LMFDB - Embedded newform 1680.2.t.i.1009.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(1009,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1680.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.t (of order $2$, degree $1$, not 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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.1
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	1680.1009
Dual form			1680.2.t.i.1009.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -5.41855 q^{11} +4.34017i q^{13} +(-0.539189 + 2.17009i) q^{15} -1.07838i q^{17} +4.34017 q^{19} +1.00000 q^{21} -6.34017i q^{23} +(4.41855 + 2.34017i) q^{25} +1.00000i q^{27} +8.83710 q^{29} +4.34017 q^{31} +5.41855i q^{33} +(0.539189 - 2.17009i) q^{35} +8.68035i q^{37} +4.34017 q^{39} +8.34017 q^{41} -6.15676i q^{43} +(2.17009 + 0.539189i) q^{45} -6.83710i q^{47} -1.00000 q^{49} -1.07838 q^{51} +6.18342i q^{53} +(11.7587 + 2.92162i) q^{55} -4.34017i q^{57} -6.83710 q^{59} -4.52359 q^{61} -1.00000i q^{63} +(2.34017 - 9.41855i) q^{65} -6.34017 q^{69} +14.0989 q^{71} -11.1773i q^{73} +(2.34017 - 4.41855i) q^{75} -5.41855i q^{77} +0.680346 q^{79} +1.00000 q^{81} +6.83710i q^{83} +(-0.581449 + 2.34017i) q^{85} -8.83710i q^{87} -6.49693 q^{89} -4.34017 q^{91} -4.34017i q^{93} +(-9.41855 - 2.34017i) q^{95} +10.4969i q^{97} +5.41855 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^5 - 6 * q^9	$ 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} + 4 q^{31} + 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} + 20 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 12 q^{71} - 8 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} - 4 q^{91} - 28 q^{95} + 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 + 4 * q^31 + 4 * q^39 + 28 * q^41 + 2 * q^45 - 6 * q^49 + 20 * q^55 + 16 * q^59 + 4 * q^61 - 8 * q^65 - 16 * q^69 + 12 * q^71 - 8 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 - 4 * q^89 - 4 * q^91 - 28 * q^95 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$-1$	$1$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−2.17009	−	0.539189i	−0.970492	−	0.241133i
$5$	−2.17009	−	0.539189i	−0.970492	−	0.241133i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.41855			−1.63375			−0.816877	−	0.576812i	$-0.804297\pi$
$11$	−5.41855			−1.63375			−0.816877	+	0.576812i	$0.804297\pi$
$12$		0			0
$13$			4.34017i			1.20375i	0.798591	+	0.601874i	$0.205579\pi$
$13$			4.34017i			1.20375i	−0.798591	+	0.601874i	$0.794421\pi$
$14$		0			0
$15$	−0.539189	+	2.17009i	−0.139218	+	0.560314i
$16$		0			0
$17$		−	1.07838i		−	0.261545i	−0.991412	−	0.130773i	$-0.958254\pi$
$17$		−	1.07838i		−	0.261545i	0.991412	−	0.130773i	$-0.0417457\pi$
$18$		0			0
$19$	4.34017			0.995704			0.497852	−	0.867262i	$-0.334122\pi$
$19$	4.34017			0.995704			0.497852	+	0.867262i	$0.334122\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$		−	6.34017i		−	1.32202i	−0.750378	−	0.661009i	$-0.770129\pi$
$23$		−	6.34017i		−	1.32202i	0.750378	−	0.661009i	$-0.229871\pi$
$24$		0			0
$25$	4.41855	+	2.34017i	0.883710	+	0.468035i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	8.83710			1.64101			0.820504	−	0.571640i	$-0.193693\pi$
$29$	8.83710			1.64101			0.820504	+	0.571640i	$0.193693\pi$
$30$		0			0
$31$	4.34017			0.779518			0.389759	−	0.920917i	$-0.372558\pi$
$31$	4.34017			0.779518			0.389759	+	0.920917i	$0.372558\pi$
$32$		0			0
$33$			5.41855i			0.943249i
$34$		0			0
$35$	0.539189	−	2.17009i	0.0911396	−	0.366812i
$36$		0			0
$37$			8.68035i			1.42704i	0.700635	+	0.713520i	$0.252900\pi$
$37$			8.68035i			1.42704i	−0.700635	+	0.713520i	$0.747100\pi$
$38$		0			0
$39$	4.34017			0.694984
$40$		0			0
$41$	8.34017			1.30252			0.651258	−	0.758856i	$-0.274242\pi$
$41$	8.34017			1.30252			0.651258	+	0.758856i	$0.274242\pi$
$42$		0			0
$43$		−	6.15676i		−	0.938896i	−0.882960	−	0.469448i	$-0.844453\pi$
$43$		−	6.15676i		−	0.938896i	0.882960	−	0.469448i	$-0.155547\pi$
$44$		0			0
$45$	2.17009	+	0.539189i	0.323497	+	0.0803775i
$46$		0			0
$47$		−	6.83710i		−	0.997294i	−0.866805	−	0.498647i	$-0.833830\pi$
$47$		−	6.83710i		−	0.997294i	0.866805	−	0.498647i	$-0.166170\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−1.07838			−0.151003
$52$		0			0
$53$			6.18342i			0.849358i	0.905344	+	0.424679i	$0.139613\pi$
$53$			6.18342i			0.849358i	−0.905344	+	0.424679i	$0.860387\pi$
$54$		0			0
$55$	11.7587	+	2.92162i	1.58555	+	0.393951i
$56$		0			0
$57$		−	4.34017i		−	0.574870i
$58$		0			0
$59$	−6.83710			−0.890115			−0.445057	−	0.895502i	$-0.646817\pi$
$59$	−6.83710			−0.890115			−0.445057	+	0.895502i	$0.646817\pi$
$60$		0			0
$61$	−4.52359			−0.579186			−0.289593	−	0.957150i	$-0.593520\pi$
$61$	−4.52359			−0.579186			−0.289593	+	0.957150i	$0.593520\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	2.34017	−	9.41855i	0.290263	−	1.16823i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−6.34017			−0.763267
$70$		0			0
$71$	14.0989			1.67323			0.836616	−	0.547790i	$-0.184531\pi$
$71$	14.0989			1.67323			0.836616	+	0.547790i	$0.184531\pi$
$72$		0			0
$73$		−	11.1773i		−	1.30820i	−0.756408	−	0.654101i	$-0.773047\pi$
$73$		−	11.1773i		−	1.30820i	0.756408	−	0.654101i	$-0.226953\pi$
$74$		0			0
$75$	2.34017	−	4.41855i	0.270220	−	0.510210i
$76$		0			0
$77$		−	5.41855i		−	0.617501i
$78$		0			0
$79$	0.680346			0.0765449			0.0382724	−	0.999267i	$-0.487815\pi$
$79$	0.680346			0.0765449			0.0382724	+	0.999267i	$0.487815\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			6.83710i			0.750469i	0.926930	+	0.375235i	$0.122438\pi$
$83$			6.83710i			0.750469i	−0.926930	+	0.375235i	$0.877562\pi$
$84$		0			0
$85$	−0.581449	+	2.34017i	−0.0630670	+	0.253827i
$86$		0			0
$87$		−	8.83710i		−	0.947437i
$88$		0			0
$89$	−6.49693			−0.688673			−0.344337	−	0.938846i	$-0.611896\pi$
$89$	−6.49693			−0.688673			−0.344337	+	0.938846i	$0.611896\pi$
$90$		0			0
$91$	−4.34017			−0.454974
$92$		0			0
$93$		−	4.34017i		−	0.450055i
$94$		0			0
$95$	−9.41855	−	2.34017i	−0.966323	−	0.240097i
$96$		0			0
$97$			10.4969i			1.06580i	0.846178	+	0.532901i	$0.178898\pi$
$97$			10.4969i			1.06580i	−0.846178	+	0.532901i	$0.821102\pi$
$98$		0			0
$99$	5.41855			0.544585
$100$		0			0
$101$	18.8638			1.87701			0.938507	−	0.345259i	$-0.112209\pi$
$101$	18.8638			1.87701			0.938507	+	0.345259i	$0.112209\pi$
$102$		0			0
$103$		−	10.1568i		−	1.00077i	−0.865802	−	0.500387i	$-0.833191\pi$
$103$		−	10.1568i		−	1.00077i	0.865802	−	0.500387i	$-0.166809\pi$
$104$		0			0
$105$	−2.17009	−	0.539189i	−0.211779	−	0.0526194i
$106$		0			0
$107$			14.6537i			1.41663i	0.705899	+	0.708313i	$0.250543\pi$
$107$			14.6537i			1.41663i	−0.705899	+	0.708313i	$0.749457\pi$
$108$		0			0
$109$	12.8371			1.22957			0.614786	−	0.788694i	$-0.289243\pi$
$109$	12.8371			1.22957			0.614786	+	0.788694i	$0.289243\pi$
$110$		0			0
$111$	8.68035			0.823902
$112$		0			0
$113$			1.50307i			0.141397i	0.997498	+	0.0706985i	$0.0225228\pi$
$113$			1.50307i			0.141397i	−0.997498	+	0.0706985i	$0.977477\pi$
$114$		0			0
$115$	−3.41855	+	13.7587i	−0.318781	+	1.28301i
$116$		0			0
$117$		−	4.34017i		−	0.401249i
$118$		0			0
$119$	1.07838			0.0988547
$120$		0			0
$121$	18.3607			1.66915
$122$		0			0
$123$		−	8.34017i		−	0.752008i
$124$		0			0
$125$	−8.32684	−	7.46081i	−0.744775	−	0.667315i
$126$		0			0
$127$			19.2039i			1.70407i	0.523482	+	0.852037i	$0.324633\pi$
$127$			19.2039i			1.70407i	−0.523482	+	0.852037i	$0.675367\pi$
$128$		0			0
$129$	−6.15676			−0.542072
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$		0			0
$133$			4.34017i			0.376341i
$134$		0			0
$135$	0.539189	−	2.17009i	0.0464060	−	0.186771i
$136$		0			0
$137$		−	8.65368i		−	0.739334i	−0.929164	−	0.369667i	$-0.879472\pi$
$137$		−	8.65368i		−	0.739334i	0.929164	−	0.369667i	$-0.120528\pi$
$138$		0			0
$139$	−6.18342			−0.524471			−0.262235	−	0.965004i	$-0.584460\pi$
$139$	−6.18342			−0.524471			−0.262235	+	0.965004i	$0.584460\pi$
$140$		0			0
$141$	−6.83710			−0.575788
$142$		0			0
$143$		−	23.5174i		−	1.96663i
$144$		0			0
$145$	−19.1773	−	4.76487i	−1.59259	−	0.395701i
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	13.2039			1.08171			0.540854	−	0.841116i	$-0.318101\pi$
$149$	13.2039			1.08171			0.540854	+	0.841116i	$0.318101\pi$
$150$		0			0
$151$	18.1568			1.47758			0.738788	−	0.673938i	$-0.235399\pi$
$151$	18.1568			1.47758			0.738788	+	0.673938i	$0.235399\pi$
$152$		0			0
$153$			1.07838i			0.0871817i
$154$		0			0
$155$	−9.41855	−	2.34017i	−0.756516	−	0.187967i
$156$		0			0
$157$		−	15.1773i		−	1.21128i	−0.795739	−	0.605639i	$-0.792917\pi$
$157$		−	15.1773i		−	1.21128i	0.795739	−	0.605639i	$-0.207083\pi$
$158$		0			0
$159$	6.18342			0.490377
$160$		0			0
$161$	6.34017			0.499676
$162$		0			0
$163$			2.83710i			0.222219i	0.993808	+	0.111109i	$0.0354404\pi$
$163$			2.83710i			0.222219i	−0.993808	+	0.111109i	$0.964560\pi$
$164$		0			0
$165$	2.92162	−	11.7587i	0.227448	−	0.915415i
$166$		0			0
$167$			13.3607i			1.03388i	0.856021	+	0.516941i	$0.172929\pi$
$167$			13.3607i			1.03388i	−0.856021	+	0.516941i	$0.827071\pi$
$168$		0			0
$169$	−5.83710			−0.449008
$170$		0			0
$171$	−4.34017			−0.331901
$172$		0			0
$173$			2.55479i			0.194237i	0.995273	+	0.0971184i	$0.0309626\pi$
$173$			2.55479i			0.194237i	−0.995273	+	0.0971184i	$0.969037\pi$
$174$		0			0
$175$	−2.34017	+	4.41855i	−0.176900	+	0.334011i
$176$		0			0
$177$			6.83710i			0.513908i
$178$		0			0
$179$	−11.9421			−0.892598			−0.446299	−	0.894884i	$-0.647258\pi$
$179$	−11.9421			−0.892598			−0.446299	+	0.894884i	$0.647258\pi$
$180$		0			0
$181$	4.15676			0.308969			0.154485	−	0.987995i	$-0.450628\pi$
$181$	4.15676			0.308969			0.154485	+	0.987995i	$0.450628\pi$
$182$		0			0
$183$			4.52359i			0.334393i
$184$		0			0
$185$	4.68035	−	18.8371i	0.344106	−	1.38493i
$186$		0			0
$187$			5.84324i			0.427300i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	−6.09890			−0.441301			−0.220650	−	0.975353i	$-0.570818\pi$
$191$	−6.09890			−0.441301			−0.220650	+	0.975353i	$0.570818\pi$
$192$		0			0
$193$			12.6803i			0.912751i	0.889787	+	0.456376i	$0.150853\pi$
$193$			12.6803i			0.912751i	−0.889787	+	0.456376i	$0.849147\pi$
$194$		0			0
$195$	−9.41855	−	2.34017i	−0.674476	−	0.167583i
$196$		0			0
$197$			11.8576i			0.844820i	0.906405	+	0.422410i	$0.138816\pi$
$197$			11.8576i			0.844820i	−0.906405	+	0.422410i	$0.861184\pi$
$198$		0			0
$199$	−5.50307			−0.390102			−0.195051	−	0.980793i	$-0.562487\pi$
$199$	−5.50307			−0.390102			−0.195051	+	0.980793i	$0.562487\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			8.83710i			0.620243i
$204$		0			0
$205$	−18.0989	−	4.49693i	−1.26408	−	0.314079i
$206$		0			0
$207$			6.34017i			0.440672i
$208$		0			0
$209$	−23.5174			−1.62674
$210$		0			0
$211$	19.1506			1.31838			0.659191	−	0.751975i	$-0.270899\pi$
$211$	19.1506			1.31838			0.659191	+	0.751975i	$0.270899\pi$
$212$		0			0
$213$		−	14.0989i		−	0.966040i
$214$		0			0
$215$	−3.31965	+	13.3607i	−0.226399	+	0.911192i
$216$		0			0
$217$			4.34017i			0.294630i
$218$		0			0
$219$	−11.1773			−0.755290
$220$		0			0
$221$	4.68035			0.314834
$222$		0			0
$223$		−	12.3135i		−	0.824574i	−0.911054	−	0.412287i	$-0.864730\pi$
$223$		−	12.3135i		−	0.824574i	0.911054	−	0.412287i	$-0.135270\pi$
$224$		0			0
$225$	−4.41855	−	2.34017i	−0.294570	−	0.156012i
$226$		0			0
$227$		−	15.2039i		−	1.00912i	−0.863376	−	0.504560i	$-0.831655\pi$
$227$		−	15.2039i		−	1.00912i	0.863376	−	0.504560i	$-0.168345\pi$
$228$		0			0
$229$	−5.20394			−0.343886			−0.171943	−	0.985107i	$-0.555004\pi$
$229$	−5.20394			−0.343886			−0.171943	+	0.985107i	$0.555004\pi$
$230$		0			0
$231$	−5.41855			−0.356514
$232$		0			0
$233$			11.6598i			0.763861i	0.924191	+	0.381930i	$0.124741\pi$
$233$			11.6598i			0.763861i	−0.924191	+	0.381930i	$0.875259\pi$
$234$		0			0
$235$	−3.68649	+	14.8371i	−0.240480	+	0.967866i
$236$		0			0
$237$		−	0.680346i		−	0.0441932i
$238$		0			0
$239$	20.6225			1.33396			0.666979	−	0.745077i	$-0.267587\pi$
$239$	20.6225			1.33396			0.666979	+	0.745077i	$0.267587\pi$
$240$		0			0
$241$	−20.3545			−1.31115			−0.655576	−	0.755129i	$-0.727574\pi$
$241$	−20.3545			−1.31115			−0.655576	+	0.755129i	$0.727574\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	2.17009	+	0.539189i	0.138642	+	0.0344475i
$246$		0			0
$247$			18.8371i			1.19858i
$248$		0			0
$249$	6.83710			0.433284
$250$		0			0
$251$	−10.5236			−0.664243			−0.332122	−	0.943237i	$-0.607764\pi$
$251$	−10.5236			−0.664243			−0.332122	+	0.943237i	$0.607764\pi$
$252$		0			0
$253$			34.3545i			2.15985i
$254$		0			0
$255$	2.34017	+	0.581449i	0.146547	+	0.0364118i
$256$		0			0
$257$		−	22.8059i		−	1.42259i	−0.702892	−	0.711297i	$-0.748108\pi$
$257$		−	22.8059i		−	1.42259i	0.702892	−	0.711297i	$-0.251892\pi$
$258$		0			0
$259$	−8.68035			−0.539370
$260$		0			0
$261$	−8.83710			−0.547003
$262$		0			0
$263$			28.0144i			1.72744i	0.503972	+	0.863720i	$0.331872\pi$
$263$			28.0144i			1.72744i	−0.503972	+	0.863720i	$0.668128\pi$
$264$		0			0
$265$	3.33403	−	13.4186i	0.204808	−	0.824295i
$266$		0			0
$267$			6.49693i			0.397606i
$268$		0			0
$269$	−18.4969			−1.12778			−0.563889	−	0.825851i	$-0.690695\pi$
$269$	−18.4969			−1.12778			−0.563889	+	0.825851i	$0.690695\pi$
$270$		0			0
$271$	29.0205			1.76287			0.881435	−	0.472304i	$-0.156578\pi$
$271$	29.0205			1.76287			0.881435	+	0.472304i	$0.156578\pi$
$272$		0			0
$273$			4.34017i			0.262679i
$274$		0			0
$275$	−23.9421	−	12.6803i	−1.44377	−	0.764654i
$276$		0			0
$277$			8.68035i			0.521551i	0.965399	+	0.260776i	$0.0839783\pi$
$277$			8.68035i			0.521551i	−0.965399	+	0.260776i	$0.916022\pi$
$278$		0			0
$279$	−4.34017			−0.259839
$280$		0			0
$281$	5.63317			0.336046			0.168023	−	0.985783i	$-0.446262\pi$
$281$	5.63317			0.336046			0.168023	+	0.985783i	$0.446262\pi$
$282$		0			0
$283$			2.47027i			0.146842i	0.997301	+	0.0734210i	$0.0233917\pi$
$283$			2.47027i			0.146842i	−0.997301	+	0.0734210i	$0.976608\pi$
$284$		0			0
$285$	−2.34017	+	9.41855i	−0.138620	+	0.557907i
$286$		0			0
$287$			8.34017i			0.492305i
$288$		0			0
$289$	15.8371			0.931594
$290$		0			0
$291$	10.4969			0.615341
$292$		0			0
$293$		−	7.60197i		−	0.444112i	−0.975034	−	0.222056i	$-0.928723\pi$
$293$		−	7.60197i		−	0.444112i	0.975034	−	0.222056i	$-0.0712767\pi$
$294$		0			0
$295$	14.8371	+	3.68649i	0.863849	+	0.214636i
$296$		0			0
$297$		−	5.41855i		−	0.314416i
$298$		0			0
$299$	27.5174			1.59138
$300$		0			0
$301$	6.15676			0.354869
$302$		0			0
$303$		−	18.8638i		−	1.08369i
$304$		0			0
$305$	9.81658	+	2.43907i	0.562096	+	0.139661i
$306$		0			0
$307$			6.15676i			0.351385i	0.984445	+	0.175692i	$0.0562164\pi$
$307$			6.15676i			0.351385i	−0.984445	+	0.175692i	$0.943784\pi$
$308$		0			0
$309$	−10.1568			−0.577798
$310$		0			0
$311$	1.52973			0.0867432			0.0433716	−	0.999059i	$-0.486190\pi$
$311$	1.52973			0.0867432			0.0433716	+	0.999059i	$0.486190\pi$
$312$		0			0
$313$			11.9733i			0.676773i	0.941007	+	0.338387i	$0.109881\pi$
$313$			11.9733i			0.676773i	−0.941007	+	0.338387i	$0.890119\pi$
$314$		0			0
$315$	−0.539189	+	2.17009i	−0.0303799	+	0.122271i
$316$		0			0
$317$		−	23.5441i		−	1.32237i	−0.750223	−	0.661184i	$-0.770054\pi$
$317$		−	23.5441i		−	1.32237i	0.750223	−	0.661184i	$-0.229946\pi$
$318$		0			0
$319$	−47.8843			−2.68101
$320$		0			0
$321$	14.6537			0.817889
$322$		0			0
$323$		−	4.68035i		−	0.260421i
$324$		0			0
$325$	−10.1568	+	19.1773i	−0.563395	+	1.06376i
$326$		0			0
$327$		−	12.8371i		−	0.709893i
$328$		0			0
$329$	6.83710			0.376942
$330$		0			0
$331$	9.16290			0.503638			0.251819	−	0.967774i	$-0.418971\pi$
$331$	9.16290			0.503638			0.251819	+	0.967774i	$0.418971\pi$
$332$		0			0
$333$		−	8.68035i		−	0.475680i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	16.0000i		−	0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$		−	16.0000i		−	0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$		0			0
$339$	1.50307			0.0816356
$340$		0			0
$341$	−23.5174			−1.27354
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	13.7587	+	3.41855i	0.740745	+	0.184049i
$346$		0			0
$347$		−	23.0205i		−	1.23581i	−0.786254	−	0.617903i	$-0.787982\pi$
$347$		−	23.0205i		−	1.23581i	0.786254	−	0.617903i	$-0.212018\pi$
$348$		0			0
$349$	3.78992			0.202870			0.101435	−	0.994842i	$-0.467657\pi$
$349$	3.78992			0.202870			0.101435	+	0.994842i	$0.467657\pi$
$350$		0			0
$351$	−4.34017			−0.231661
$352$		0			0
$353$			28.5958i			1.52200i	0.648751	+	0.761001i	$0.275292\pi$
$353$			28.5958i			1.52200i	−0.648751	+	0.761001i	$0.724708\pi$
$354$		0			0
$355$	−30.5958	−	7.60197i	−1.62386	−	0.403471i
$356$		0			0
$357$		−	1.07838i		−	0.0570738i
$358$		0			0
$359$	−11.2618			−0.594375			−0.297187	−	0.954819i	$-0.596049\pi$
$359$	−11.2618			−0.594375			−0.297187	+	0.954819i	$0.596049\pi$
$360$		0			0
$361$	−0.162899			−0.00857361
$362$		0			0
$363$		−	18.3607i		−	0.963686i
$364$		0			0
$365$	−6.02666	+	24.2557i	−0.315450	+	1.26960i
$366$		0			0
$367$		−	25.3607i		−	1.32382i	−0.749584	−	0.661909i	$-0.769747\pi$
$367$		−	25.3607i		−	1.32382i	0.749584	−	0.661909i	$-0.230253\pi$
$368$		0			0
$369$	−8.34017			−0.434172
$370$		0			0
$371$	−6.18342			−0.321027
$372$		0			0
$373$		−	21.3074i		−	1.10325i	−0.834091	−	0.551627i	$-0.814007\pi$
$373$		−	21.3074i		−	1.10325i	0.834091	−	0.551627i	$-0.185993\pi$
$374$		0			0
$375$	−7.46081	+	8.32684i	−0.385275	+	0.429996i
$376$		0			0
$377$			38.3545i			1.97536i
$378$		0			0
$379$	−1.84324			−0.0946811			−0.0473406	−	0.998879i	$-0.515075\pi$
$379$	−1.84324			−0.0946811			−0.0473406	+	0.998879i	$0.515075\pi$
$380$		0			0
$381$	19.2039			0.983847
$382$		0			0
$383$		−	4.99386i		−	0.255174i	−0.991827	−	0.127587i	$-0.959277\pi$
$383$		−	4.99386i		−	0.255174i	0.991827	−	0.127587i	$-0.0407232\pi$
$384$		0			0
$385$	−2.92162	+	11.7587i	−0.148900	+	0.599280i
$386$		0			0
$387$			6.15676i			0.312965i
$388$		0			0
$389$	−16.8371			−0.853675			−0.426837	−	0.904328i	$-0.640372\pi$
$389$	−16.8371			−0.853675			−0.426837	+	0.904328i	$0.640372\pi$
$390$		0			0
$391$	−6.83710			−0.345767
$392$		0			0
$393$		−	4.00000i		−	0.201773i
$394$		0			0
$395$	−1.47641	−	0.366835i	−0.0742862	−	0.0184575i
$396$		0			0
$397$			36.8515i			1.84952i	0.380548	+	0.924761i	$0.375736\pi$
$397$			36.8515i			1.84952i	−0.380548	+	0.924761i	$0.624264\pi$
$398$		0			0
$399$	4.34017			0.217280
$400$		0			0
$401$	−24.3545			−1.21621			−0.608104	−	0.793857i	$-0.708070\pi$
$401$	−24.3545			−1.21621			−0.608104	+	0.793857i	$0.708070\pi$
$402$		0			0
$403$			18.8371i			0.938343i
$404$		0			0
$405$	−2.17009	−	0.539189i	−0.107832	−	0.0267925i
$406$		0			0
$407$		−	47.0349i		−	2.33143i
$408$		0			0
$409$	28.0410			1.38654			0.693270	−	0.720678i	$-0.256169\pi$
$409$	28.0410			1.38654			0.693270	+	0.720678i	$0.256169\pi$
$410$		0			0
$411$	−8.65368			−0.426855
$412$		0			0
$413$		−	6.83710i		−	0.336432i
$414$		0			0
$415$	3.68649	−	14.8371i	0.180963	−	0.728325i
$416$		0			0
$417$			6.18342i			0.302803i
$418$		0			0
$419$	0.482553			0.0235742			0.0117871	−	0.999931i	$-0.496248\pi$
$419$	0.482553			0.0235742			0.0117871	+	0.999931i	$0.496248\pi$
$420$		0			0
$421$	21.1506			1.03082			0.515409	−	0.856944i	$-0.327640\pi$
$421$	21.1506			1.03082			0.515409	+	0.856944i	$0.327640\pi$
$422$		0			0
$423$			6.83710i			0.332431i
$424$		0			0
$425$	2.52359	−	4.76487i	0.122412	−	0.231130i
$426$		0			0
$427$		−	4.52359i		−	0.218912i
$428$		0			0
$429$	−23.5174			−1.13543
$430$		0			0
$431$	28.7382			1.38427			0.692135	−	0.721768i	$-0.256670\pi$
$431$	28.7382			1.38427			0.692135	+	0.721768i	$0.256670\pi$
$432$		0			0
$433$			9.02052i			0.433498i	0.976227	+	0.216749i	$0.0695454\pi$
$433$			9.02052i			0.433498i	−0.976227	+	0.216749i	$0.930455\pi$
$434$		0			0
$435$	−4.76487	+	19.1773i	−0.228458	+	0.919480i
$436$		0			0
$437$		−	27.5174i		−	1.31634i
$438$		0			0
$439$	−17.8166			−0.850339			−0.425170	−	0.905114i	$-0.639786\pi$
$439$	−17.8166			−0.850339			−0.425170	+	0.905114i	$0.639786\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	11.7009i		−	0.555925i	−0.960592	−	0.277962i	$-0.910341\pi$
$443$		−	11.7009i		−	0.555925i	0.960592	−	0.277962i	$-0.0896591\pi$
$444$		0			0
$445$	14.0989	+	3.50307i	0.668352	+	0.166062i
$446$		0			0
$447$		−	13.2039i		−	0.624525i
$448$		0			0
$449$	−10.7337			−0.506553			−0.253277	−	0.967394i	$-0.581508\pi$
$449$	−10.7337			−0.506553			−0.253277	+	0.967394i	$0.581508\pi$
$450$		0			0
$451$	−45.1917			−2.12799
$452$		0			0
$453$		−	18.1568i		−	0.853079i
$454$		0			0
$455$	9.41855	+	2.34017i	0.441548	+	0.109709i
$456$		0			0
$457$		−	20.9939i		−	0.982051i	−0.871145	−	0.491026i	$-0.836622\pi$
$457$		−	20.9939i		−	0.982051i	0.871145	−	0.491026i	$-0.163378\pi$
$458$		0			0
$459$	1.07838			0.0503344
$460$		0			0
$461$	−15.3751			−0.716088			−0.358044	−	0.933705i	$-0.616556\pi$
$461$	−15.3751			−0.716088			−0.358044	+	0.933705i	$0.616556\pi$
$462$		0			0
$463$			29.1917i			1.35665i	0.734762	+	0.678326i	$0.237294\pi$
$463$			29.1917i			1.35665i	−0.734762	+	0.678326i	$0.762706\pi$
$464$		0			0
$465$	−2.34017	+	9.41855i	−0.108523	+	0.436775i
$466$		0			0
$467$			15.2039i			0.703554i	0.936084	+	0.351777i	$0.114423\pi$
$467$			15.2039i			0.703554i	−0.936084	+	0.351777i	$0.885577\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−15.1773			−0.699332
$472$		0			0
$473$			33.3607i			1.53393i
$474$		0			0
$475$	19.1773	+	10.1568i	0.879914	+	0.466024i
$476$		0			0
$477$		−	6.18342i		−	0.283119i
$478$		0			0
$479$	−18.7838			−0.858253			−0.429126	−	0.903244i	$-0.641178\pi$
$479$	−18.7838			−0.858253			−0.429126	+	0.903244i	$0.641178\pi$
$480$		0			0
$481$	−37.6742			−1.71780
$482$		0			0
$483$		−	6.34017i		−	0.288488i
$484$		0			0
$485$	5.65983	−	22.7792i	0.257000	−	1.03435i
$486$		0			0
$487$			16.5113i			0.748199i	0.927389	+	0.374099i	$0.122048\pi$
$487$			16.5113i			0.748199i	−0.927389	+	0.374099i	$0.877952\pi$
$488$		0			0
$489$	2.83710			0.128298
$490$		0			0
$491$	−39.4063			−1.77838			−0.889190	−	0.457538i	$-0.848731\pi$
$491$	−39.4063			−1.77838			−0.889190	+	0.457538i	$0.848731\pi$
$492$		0			0
$493$		−	9.52973i		−	0.429198i
$494$		0			0
$495$	−11.7587	−	2.92162i	−0.528515	−	0.131317i
$496$		0			0
$497$			14.0989i			0.632422i
$498$		0			0
$499$	−31.5174			−1.41091			−0.705457	−	0.708752i	$-0.749258\pi$
$499$	−31.5174			−1.41091			−0.705457	+	0.708752i	$0.749258\pi$
$500$		0			0
$501$	13.3607			0.596912
$502$		0			0
$503$			20.3668i			0.908112i	0.890973	+	0.454056i	$0.150023\pi$
$503$			20.3668i			0.908112i	−0.890973	+	0.454056i	$0.849977\pi$
$504$		0			0
$505$	−40.9360	−	10.1711i	−1.82163	−	0.452609i
$506$		0			0
$507$			5.83710i			0.259235i
$508$		0			0
$509$	−11.1773			−0.495424			−0.247712	−	0.968834i	$-0.579679\pi$
$509$	−11.1773			−0.495424			−0.247712	+	0.968834i	$0.579679\pi$
$510$		0			0
$511$	11.1773			0.494454
$512$		0			0
$513$			4.34017i			0.191623i
$514$		0			0
$515$	−5.47641	+	22.0410i	−0.241319	+	0.971244i
$516$		0			0
$517$			37.0472i			1.62933i
$518$		0			0
$519$	2.55479			0.112143
$520$		0			0
$521$	32.6537			1.43058			0.715292	−	0.698826i	$-0.246294\pi$
$521$	32.6537			1.43058			0.715292	+	0.698826i	$0.246294\pi$
$522$		0			0
$523$		−	15.6865i		−	0.685922i	−0.939350	−	0.342961i	$-0.888570\pi$
$523$		−	15.6865i		−	0.685922i	0.939350	−	0.342961i	$-0.111430\pi$
$524$		0			0
$525$	4.41855	+	2.34017i	0.192841	+	0.102134i
$526$		0			0
$527$		−	4.68035i		−	0.203879i
$528$		0			0
$529$	−17.1978			−0.747730
$530$		0			0
$531$	6.83710			0.296705
$532$		0			0
$533$			36.1978i			1.56790i
$534$		0			0
$535$	7.90110	−	31.7998i	0.341594	−	1.37482i
$536$		0			0
$537$			11.9421i			0.515341i
$538$		0			0
$539$	5.41855			0.233394
$540$		0			0
$541$	30.1978			1.29830			0.649152	−	0.760658i	$-0.275124\pi$
$541$	30.1978			1.29830			0.649152	+	0.760658i	$0.275124\pi$
$542$		0			0
$543$		−	4.15676i		−	0.178383i
$544$		0			0
$545$	−27.8576	−	6.92162i	−1.19329	−	0.296490i
$546$		0			0
$547$			9.36069i			0.400234i	0.979772	+	0.200117i	$0.0641323\pi$
$547$			9.36069i			0.400234i	−0.979772	+	0.200117i	$0.935868\pi$
$548$		0			0
$549$	4.52359			0.193062
$550$		0			0
$551$	38.3545			1.63396
$552$		0			0
$553$			0.680346i			0.0289313i
$554$		0			0
$555$	−18.8371	−	4.68035i	−0.799590	−	0.198670i
$556$		0			0
$557$			7.49079i			0.317395i	0.987327	+	0.158697i	$0.0507294\pi$
$557$			7.49079i			0.317395i	−0.987327	+	0.158697i	$0.949271\pi$
$558$		0			0
$559$	26.7214			1.13019
$560$		0			0
$561$	5.84324			0.246702
$562$		0			0
$563$		−	31.7152i		−	1.33664i	−0.743875	−	0.668319i	$-0.767014\pi$
$563$		−	31.7152i		−	1.33664i	0.743875	−	0.668319i	$-0.232986\pi$
$564$		0			0
$565$	0.810439	−	3.26180i	0.0340954	−	0.137225i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	−30.6803			−1.28619			−0.643094	−	0.765788i	$-0.722349\pi$
$569$	−30.6803			−1.28619			−0.643094	+	0.765788i	$0.722349\pi$
$570$		0			0
$571$	10.6393			0.445241			0.222621	−	0.974905i	$-0.428539\pi$
$571$	10.6393			0.445241			0.222621	+	0.974905i	$0.428539\pi$
$572$		0			0
$573$			6.09890i			0.254785i
$574$		0			0
$575$	14.8371	−	28.0144i	0.618750	−	1.16828i
$576$		0			0
$577$		−	30.0677i		−	1.25173i	−0.779930	−	0.625867i	$-0.784745\pi$
$577$		−	30.0677i		−	1.25173i	0.779930	−	0.625867i	$-0.215255\pi$
$578$		0			0
$579$	12.6803			0.526977
$580$		0			0
$581$	−6.83710			−0.283651
$582$		0			0
$583$		−	33.5052i		−	1.38764i
$584$		0			0
$585$	−2.34017	+	9.41855i	−0.0967542	+	0.389409i
$586$		0			0
$587$			10.0410i			0.414438i	0.978295	+	0.207219i	$0.0664413\pi$
$587$			10.0410i			0.414438i	−0.978295	+	0.207219i	$0.933559\pi$
$588$		0			0
$589$	18.8371			0.776169
$590$		0			0
$591$	11.8576			0.487757
$592$		0			0
$593$		−	24.2823i		−	0.997155i	−0.866845	−	0.498578i	$-0.833856\pi$
$593$		−	24.2823i		−	0.997155i	0.866845	−	0.498578i	$-0.166144\pi$
$594$		0			0
$595$	−2.34017	−	0.581449i	−0.0959377	−	0.0238371i
$596$		0			0
$597$			5.50307i			0.225226i
$598$		0			0
$599$	−9.90110			−0.404548			−0.202274	−	0.979329i	$-0.564833\pi$
$599$	−9.90110			−0.404548			−0.202274	+	0.979329i	$0.564833\pi$
$600$		0			0
$601$	17.6865			0.721447			0.360723	−	0.932673i	$-0.382530\pi$
$601$	17.6865			0.721447			0.360723	+	0.932673i	$0.382530\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−39.8443	−	9.89988i	−1.61990	−	0.402487i
$606$		0			0
$607$			28.3135i			1.14921i	0.818431	+	0.574605i	$0.194844\pi$
$607$			28.3135i			1.14921i	−0.818431	+	0.574605i	$0.805156\pi$
$608$		0			0
$609$	8.83710			0.358097
$610$		0			0
$611$	29.6742			1.20049
$612$		0			0
$613$		−	23.6865i		−	0.956688i	−0.878172	−	0.478344i	$-0.841237\pi$
$613$		−	23.6865i		−	0.956688i	0.878172	−	0.478344i	$-0.158763\pi$
$614$		0			0
$615$	−4.49693	+	18.0989i	−0.181334	+	0.729818i
$616$		0			0
$617$			13.7009i			0.551576i	0.961218	+	0.275788i	$0.0889388\pi$
$617$			13.7009i			0.551576i	−0.961218	+	0.275788i	$0.911061\pi$
$618$		0			0
$619$	−2.49693			−0.100360			−0.0501800	−	0.998740i	$-0.515980\pi$
$619$	−2.49693			−0.100360			−0.0501800	+	0.998740i	$0.515980\pi$
$620$		0			0
$621$	6.34017			0.254422
$622$		0			0
$623$		−	6.49693i		−	0.260294i
$624$		0			0
$625$	14.0472	+	20.6803i	0.561887	+	0.827214i
$626$		0			0
$627$			23.5174i			0.939196i
$628$		0			0
$629$	9.36069			0.373235
$630$		0			0
$631$	−8.68035			−0.345559			−0.172780	−	0.984961i	$-0.555275\pi$
$631$	−8.68035			−0.345559			−0.172780	+	0.984961i	$0.555275\pi$
$632$		0			0
$633$		−	19.1506i		−	0.761169i
$634$		0			0
$635$	10.3545	−	41.6742i	0.410908	−	1.65379i
$636$		0			0
$637$		−	4.34017i		−	0.171964i
$638$		0			0
$639$	−14.0989			−0.557744
$640$		0			0
$641$	3.30737			0.130633			0.0653166	−	0.997865i	$-0.479194\pi$
$641$	3.30737			0.130633			0.0653166	+	0.997865i	$0.479194\pi$
$642$		0			0
$643$			6.15676i			0.242799i	0.992604	+	0.121399i	$0.0387382\pi$
$643$			6.15676i			0.242799i	−0.992604	+	0.121399i	$0.961262\pi$
$644$		0			0
$645$	13.3607	+	3.31965i	0.526077	+	0.130711i
$646$		0			0
$647$		−	6.95282i		−	0.273344i	−0.990616	−	0.136672i	$-0.956359\pi$
$647$		−	6.95282i		−	0.273344i	0.990616	−	0.136672i	$-0.0436406\pi$
$648$		0			0
$649$	37.0472			1.45423
$650$		0			0
$651$	4.34017			0.170105
$652$		0			0
$653$		−	38.7480i		−	1.51633i	−0.652064	−	0.758164i	$-0.726097\pi$
$653$		−	38.7480i		−	1.51633i	0.652064	−	0.758164i	$-0.273903\pi$
$654$		0			0
$655$	−8.68035	−	2.15676i	−0.339169	−	0.0842714i
$656$		0			0
$657$			11.1773i			0.436067i
$658$		0			0
$659$	9.22076			0.359190			0.179595	−	0.983741i	$-0.442521\pi$
$659$	9.22076			0.359190			0.179595	+	0.983741i	$0.442521\pi$
$660$		0			0
$661$	25.8843			1.00678			0.503391	−	0.864059i	$-0.332086\pi$
$661$	25.8843			1.00678			0.503391	+	0.864059i	$0.332086\pi$
$662$		0			0
$663$		−	4.68035i		−	0.181770i
$664$		0			0
$665$	2.34017	−	9.41855i	0.0907480	−	0.365236i
$666$		0			0
$667$		−	56.0288i		−	2.16944i
$668$		0			0
$669$	−12.3135			−0.476068
$670$		0			0
$671$	24.5113			0.946248
$672$		0			0
$673$		−	40.0821i		−	1.54505i	−0.634984	−	0.772525i	$-0.718993\pi$
$673$		−	40.0821i		−	1.54505i	0.634984	−	0.772525i	$-0.281007\pi$
$674$		0			0
$675$	−2.34017	+	4.41855i	−0.0900733	+	0.170070i
$676$		0			0
$677$		−	33.5897i		−	1.29096i	−0.763779	−	0.645478i	$-0.776658\pi$
$677$		−	33.5897i		−	1.29096i	0.763779	−	0.645478i	$-0.223342\pi$
$678$		0			0
$679$	−10.4969			−0.402835
$680$		0			0
$681$	−15.2039			−0.582616
$682$		0			0
$683$			18.7070i			0.715804i	0.933759	+	0.357902i	$0.116508\pi$
$683$			18.7070i			0.715804i	−0.933759	+	0.357902i	$0.883492\pi$
$684$		0			0
$685$	−4.66597	+	18.7792i	−0.178278	+	0.717518i
$686$		0			0
$687$			5.20394i			0.198543i
$688$		0			0
$689$	−26.8371			−1.02241
$690$		0			0
$691$	19.1773			0.729538			0.364769	−	0.931098i	$-0.381148\pi$
$691$	19.1773			0.729538			0.364769	+	0.931098i	$0.381148\pi$
$692$		0			0
$693$			5.41855i			0.205834i
$694$		0			0
$695$	13.4186	+	3.33403i	0.508995	+	0.126467i
$696$		0			0
$697$		−	8.99386i		−	0.340667i
$698$		0			0
$699$	11.6598			0.441015
$700$		0			0
$701$	21.4641			0.810689			0.405344	−	0.914164i	$-0.367152\pi$
$701$	21.4641			0.810689			0.405344	+	0.914164i	$0.367152\pi$
$702$		0			0
$703$			37.6742i			1.42091i
$704$		0			0
$705$	14.8371	+	3.68649i	0.558798	+	0.138841i
$706$		0			0
$707$			18.8638i			0.709445i
$708$		0			0
$709$	−2.62702			−0.0986599			−0.0493299	−	0.998783i	$-0.515709\pi$
$709$	−2.62702			−0.0986599			−0.0493299	+	0.998783i	$0.515709\pi$
$710$		0			0
$711$	−0.680346			−0.0255150
$712$		0			0
$713$		−	27.5174i		−	1.03054i
$714$		0			0
$715$	−12.6803	+	51.0349i	−0.474218	+	1.90860i
$716$		0			0
$717$		−	20.6225i		−	0.770161i
$718$		0			0
$719$	−4.36683			−0.162855			−0.0814277	−	0.996679i	$-0.525948\pi$
$719$	−4.36683			−0.162855			−0.0814277	+	0.996679i	$0.525948\pi$
$720$		0			0
$721$	10.1568			0.378257
$722$		0			0
$723$			20.3545i			0.756994i
$724$		0			0
$725$	39.0472	+	20.6803i	1.45018	+	0.768049i
$726$		0			0
$727$		−	28.1445i		−	1.04382i	−0.853000	−	0.521910i	$-0.825220\pi$
$727$		−	28.1445i		−	1.04382i	0.853000	−	0.521910i	$-0.174780\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−6.63931			−0.245564
$732$		0			0
$733$			26.7480i			0.987962i	0.869473	+	0.493981i	$0.164459\pi$
$733$			26.7480i			0.987962i	−0.869473	+	0.493981i	$0.835541\pi$
$734$		0			0
$735$	0.539189	−	2.17009i	0.0198883	−	0.0800448i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	19.0349			0.700210			0.350105	−	0.936710i	$-0.386146\pi$
$739$	19.0349			0.700210			0.350105	+	0.936710i	$0.386146\pi$
$740$		0			0
$741$	18.8371			0.691998
$742$		0			0
$743$			33.6598i			1.23486i	0.786626	+	0.617430i	$0.211826\pi$
$743$			33.6598i			1.23486i	−0.786626	+	0.617430i	$0.788174\pi$
$744$		0			0
$745$	−28.6537	−	7.11942i	−1.04979	−	0.260835i
$746$		0			0
$747$		−	6.83710i		−	0.250156i
$748$		0			0
$749$	−14.6537			−0.535434
$750$		0			0
$751$	49.8720			1.81985			0.909927	−	0.414767i	$-0.136137\pi$
$751$	49.8720			1.81985			0.909927	+	0.414767i	$0.136137\pi$
$752$		0			0
$753$			10.5236i			0.383501i
$754$		0			0
$755$	−39.4017	−	9.78992i	−1.43398	−	0.356292i
$756$		0			0
$757$		−	7.63317i		−	0.277432i	−0.990332	−	0.138716i	$-0.955702\pi$
$757$		−	7.63317i		−	0.277432i	0.990332	−	0.138716i	$-0.0442975\pi$
$758$		0			0
$759$	34.3545			1.24699
$760$		0			0
$761$	19.5974			0.710406			0.355203	−	0.934789i	$-0.384412\pi$
$761$	19.5974			0.710406			0.355203	+	0.934789i	$0.384412\pi$
$762$		0			0
$763$			12.8371i			0.464734i
$764$		0			0
$765$	0.581449	−	2.34017i	0.0210223	−	0.0846091i
$766$		0			0
$767$		−	29.6742i		−	1.07147i
$768$		0			0
$769$	32.4079			1.16866			0.584329	−	0.811517i	$-0.301358\pi$
$769$	32.4079			1.16866			0.584329	+	0.811517i	$0.301358\pi$
$770$		0			0
$771$	−22.8059			−0.821335
$772$		0			0
$773$			12.0845i			0.434650i	0.976099	+	0.217325i	$0.0697331\pi$
$773$			12.0845i			0.434650i	−0.976099	+	0.217325i	$0.930267\pi$
$774$		0			0
$775$	19.1773	+	10.1568i	0.688868	+	0.364841i
$776$		0			0
$777$			8.68035i			0.311406i
$778$		0			0
$779$	36.1978			1.29692
$780$		0			0
$781$	−76.3956			−2.73365
$782$		0			0
$783$			8.83710i			0.315812i
$784$		0			0
$785$	−8.18342	+	32.9360i	−0.292079	+	1.17554i
$786$		0			0
$787$		−	25.0472i		−	0.892836i	−0.894825	−	0.446418i	$-0.852700\pi$
$787$		−	25.0472i		−	0.892836i	0.894825	−	0.446418i	$-0.147300\pi$
$788$		0			0
$789$	28.0144			0.997338
$790$		0			0
$791$	−1.50307			−0.0534431
$792$		0			0
$793$		−	19.6332i		−	0.697194i
$794$		0			0
$795$	−13.4186	−	3.33403i	−0.475907	−	0.118246i
$796$		0			0
$797$		−	21.2762i		−	0.753641i	−0.926286	−	0.376820i	$-0.877017\pi$
$797$		−	21.2762i		−	0.753641i	0.926286	−	0.376820i	$-0.122983\pi$
$798$		0			0
$799$	−7.37298			−0.260837
$800$		0			0
$801$	6.49693			0.229558
$802$		0			0
$803$			60.5646i			2.13728i
$804$		0			0
$805$	−13.7587	−	3.41855i	−0.484931	−	0.120488i
$806$		0			0
$807$			18.4969i			0.651123i
$808$		0			0
$809$	49.6619			1.74602			0.873010	−	0.487702i	$-0.162165\pi$
$809$	49.6619			1.74602			0.873010	+	0.487702i	$0.162165\pi$
$810$		0			0
$811$	−25.8166			−0.906543			−0.453271	−	0.891373i	$-0.649743\pi$
$811$	−25.8166			−0.906543			−0.453271	+	0.891373i	$0.649743\pi$
$812$		0			0
$813$		−	29.0205i		−	1.01779i
$814$		0			0
$815$	1.52973	−	6.15676i	0.0535842	−	0.215662i
$816$		0			0
$817$		−	26.7214i		−	0.934863i
$818$		0			0
$819$	4.34017			0.151658
$820$		0			0
$821$	24.8904			0.868682			0.434341	−	0.900749i	$-0.356981\pi$
$821$	24.8904			0.868682			0.434341	+	0.900749i	$0.356981\pi$
$822$		0			0
$823$			18.4703i			0.643833i	0.946768	+	0.321917i	$0.104327\pi$
$823$			18.4703i			0.643833i	−0.946768	+	0.321917i	$0.895673\pi$
$824$		0			0
$825$	−12.6803	+	23.9421i	−0.441473	+	0.833558i
$826$		0			0
$827$			21.6886i			0.754186i	0.926175	+	0.377093i	$0.123076\pi$
$827$			21.6886i			0.754186i	−0.926175	+	0.377093i	$0.876924\pi$
$828$		0			0
$829$	−47.5052			−1.64992			−0.824961	−	0.565189i	$-0.808803\pi$
$829$	−47.5052			−1.64992			−0.824961	+	0.565189i	$0.808803\pi$
$830$		0			0
$831$	8.68035			0.301118
$832$		0			0
$833$			1.07838i			0.0373636i
$834$		0			0
$835$	7.20394	−	28.9939i	0.249302	−	1.00337i
$836$		0			0
$837$			4.34017i			0.150018i
$838$		0			0
$839$	11.4641			0.395785			0.197893	−	0.980224i	$-0.436590\pi$
$839$	11.4641			0.395785			0.197893	+	0.980224i	$0.436590\pi$
$840$		0			0
$841$	49.0944			1.69291
$842$		0			0
$843$		−	5.63317i		−	0.194017i
$844$		0			0
$845$	12.6670	+	3.14730i	0.435759	+	0.108270i
$846$		0			0
$847$			18.3607i			0.630881i
$848$		0			0
$849$	2.47027			0.0847793
$850$		0			0
$851$	55.0349			1.88657
$852$		0			0
$853$		−	39.8043i		−	1.36287i	−0.731877	−	0.681437i	$-0.761356\pi$
$853$		−	39.8043i		−	1.36287i	0.731877	−	0.681437i	$-0.238644\pi$
$854$		0			0
$855$	9.41855	+	2.34017i	0.322108	+	0.0800322i
$856$		0			0
$857$			16.9627i			0.579433i	0.957112	+	0.289717i	$0.0935611\pi$
$857$			16.9627i			0.579433i	−0.957112	+	0.289717i	$0.906439\pi$
$858$		0			0
$859$	−41.0493			−1.40058			−0.700292	−	0.713857i	$-0.746947\pi$
$859$	−41.0493			−1.40058			−0.700292	+	0.713857i	$0.746947\pi$
$860$		0			0
$861$	8.34017			0.284232
$862$		0			0
$863$			50.0554i			1.70391i	0.523618	+	0.851953i	$0.324582\pi$
$863$			50.0554i			1.70391i	−0.523618	+	0.851953i	$0.675418\pi$
$864$		0			0
$865$	1.37751	−	5.54411i	0.0468368	−	0.188505i
$866$		0			0
$867$		−	15.8371i		−	0.537856i
$868$		0			0
$869$	−3.68649			−0.125056
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	10.4969i		−	0.355267i
$874$		0			0
$875$	7.46081	−	8.32684i	0.252221	−	0.281499i
$876$		0			0
$877$			6.32580i			0.213607i	0.994280	+	0.106803i	$0.0340616\pi$
$877$			6.32580i			0.213607i	−0.994280	+	0.106803i	$0.965938\pi$
$878$		0			0
$879$	−7.60197			−0.256408
$880$		0			0
$881$	20.5380			0.691942			0.345971	−	0.938245i	$-0.387550\pi$
$881$	20.5380			0.691942			0.345971	+	0.938245i	$0.387550\pi$
$882$		0			0
$883$		−	5.30737i		−	0.178607i	−0.996004	−	0.0893036i	$-0.971536\pi$
$883$		−	5.30737i		−	0.178607i	0.996004	−	0.0893036i	$-0.0284641\pi$
$884$		0			0
$885$	3.68649	−	14.8371i	0.123920	−	0.498744i
$886$		0			0
$887$			3.15061i			0.105787i	0.998600	+	0.0528936i	$0.0168444\pi$
$887$			3.15061i			0.105787i	−0.998600	+	0.0528936i	$0.983156\pi$
$888$		0			0
$889$	−19.2039			−0.644079
$890$		0			0
$891$	−5.41855			−0.181528
$892$		0			0
$893$		−	29.6742i		−	0.993009i
$894$		0			0
$895$	25.9155	+	6.43907i	0.866259	+	0.215234i
$896$		0			0
$897$		−	27.5174i		−	0.918781i
$898$		0			0
$899$	38.3545			1.27920
$900$		0			0
$901$	6.66806			0.222145
$902$		0			0
$903$		−	6.15676i		−	0.204884i
$904$		0			0
$905$	−9.02052	−	2.24128i	−0.299852	−	0.0745025i
$906$		0			0
$907$		−	11.9467i		−	0.396683i	−0.980133	−	0.198341i	$-0.936445\pi$
$907$		−	11.9467i		−	0.396683i	0.980133	−	0.198341i	$-0.0635555\pi$
$908$		0			0
$909$	−18.8638			−0.625672
$910$		0			0
$911$	−46.0989			−1.52732			−0.763662	−	0.645616i	$-0.776601\pi$
$911$	−46.0989			−1.52732			−0.763662	+	0.645616i	$0.776601\pi$
$912$		0			0
$913$		−	37.0472i		−	1.22608i
$914$		0			0
$915$	2.43907	−	9.81658i	0.0806332	−	0.324526i
$916$		0			0
$917$			4.00000i			0.132092i
$918$		0			0
$919$	7.25726			0.239395			0.119697	−	0.992810i	$-0.461808\pi$
$919$	7.25726			0.239395			0.119697	+	0.992810i	$0.461808\pi$
$920$		0			0
$921$	6.15676			0.202872
$922$		0			0
$923$			61.1917i			2.01415i
$924$		0			0
$925$	−20.3135	+	38.3545i	−0.667904	+	1.26109i
$926$		0			0
$927$			10.1568i			0.333592i
$928$		0			0
$929$	44.9048			1.47328			0.736639	−	0.676286i	$-0.236412\pi$
$929$	44.9048			1.47328			0.736639	+	0.676286i	$0.236412\pi$
$930$		0			0
$931$	−4.34017			−0.142243
$932$		0			0
$933$		−	1.52973i		−	0.0500812i
$934$		0			0
$935$	3.15061	−	12.6803i	0.103036	−	0.414692i
$936$		0			0
$937$		−	43.6886i		−	1.42724i	−0.700531	−	0.713622i	$-0.747054\pi$
$937$		−	43.6886i		−	1.42724i	0.700531	−	0.713622i	$-0.252946\pi$
$938$		0			0
$939$	11.9733			0.390735
$940$		0			0
$941$	14.6660			0.478097			0.239048	−	0.971008i	$-0.423165\pi$
$941$	14.6660			0.478097			0.239048	+	0.971008i	$0.423165\pi$
$942$		0			0
$943$		−	52.8781i		−	1.72195i
$944$		0			0
$945$	2.17009	+	0.539189i	0.0705929	+	0.0175398i
$946$		0			0
$947$		−	2.96719i		−	0.0964209i	−0.998837	−	0.0482104i	$-0.984648\pi$
$947$		−	2.96719i		−	0.0964209i	0.998837	−	0.0482104i	$-0.0153518\pi$
$948$		0			0
$949$	48.5113			1.57474
$950$		0			0
$951$	−23.5441			−0.763470
$952$		0			0
$953$			45.5851i			1.47665i	0.674446	+	0.738324i	$0.264382\pi$
$953$			45.5851i			1.47665i	−0.674446	+	0.738324i	$0.735618\pi$
$954$		0			0
$955$	13.2351	+	3.28846i	0.428279	+	0.106412i
$956$		0			0
$957$			47.8843i			1.54788i
$958$		0			0
$959$	8.65368			0.279442
$960$		0			0
$961$	−12.1629			−0.392352
$962$		0			0
$963$		−	14.6537i		−	0.472208i
$964$		0			0
$965$	6.83710	−	27.5174i	0.220094	−	0.885818i
$966$		0			0
$967$		−	32.3668i		−	1.04085i	−0.853908	−	0.520424i	$-0.825774\pi$
$967$		−	32.3668i		−	1.04085i	0.853908	−	0.520424i	$-0.174226\pi$
$968$		0			0
$969$	−4.68035			−0.150354
$970$		0			0
$971$	11.3197			0.363265			0.181632	−	0.983366i	$-0.441862\pi$
$971$	11.3197			0.363265			0.181632	+	0.983366i	$0.441862\pi$
$972$		0			0
$973$		−	6.18342i		−	0.198231i
$974$		0			0
$975$	19.1773	+	10.1568i	0.614164	+	0.325277i
$976$		0			0
$977$			39.6886i			1.26975i	0.772615	+	0.634875i	$0.218948\pi$
$977$			39.6886i			1.26975i	−0.772615	+	0.634875i	$0.781052\pi$
$978$		0			0
$979$	35.2039			1.12512
$980$		0			0
$981$	−12.8371			−0.409857
$982$		0			0
$983$			43.5174i			1.38799i	0.719979	+	0.693996i	$0.244151\pi$
$983$			43.5174i			1.38799i	−0.719979	+	0.693996i	$0.755849\pi$
$984$		0			0
$985$	6.39350	−	25.7321i	0.203714	−	0.819892i
$986$		0			0
$987$		−	6.83710i		−	0.217627i
$988$		0			0
$989$	−39.0349			−1.24124
$990$		0			0
$991$	3.80221			0.120781			0.0603905	−	0.998175i	$-0.480765\pi$
$991$	3.80221			0.120781			0.0603905	+	0.998175i	$0.480765\pi$
$992$		0			0
$993$		−	9.16290i		−	0.290776i
$994$		0			0
$995$	11.9421	+	2.96719i	0.378591	+	0.0940664i
$996$		0			0
$997$			23.8043i			0.753890i	0.926236	+	0.376945i	$0.123025\pi$
$997$			23.8043i			0.753890i	−0.926236	+	0.376945i	$0.876975\pi$
$998$		0			0
$999$	−8.68035			−0.274634

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.t.i.1009.1		6
3.2	odd	2		5040.2.t.ba.1009.6		6
4.3	odd	2		840.2.t.e.169.4	yes	6
5.2	odd	4		8400.2.a.dh.1.1		3
5.3	odd	4		8400.2.a.dk.1.1		3
5.4	even	2	inner	1680.2.t.i.1009.4		6
12.11	even	2		2520.2.t.j.1009.6		6
15.14	odd	2		5040.2.t.ba.1009.5		6
20.3	even	4		4200.2.a.bo.1.3		3
20.7	even	4		4200.2.a.bq.1.3		3
20.19	odd	2		840.2.t.e.169.1	✓	6
60.59	even	2		2520.2.t.j.1009.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.e.169.1	✓	6	20.19	odd	2
840.2.t.e.169.4	yes	6	4.3	odd	2
1680.2.t.i.1009.1		6	1.1	even	1	trivial
1680.2.t.i.1009.4		6	5.4	even	2	inner
2520.2.t.j.1009.5		6	60.59	even	2
2520.2.t.j.1009.6		6	12.11	even	2
4200.2.a.bo.1.3		3	20.3	even	4
4200.2.a.bq.1.3		3	20.7	even	4
5040.2.t.ba.1009.5		6	15.14	odd	2
5040.2.t.ba.1009.6		6	3.2	odd	2
8400.2.a.dh.1.1		3	5.2	odd	4
8400.2.a.dk.1.1		3	5.3	odd	4

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-2.17009 - 0.539189i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -5.41855 q^{11} +4.34017i q^{13} +(-0.539189 + 2.17009i) q^{15} -1.07838i q^{17} +4.34017 q^{19} +1.00000 q^{21} -6.34017i q^{23} +(4.41855 + 2.34017i) q^{25} +1.00000i q^{27} +8.83710 q^{29} +4.34017 q^{31} +5.41855i q^{33} +(0.539189 - 2.17009i) q^{35} +8.68035i q^{37} +4.34017 q^{39} +8.34017 q^{41} -6.15676i q^{43} +(2.17009 + 0.539189i) q^{45} -6.83710i q^{47} -1.00000 q^{49} -1.07838 q^{51} +6.18342i q^{53} +(11.7587 + 2.92162i) q^{55} -4.34017i q^{57} -6.83710 q^{59} -4.52359 q^{61} -1.00000i q^{63} +(2.34017 - 9.41855i) q^{65} -6.34017 q^{69} +14.0989 q^{71} -11.1773i q^{73} +(2.34017 - 4.41855i) q^{75} -5.41855i q^{77} +0.680346 q^{79} +1.00000 q^{81} +6.83710i q^{83} +(-0.581449 + 2.34017i) q^{85} -8.83710i q^{87} -6.49693 q^{89} -4.34017 q^{91} -4.34017i q^{93} +(-9.41855 - 2.34017i) q^{95} +10.4969i q^{97} +5.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^5 - 6 * q^9	\( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} + 4 q^{31} + 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} + 20 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 12 q^{71} - 8 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} - 4 q^{91} - 28 q^{95} + 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 + 4 * q^31 + 4 * q^39 + 28 * q^41 + 2 * q^45 - 6 * q^49 + 20 * q^55 + 16 * q^59 + 4 * q^61 - 8 * q^65 - 16 * q^69 + 12 * q^71 - 8 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 - 4 * q^89 - 4 * q^91 - 28 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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