	−	9.48683i	0.510754	−	1.14208i
$70$		0			0
$71$			8.00000i			0.949425i	0.880141	+	0.474713i	$0.157448\pi$
$71$			8.00000i			0.949425i	−0.880141	+	0.474713i	$0.842552\pi$
$72$		0			0
$73$			12.6491i			1.48047i	0.672350	+	0.740233i	$0.265285\pi$
$73$			12.6491i			1.48047i	−0.672350	+	0.740233i	$0.734715\pi$
$74$		0			0
$75$	−4.74342	−	2.12132i	−0.547723	−	0.244949i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−13.4164			−1.50946			−0.754732	−	0.656033i	$-0.772233\pi$
$79$	−13.4164			−1.50946			−0.754732	+	0.656033i	$0.772233\pi$
$80$		0			0
$81$	−1.00000	+	8.94427i	−0.111111	+	0.993808i
$82$		0			0
$83$	3.16228			0.347105			0.173553	−	0.984825i	$-0.444475\pi$
$83$	3.16228			0.347105			0.173553	+	0.984825i	$0.444475\pi$
$84$		0			0
$85$	−4.00000			−0.433861
$86$		0			0
$87$	3.16228	−	7.07107i	0.339032	−	0.758098i
$88$		0			0
$89$	−11.3137			−1.19925			−0.599625	−	0.800281i	$-0.704684\pi$
$89$	−11.3137			−1.19925			−0.599625	+	0.800281i	$0.704684\pi$
$90$		0			0
$91$	4.47214	−	7.07107i	0.468807	−	0.741249i
$92$		0			0
$93$	−4.00000	+	8.94427i	−0.414781	+	0.927478i
$94$		0			0
$95$			6.00000i			0.615587i
$96$		0			0
$97$		−	18.9737i		−	1.92648i	−0.268635	−	0.963242i	$-0.586573\pi$
$97$		−	18.9737i		−	1.92648i	0.268635	−	0.963242i	$-0.413427\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	12.7279			1.26648			0.633238	−	0.773957i	$-0.281726\pi$
$101$	12.7279			1.26648			0.633238	+	0.773957i	$0.281726\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	3.58579	+	5.39835i	0.349937	+	0.526825i
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$	−3.16228	−	1.41421i	−0.300150	−	0.134231i
$112$		0			0
$113$		−	4.47214i		−	0.420703i	−0.977626	−	0.210352i	$-0.932539\pi$
$113$		−	4.47214i		−	0.420703i	0.977626	−	0.210352i	$-0.0674609\pi$
$114$		0			0
$115$		−	8.48528i		−	0.791257i
$116$		0			0
$117$	7.07107	−	6.32456i	0.653720	−	0.584705i
$118$		0			0
$119$	−6.32456	−	4.00000i	−0.579771	−	0.366679i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	−13.4164	−	6.00000i	−1.20972	−	0.541002i
$124$		0			0
$125$	−11.3137			−1.01193
$126$		0			0
$127$	−8.94427			−0.793676			−0.396838	−	0.917889i	$-0.629892\pi$
$127$	−8.94427			−0.793676			−0.396838	+	0.917889i	$0.629892\pi$
$128$		0			0
$129$	14.1421	+	6.32456i	1.24515	+	0.556846i
$130$		0			0
$131$	−15.8114			−1.38145			−0.690724	−	0.723119i	$-0.742708\pi$
$131$	−15.8114			−1.38145			−0.690724	+	0.723119i	$0.742708\pi$
$132$		0			0
$133$	−6.00000	+	9.48683i	−0.520266	+	0.822613i
$134$		0			0
$135$	2.23607	+	7.00000i	0.192450	+	0.602464i
$136$		0			0
$137$			8.94427i			0.764161i	0.924129	+	0.382080i	$0.124792\pi$
$137$			8.94427i			0.764161i	−0.924129	+	0.382080i	$0.875208\pi$
$138$		0			0
$139$		−	18.3848i		−	1.55938i	−0.626168	−	0.779688i	$-0.715378\pi$
$139$		−	18.3848i		−	1.55938i	0.626168	−	0.779688i	$-0.284622\pi$
$140$		0			0
$141$	10.0000	+	4.47214i	0.842152	+	0.376622i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	6.32456i		−	0.525226i
$146$		0			0
$147$	0.271281	+	12.1213i	0.0223748	+	0.999750i
$148$		0			0
$149$		−	4.47214i		−	0.366372i	−0.983078	−	0.183186i	$-0.941359\pi$
$149$		−	4.47214i		−	0.366372i	0.983078	−	0.183186i	$-0.0586410\pi$
$150$		0			0
$151$	−8.94427			−0.727875			−0.363937	−	0.931423i	$-0.618568\pi$
$151$	−8.94427			−0.727875			−0.363937	+	0.931423i	$0.618568\pi$
$152$		0			0
$153$	−5.65685	−	6.32456i	−0.457330	−	0.511310i
$154$		0			0
$155$			8.00000i			0.642575i
$156$		0			0
$157$			15.8114i			1.26189i	0.775829	+	0.630943i	$0.217332\pi$
$157$			15.8114i			1.26189i	−0.775829	+	0.630943i	$0.782668\pi$
$158$		0			0
$159$	9.48683	−	21.2132i	0.752355	−	1.68232i
$160$		0			0
$161$	8.48528	−	13.4164i	0.668734	−	1.05736i
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−18.9737			−1.46823			−0.734113	−	0.679027i	$-0.762402\pi$
$167$	−18.9737			−1.46823			−0.734113	+	0.679027i	$0.762402\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	−9.48683	+	8.48528i	−0.725476	+	0.648886i
$172$		0			0
$173$	−15.5563			−1.18273			−0.591364	−	0.806405i	$-0.701410\pi$
$173$	−15.5563			−1.18273			−0.591364	+	0.806405i	$0.701410\pi$
$174$		0			0
$175$	−6.70820	−	4.24264i	−0.507093	−	0.320713i
$176$		0			0
$177$	−5.00000	−	2.23607i	−0.375823	−	0.168073i
$178$		0			0
$179$			16.0000i			1.19590i	0.801535	+	0.597948i	$0.204017\pi$
$179$			16.0000i			1.19590i	−0.801535	+	0.597948i	$0.795983\pi$
$180$		0			0
$181$		−	22.1359i		−	1.64535i	−0.568511	−	0.822676i	$-0.692480\pi$
$181$		−	22.1359i		−	1.64535i	0.568511	−	0.822676i	$-0.307520\pi$
$182$		0			0
$183$	2.23607	−	5.00000i	0.165295	−	0.369611i
$184$		0			0
$185$	−2.82843			−0.207950
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−3.46447	+	13.3040i	−0.252003	+	0.967726i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−16.0000			−1.15171			−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000			−1.15171			−0.575853	+	0.817554i	$0.695330\pi$
$194$		0			0
$195$	3.16228	−	7.07107i	0.226455	−	0.506370i
$196$		0			0
$197$		−	4.47214i		−	0.318626i	−0.987228	−	0.159313i	$-0.949072\pi$
$197$		−	4.47214i		−	0.318626i	0.987228	−	0.159313i	$-0.0509280\pi$
$198$		0			0
$199$		−	11.3137i		−	0.802008i	−0.916076	−	0.401004i	$-0.868661\pi$
$199$		−	11.3137i		−	0.802008i	0.916076	−	0.401004i	$-0.131339\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.32456	−	10.0000i	0.443897	−	0.701862i
$204$		0			0
$205$	−12.0000			−0.838116
$206$		0			0
$207$	13.4164	−	12.0000i	0.932505	−	0.834058i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$	−5.65685	+	12.6491i	−0.387601	+	0.866703i
$214$		0			0
$215$	12.6491			0.862662
$216$		0			0
$217$	−8.00000	+	12.6491i	−0.543075	+	0.858678i
$218$		0			0
$219$	−8.94427	+	20.0000i	−0.604398	+	1.35147i
$220$		0			0
$221$			8.94427i			0.601657i
$222$		0			0
$223$			14.1421i			0.947027i	0.880786	+	0.473514i	$0.157015\pi$
$223$			14.1421i			0.947027i	−0.880786	+	0.473514i	$0.842985\pi$
$224$		0			0
$225$	−6.00000	−	6.70820i	−0.400000	−	0.447214i
$226$		0			0
$227$	3.16228			0.209888			0.104944	−	0.994478i	$-0.466534\pi$
$227$	3.16228			0.209888			0.104944	+	0.994478i	$0.466534\pi$
$228$		0			0
$229$		−	3.16228i		−	0.208969i	−0.994527	−	0.104485i	$-0.966681\pi$
$229$		−	3.16228i		−	0.208969i	0.994527	−	0.104485i	$-0.0333193\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	17.8885i		−	1.17192i	−0.810341	−	0.585959i	$-0.800718\pi$
$233$		−	17.8885i		−	1.17192i	0.810341	−	0.585959i	$-0.199282\pi$
$234$		0			0
$235$	8.94427			0.583460
$236$		0			0
$237$	−21.2132	−	9.48683i	−1.37795	−	0.616236i
$238$		0			0
$239$		−	10.0000i		−	0.646846i	−0.946254	−	0.323423i	$-0.895166\pi$
$239$		−	10.0000i		−	0.646846i	0.946254	−	0.323423i	$-0.104834\pi$
$240$		0			0
$241$			6.32456i			0.407400i	0.979033	+	0.203700i	$0.0652968\pi$
$241$			6.32456i			0.407400i	−0.979033	+	0.203700i	$0.934703\pi$
$242$		0			0
$243$	−7.90569	+	13.4350i	−0.507151	+	0.861858i
$244$		0			0
$245$	4.24264	+	8.94427i	0.271052	+	0.571429i
$246$		0			0
$247$	13.4164			0.853666
$248$		0			0
$249$	5.00000	+	2.23607i	0.316862	+	0.141705i
$250$		0			0
$251$	28.4605			1.79641			0.898205	−	0.439576i	$-0.144871\pi$
$251$	28.4605			1.79641			0.898205	+	0.439576i	$0.144871\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−6.32456	−	2.82843i	−0.396059	−	0.177123i
$256$		0			0
$257$	28.2843			1.76432			0.882162	−	0.470946i	$-0.156087\pi$
$257$	28.2843			1.76432			0.882162	+	0.470946i	$0.156087\pi$
$258$		0			0
$259$	−4.47214	−	2.82843i	−0.277885	−	0.175750i
$260$		0			0
$261$	10.0000	−	8.94427i	0.618984	−	0.553637i
$262$		0			0
$263$			16.0000i			0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$		0			0
$265$		−	18.9737i		−	1.16554i
$266$		0			0
$267$	−17.8885	−	8.00000i	−1.09476	−	0.489592i
$268$		0			0
$269$	−9.89949			−0.603583			−0.301791	−	0.953374i	$-0.597585\pi$
$269$	−9.89949			−0.603583			−0.301791	+	0.953374i	$0.597585\pi$
$270$		0			0
$271$			8.48528i			0.515444i	0.966219	+	0.257722i	$0.0829719\pi$
$271$			8.48528i			0.515444i	−0.966219	+	0.257722i	$0.917028\pi$
$272$		0			0
$273$	12.0711	−	8.01806i	0.730574	−	0.485275i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	18.0000			1.08152			0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000			1.08152			0.540758	+	0.841178i	$0.318138\pi$
$278$		0			0
$279$	−12.6491	+	11.3137i	−0.757282	+	0.677334i
$280$		0			0
$281$			17.8885i			1.06714i	0.845756	+	0.533571i	$0.179150\pi$
$281$			17.8885i			1.06714i	−0.845756	+	0.533571i	$0.820850\pi$
$282$		0			0
$283$			21.2132i			1.26099i	0.776192	+	0.630497i	$0.217149\pi$
$283$			21.2132i			1.26099i	−0.776192	+	0.630497i	$0.782851\pi$
$284$		0			0
$285$	−4.24264	+	9.48683i	−0.251312	+	0.561951i
$286$		0			0
$287$	−18.9737	−	12.0000i	−1.11998	−	0.708338i
$288$		0			0
$289$	−9.00000			−0.529412
$290$		0			0
$291$	13.4164	−	30.0000i	0.786484	−	1.75863i
$292$		0			0
$293$	−7.07107			−0.413096			−0.206548	−	0.978436i	$-0.566223\pi$
$293$	−7.07107			−0.413096			−0.206548	+	0.978436i	$0.566223\pi$
$294$		0			0
$295$	−4.47214			−0.260378
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−18.9737			−1.09728
$300$		0			0
$301$	20.0000	+	12.6491i	1.15278	+	0.729083i
$302$		0			0
$303$	20.1246	+	9.00000i	1.15613	+	0.517036i
$304$		0			0
$305$		−	4.47214i		−	0.256074i
$306$		0			0
$307$			4.24264i			0.242140i	0.992644	+	0.121070i	$0.0386326\pi$
$307$			4.24264i			0.242140i	−0.992644	+	0.121070i	$0.961367\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	25.2982			1.43453			0.717265	−	0.696800i	$-0.245394\pi$
$311$	25.2982			1.43453			0.717265	+	0.696800i	$0.245394\pi$
$312$		0			0
$313$		−	6.32456i		−	0.357485i	−0.983896	−	0.178743i	$-0.942797\pi$
$313$		−	6.32456i		−	0.357485i	0.983896	−	0.178743i	$-0.0572029\pi$
$314$		0			0
$315$	1.85242	+	11.0711i	0.104372	+	0.623784i
$316$		0			0
$317$			31.3050i			1.75826i	0.476581	+	0.879131i	$0.341876\pi$
$317$			31.3050i			1.75826i	−0.476581	+	0.879131i	$0.658124\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	8.48528	−	18.9737i	0.473602	−	1.05901i
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$			9.48683i			0.526235i
$326$		0			0
$327$	15.8114	+	7.07107i	0.874372	+	0.391031i
$328$		0			0
$329$	14.1421	+	8.94427i	0.779681	+	0.493114i
$330$		0			0
$331$	35.7771			1.96649			0.983243	−	0.182298i	$-0.0583536\pi$
$331$	35.7771			1.96649			0.983243	+	0.182298i	$0.0583536\pi$
$332$		0			0
$333$	−4.00000	−	4.47214i	−0.219199	−	0.245072i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	12.0000			0.653682			0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000			0.653682			0.326841	+	0.945079i	$0.394016\pi$
$338$		0			0
$339$	3.16228	−	7.07107i	0.171751	−	0.384048i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.23607	+	18.3848i	−0.120736	+	0.992685i
$344$		0			0
$345$	6.00000	−	13.4164i	0.323029	−	0.722315i
$346$		0			0
$347$			28.0000i			1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$			28.0000i			1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$		0			0
$349$		−	9.48683i		−	0.507819i	−0.967228	−	0.253909i	$-0.918284\pi$
$349$		−	9.48683i		−	0.507819i	0.967228	−	0.253909i	$-0.0817165\pi$
$350$		0			0
$351$	15.6525	−	5.00000i	0.835467	−	0.266880i
$352$		0			0
$353$	28.2843			1.50542			0.752710	−	0.658352i	$-0.228746\pi$
$353$	28.2843			1.50542			0.752710	+	0.658352i	$0.228746\pi$
$354$		0			0
$355$			11.3137i			0.600469i
$356$		0			0
$357$	−7.17157	−	10.7967i	−0.379560	−	0.571421i
$358$		0			0
$359$		−	10.0000i		−	0.527780i	−0.964553	−	0.263890i	$-0.914994\pi$
$359$		−	10.0000i		−	0.527780i	0.964553	−	0.263890i	$-0.0850056\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$	17.3925	+	7.77817i	0.912871	+	0.408248i
$364$		0			0
$365$			17.8885i			0.936329i
$366$		0			0
$367$			14.1421i			0.738213i	0.929387	+	0.369107i	$0.120336\pi$
$367$			14.1421i			0.738213i	−0.929387	+	0.369107i	$0.879664\pi$
$368$		0			0
$369$	−16.9706	−	18.9737i	−0.883452	−	0.987730i
$370$		0			0
$371$	18.9737	−	30.0000i	0.985064	−	1.55752i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$	−17.8885	−	8.00000i	−0.923760	−	0.413118i
$376$		0			0
$377$	−14.1421			−0.728357
$378$		0			0
$379$	−8.94427			−0.459436			−0.229718	−	0.973257i	$-0.573780\pi$
$379$	−8.94427			−0.459436			−0.229718	+	0.973257i	$0.573780\pi$
$380$		0			0
$381$	−14.1421	−	6.32456i	−0.724524	−	0.324017i
$382$		0			0
$383$	31.6228			1.61585			0.807924	−	0.589286i	$-0.200591\pi$
$383$	31.6228			1.61585			0.807924	+	0.589286i	$0.200591\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	17.8885	+	20.0000i	0.909326	+	1.01666i
$388$		0			0
$389$			13.4164i			0.680239i	0.940382	+	0.340119i	$0.110468\pi$
$389$			13.4164i			0.680239i	−0.940382	+	0.340119i	$0.889532\pi$
$390$		0			0
$391$			16.9706i			0.858238i
$392$		0			0
$393$	−25.0000	−	11.1803i	−1.26108	−	0.563974i
$394$		0			0
$395$	−18.9737			−0.954669
$396$		0			0
$397$			3.16228i			0.158710i	0.996846	+	0.0793551i	$0.0252861\pi$
$397$			3.16228i			0.158710i	−0.996846	+	0.0793551i	$0.974714\pi$
$398$		0			0
$399$	−16.1950	+	10.7574i	−0.810766	+	0.538542i
$400$		0			0
$401$			31.3050i			1.56329i	0.623721	+	0.781647i	$0.285620\pi$
$401$			31.3050i			1.56329i	−0.623721	+	0.781647i	$0.714380\pi$
$402$		0			0
$403$	17.8885			0.891092
$404$		0			0
$405$	−1.41421	+	12.6491i	−0.0702728	+	0.628539i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			6.32456i			0.312729i	0.987699	+	0.156365i	$0.0499775\pi$
$409$			6.32456i			0.312729i	−0.987699	+	0.156365i	$0.950023\pi$
$410$		0			0
$411$	−6.32456	+	14.1421i	−0.311967	+	0.697580i
$412$		0			0
$413$	−7.07107	−	4.47214i	−0.347945	−	0.220059i
$414$		0			0
$415$	4.47214			0.219529
$416$		0			0
$417$	13.0000	−	29.0689i	0.636613	−	1.42351i
$418$		0			0
$419$	−9.48683			−0.463462			−0.231731	−	0.972780i	$-0.574439\pi$
$419$	−9.48683			−0.463462			−0.231731	+	0.972780i	$0.574439\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$	12.6491	+	14.1421i	0.615021	+	0.687614i
$424$		0			0
$425$	8.48528			0.411597
$426$		0			0
$427$	4.47214	−	7.07107i	0.216422	−	0.342193i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			18.0000i			0.867029i	0.901146	+	0.433515i	$0.142727\pi$
$431$			18.0000i			0.867029i	−0.901146	+	0.433515i	$0.857273\pi$
$432$		0			0
$433$			6.32456i			0.303939i	0.988385	+	0.151969i	$0.0485615\pi$
$433$			6.32456i			0.303939i	−0.988385	+	0.151969i	$0.951438\pi$
$434$		0			0
$435$	4.47214	−	10.0000i	0.214423	−	0.479463i
$436$		0			0
$437$	25.4558			1.21772
$438$		0			0
$439$		−	25.4558i		−	1.21494i	−0.794342	−	0.607471i	$-0.792184\pi$
$439$		−	25.4558i		−	1.21494i	0.794342	−	0.607471i	$-0.207816\pi$
$440$		0			0
$441$	−8.14214	+	19.3573i	−0.387721	+	0.921777i
$442$		0			0
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$	−16.0000			−0.758473
$446$		0			0
$447$	3.16228	−	7.07107i	0.149571	−	0.334450i
$448$		0			0
$449$		−	26.8328i		−	1.26632i	−0.774021	−	0.633159i	$-0.781758\pi$
$449$		−	26.8328i		−	1.26632i	0.774021	−	0.633159i	$-0.218242\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−14.1421	−	6.32456i	−0.664455	−	0.297154i
$454$		0			0
$455$	6.32456	−	10.0000i	0.296500	−	0.468807i
$456$		0			0
$457$	8.00000			0.374224			0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000			0.374224			0.187112	+	0.982339i	$0.440087\pi$
$458$		0			0
$459$	−4.47214	−	14.0000i	−0.208741	−	0.653464i
$460$		0			0
$461$	15.5563			0.724531			0.362266	−	0.932075i	$-0.382003\pi$
$461$	15.5563			0.724531			0.362266	+	0.932075i	$0.382003\pi$
$462$		0			0
$463$	4.47214			0.207838			0.103919	−	0.994586i	$-0.466862\pi$
$463$	4.47214			0.207838			0.103919	+	0.994586i	$0.466862\pi$
$464$		0			0
$465$	−5.65685	+	12.6491i	−0.262330	+	0.586588i
$466$		0			0
$467$	−22.1359			−1.02433			−0.512165	−	0.858887i	$-0.671156\pi$
$467$	−22.1359			−1.02433			−0.512165	+	0.858887i	$0.671156\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−11.1803	+	25.0000i	−0.515163	+	1.15194i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	12.7279i		−	0.583997i
$476$		0			0
$477$	30.0000	−	26.8328i	1.37361	−	1.22859i
$478$		0			0
$479$	−18.9737			−0.866929			−0.433464	−	0.901171i	$-0.642709\pi$
$479$	−18.9737			−0.866929			−0.433464	+	0.901171i	$0.642709\pi$
$480$		0			0
$481$			6.32456i			0.288375i
$482$		0			0
$483$	22.9032	−	15.2132i	1.04213	−	0.692225i
$484$		0			0
$485$		−	26.8328i		−	1.21842i
$486$		0			0
$487$	8.94427			0.405304			0.202652	−	0.979251i	$-0.435044\pi$
$487$	8.94427			0.405304			0.202652	+	0.979251i	$0.435044\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			8.00000i			0.361035i	0.983572	+	0.180517i	$0.0577772\pi$
$491$			8.00000i			0.361035i	−0.983572	+	0.180517i	$0.942223\pi$
$492$		0			0
$493$			12.6491i			0.569687i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.3137	+	17.8885i	−0.507489	+	0.802411i
$498$		0			0
$499$	−26.8328			−1.20120			−0.600601	−	0.799549i	$-0.705072\pi$
$499$	−26.8328			−1.20120			−0.600601	+	0.799549i	$0.705072\pi$
$500$		0			0
$501$	−30.0000	−	13.4164i	−1.34030	−	0.599401i
$502$		0			0
$503$	−25.2982			−1.12799			−0.563996	−	0.825778i	$-0.690737\pi$
$503$	−25.2982			−1.12799			−0.563996	+	0.825778i	$0.690737\pi$
$504$		0			0
$505$	18.0000			0.800989
$506$		0			0
$507$	4.74342	+	2.12132i	0.210663	+	0.0942111i
$508$		0			0
$509$	−4.24264			−0.188052			−0.0940259	−	0.995570i	$-0.529974\pi$
$509$	−4.24264			−0.188052			−0.0940259	+	0.995570i	$0.529974\pi$
$510$		0			0
$511$	−17.8885	+	28.2843i	−0.791343	+	1.25122i
$512$		0			0
$513$	−21.0000	+	6.70820i	−0.927173	+	0.296174i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−24.5967	−	11.0000i	−1.07968	−	0.482846i
$520$		0			0
$521$	−8.48528			−0.371747			−0.185873	−	0.982574i	$-0.559511\pi$
$521$	−8.48528			−0.371747			−0.185873	+	0.982574i	$0.559511\pi$
$522$		0			0
$523$			7.07107i			0.309196i	0.987977	+	0.154598i	$0.0494083\pi$
$523$			7.07107i			0.309196i	−0.987977	+	0.154598i	$0.950592\pi$
$524$		0			0
$525$	−7.60660	−	11.4516i	−0.331979	−	0.499790i
$526$		0			0
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	−6.32456	−	7.07107i	−0.274462	−	0.306858i
$532$		0			0
$533$			26.8328i			1.16226i
$534$		0			0
$535$		−	16.9706i		−	0.733701i
$536$		0			0
$537$	−11.3137	+	25.2982i	−0.488223	+	1.09170i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	10.0000			0.429934			0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000			0.429934			0.214967	+	0.976621i	$0.431036\pi$
$542$		0			0
$543$	15.6525	−	35.0000i	0.671712	−	1.50199i
$544$		0			0
$545$	14.1421			0.605783
$546$		0			0
$547$	−17.8885			−0.764859			−0.382429	−	0.923985i	$-0.624912\pi$
$547$	−17.8885			−0.764859			−0.382429	+	0.923985i	$0.624912\pi$
$548$		0			0
$549$	7.07107	−	6.32456i	0.301786	−	0.269925i
$550$		0			0
$551$	18.9737			0.808305
$552$		0			0
$553$	−30.0000	−	18.9737i	−1.27573	−	0.806842i
$554$		0			0
$555$	−4.47214	−	2.00000i	−0.189832	−	0.0848953i
$556$		0			0
$557$		−	4.47214i		−	0.189490i	−0.995502	−	0.0947452i	$-0.969796\pi$
$557$		−	4.47214i		−	0.189490i	0.995502	−	0.0947452i	$-0.0302037\pi$
$558$		0			0
$559$		−	28.2843i		−	1.19630i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.48683			0.399822			0.199911	−	0.979814i	$-0.435935\pi$
$563$	9.48683			0.399822			0.199911	+	0.979814i	$0.435935\pi$
$564$		0			0
$565$		−	6.32456i		−	0.266076i
$566$		0			0
$567$	−14.8852	+	18.5858i	−0.625119	+	0.780530i
$568$		0			0
$569$			22.3607i			0.937408i	0.883355	+	0.468704i	$0.155279\pi$
$569$			22.3607i			0.937408i	−0.883355	+	0.468704i	$0.844721\pi$
$570$		0			0
$571$	−17.8885			−0.748612			−0.374306	−	0.927305i	$-0.622119\pi$
$571$	−17.8885			−0.748612			−0.374306	+	0.927305i	$0.622119\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			18.0000i			0.750652i
$576$		0			0
$577$		−	25.2982i		−	1.05318i	−0.850120	−	0.526589i	$-0.823471\pi$
$577$		−	25.2982i		−	1.05318i	0.850120	−	0.526589i	$-0.176529\pi$
$578$		0			0
$579$	−25.2982	−	11.3137i	−1.05136	−	0.470182i
$580$		0			0
$581$	7.07107	+	4.47214i	0.293357	+	0.185535i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	10.0000	−	8.94427i	0.413449	−	0.369800i
$586$		0			0
$587$	28.4605			1.17469			0.587345	−	0.809336i	$-0.300173\pi$
$587$	28.4605			1.17469			0.587345	+	0.809336i	$0.300173\pi$
$588$		0			0
$589$	−24.0000			−0.988903
$590$		0			0
$591$	3.16228	−	7.07107i	0.130079	−	0.290865i
$592$		0			0
$593$	−22.6274			−0.929197			−0.464598	−	0.885522i	$-0.653801\pi$
$593$	−22.6274			−0.929197			−0.464598	+	0.885522i	$0.653801\pi$
$594$		0			0
$595$	−8.94427	−	5.65685i	−0.366679	−	0.231908i
$596$		0			0
$597$	8.00000	−	17.8885i	0.327418	−	0.732129i
$598$		0			0
$599$			40.0000i			1.63436i	0.576386	+	0.817178i	$0.304463\pi$
$599$			40.0000i			1.63436i	−0.576386	+	0.817178i	$0.695537\pi$
$600$		0			0
$601$			25.2982i			1.03194i	0.856608	+	0.515968i	$0.172568\pi$
$601$			25.2982i			1.03194i	−0.856608	+	0.515968i	$0.827432\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.5563			0.632456
$606$		0			0
$607$		−	42.4264i		−	1.72203i	−0.508576	−	0.861017i	$-0.669828\pi$
$607$		−	42.4264i		−	1.72203i	0.508576	−	0.861017i	$-0.330172\pi$
$608$		0			0
$609$	17.0711	−	11.3393i	0.691755	−	0.459490i
$610$		0			0
$611$		−	20.0000i		−	0.809113i
$612$		0			0
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$		0			0
$615$	−18.9737	−	8.48528i	−0.765092	−	0.342160i
$616$		0			0
$617$			22.3607i			0.900207i	0.892976	+	0.450104i	$0.148613\pi$
$617$			22.3607i			0.900207i	−0.892976	+	0.450104i	$0.851387\pi$
$618$		0			0
$619$		−	38.1838i		−	1.53474i	−0.641207	−	0.767368i	$-0.721566\pi$
$619$		−	38.1838i		−	1.53474i	0.641207	−	0.767368i	$-0.278434\pi$
$620$		0			0
$621$	29.6985	−	9.48683i	1.19176	−	0.380693i
$622$		0			0
$623$	−25.2982	−	16.0000i	−1.01355	−	0.641026i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.65685			0.225554
$630$		0			0
$631$	31.3050			1.24623			0.623115	−	0.782130i	$-0.285867\pi$
$631$	31.3050			1.24623			0.623115	+	0.782130i	$0.285867\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.6491			−0.501965
$636$		0			0
$637$	20.0000	−	9.48683i	0.792429	−	0.375882i
$638$		0			0
$639$	−17.8885	+	16.0000i	−0.707660	+	0.632950i
$640$		0			0
$641$			4.47214i			0.176639i	0.996092	+	0.0883194i	$0.0281496\pi$
$641$			4.47214i			0.176639i	−0.996092	+	0.0883194i	$0.971850\pi$
$642$		0			0
$643$			26.8701i			1.05965i	0.848106	+	0.529826i	$0.177743\pi$
$643$			26.8701i			1.05965i	−0.848106	+	0.529826i	$0.822257\pi$
$644$		0			0
$645$	20.0000	+	8.94427i	0.787499	+	0.352180i
$646$		0			0
$647$	−6.32456			−0.248644			−0.124322	−	0.992242i	$-0.539676\pi$
$647$	−6.32456			−0.248644			−0.124322	+	0.992242i	$0.539676\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−21.5934	+	14.3431i	−0.846311	+	0.562152i
$652$		0			0
$653$			4.47214i			0.175008i	0.996164	+	0.0875041i	$0.0278891\pi$
$653$			4.47214i			0.175008i	−0.996164	+	0.0875041i	$0.972111\pi$
$654$		0			0
$655$	−22.3607			−0.873704
$656$		0			0
$657$	−28.2843	+	25.2982i	−1.10347	+	0.986978i
$658$		0			0
$659$			44.0000i			1.71400i	0.515319	+	0.856998i	$0.327673\pi$
$659$			44.0000i			1.71400i	−0.515319	+	0.856998i	$0.672327\pi$
$660$		0			0
$661$		−	34.7851i		−	1.35298i	−0.736451	−	0.676491i	$-0.763500\pi$
$661$		−	34.7851i		−	1.35298i	0.736451	−	0.676491i	$-0.236500\pi$
$662$		0			0
$663$	−6.32456	+	14.1421i	−0.245625	+	0.549235i
$664$		0			0
$665$	−8.48528	+	13.4164i	−0.329045	+	0.520266i
$666$		0			0
$667$	−26.8328			−1.03897
$668$		0			0
$669$	−10.0000	+	22.3607i	−0.386622	+	0.864514i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$		0			0
$675$	−4.74342	−	14.8492i	−0.182574	−	0.571548i
$676$		0			0
$677$	38.1838			1.46752			0.733761	−	0.679408i	$-0.237763\pi$
$677$	38.1838			1.46752			0.733761	+	0.679408i	$0.237763\pi$
$678$		0			0
$679$	26.8328	−	42.4264i	1.02975	−	1.62818i
$680$		0			0
$681$	5.00000	+	2.23607i	0.191600	+	0.0856863i
$682$		0			0
$683$			24.0000i			0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$			24.0000i			0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$			12.6491i			0.483298i
$686$		0			0
$687$	2.23607	−	5.00000i	0.0853113	−	0.190762i
$688$		0			0
$689$	−42.4264			−1.61632
$690$		0			0
$691$		−	1.41421i		−	0.0537992i	−0.999638	−	0.0268996i	$-0.991437\pi$
$691$		−	1.41421i		−	0.0537992i	0.999638	−	0.0268996i	$-0.00856344\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	26.0000i		−	0.986236i
$696$		0			0
$697$	24.0000			0.909065
$698$		0			0
$699$	12.6491	−	28.2843i	0.478433	−	1.06981i
$700$		0			0
$701$		−	49.1935i		−	1.85801i	−0.370064	−	0.929006i	$-0.620664\pi$
$701$		−	49.1935i		−	1.85801i	0.370064	−	0.929006i	$-0.379336\pi$
$702$		0			0
$703$		−	8.48528i		−	0.320028i
$704$		0			0
$705$	14.1421	+	6.32456i	0.532624	+	0.238197i
$706$		0			0
$707$	28.4605	+	18.0000i	1.07037	+	0.676960i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$	−26.8328	−	30.0000i	−1.00631	−	1.12509i
$712$		0			0
$713$	33.9411			1.27111
$714$		0			0
$715$		0			0
$716$		0			0
$717$	7.07107	−	15.8114i	0.264074	−	0.590487i
$718$		0			0
$719$	6.32456			0.235866			0.117933	−	0.993022i	$-0.462373\pi$
$719$	6.32456			0.235866			0.117933	+	0.993022i	$0.462373\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−4.47214	+	10.0000i	−0.166321	+	0.371904i
$724$		0			0
$725$			13.4164i			0.498273i
$726$		0			0
$727$			11.3137i			0.419602i	0.977744	+	0.209801i	$0.0672817\pi$
$727$			11.3137i			0.419602i	−0.977744	+	0.209801i	$0.932718\pi$
$728$		0			0
$729$	−22.0000	+	15.6525i	−0.814815	+	0.579721i
$730$		0			0
$731$	−25.2982			−0.935689
$732$		0			0
$733$		−	9.48683i		−	0.350404i	−0.984532	−	0.175202i	$-0.943942\pi$
$733$		−	9.48683i		−	0.350404i	0.984532	−	0.175202i	$-0.0560579\pi$
$734$		0			0
$735$	0.383649	+	17.1421i	0.0141511	+	0.632297i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−44.7214			−1.64510			−0.822551	−	0.568691i	$-0.807450\pi$
$739$	−44.7214			−1.64510			−0.822551	+	0.568691i	$0.807450\pi$
$740$		0			0
$741$	21.2132	+	9.48683i	0.779287	+	0.348508i
$742$		0			0
$743$		−	6.00000i		−	0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$		−	6.00000i		−	0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$		0			0
$745$		−	6.32456i		−	0.231714i
$746$		0			0
$747$	6.32456	+	7.07107i	0.231403	+	0.258717i
$748$		0			0
$749$	16.9706	−	26.8328i	0.620091	−	0.980450i
$750$		0			0
$751$	−8.94427			−0.326381			−0.163191	−	0.986595i	$-0.552179\pi$
$751$	−8.94427			−0.326381			−0.163191	+	0.986595i	$0.552179\pi$
$752$		0			0
$753$	45.0000	+	20.1246i	1.63989	+	0.733382i
$754$		0			0
$755$	−12.6491			−0.460348
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19.7990			−0.717713			−0.358856	−	0.933393i	$-0.616833\pi$
$761$	−19.7990			−0.717713			−0.358856	+	0.933393i	$0.616833\pi$
$762$		0			0
$763$	22.3607	+	14.1421i	0.809511	+	0.511980i
$764$		0			0
$765$	−8.00000	−	8.94427i	−0.289241	−	0.323381i
$766$		0			0
$767$			10.0000i			0.361079i
$768$		0			0
$769$			31.6228i			1.14035i	0.821524	+	0.570173i	$0.193124\pi$
$769$			31.6228i			1.14035i	−0.821524	+	0.570173i	$0.806876\pi$
$770$		0			0
$771$	44.7214	+	20.0000i	1.61060	+	0.720282i
$772$		0			0
$773$	−49.4975			−1.78030			−0.890150	−	0.455667i	$-0.849401\pi$
$773$	−49.4975			−1.78030			−0.890150	+	0.455667i	$0.849401\pi$
$774$		0			0
$775$		−	16.9706i		−	0.609601i
$776$		0			0
$777$	−5.07107	−	7.63441i	−0.181924	−	0.273883i
$778$		0			0
$779$		−	36.0000i		−	1.28983i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	22.1359	−	7.07107i	0.791074	−	0.252699i
$784$		0			0
$785$			22.3607i			0.798087i
$786$		0			0
$787$			35.3553i			1.26028i	0.776481	+	0.630141i	$0.217003\pi$
$787$			35.3553i			1.26028i	−0.776481	+	0.630141i	$0.782997\pi$
$788$		0			0
$789$	−11.3137	+	25.2982i	−0.402779	+	0.900641i
$790$		0			0
$791$	6.32456	−	10.0000i	0.224875	−	0.355559i
$792$		0			0
$793$	−10.0000			−0.355110
$794$		0			0
$795$	13.4164	−	30.0000i	0.475831	−	1.06399i
$796$		0			0
$797$	32.5269			1.15216			0.576081	−	0.817392i	$-0.304581\pi$
$797$	32.5269			1.15216			0.576081	+	0.817392i	$0.304581\pi$
$798$		0			0
$799$	−17.8885			−0.632851
$800$		0			0
$801$	−22.6274	−	25.2982i	−0.799500	−	0.893869i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	12.0000	−	18.9737i	0.422944	−	0.668734i
$806$		0			0
$807$	−15.6525	−	7.00000i	−0.550993	−	0.246412i
$808$		0			0
$809$		−	49.1935i		−	1.72955i	−0.502159	−	0.864776i	$-0.667461\pi$
$809$		−	49.1935i		−	1.72955i	0.502159	−	0.864776i	$-0.332539\pi$
$810$		0			0
$811$		−	1.41421i		−	0.0496598i	−0.999692	−	0.0248299i	$-0.992096\pi$
$811$		−	1.41421i		−	0.0496598i	0.999692	−	0.0248299i	$-0.00790441\pi$
$812$		0			0
$813$	−6.00000	+	13.4164i	−0.210429	+	0.470534i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			37.9473i			1.32761i
$818$		0			0
$819$	24.7557	−	4.14214i	0.865033	−	0.144738i
$820$		0			0
$821$		−	22.3607i		−	0.780393i	−0.920732	−	0.390197i	$-0.872407\pi$
$821$		−	22.3607i		−	0.780393i	0.920732	−	0.390197i	$-0.127593\pi$
$822$		0			0
$823$	40.2492			1.40300			0.701500	−	0.712670i	$-0.252514\pi$
$823$	40.2492			1.40300			0.701500	+	0.712670i	$0.252514\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	28.0000i		−	0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$		−	28.0000i		−	0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$		0			0
$829$		−	28.4605i		−	0.988474i	−0.869327	−	0.494237i	$-0.835448\pi$
$829$		−	28.4605i		−	0.988474i	0.869327	−	0.494237i	$-0.164552\pi$
$830$		0			0
$831$	28.4605	+	12.7279i	0.987284	+	0.441527i
$832$		0			0
$833$	−8.48528	−	17.8885i	−0.293998	−	0.619801i
$834$		0			0
$835$	−26.8328			−0.928588
$836$		0			0
$837$	−28.0000	+	8.94427i	−0.967822	+	0.309159i
$838$		0			0
$839$	−31.6228			−1.09174			−0.545870	−	0.837870i	$-0.683801\pi$
$839$	−31.6228			−1.09174			−0.545870	+	0.837870i	$0.683801\pi$
$840$		0			0
$841$	9.00000			0.310345
$842$		0			0
$843$	−12.6491	+	28.2843i	−0.435659	+	0.974162i
$844$		0			0
$845$	4.24264			0.145951
$846$		0			0
$847$	24.5967	+	15.5563i	0.845154	+	0.534522i
$848$		0			0
$849$	−15.0000	+	33.5410i	−0.514799	+	1.15112i
$850$		0			0
$851$			12.0000i			0.411355i
$852$		0			0
$853$			47.4342i			1.62411i	0.583578	+	0.812057i	$0.301652\pi$
$853$			47.4342i			1.62411i	−0.583578	+	0.812057i	$0.698348\pi$
$854$		0			0
$855$	−13.4164	+	12.0000i	−0.458831	+	0.410391i
$856$		0			0
$857$	−14.1421			−0.483086			−0.241543	−	0.970390i	$-0.577654\pi$
$857$	−14.1421			−0.483086			−0.241543	+	0.970390i	$0.577654\pi$
$858$		0			0
$859$		−	4.24264i		−	0.144757i	−0.997377	−	0.0723785i	$-0.976941\pi$
$859$		−	4.24264i		−	0.144757i	0.997377	−	0.0723785i	$-0.0230590\pi$
$860$		0			0
$861$	−21.5147	−	32.3901i	−0.733220	−	1.10385i
$862$		0			0
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	−22.0000			−0.748022
$866$		0			0
$867$	−14.2302	−	6.36396i	−0.483285	−	0.216131i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	42.4264	−	37.9473i	1.43592	−	1.28432i
$874$		0			0
$875$	−25.2982	−	16.0000i	−0.855236	−	0.540899i
$876$		0			0
$877$	−38.0000			−1.28317			−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000			−1.28317			−0.641584	+	0.767052i	$0.721723\pi$
$878$		0			0
$879$	−11.1803	−	5.00000i	−0.377104	−	0.168646i
$880$		0			0
$881$	−50.9117			−1.71526			−0.857629	−	0.514269i	$-0.828063\pi$
$881$	−50.9117			−1.71526			−0.857629	+	0.514269i	$0.828063\pi$
$882$		0			0
$883$	17.8885			0.601997			0.300999	−	0.953625i	$-0.402680\pi$
$883$	17.8885			0.601997			0.300999	+	0.953625i	$0.402680\pi$
$884$		0			0
$885$	−7.07107	−	3.16228i	−0.237691	−	0.106299i
$886$		0			0
$887$	−18.9737			−0.637073			−0.318537	−	0.947911i	$-0.603191\pi$
$887$	−18.9737			−0.637073			−0.318537	+	0.947911i	$0.603191\pi$
$888$		0			0
$889$	−20.0000	−	12.6491i	−0.670778	−	0.424238i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			26.8328i			0.897926i
$894$		0			0
$895$			22.6274i			0.756351i
$896$		0			0
$897$	−30.0000	−	13.4164i	−1.00167	−	0.447961i
$898$		0			0
$899$	25.2982			0.843743
$900$		0			0
$901$			37.9473i			1.26421i
$902$		0			0
$903$	22.6785	+	34.1421i	0.754693	+	1.13618i
$904$		0			0
$905$		−	31.3050i		−	1.04061i
$906$		0			0
$907$	26.8328			0.890969			0.445485	−	0.895290i	$-0.353031\pi$
$907$	26.8328			0.890969			0.445485	+	0.895290i	$0.353031\pi$
$908$		0			0
$909$	25.4558	+	28.4605i	0.844317	+	0.943975i
$910$		0			0
$911$			50.0000i			1.65657i	0.560304	+	0.828287i	$0.310684\pi$
$911$			50.0000i			1.65657i	−0.560304	+	0.828287i	$0.689316\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	3.16228	−	7.07107i	0.104542	−	0.233762i
$916$		0			0
$917$	−35.3553	−	22.3607i	−1.16754	−	0.738415i
$918$		0			0
$919$	13.4164			0.442566			0.221283	−	0.975210i	$-0.428975\pi$
$919$	13.4164			0.442566			0.221283	+	0.975210i	$0.428975\pi$
$920$		0			0
$921$	−3.00000	+	6.70820i	−0.0988534	+	0.221043i
$922$		0			0
$923$	25.2982			0.832701
$924$		0			0
$925$	6.00000			0.197279
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.82843			0.0927977			0.0463988	−	0.998923i	$-0.485225\pi$
$929$	2.82843			0.0927977			0.0463988	+	0.998923i	$0.485225\pi$
$930$		0			0
$931$	−26.8328	+	12.7279i	−0.879410	+	0.417141i
$932$		0			0
$933$	40.0000	+	17.8885i	1.30954	+	0.585645i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			25.2982i			0.826457i	0.910627	+	0.413228i	$0.135599\pi$
$937$			25.2982i			0.826457i	−0.910627	+	0.413228i	$0.864401\pi$
$938$		0			0
$939$	4.47214	−	10.0000i	0.145943	−	0.326338i
$940$		0			0
$941$	−1.41421			−0.0461020			−0.0230510	−	0.999734i	$-0.507338\pi$
$941$	−1.41421			−0.0461020			−0.0230510	+	0.999734i	$0.507338\pi$
$942$		0			0
$943$			50.9117i			1.65791i
$944$		0			0
$945$	−4.89949	+	18.8148i	−0.159381	+	0.612044i
$946$		0			0
$947$		−	28.0000i		−	0.909878i	−0.890523	−	0.454939i	$-0.849661\pi$
$947$		−	28.0000i		−	0.909878i	0.890523	−	0.454939i	$-0.150339\pi$
$948$		0			0
$949$	40.0000			1.29845
$950$		0			0
$951$	−22.1359	+	49.4975i	−0.717807	+	1.60507i
$952$		0			0
$953$		−	44.7214i		−	1.44867i	−0.689450	−	0.724333i	$-0.742148\pi$
$953$		−	44.7214i		−	1.44867i	0.689450	−	0.724333i	$-0.257852\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.6491	+	20.0000i	−0.408461	+	0.645834i
$960$		0			0
$961$	−1.00000			−0.0322581
$962$		0			0
$963$	26.8328	−	24.0000i	0.864675	−	0.773389i
$964$		0			0
$965$	−22.6274			−0.728402
$966$		0			0
$967$	26.8328			0.862885			0.431443	−	0.902140i	$-0.358005\pi$
$967$	26.8328			0.862885			0.431443	+	0.902140i	$0.358005\pi$
$968$		0			0
$969$	8.48528	−	18.9737i	0.272587	−	0.609522i
$970$		0			0
$971$	9.48683			0.304447			0.152223	−	0.988346i	$-0.451357\pi$
$971$	9.48683			0.304447			0.152223	+	0.988346i	$0.451357\pi$
$972$		0			0
$973$	26.0000	−	41.1096i	0.833522	−	1.31791i
$974$		0			0
$975$	−6.70820	+	15.0000i	−0.214834	+	0.480384i
$976$		0			0
$977$			26.8328i			0.858458i	0.903196	+	0.429229i	$0.141215\pi$
$977$			26.8328i			0.858458i	−0.903196	+	0.429229i	$0.858785\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	20.0000	+	22.3607i	0.638551	+	0.713922i
$982$		0			0
$983$	56.9210			1.81550			0.907749	−	0.419514i	$-0.137799\pi$
$983$	56.9210			1.81550			0.907749	+	0.419514i	$0.137799\pi$
$984$		0			0
$985$		−	6.32456i		−	0.201517i
$986$		0			0
$987$	16.0361	+	24.1421i	0.510435	+	0.768453i
$988$		0			0
$989$		−	53.6656i		−	1.70647i
$990$		0			0
$991$	13.4164			0.426186			0.213093	−	0.977032i	$-0.431646\pi$
$991$	13.4164			0.426186			0.213093	+	0.977032i	$0.431646\pi$
$992$		0			0
$993$	56.5685	+	25.2982i	1.79515	+	0.802815i
$994$		0			0
$995$		−	16.0000i		−	0.507234i
$996$		0			0
$997$		−	22.1359i		−	0.701052i	−0.936553	−	0.350526i	$-0.886003\pi$
$997$		−	22.1359i		−	0.701052i	0.936553	−	0.350526i	$-0.113997\pi$
$998$		0			0
$999$	−3.16228	−	9.89949i	−0.100050	−	0.313206i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.k.c.545.8	yes	8
3.2	odd	2	inner	672.2.k.c.545.3	yes	8
4.3	odd	2	inner	672.2.k.c.545.2	yes	8
7.6	odd	2	inner	672.2.k.c.545.1	✓	8
8.3	odd	2		1344.2.k.h.1217.7		8
8.5	even	2		1344.2.k.h.1217.1		8
12.11	even	2	inner	672.2.k.c.545.5	yes	8
21.20	even	2	inner	672.2.k.c.545.6	yes	8
24.5	odd	2		1344.2.k.h.1217.6		8
24.11	even	2		1344.2.k.h.1217.4		8
28.27	even	2	inner	672.2.k.c.545.7	yes	8
56.13	odd	2		1344.2.k.h.1217.8		8
56.27	even	2		1344.2.k.h.1217.2		8
84.83	odd	2	inner	672.2.k.c.545.4	yes	8
168.83	odd	2		1344.2.k.h.1217.5		8
168.125	even	2		1344.2.k.h.1217.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.k.c.545.1	✓	8	7.6	odd	2	inner
672.2.k.c.545.2	yes	8	4.3	odd	2	inner
672.2.k.c.545.3	yes	8	3.2	odd	2	inner
672.2.k.c.545.4	yes	8	84.83	odd	2	inner
672.2.k.c.545.5	yes	8	12.11	even	2	inner
672.2.k.c.545.6	yes	8	21.20	even	2	inner
672.2.k.c.545.7	yes	8	28.27	even	2	inner
672.2.k.c.545.8	yes	8	1.1	even	1	trivial
1344.2.k.h.1217.1		8	8.5	even	2
1344.2.k.h.1217.2		8	56.27	even	2
1344.2.k.h.1217.3		8	168.125	even	2
1344.2.k.h.1217.4		8	24.11	even	2
1344.2.k.h.1217.5		8	168.83	odd	2
1344.2.k.h.1217.6		8	24.5	odd	2
1344.2.k.h.1217.7		8	8.3	odd	2
1344.2.k.h.1217.8		8	56.13	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.58114 + 0.707107i) q^{3} +1.41421 q^{5} +(2.23607 + 1.41421i) q^{7} +(2.00000 + 2.23607i) q^{9} +O(q^{10})\)	\(q+(1.58114 + 0.707107i) q^{3} +1.41421 q^{5} +(2.23607 + 1.41421i) q^{7} +(2.00000 + 2.23607i) q^{9} -3.16228i q^{13} +(2.23607 + 1.00000i) q^{15} -2.82843 q^{17} +4.24264i q^{19} +(2.53553 + 3.81721i) q^{21} -6.00000i q^{23} -3.00000 q^{25} +(1.58114 + 4.94975i) q^{27} -4.47214i q^{29} +5.65685i q^{31} +(3.16228 + 2.00000i) q^{35} -2.00000 q^{37} +(2.23607 - 5.00000i) q^{39} -8.48528 q^{41} +8.94427 q^{43} +(2.82843 + 3.16228i) q^{45} +6.32456 q^{47} +(3.00000 + 6.32456i) q^{49} +(-4.47214 - 2.00000i) q^{51} -13.4164i q^{53} +(-3.00000 + 6.70820i) q^{57} -3.16228 q^{59} -3.16228i q^{61} +(1.30986 + 7.82843i) q^{63} -4.47214i q^{65} +(4.24264 - 9.48683i) q^{69} +8.00000i q^{71} +12.6491i q^{73} +(-4.74342 - 2.12132i) q^{75} -13.4164 q^{79} +(-1.00000 + 8.94427i) q^{81} +3.16228 q^{83} -4.00000 q^{85} +(3.16228 - 7.07107i) q^{87} -11.3137 q^{89} +(4.47214 - 7.07107i) q^{91} +(-4.00000 + 8.94427i) q^{93} +6.00000i q^{95} -18.9737i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} - 8 q^{21} - 24 q^{25} - 16 q^{37} + 24 q^{49} - 24 q^{57} - 8 q^{81} - 32 q^{85} - 32 q^{93}+O(q^{100}) \) 8 * q + 16 * q^9 - 8 * q^21 - 24 * q^25 - 16 * q^37 + 24 * q^49 - 24 * q^57 - 8 * q^81 - 32 * q^85 - 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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