LMFDB - Embedded newform 5040.2.t.ba.1009.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(1009,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, 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("5040.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.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:	$40.2446026187$
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			1009.6
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	5040.1009
Dual form			5040.2.t.ba.1009.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		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
$9$		0			0
$10$		0			0
$11$	5.41855			1.63375			0.816877	−	0.576812i	$-0.195703\pi$
$11$	5.41855			1.63375			0.816877	+	0.576812i	$0.195703\pi$
$12$		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			0
$16$		0			0
$17$			1.07838i			0.261545i	0.991412	+	0.130773i	$0.0417457\pi$
$17$			1.07838i			0.261545i	−0.991412	+	0.130773i	$0.958254\pi$
$18$		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
$21$		0			0
$22$		0			0
$23$			6.34017i			1.32202i	0.750378	+	0.661009i	$0.229871\pi$
$23$			6.34017i			1.32202i	−0.750378	+	0.661009i	$0.770129\pi$
$24$		0			0
$25$	4.41855	+	2.34017i	0.883710	+	0.468035i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.83710			−1.64101			−0.820504	−	0.571640i	$-0.806307\pi$
$29$	−8.83710			−1.64101			−0.820504	+	0.571640i	$0.806307\pi$
$30$		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
$33$		0			0
$34$		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
$39$		0			0
$40$		0			0
$41$	−8.34017			−1.30252			−0.651258	−	0.758856i	$-0.725758\pi$
$41$	−8.34017			−1.30252			−0.651258	+	0.758856i	$0.725758\pi$
$42$		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
$45$		0			0
$46$		0			0
$47$			6.83710i			0.997294i	0.866805	+	0.498647i	$0.166170\pi$
$47$			6.83710i			0.997294i	−0.866805	+	0.498647i	$0.833830\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.18342i		−	0.849358i	−0.905344	−	0.424679i	$-0.860387\pi$
$53$		−	6.18342i		−	0.849358i	0.905344	−	0.424679i	$-0.139613\pi$
$54$		0			0
$55$	11.7587	+	2.92162i	1.58555	+	0.393951i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.83710			0.890115			0.445057	−	0.895502i	$-0.353183\pi$
$59$	6.83710			0.890115			0.445057	+	0.895502i	$0.353183\pi$
$60$		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
$63$		0			0
$64$		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
$69$		0			0
$70$		0			0
$71$	−14.0989			−1.67323			−0.836616	−	0.547790i	$-0.815469\pi$
$71$	−14.0989			−1.67323			−0.836616	+	0.547790i	$0.815469\pi$
$72$		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
$75$		0			0
$76$		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
$81$		0			0
$82$		0			0
$83$		−	6.83710i		−	0.750469i	−0.926930	−	0.375235i	$-0.877562\pi$
$83$		−	6.83710i		−	0.750469i	0.926930	−	0.375235i	$-0.122438\pi$
$84$		0			0
$85$	−0.581449	+	2.34017i	−0.0630670	+	0.253827i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.49693			0.688673			0.344337	−	0.938846i	$-0.388104\pi$
$89$	6.49693			0.688673			0.344337	+	0.938846i	$0.388104\pi$
$90$		0			0
$91$	−4.34017			−0.454974
$92$		0			0
$93$		0			0
$94$		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
$99$		0			0
$100$		0			0
$101$	−18.8638			−1.87701			−0.938507	−	0.345259i	$-0.887791\pi$
$101$	−18.8638			−1.87701			−0.938507	+	0.345259i	$0.887791\pi$
$102$		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
$105$		0			0
$106$		0			0
$107$		−	14.6537i		−	1.41663i	−0.705899	−	0.708313i	$-0.749457\pi$
$107$		−	14.6537i		−	1.41663i	0.705899	−	0.708313i	$-0.250543\pi$
$108$		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
$111$		0			0
$112$		0			0
$113$		−	1.50307i		−	0.141397i	−0.997498	−	0.0706985i	$-0.977477\pi$
$113$		−	1.50307i		−	0.141397i	0.997498	−	0.0706985i	$-0.0225228\pi$
$114$		0			0
$115$	−3.41855	+	13.7587i	−0.318781	+	1.28301i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.07838			−0.0988547
$120$		0			0
$121$	18.3607			1.66915
$122$		0			0
$123$		0			0
$124$		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
$129$		0			0
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$			4.34017i			0.376341i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.65368i			0.739334i	0.929164	+	0.369667i	$0.120528\pi$
$137$			8.65368i			0.739334i	−0.929164	+	0.369667i	$0.879472\pi$
$138$		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
$141$		0			0
$142$		0			0
$143$			23.5174i			1.96663i
$144$		0			0
$145$	−19.1773	−	4.76487i	−1.59259	−	0.395701i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.2039			−1.08171			−0.540854	−	0.841116i	$-0.681899\pi$
$149$	−13.2039			−1.08171			−0.540854	+	0.841116i	$0.681899\pi$
$150$		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
$153$		0			0
$154$		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
$159$		0			0
$160$		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
$165$		0			0
$166$		0			0
$167$		−	13.3607i		−	1.03388i	−0.856021	−	0.516941i	$-0.827071\pi$
$167$		−	13.3607i		−	1.03388i	0.856021	−	0.516941i	$-0.172929\pi$
$168$		0			0
$169$	−5.83710			−0.449008
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2.55479i		−	0.194237i	−0.995273	−	0.0971184i	$-0.969037\pi$
$173$		−	2.55479i		−	0.194237i	0.995273	−	0.0971184i	$-0.0309626\pi$
$174$		0			0
$175$	−2.34017	+	4.41855i	−0.176900	+	0.334011i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.9421			0.892598			0.446299	−	0.894884i	$-0.352742\pi$
$179$	11.9421			0.892598			0.446299	+	0.894884i	$0.352742\pi$
$180$		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
$183$		0			0
$184$		0			0
$185$	−4.68035	+	18.8371i	−0.344106	+	1.38493i
$186$		0			0
$187$			5.84324i			0.427300i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.09890			0.441301			0.220650	−	0.975353i	$-0.429182\pi$
$191$	6.09890			0.441301			0.220650	+	0.975353i	$0.429182\pi$
$192$		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
$195$		0			0
$196$		0			0
$197$		−	11.8576i		−	0.844820i	−0.906405	−	0.422410i	$-0.861184\pi$
$197$		−	11.8576i		−	0.844820i	0.906405	−	0.422410i	$-0.138816\pi$
$198$		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
$207$		0			0
$208$		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
$213$		0			0
$214$		0			0
$215$	3.31965	−	13.3607i	0.226399	−	0.911192i
$216$		0			0
$217$			4.34017i			0.294630i
$218$		0			0
$219$		0			0
$220$		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
$225$		0			0
$226$		0			0
$227$			15.2039i			1.00912i	0.863376	+	0.504560i	$0.168345\pi$
$227$			15.2039i			1.00912i	−0.863376	+	0.504560i	$0.831655\pi$
$228$		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
$231$		0			0
$232$		0			0
$233$		−	11.6598i		−	0.763861i	−0.924191	−	0.381930i	$-0.875259\pi$
$233$		−	11.6598i		−	0.763861i	0.924191	−	0.381930i	$-0.124741\pi$
$234$		0			0
$235$	−3.68649	+	14.8371i	−0.240480	+	0.967866i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−20.6225			−1.33396			−0.666979	−	0.745077i	$-0.732413\pi$
$239$	−20.6225			−1.33396			−0.666979	+	0.745077i	$0.732413\pi$
$240$		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
$243$		0			0
$244$		0			0
$245$	−2.17009	−	0.539189i	−0.138642	−	0.0344475i
$246$		0			0
$247$			18.8371i			1.19858i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.5236			0.664243			0.332122	−	0.943237i	$-0.392236\pi$
$251$	10.5236			0.664243			0.332122	+	0.943237i	$0.392236\pi$
$252$		0			0
$253$			34.3545i			2.15985i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			22.8059i			1.42259i	0.702892	+	0.711297i	$0.251892\pi$
$257$			22.8059i			1.42259i	−0.702892	+	0.711297i	$0.748108\pi$
$258$		0			0
$259$	−8.68035			−0.539370
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	28.0144i		−	1.72744i	−0.503972	−	0.863720i	$-0.668128\pi$
$263$		−	28.0144i		−	1.72744i	0.503972	−	0.863720i	$-0.331872\pi$
$264$		0			0
$265$	3.33403	−	13.4186i	0.204808	−	0.824295i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.4969			1.12778			0.563889	−	0.825851i	$-0.309305\pi$
$269$	18.4969			1.12778			0.563889	+	0.825851i	$0.309305\pi$
$270$		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
$273$		0			0
$274$		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
$279$		0			0
$280$		0			0
$281$	−5.63317			−0.336046			−0.168023	−	0.985783i	$-0.553738\pi$
$281$	−5.63317			−0.336046			−0.168023	+	0.985783i	$0.553738\pi$
$282$		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
$285$		0			0
$286$		0			0
$287$		−	8.34017i		−	0.492305i
$288$		0			0
$289$	15.8371			0.931594
$290$		0			0
$291$		0			0
$292$		0			0
$293$			7.60197i			0.444112i	0.975034	+	0.222056i	$0.0712767\pi$
$293$			7.60197i			0.444112i	−0.975034	+	0.222056i	$0.928723\pi$
$294$		0			0
$295$	14.8371	+	3.68649i	0.863849	+	0.214636i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−27.5174			−1.59138
$300$		0			0
$301$	6.15676			0.354869
$302$		0			0
$303$		0			0
$304$		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
$309$		0			0
$310$		0			0
$311$	−1.52973			−0.0867432			−0.0433716	−	0.999059i	$-0.513810\pi$
$311$	−1.52973			−0.0867432			−0.0433716	+	0.999059i	$0.513810\pi$
$312$		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			0
$316$		0			0
$317$			23.5441i			1.32237i	0.750223	+	0.661184i	$0.229946\pi$
$317$			23.5441i			1.32237i	−0.750223	+	0.661184i	$0.770054\pi$
$318$		0			0
$319$	−47.8843			−2.68101
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4.68035i			0.260421i
$324$		0			0
$325$	−10.1568	+	19.1773i	−0.563395	+	1.06376i
$326$		0			0
$327$		0			0
$328$		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
$333$		0			0
$334$		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
$339$		0			0
$340$		0			0
$341$	23.5174			1.27354
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			23.0205i			1.23581i	0.786254	+	0.617903i	$0.212018\pi$
$347$			23.0205i			1.23581i	−0.786254	+	0.617903i	$0.787982\pi$
$348$		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
$351$		0			0
$352$		0			0
$353$		−	28.5958i		−	1.52200i	−0.648751	−	0.761001i	$-0.724708\pi$
$353$		−	28.5958i		−	1.52200i	0.648751	−	0.761001i	$-0.275292\pi$
$354$		0			0
$355$	−30.5958	−	7.60197i	−1.62386	−	0.403471i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	11.2618			0.594375			0.297187	−	0.954819i	$-0.403951\pi$
$359$	11.2618			0.594375			0.297187	+	0.954819i	$0.403951\pi$
$360$		0			0
$361$	−0.162899			−0.00857361
$362$		0			0
$363$		0			0
$364$		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
$369$		0			0
$370$		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
$375$		0			0
$376$		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
$381$		0			0
$382$		0			0
$383$			4.99386i			0.255174i	0.991827	+	0.127587i	$0.0407232\pi$
$383$			4.99386i			0.255174i	−0.991827	+	0.127587i	$0.959277\pi$
$384$		0			0
$385$	−2.92162	+	11.7587i	−0.148900	+	0.599280i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.8371			0.853675			0.426837	−	0.904328i	$-0.359628\pi$
$389$	16.8371			0.853675			0.426837	+	0.904328i	$0.359628\pi$
$390$		0			0
$391$	−6.83710			−0.345767
$392$		0			0
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$	24.3545			1.21621			0.608104	−	0.793857i	$-0.291930\pi$
$401$	24.3545			1.21621			0.608104	+	0.793857i	$0.291930\pi$
$402$		0			0
$403$			18.8371i			0.938343i
$404$		0			0
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$			6.83710i			0.336432i
$414$		0			0
$415$	3.68649	−	14.8371i	0.180963	−	0.728325i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.482553			−0.0235742			−0.0117871	−	0.999931i	$-0.503752\pi$
$419$	−0.482553			−0.0235742			−0.0117871	+	0.999931i	$0.503752\pi$
$420$		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
$423$		0			0
$424$		0			0
$425$	−2.52359	+	4.76487i	−0.122412	+	0.231130i
$426$		0			0
$427$		−	4.52359i		−	0.218912i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−28.7382			−1.38427			−0.692135	−	0.721768i	$-0.743330\pi$
$431$	−28.7382			−1.38427			−0.692135	+	0.721768i	$0.743330\pi$
$432$		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
$435$		0			0
$436$		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
$441$		0			0
$442$		0			0
$443$			11.7009i			0.555925i	0.960592	+	0.277962i	$0.0896591\pi$
$443$			11.7009i			0.555925i	−0.960592	+	0.277962i	$0.910341\pi$
$444$		0			0
$445$	14.0989	+	3.50307i	0.668352	+	0.166062i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.7337			0.506553			0.253277	−	0.967394i	$-0.418492\pi$
$449$	10.7337			0.506553			0.253277	+	0.967394i	$0.418492\pi$
$450$		0			0
$451$	−45.1917			−2.12799
$452$		0			0
$453$		0			0
$454$		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
$459$		0			0
$460$		0			0
$461$	15.3751			0.716088			0.358044	−	0.933705i	$-0.383444\pi$
$461$	15.3751			0.716088			0.358044	+	0.933705i	$0.383444\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$		−	15.2039i		−	0.703554i	−0.936084	−	0.351777i	$-0.885577\pi$
$467$		−	15.2039i		−	0.703554i	0.936084	−	0.351777i	$-0.114423\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	33.3607i		−	1.53393i
$474$		0			0
$475$	19.1773	+	10.1568i	0.879914	+	0.466024i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.7838			0.858253			0.429126	−	0.903244i	$-0.358822\pi$
$479$	18.7838			0.858253			0.429126	+	0.903244i	$0.358822\pi$
$480$		0			0
$481$	−37.6742			−1.71780
$482$		0			0
$483$		0			0
$484$		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
$489$		0			0
$490$		0			0
$491$	39.4063			1.77838			0.889190	−	0.457538i	$-0.151269\pi$
$491$	39.4063			1.77838			0.889190	+	0.457538i	$0.151269\pi$
$492$		0			0
$493$		−	9.52973i		−	0.429198i
$494$		0			0
$495$		0			0
$496$		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
$501$		0			0
$502$		0			0
$503$		−	20.3668i		−	0.908112i	−0.890973	−	0.454056i	$-0.849977\pi$
$503$		−	20.3668i		−	0.908112i	0.890973	−	0.454056i	$-0.150023\pi$
$504$		0			0
$505$	−40.9360	−	10.1711i	−1.82163	−	0.452609i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.1773			0.495424			0.247712	−	0.968834i	$-0.420321\pi$
$509$	11.1773			0.495424			0.247712	+	0.968834i	$0.420321\pi$
$510$		0			0
$511$	11.1773			0.494454
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.47641	−	22.0410i	0.241319	−	0.971244i
$516$		0			0
$517$			37.0472i			1.62933i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−32.6537			−1.43058			−0.715292	−	0.698826i	$-0.753706\pi$
$521$	−32.6537			−1.43058			−0.715292	+	0.698826i	$0.753706\pi$
$522$		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
$525$		0			0
$526$		0			0
$527$			4.68035i			0.203879i
$528$		0			0
$529$	−17.1978			−0.747730
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	36.1978i		−	1.56790i
$534$		0			0
$535$	7.90110	−	31.7998i	0.341594	−	1.37482i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		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
$549$		0			0
$550$		0			0
$551$	−38.3545			−1.63396
$552$		0			0
$553$			0.680346i			0.0289313i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	7.49079i		−	0.317395i	−0.987327	−	0.158697i	$-0.949271\pi$
$557$		−	7.49079i		−	0.317395i	0.987327	−	0.158697i	$-0.0507294\pi$
$558$		0			0
$559$	26.7214			1.13019
$560$		0			0
$561$		0			0
$562$		0			0
$563$			31.7152i			1.33664i	0.743875	+	0.668319i	$0.232986\pi$
$563$			31.7152i			1.33664i	−0.743875	+	0.668319i	$0.767014\pi$
$564$		0			0
$565$	0.810439	−	3.26180i	0.0340954	−	0.137225i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	30.6803			1.28619			0.643094	−	0.765788i	$-0.277651\pi$
$569$	30.6803			1.28619			0.643094	+	0.765788i	$0.277651\pi$
$570$		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
$573$		0			0
$574$		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
$579$		0			0
$580$		0			0
$581$	6.83710			0.283651
$582$		0			0
$583$		−	33.5052i		−	1.38764i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	10.0410i		−	0.414438i	−0.978295	−	0.207219i	$-0.933559\pi$
$587$		−	10.0410i		−	0.414438i	0.978295	−	0.207219i	$-0.0664413\pi$
$588$		0			0
$589$	18.8371			0.776169
$590$		0			0
$591$		0			0
$592$		0			0
$593$			24.2823i			0.997155i	0.866845	+	0.498578i	$0.166144\pi$
$593$			24.2823i			0.997155i	−0.866845	+	0.498578i	$0.833856\pi$
$594$		0			0
$595$	−2.34017	−	0.581449i	−0.0959377	−	0.0238371i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.90110			0.404548			0.202274	−	0.979329i	$-0.435167\pi$
$599$	9.90110			0.404548			0.202274	+	0.979329i	$0.435167\pi$
$600$		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
$609$		0			0
$610$		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
$615$		0			0
$616$		0			0
$617$		−	13.7009i		−	0.551576i	−0.961218	−	0.275788i	$-0.911061\pi$
$617$		−	13.7009i		−	0.551576i	0.961218	−	0.275788i	$-0.0889388\pi$
$618$		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
$621$		0			0
$622$		0			0
$623$			6.49693i			0.260294i
$624$		0			0
$625$	14.0472	+	20.6803i	0.561887	+	0.827214i
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$	−10.3545	+	41.6742i	−0.410908	+	1.65379i
$636$		0			0
$637$		−	4.34017i		−	0.171964i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.30737			−0.130633			−0.0653166	−	0.997865i	$-0.520806\pi$
$641$	−3.30737			−0.130633			−0.0653166	+	0.997865i	$0.520806\pi$
$642$		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
$645$		0			0
$646$		0			0
$647$			6.95282i			0.273344i	0.990616	+	0.136672i	$0.0436406\pi$
$647$			6.95282i			0.273344i	−0.990616	+	0.136672i	$0.956359\pi$
$648$		0			0
$649$	37.0472			1.45423
$650$		0			0
$651$		0			0
$652$		0			0
$653$			38.7480i			1.51633i	0.652064	+	0.758164i	$0.273903\pi$
$653$			38.7480i			1.51633i	−0.652064	+	0.758164i	$0.726097\pi$
$654$		0			0
$655$	−8.68035	−	2.15676i	−0.339169	−	0.0842714i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.22076			−0.359190			−0.179595	−	0.983741i	$-0.557479\pi$
$659$	−9.22076			−0.359190			−0.179595	+	0.983741i	$0.557479\pi$
$660$		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
$663$		0			0
$664$		0			0
$665$	−2.34017	+	9.41855i	−0.0907480	+	0.365236i
$666$		0			0
$667$		−	56.0288i		−	2.16944i
$668$		0			0
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$			33.5897i			1.29096i	0.763779	+	0.645478i	$0.223342\pi$
$677$			33.5897i			1.29096i	−0.763779	+	0.645478i	$0.776658\pi$
$678$		0			0
$679$	−10.4969			−0.402835
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	18.7070i		−	0.715804i	−0.933759	−	0.357902i	$-0.883492\pi$
$683$		−	18.7070i		−	0.715804i	0.933759	−	0.357902i	$-0.116508\pi$
$684$		0			0
$685$	−4.66597	+	18.7792i	−0.178278	+	0.717518i
$686$		0			0
$687$		0			0
$688$		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
$693$		0			0
$694$		0			0
$695$	−13.4186	−	3.33403i	−0.508995	−	0.126467i
$696$		0			0
$697$		−	8.99386i		−	0.340667i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21.4641			−0.810689			−0.405344	−	0.914164i	$-0.632848\pi$
$701$	−21.4641			−0.810689			−0.405344	+	0.914164i	$0.632848\pi$
$702$		0			0
$703$			37.6742i			1.42091i
$704$		0			0
$705$		0			0
$706$		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			0
$712$		0			0
$713$			27.5174i			1.03054i
$714$		0			0
$715$	−12.6803	+	51.0349i	−0.474218	+	1.90860i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.36683			0.162855			0.0814277	−	0.996679i	$-0.474052\pi$
$719$	4.36683			0.162855			0.0814277	+	0.996679i	$0.474052\pi$
$720$		0			0
$721$	10.1568			0.378257
$722$		0			0
$723$		0			0
$724$		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
$729$		0			0
$730$		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			0
$736$		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
$741$		0			0
$742$		0			0
$743$		−	33.6598i		−	1.23486i	−0.786626	−	0.617430i	$-0.788174\pi$
$743$		−	33.6598i		−	1.23486i	0.786626	−	0.617430i	$-0.211826\pi$
$744$		0			0
$745$	−28.6537	−	7.11942i	−1.04979	−	0.260835i
$746$		0			0
$747$		0			0
$748$		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
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$	−19.5974			−0.710406			−0.355203	−	0.934789i	$-0.615588\pi$
$761$	−19.5974			−0.710406			−0.355203	+	0.934789i	$0.615588\pi$
$762$		0			0
$763$			12.8371i			0.464734i
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$		−	12.0845i		−	0.434650i	−0.976099	−	0.217325i	$-0.930267\pi$
$773$		−	12.0845i		−	0.434650i	0.976099	−	0.217325i	$-0.0697331\pi$
$774$		0			0
$775$	19.1773	+	10.1568i	0.688868	+	0.364841i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−36.1978			−1.29692
$780$		0			0
$781$	−76.3956			−2.73365
$782$		0			0
$783$		0			0
$784$		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
$789$		0			0
$790$		0			0
$791$	1.50307			0.0534431
$792$		0			0
$793$		−	19.6332i		−	0.697194i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			21.2762i			0.753641i	0.926286	+	0.376820i	$0.122983\pi$
$797$			21.2762i			0.753641i	−0.926286	+	0.376820i	$0.877017\pi$
$798$		0			0
$799$	−7.37298			−0.260837
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	60.5646i		−	2.13728i
$804$		0			0
$805$	−13.7587	−	3.41855i	−0.484931	−	0.120488i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−49.6619			−1.74602			−0.873010	−	0.487702i	$-0.837835\pi$
$809$	−49.6619			−1.74602			−0.873010	+	0.487702i	$0.837835\pi$
$810$		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
$813$		0			0
$814$		0			0
$815$	−1.52973	+	6.15676i	−0.0535842	+	0.215662i
$816$		0			0
$817$		−	26.7214i		−	0.934863i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−24.8904			−0.868682			−0.434341	−	0.900749i	$-0.643019\pi$
$821$	−24.8904			−0.868682			−0.434341	+	0.900749i	$0.643019\pi$
$822$		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
$825$		0			0
$826$		0			0
$827$		−	21.6886i		−	0.754186i	−0.926175	−	0.377093i	$-0.876924\pi$
$827$		−	21.6886i		−	0.754186i	0.926175	−	0.377093i	$-0.123076\pi$
$828$		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
$831$		0			0
$832$		0			0
$833$		−	1.07838i		−	0.0373636i
$834$		0			0
$835$	7.20394	−	28.9939i	0.249302	−	1.00337i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.4641			−0.395785			−0.197893	−	0.980224i	$-0.563410\pi$
$839$	−11.4641			−0.395785			−0.197893	+	0.980224i	$0.563410\pi$
$840$		0			0
$841$	49.0944			1.69291
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.6670	−	3.14730i	−0.435759	−	0.108270i
$846$		0			0
$847$			18.3607i			0.630881i
$848$		0			0
$849$		0			0
$850$		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
$855$		0			0
$856$		0			0
$857$		−	16.9627i		−	0.579433i	−0.957112	−	0.289717i	$-0.906439\pi$
$857$		−	16.9627i		−	0.579433i	0.957112	−	0.289717i	$-0.0935611\pi$
$858$		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
$861$		0			0
$862$		0			0
$863$		−	50.0554i		−	1.70391i	−0.523618	−	0.851953i	$-0.675418\pi$
$863$		−	50.0554i		−	1.70391i	0.523618	−	0.851953i	$-0.324582\pi$
$864$		0			0
$865$	1.37751	−	5.54411i	0.0468368	−	0.188505i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.68649			0.125056
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		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
$879$		0			0
$880$		0			0
$881$	−20.5380			−0.691942			−0.345971	−	0.938245i	$-0.612450\pi$
$881$	−20.5380			−0.691942			−0.345971	+	0.938245i	$0.612450\pi$
$882$		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
$885$		0			0
$886$		0			0
$887$		−	3.15061i		−	0.105787i	−0.998600	−	0.0528936i	$-0.983156\pi$
$887$		−	3.15061i		−	0.105787i	0.998600	−	0.0528936i	$-0.0168444\pi$
$888$		0			0
$889$	−19.2039			−0.644079
$890$		0			0
$891$		0			0
$892$		0			0
$893$			29.6742i			0.993009i
$894$		0			0
$895$	25.9155	+	6.43907i	0.866259	+	0.215234i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−38.3545			−1.27920
$900$		0			0
$901$	6.66806			0.222145
$902$		0			0
$903$		0			0
$904$		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
$909$		0			0
$910$		0			0
$911$	46.0989			1.52732			0.763662	−	0.645616i	$-0.223399\pi$
$911$	46.0989			1.52732			0.763662	+	0.645616i	$0.223399\pi$
$912$		0			0
$913$		−	37.0472i		−	1.22608i
$914$		0			0
$915$		0			0
$916$		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
$921$		0			0
$922$		0			0
$923$		−	61.1917i		−	2.01415i
$924$		0			0
$925$	−20.3135	+	38.3545i	−0.667904	+	1.26109i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−44.9048			−1.47328			−0.736639	−	0.676286i	$-0.763588\pi$
$929$	−44.9048			−1.47328			−0.736639	+	0.676286i	$0.763588\pi$
$930$		0			0
$931$	−4.34017			−0.142243
$932$		0			0
$933$		0			0
$934$		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
$939$		0			0
$940$		0			0
$941$	−14.6660			−0.478097			−0.239048	−	0.971008i	$-0.576835\pi$
$941$	−14.6660			−0.478097			−0.239048	+	0.971008i	$0.576835\pi$
$942$		0			0
$943$		−	52.8781i		−	1.72195i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			2.96719i			0.0964209i	0.998837	+	0.0482104i	$0.0153518\pi$
$947$			2.96719i			0.0964209i	−0.998837	+	0.0482104i	$0.984648\pi$
$948$		0			0
$949$	48.5113			1.57474
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	45.5851i		−	1.47665i	−0.674446	−	0.738324i	$-0.735618\pi$
$953$		−	45.5851i		−	1.47665i	0.674446	−	0.738324i	$-0.264382\pi$
$954$		0			0
$955$	13.2351	+	3.28846i	0.428279	+	0.106412i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.65368			−0.279442
$960$		0			0
$961$	−12.1629			−0.392352
$962$		0			0
$963$		0			0
$964$		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
$969$		0			0
$970$		0			0
$971$	−11.3197			−0.363265			−0.181632	−	0.983366i	$-0.558138\pi$
$971$	−11.3197			−0.363265			−0.181632	+	0.983366i	$0.558138\pi$
$972$		0			0
$973$		−	6.18342i		−	0.198231i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	39.6886i		−	1.26975i	−0.772615	−	0.634875i	$-0.781052\pi$
$977$		−	39.6886i		−	1.26975i	0.772615	−	0.634875i	$-0.218948\pi$
$978$		0			0
$979$	35.2039			1.12512
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	43.5174i		−	1.38799i	−0.719979	−	0.693996i	$-0.755849\pi$
$983$		−	43.5174i		−	1.38799i	0.719979	−	0.693996i	$-0.244151\pi$
$984$		0			0
$985$	6.39350	−	25.7321i	0.203714	−	0.819892i
$986$		0			0
$987$		0			0
$988$		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
$993$		0			0
$994$		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
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.e.169.1	✓	6	60.59	even	2
840.2.t.e.169.4	yes	6	12.11	even	2
1680.2.t.i.1009.1		6	3.2	odd	2
1680.2.t.i.1009.4		6	15.14	odd	2
2520.2.t.j.1009.5		6	20.19	odd	2
2520.2.t.j.1009.6		6	4.3	odd	2
4200.2.a.bo.1.3		3	60.23	odd	4
4200.2.a.bq.1.3		3	60.47	odd	4
5040.2.t.ba.1009.5		6	5.4	even	2	inner
5040.2.t.ba.1009.6		6	1.1	even	1	trivial
8400.2.a.dh.1.1		3	15.2	even	4
8400.2.a.dk.1.1		3	15.8	even	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:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.t (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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