LMFDB - Embedded newform 1368.3.o.d.721.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,3,Mod(721,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1368.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1368.o (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:	$37.2753001645$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 264 x^{18} + 28274 x^{16} - 1545308 x^{14} + 45358441 x^{12} - 637328868 x^{10} + \cdots + 194396337216 $ x^20 - 264x^18 + 28274x^16 - 1545308x^14 + 45358441x^12 - 637328868x^10 + 1825819356x^8 + 32794262368x^6 + 135580415344x^4 + 245217530816*x^2 + 194396337216
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.1
Root			$8.89263 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	1368.721
Dual form			1368.3.o.d.721.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.89263 q^{5} +4.08883 q^{7} +O(q^{10})$	$q-8.89263 q^{5} +4.08883 q^{7} -4.37788 q^{11} -13.7670i q^{13} +14.0927 q^{17} +(11.6112 - 15.0393i) q^{19} -6.99929 q^{23} +54.0788 q^{25} +31.8337i q^{29} -12.3543i q^{31} -36.3604 q^{35} -4.89820i q^{37} +73.1184i q^{41} -23.8862 q^{43} -43.3597 q^{47} -32.2815 q^{49} -65.4462i q^{53} +38.9309 q^{55} -62.3993i q^{59} -70.1235 q^{61} +122.425i q^{65} +126.526i q^{67} -44.9262i q^{71} -0.118545 q^{73} -17.9004 q^{77} -19.5093i q^{79} -97.8163 q^{83} -125.321 q^{85} +15.6943i q^{89} -56.2911i q^{91} +(-103.254 + 133.739i) q^{95} -6.98703i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 16 q^{7}+O(q^{10}) $ 20 * q - 16 * q^7	$ 20 q - 16 q^{7} - 8 q^{19} + 68 q^{25} - 128 q^{43} + 116 q^{49} + 144 q^{55} - 104 q^{61} - 88 q^{73} - 280 q^{85}+O(q^{100}) $ 20 * q - 16 * q^7 - 8 * q^19 + 68 * q^25 - 128 * q^43 + 116 * q^49 + 144 * q^55 - 104 * q^61 - 88 * q^73 - 280 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−8.89263			−1.77853			−0.889263	−	0.457397i	$-0.848782\pi$
$5$	−8.89263			−1.77853			−0.889263	+	0.457397i	$0.848782\pi$
$6$		0			0
$7$	4.08883			0.584118			0.292059	−	0.956400i	$-0.405660\pi$
$7$	4.08883			0.584118			0.292059	+	0.956400i	$0.405660\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.37788			−0.397989			−0.198995	−	0.980001i	$-0.563768\pi$
$11$	−4.37788			−0.397989			−0.198995	+	0.980001i	$0.563768\pi$
$12$		0			0
$13$		−	13.7670i		−	1.05900i	−0.848309	−	0.529502i	$-0.822379\pi$
$13$		−	13.7670i		−	1.05900i	0.848309	−	0.529502i	$-0.177621\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.0927			0.828981			0.414490	−	0.910054i	$-0.363960\pi$
$17$	14.0927			0.828981			0.414490	+	0.910054i	$0.363960\pi$
$18$		0			0
$19$	11.6112	−	15.0393i	0.611115	−	0.791542i
$19$	11.6112	−	15.0393i	0.611115	−	0.791542i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.99929			−0.304317			−0.152158	−	0.988356i	$-0.548622\pi$
$23$	−6.99929			−0.304317			−0.152158	+	0.988356i	$0.548622\pi$
$24$		0			0
$25$	54.0788			2.16315
$26$		0			0
$27$		0			0
$28$		0			0
$29$			31.8337i			1.09771i	0.835916	+	0.548857i	$0.184937\pi$
$29$			31.8337i			1.09771i	−0.835916	+	0.548857i	$0.815063\pi$
$30$		0			0
$31$		−	12.3543i		−	0.398525i	−0.979946	−	0.199262i	$-0.936145\pi$
$31$		−	12.3543i		−	0.398525i	0.979946	−	0.199262i	$-0.0638546\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−36.3604			−1.03887
$36$		0			0
$37$		−	4.89820i		−	0.132384i	−0.997807	−	0.0661918i	$-0.978915\pi$
$37$		−	4.89820i		−	0.132384i	0.997807	−	0.0661918i	$-0.0210849\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			73.1184i			1.78337i	0.452652	+	0.891687i	$0.350478\pi$
$41$			73.1184i			1.78337i	−0.452652	+	0.891687i	$0.649522\pi$
$42$		0			0
$43$	−23.8862			−0.555492			−0.277746	−	0.960655i	$-0.589587\pi$
$43$	−23.8862			−0.555492			−0.277746	+	0.960655i	$0.589587\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−43.3597			−0.922547			−0.461274	−	0.887258i	$-0.652607\pi$
$47$	−43.3597			−0.922547			−0.461274	+	0.887258i	$0.652607\pi$
$48$		0			0
$49$	−32.2815			−0.658806
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	65.4462i		−	1.23483i	−0.786636	−	0.617417i	$-0.788179\pi$
$53$		−	65.4462i		−	1.23483i	0.786636	−	0.617417i	$-0.211821\pi$
$54$		0			0
$55$	38.9309			0.707834
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	62.3993i		−	1.05761i	−0.848742	−	0.528807i	$-0.822639\pi$
$59$		−	62.3993i		−	1.05761i	0.848742	−	0.528807i	$-0.177361\pi$
$60$		0			0
$61$	−70.1235			−1.14957			−0.574783	−	0.818306i	$-0.694914\pi$
$61$	−70.1235			−1.14957			−0.574783	+	0.818306i	$0.694914\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			122.425i			1.88346i
$66$		0			0
$67$			126.526i			1.88844i	0.329311	+	0.944222i	$0.393184\pi$
$67$			126.526i			1.88844i	−0.329311	+	0.944222i	$0.606816\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	44.9262i		−	0.632763i	−0.948632	−	0.316382i	$-0.897532\pi$
$71$		−	44.9262i		−	0.632763i	0.948632	−	0.316382i	$-0.102468\pi$
$72$		0			0
$73$	−0.118545			−0.00162390			−0.000811950	−	1.00000i	$-0.500258\pi$
$73$	−0.118545			−0.00162390			−0.000811950		1.00000i	$0.500258\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−17.9004			−0.232473
$78$		0			0
$79$		−	19.5093i		−	0.246953i	−0.992347	−	0.123477i	$-0.960596\pi$
$79$		−	19.5093i		−	0.246953i	0.992347	−	0.123477i	$-0.0394044\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−97.8163			−1.17851			−0.589255	−	0.807947i	$-0.700579\pi$
$83$	−97.8163			−1.17851			−0.589255	+	0.807947i	$0.700579\pi$
$84$		0			0
$85$	−125.321			−1.47436
$86$		0			0
$87$		0			0
$88$		0			0
$89$			15.6943i			0.176340i	0.996105	+	0.0881702i	$0.0281019\pi$
$89$			15.6943i			0.176340i	−0.996105	+	0.0881702i	$0.971898\pi$
$90$		0			0
$91$		−	56.2911i		−	0.618583i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−103.254	+	133.739i	−1.08688	+	1.40778i
$96$		0			0
$97$		−	6.98703i		−	0.0720312i	−0.999351	−	0.0360156i	$-0.988533\pi$
$97$		−	6.98703i		−	0.0720312i	0.999351	−	0.0360156i	$-0.0114666\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−75.8791			−0.751279			−0.375639	−	0.926766i	$-0.622577\pi$
$101$	−75.8791			−0.751279			−0.375639	+	0.926766i	$0.622577\pi$
$102$		0			0
$103$			172.570i			1.67544i	0.546100	+	0.837720i	$0.316112\pi$
$103$			172.570i			1.67544i	−0.546100	+	0.837720i	$0.683888\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			169.804i			1.58695i	0.608602	+	0.793475i	$0.291730\pi$
$107$			169.804i			1.58695i	−0.608602	+	0.793475i	$0.708270\pi$
$108$		0			0
$109$			23.5902i			0.216423i	0.994128	+	0.108212i	$0.0345124\pi$
$109$			23.5902i			0.216423i	−0.994128	+	0.108212i	$0.965488\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			68.2272i			0.603781i	0.953343	+	0.301890i	$0.0976177\pi$
$113$			68.2272i			0.603781i	−0.953343	+	0.301890i	$0.902382\pi$
$114$		0			0
$115$	62.2421			0.541235
$116$		0			0
$117$		0			0
$118$		0			0
$119$	57.6225			0.484223
$120$		0			0
$121$	−101.834			−0.841605
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−258.587			−2.06870
$126$		0			0
$127$			4.69781i			0.0369906i	0.999829	+	0.0184953i	$0.00588758\pi$
$127$			4.69781i			0.0369906i	−0.999829	+	0.0184953i	$0.994112\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−24.4766			−0.186844			−0.0934220	−	0.995627i	$-0.529781\pi$
$131$	−24.4766			−0.186844			−0.0934220	+	0.995627i	$0.529781\pi$
$132$		0			0
$133$	47.4761	−	61.4931i	0.356963	−	0.462354i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	66.9886			0.488968			0.244484	−	0.969653i	$-0.421381\pi$
$137$	66.9886			0.488968			0.244484	+	0.969653i	$0.421381\pi$
$138$		0			0
$139$	−182.137			−1.31034			−0.655169	−	0.755482i	$-0.727403\pi$
$139$	−182.137			−1.31034			−0.655169	+	0.755482i	$0.727403\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			60.2705i			0.421472i
$144$		0			0
$145$		−	283.086i		−	1.95231i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−203.480			−1.36564			−0.682818	−	0.730589i	$-0.739246\pi$
$149$	−203.480			−1.36564			−0.682818	+	0.730589i	$0.739246\pi$
$150$		0			0
$151$			225.098i			1.49071i	0.666666	+	0.745357i	$0.267721\pi$
$151$			225.098i			1.49071i	−0.666666	+	0.745357i	$0.732279\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			109.862i			0.708786i
$156$		0			0
$157$	72.8209			0.463827			0.231914	−	0.972736i	$-0.425501\pi$
$157$	72.8209			0.463827			0.231914	+	0.972736i	$0.425501\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−28.6189			−0.177757
$162$		0			0
$163$	274.540			1.68430			0.842148	−	0.539246i	$-0.181291\pi$
$163$	274.540			1.68430			0.842148	+	0.539246i	$0.181291\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			162.610i			0.973711i	0.873483	+	0.486855i	$0.161856\pi$
$167$			162.610i			0.973711i	−0.873483	+	0.486855i	$0.838144\pi$
$168$		0			0
$169$	−20.5314			−0.121488
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	180.593i		−	1.04389i	−0.852979	−	0.521946i	$-0.825206\pi$
$173$		−	180.593i		−	1.04389i	0.852979	−	0.521946i	$-0.174794\pi$
$174$		0			0
$175$	221.119			1.26354
$176$		0			0
$177$		0			0
$178$		0			0
$179$			24.9660i			0.139475i	0.997565	+	0.0697375i	$0.0222162\pi$
$179$			24.9660i			0.139475i	−0.997565	+	0.0697375i	$0.977784\pi$
$180$		0			0
$181$			73.8968i			0.408269i	0.978943	+	0.204135i	$0.0654380\pi$
$181$			73.8968i			0.408269i	−0.978943	+	0.204135i	$0.934562\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			43.5578i			0.235448i
$186$		0			0
$187$	−61.6961			−0.329925
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−199.072			−1.04226			−0.521130	−	0.853477i	$-0.674489\pi$
$191$	−199.072			−1.04226			−0.521130	+	0.853477i	$0.674489\pi$
$192$		0			0
$193$		−	225.689i		−	1.16937i	−0.811259	−	0.584687i	$-0.801217\pi$
$193$		−	225.689i		−	1.16937i	0.811259	−	0.584687i	$-0.198783\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−184.669			−0.937405			−0.468703	−	0.883356i	$-0.655278\pi$
$197$	−184.669			−0.937405			−0.468703	+	0.883356i	$0.655278\pi$
$198$		0			0
$199$	341.471			1.71594			0.857968	−	0.513703i	$-0.171727\pi$
$199$	341.471			1.71594			0.857968	+	0.513703i	$0.171727\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			130.163i			0.641196i
$204$		0			0
$205$		−	650.214i		−	3.17178i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−50.8324	+	65.8403i	−0.243217	+	0.315025i
$210$		0			0
$211$		−	11.7087i		−	0.0554916i	−0.999615	−	0.0277458i	$-0.991167\pi$
$211$		−	11.7087i		−	0.0554916i	0.999615	−	0.0277458i	$-0.00883290\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	212.411			0.987957
$216$		0			0
$217$		−	50.5145i		−	0.232786i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	194.014i		−	0.877893i
$222$		0			0
$223$			107.679i			0.482866i	0.970418	+	0.241433i	$0.0776174\pi$
$223$			107.679i			0.482866i	−0.970418	+	0.241433i	$0.922383\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	231.845i		−	1.02134i	−0.859776	−	0.510671i	$-0.829397\pi$
$227$		−	231.845i		−	1.02134i	0.859776	−	0.510671i	$-0.170603\pi$
$228$		0			0
$229$	−94.0527			−0.410710			−0.205355	−	0.978688i	$-0.565835\pi$
$229$	−94.0527			−0.410710			−0.205355	+	0.978688i	$0.565835\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−64.2586			−0.275788			−0.137894	−	0.990447i	$-0.544033\pi$
$233$	−64.2586			−0.275788			−0.137894	+	0.990447i	$0.544033\pi$
$234$		0			0
$235$	385.582			1.64077
$236$		0			0
$237$		0			0
$238$		0			0
$239$	254.410			1.06448			0.532239	−	0.846594i	$-0.321351\pi$
$239$	254.410			1.06448			0.532239	+	0.846594i	$0.321351\pi$
$240$		0			0
$241$			360.169i			1.49448i	0.664557	+	0.747238i	$0.268620\pi$
$241$			360.169i			1.49448i	−0.664557	+	0.747238i	$0.731380\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	287.067			1.17170
$246$		0			0
$247$	−207.047	−	159.852i	−0.838246	−	0.647172i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	190.633			0.759494			0.379747	−	0.925090i	$-0.376011\pi$
$251$	190.633			0.759494			0.379747	+	0.925090i	$0.376011\pi$
$252$		0			0
$253$	30.6421			0.121115
$254$		0			0
$255$		0			0
$256$		0			0
$257$			26.0030i			0.101179i	0.998720	+	0.0505895i	$0.0161100\pi$
$257$			26.0030i			0.101179i	−0.998720	+	0.0505895i	$0.983890\pi$
$258$		0			0
$259$		−	20.0279i		−	0.0773278i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−187.438			−0.712693			−0.356346	−	0.934354i	$-0.615978\pi$
$263$	−187.438			−0.712693			−0.356346	+	0.934354i	$0.615978\pi$
$264$		0			0
$265$			581.989i			2.19618i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	345.243i		−	1.28343i	−0.766943	−	0.641715i	$-0.778223\pi$
$269$		−	345.243i		−	1.28343i	0.766943	−	0.641715i	$-0.221777\pi$
$270$		0			0
$271$	−196.509			−0.725124			−0.362562	−	0.931960i	$-0.618098\pi$
$271$	−196.509			−0.725124			−0.362562	+	0.931960i	$0.618098\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−236.751			−0.860911
$276$		0			0
$277$	−501.779			−1.81148			−0.905738	−	0.423839i	$-0.860682\pi$
$277$	−501.779			−1.81148			−0.905738	+	0.423839i	$0.860682\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			448.171i			1.59492i	0.603375	+	0.797458i	$0.293822\pi$
$281$			448.171i			1.59492i	−0.603375	+	0.797458i	$0.706178\pi$
$282$		0			0
$283$	−186.965			−0.660655			−0.330328	−	0.943866i	$-0.607159\pi$
$283$	−186.965			−0.660655			−0.330328	+	0.943866i	$0.607159\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			298.969i			1.04170i
$288$		0			0
$289$	−90.3966			−0.312791
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	152.099i		−	0.519111i	−0.965728	−	0.259555i	$-0.916424\pi$
$293$		−	152.099i		−	0.519111i	0.965728	−	0.259555i	$-0.0835760\pi$
$294$		0			0
$295$			554.893i			1.88099i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			96.3595i			0.322273i
$300$		0			0
$301$	−97.6665			−0.324473
$302$		0			0
$303$		0			0
$304$		0			0
$305$	623.582			2.04453
$306$		0			0
$307$		−	223.721i		−	0.728733i	−0.931256	−	0.364367i	$-0.881286\pi$
$307$		−	223.721i		−	0.728733i	0.931256	−	0.364367i	$-0.118714\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−107.817			−0.346678			−0.173339	−	0.984862i	$-0.555456\pi$
$311$	−107.817			−0.346678			−0.173339	+	0.984862i	$0.555456\pi$
$312$		0			0
$313$	−18.9501			−0.0605435			−0.0302718	−	0.999542i	$-0.509637\pi$
$313$	−18.9501			−0.0605435			−0.0302718	+	0.999542i	$0.509637\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			316.377i			0.998034i	0.866592	+	0.499017i	$0.166306\pi$
$317$			316.377i			0.998034i	−0.866592	+	0.499017i	$0.833694\pi$
$318$		0			0
$319$		−	139.364i		−	0.436879i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	163.633	−	211.944i	0.506602	−	0.656173i
$324$		0			0
$325$		−	744.505i		−	2.29079i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−177.291			−0.538877
$330$		0			0
$331$			562.873i			1.70052i	0.526362	+	0.850261i	$0.323556\pi$
$331$			562.873i			1.70052i	−0.526362	+	0.850261i	$0.676444\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1125.15i		−	3.35864i
$336$		0			0
$337$			335.863i			0.996628i	0.866997	+	0.498314i	$0.166047\pi$
$337$			335.863i			0.996628i	−0.866997	+	0.498314i	$0.833953\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			54.0855i			0.158609i
$342$		0			0
$343$	−332.346			−0.968939
$344$		0			0
$345$		0			0
$346$		0			0
$347$	288.891			0.832538			0.416269	−	0.909241i	$-0.363337\pi$
$347$	288.891			0.832538			0.416269	+	0.909241i	$0.363337\pi$
$348$		0			0
$349$	−332.076			−0.951508			−0.475754	−	0.879578i	$-0.657825\pi$
$349$	−332.076			−0.951508			−0.475754	+	0.879578i	$0.657825\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−100.942			−0.285954			−0.142977	−	0.989726i	$-0.545667\pi$
$353$	−100.942			−0.285954			−0.142977	+	0.989726i	$0.545667\pi$
$354$		0			0
$355$			399.512i			1.12539i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−126.478			−0.352306			−0.176153	−	0.984363i	$-0.556365\pi$
$359$	−126.478			−0.352306			−0.176153	+	0.984363i	$0.556365\pi$
$360$		0			0
$361$	−91.3610	−	349.248i	−0.253078	−	0.967446i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.05417			0.00288815
$366$		0			0
$367$	385.079			1.04926			0.524631	−	0.851330i	$-0.324203\pi$
$367$	385.079			1.04926			0.524631	+	0.851330i	$0.324203\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	267.598i		−	0.721290i
$372$		0			0
$373$		−	188.228i		−	0.504632i	−0.967645	−	0.252316i	$-0.918808\pi$
$373$		−	188.228i		−	0.504632i	0.967645	−	0.252316i	$-0.0811922\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	438.256			1.16248
$378$		0			0
$379$			694.899i			1.83351i	0.399453	+	0.916754i	$0.369200\pi$
$379$			694.899i			1.83351i	−0.399453	+	0.916754i	$0.630800\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	496.495i		−	1.29633i	−0.761500	−	0.648165i	$-0.775537\pi$
$383$		−	496.495i		−	1.29633i	0.761500	−	0.648165i	$-0.224463\pi$
$384$		0			0
$385$	159.182			0.413459
$386$		0			0
$387$		0			0
$388$		0			0
$389$	454.909			1.16943			0.584715	−	0.811239i	$-0.301206\pi$
$389$	454.909			1.16943			0.584715	+	0.811239i	$0.301206\pi$
$390$		0			0
$391$	−98.6387			−0.252273
$392$		0			0
$393$		0			0
$394$		0			0
$395$			173.489i			0.439213i
$396$		0			0
$397$	56.0875			0.141278			0.0706392	−	0.997502i	$-0.477496\pi$
$397$	56.0875			0.141278			0.0706392	+	0.997502i	$0.477496\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	21.3922i		−	0.0533470i	−0.999644	−	0.0266735i	$-0.991509\pi$
$401$		−	21.3922i		−	0.0533470i	0.999644	−	0.0266735i	$-0.00849145\pi$
$402$		0			0
$403$	−170.082			−0.422039
$404$		0			0
$405$		0			0
$406$		0			0
$407$			21.4437i			0.0526873i
$408$		0			0
$409$			259.092i			0.633476i	0.948513	+	0.316738i	$0.102588\pi$
$409$			259.092i			0.633476i	−0.948513	+	0.316738i	$0.897412\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	255.140i		−	0.617772i
$414$		0			0
$415$	869.844			2.09601
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−822.708			−1.96350			−0.981752	−	0.190166i	$-0.939097\pi$
$419$	−822.708			−1.96350			−0.981752	+	0.190166i	$0.939097\pi$
$420$		0			0
$421$		−	512.963i		−	1.21844i	−0.793002	−	0.609219i	$-0.791483\pi$
$421$		−	512.963i		−	1.21844i	0.793002	−	0.609219i	$-0.208517\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	762.115			1.79321
$426$		0			0
$427$	−286.723			−0.671483
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	217.995i		−	0.505789i	−0.967494	−	0.252894i	$-0.918617\pi$
$431$		−	217.995i		−	0.505789i	0.967494	−	0.252894i	$-0.0813826\pi$
$432$		0			0
$433$		−	474.721i		−	1.09635i	−0.836363	−	0.548177i	$-0.815322\pi$
$433$		−	474.721i		−	1.09635i	0.836363	−	0.548177i	$-0.184678\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−81.2700	+	105.264i	−0.185973	+	0.240880i
$438$		0			0
$439$		−	610.886i		−	1.39154i	−0.718264	−	0.695770i	$-0.755063\pi$
$439$		−	610.886i		−	1.39154i	0.718264	−	0.695770i	$-0.244937\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	743.578			1.67851			0.839253	−	0.543741i	$-0.182992\pi$
$443$	743.578			1.67851			0.839253	+	0.543741i	$0.182992\pi$
$444$		0			0
$445$		−	139.564i		−	0.313626i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	244.636i		−	0.544846i	−0.962178	−	0.272423i	$-0.912175\pi$
$449$		−	244.636i		−	0.544846i	0.962178	−	0.272423i	$-0.0878249\pi$
$450$		0			0
$451$		−	320.104i		−	0.709764i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			500.576i			1.10017i
$456$		0			0
$457$	456.513			0.998934			0.499467	−	0.866333i	$-0.333529\pi$
$457$	456.513			0.998934			0.499467	+	0.866333i	$0.333529\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	230.954			0.500986			0.250493	−	0.968118i	$-0.419407\pi$
$461$	230.954			0.500986			0.250493	+	0.968118i	$0.419407\pi$
$462$		0			0
$463$	−438.242			−0.946527			−0.473264	−	0.880921i	$-0.656924\pi$
$463$	−438.242			−0.946527			−0.473264	+	0.880921i	$0.656924\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	443.535			0.949753			0.474877	−	0.880052i	$-0.342493\pi$
$467$	443.535			0.949753			0.474877	+	0.880052i	$0.342493\pi$
$468$		0			0
$469$			517.342i			1.10307i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	104.571			0.221080
$474$		0			0
$475$	627.919	−	813.307i	1.32193	−	1.71223i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	263.159			0.549393			0.274696	−	0.961531i	$-0.411423\pi$
$479$	263.159			0.549393			0.274696	+	0.961531i	$0.411423\pi$
$480$		0			0
$481$	−67.4337			−0.140195
$482$		0			0
$483$		0			0
$484$		0			0
$485$			62.1331i			0.128109i
$486$		0			0
$487$			270.643i			0.555735i	0.960619	+	0.277868i	$0.0896276\pi$
$487$			270.643i			0.555735i	−0.960619	+	0.277868i	$0.910372\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	331.509			0.675172			0.337586	−	0.941295i	$-0.390390\pi$
$491$	331.509			0.675172			0.337586	+	0.941295i	$0.390390\pi$
$492$		0			0
$493$			448.622i			0.909985i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	183.696i		−	0.369609i
$498$		0			0
$499$	846.591			1.69658			0.848288	−	0.529536i	$-0.177634\pi$
$499$	846.591			1.69658			0.848288	+	0.529536i	$0.177634\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	714.432			1.42034			0.710171	−	0.704030i	$-0.248618\pi$
$503$	714.432			1.42034			0.710171	+	0.704030i	$0.248618\pi$
$504$		0			0
$505$	674.765			1.33617
$506$		0			0
$507$		0			0
$508$		0			0
$509$			171.981i			0.337880i	0.985626	+	0.168940i	$0.0540345\pi$
$509$			171.981i			0.337880i	−0.985626	+	0.168940i	$0.945966\pi$
$510$		0			0
$511$	−0.484709			−0.000948550
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	1534.60i		−	2.97981i
$516$		0			0
$517$	189.824			0.367164
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	624.308i		−	1.19829i	−0.800641	−	0.599144i	$-0.795508\pi$
$521$		−	624.308i		−	1.19829i	0.800641	−	0.599144i	$-0.204492\pi$
$522$		0			0
$523$			874.036i			1.67120i	0.549340	+	0.835599i	$0.314879\pi$
$523$			874.036i			1.67120i	−0.549340	+	0.835599i	$0.685121\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	174.105i		−	0.330369i
$528$		0			0
$529$	−480.010			−0.907391
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1006.62			1.88860
$534$		0			0
$535$		−	1510.00i		−	2.82243i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	141.324			0.262198
$540$		0			0
$541$	−382.026			−0.706149			−0.353074	−	0.935595i	$-0.614864\pi$
$541$	−382.026			−0.706149			−0.353074	+	0.935595i	$0.614864\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	209.778i		−	0.384915i
$546$		0			0
$547$		−	494.309i		−	0.903672i	−0.892101	−	0.451836i	$-0.850769\pi$
$547$		−	494.309i		−	0.903672i	0.892101	−	0.451836i	$-0.149231\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	478.757	+	369.627i	0.868888	+	0.670830i
$552$		0			0
$553$		−	79.7703i		−	0.144250i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−987.867			−1.77355			−0.886775	−	0.462201i	$-0.847060\pi$
$557$	−987.867			−1.77355			−0.886775	+	0.462201i	$0.847060\pi$
$558$		0			0
$559$			328.842i			0.588268i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			444.761i			0.789984i	0.918685	+	0.394992i	$0.129253\pi$
$563$			444.761i			0.789984i	−0.918685	+	0.394992i	$0.870747\pi$
$564$		0			0
$565$		−	606.719i		−	1.07384i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	760.247i		−	1.33611i	−0.744111	−	0.668056i	$-0.767127\pi$
$569$		−	760.247i		−	1.33611i	0.744111	−	0.668056i	$-0.232873\pi$
$570$		0			0
$571$	587.441			1.02879			0.514397	−	0.857552i	$-0.328016\pi$
$571$	587.441			1.02879			0.514397	+	0.857552i	$0.328016\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−378.513			−0.658284
$576$		0			0
$577$	470.148			0.814814			0.407407	−	0.913247i	$-0.366433\pi$
$577$	470.148			0.814814			0.407407	+	0.913247i	$0.366433\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−399.954			−0.688389
$582$		0			0
$583$			286.516i			0.491451i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−559.569			−0.953269			−0.476634	−	0.879102i	$-0.658143\pi$
$587$	−559.569			−0.953269			−0.476634	+	0.879102i	$0.658143\pi$
$588$		0			0
$589$	−185.799	−	143.448i	−0.315449	−	0.243544i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−923.095			−1.55665			−0.778326	−	0.627860i	$-0.783931\pi$
$593$	−923.095			−1.55665			−0.778326	+	0.627860i	$0.783931\pi$
$594$		0			0
$595$	−512.416			−0.861203
$596$		0			0
$597$		0			0
$598$		0			0
$599$			569.367i			0.950529i	0.879843	+	0.475264i	$0.157648\pi$
$599$			569.367i			0.950529i	−0.879843	+	0.475264i	$0.842352\pi$
$600$		0			0
$601$			575.057i			0.956833i	0.878133	+	0.478417i	$0.158789\pi$
$601$			575.057i			0.956833i	−0.878133	+	0.478417i	$0.841211\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	905.573			1.49682
$606$		0			0
$607$			941.747i			1.55148i	0.631054	+	0.775739i	$0.282623\pi$
$607$			941.747i			1.55148i	−0.631054	+	0.775739i	$0.717377\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			596.935i			0.976981i
$612$		0			0
$613$	−835.577			−1.36309			−0.681547	−	0.731774i	$-0.738693\pi$
$613$	−835.577			−1.36309			−0.681547	+	0.731774i	$0.738693\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−725.675			−1.17613			−0.588067	−	0.808812i	$-0.700111\pi$
$617$	−725.675			−1.17613			−0.588067	+	0.808812i	$0.700111\pi$
$618$		0			0
$619$	−539.760			−0.871988			−0.435994	−	0.899950i	$-0.643603\pi$
$619$	−539.760			−0.871988			−0.435994	+	0.899950i	$0.643603\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			64.1713i			0.103004i
$624$		0			0
$625$	947.548			1.51608
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	69.0287i		−	0.109744i
$630$		0			0
$631$	−1166.75			−1.84905			−0.924524	−	0.381123i	$-0.875537\pi$
$631$	−1166.75			−1.84905			−0.924524	+	0.381123i	$0.875537\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	41.7759i		−	0.0657888i
$636$		0			0
$637$			444.420i			0.697677i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1000.68i		−	1.56112i	−0.625083	−	0.780558i	$-0.714935\pi$
$641$		−	1000.68i		−	1.56112i	0.625083	−	0.780558i	$-0.285065\pi$
$642$		0			0
$643$	−857.488			−1.33357			−0.666787	−	0.745248i	$-0.732331\pi$
$643$	−857.488			−1.33357			−0.666787	+	0.745248i	$0.732331\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−83.5434			−0.129124			−0.0645621	−	0.997914i	$-0.520565\pi$
$647$	−83.5434			−0.129124			−0.0645621	+	0.997914i	$0.520565\pi$
$648$		0			0
$649$			273.177i			0.420919i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	482.731			0.739252			0.369626	−	0.929181i	$-0.379486\pi$
$653$	482.731			0.739252			0.369626	+	0.929181i	$0.379486\pi$
$654$		0			0
$655$	217.661			0.332307
$656$		0			0
$657$		0			0
$658$		0			0
$659$			292.175i			0.443360i	0.975119	+	0.221680i	$0.0711541\pi$
$659$			292.175i			0.443360i	−0.975119	+	0.221680i	$0.928846\pi$
$660$		0			0
$661$		−	524.668i		−	0.793748i	−0.917873	−	0.396874i	$-0.870095\pi$
$661$		−	524.668i		−	0.793748i	0.917873	−	0.396874i	$-0.129905\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−422.188	+	546.835i	−0.634868	+	0.822309i
$666$		0			0
$667$		−	222.813i		−	0.334053i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	306.992			0.457515
$672$		0			0
$673$			933.628i			1.38726i	0.720330	+	0.693632i	$0.243990\pi$
$673$			933.628i			1.38726i	−0.720330	+	0.693632i	$0.756010\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	221.809i		−	0.327635i	−0.986491	−	0.163818i	$-0.947619\pi$
$677$		−	221.809i		−	0.327635i	0.986491	−	0.163818i	$-0.0523809\pi$
$678$		0			0
$679$		−	28.5688i		−	0.0420748i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1042.38i			1.52618i	0.646293	+	0.763090i	$0.276319\pi$
$683$			1042.38i			1.52618i	−0.646293	+	0.763090i	$0.723681\pi$
$684$		0			0
$685$	−595.705			−0.869642
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−901.001			−1.30769
$690$		0			0
$691$	296.453			0.429020			0.214510	−	0.976722i	$-0.431185\pi$
$691$	296.453			0.429020			0.214510	+	0.976722i	$0.431185\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1619.68			2.33047
$696$		0			0
$697$			1030.43i			1.47838i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1000.32			1.42700			0.713499	−	0.700657i	$-0.247110\pi$
$701$	1000.32			1.42700			0.713499	+	0.700657i	$0.247110\pi$
$702$		0			0
$703$	−73.6654	−	56.8738i	−0.104787	−	0.0809016i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−310.257			−0.438836
$708$		0			0
$709$	−198.456			−0.279909			−0.139955	−	0.990158i	$-0.544696\pi$
$709$	−198.456			−0.279909			−0.139955	+	0.990158i	$0.544696\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			86.4711i			0.121278i
$714$		0			0
$715$		−	535.963i		−	0.749598i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−648.586			−0.902067			−0.451033	−	0.892507i	$-0.648944\pi$
$719$	−648.586			−0.902067			−0.451033	+	0.892507i	$0.648944\pi$
$720$		0			0
$721$			705.611i			0.978656i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			1721.53i			2.37452i
$726$		0			0
$727$	−1282.65			−1.76431			−0.882155	−	0.470960i	$-0.843908\pi$
$727$	−1282.65			−1.76431			−0.882155	+	0.470960i	$0.843908\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−336.620			−0.460492
$732$		0			0
$733$	892.888			1.21813			0.609064	−	0.793121i	$-0.291545\pi$
$733$	892.888			1.21813			0.609064	+	0.793121i	$0.291545\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	553.915i		−	0.751580i
$738$		0			0
$739$	−1150.31			−1.55658			−0.778290	−	0.627905i	$-0.783913\pi$
$739$	−1150.31			−1.55658			−0.778290	+	0.627905i	$0.783913\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			436.802i			0.587889i	0.955822	+	0.293945i	$0.0949681\pi$
$743$			436.802i			0.587889i	−0.955822	+	0.293945i	$0.905032\pi$
$744$		0			0
$745$	1809.47			2.42882
$746$		0			0
$747$		0			0
$748$		0			0
$749$			694.298i			0.926967i
$750$		0			0
$751$		−	491.952i		−	0.655063i	−0.944840	−	0.327531i	$-0.893783\pi$
$751$		−	491.952i		−	0.655063i	0.944840	−	0.327531i	$-0.106217\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	2001.71i		−	2.65127i
$756$		0			0
$757$	−1120.04			−1.47958			−0.739789	−	0.672839i	$-0.765075\pi$
$757$	−1120.04			−1.47958			−0.739789	+	0.672839i	$0.765075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−235.547			−0.309523			−0.154761	−	0.987952i	$-0.549461\pi$
$761$	−235.547			−0.309523			−0.154761	+	0.987952i	$0.549461\pi$
$762$		0			0
$763$			96.4561i			0.126417i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−859.053			−1.12002
$768$		0			0
$769$	558.287			0.725991			0.362996	−	0.931791i	$-0.381754\pi$
$769$	558.287			0.725991			0.362996	+	0.931791i	$0.381754\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	964.213i		−	1.24737i	−0.781678	−	0.623683i	$-0.785636\pi$
$773$		−	964.213i		−	1.24737i	0.781678	−	0.623683i	$-0.214364\pi$
$774$		0			0
$775$		−	668.104i		−	0.862070i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1099.65	+	848.991i	1.41162	+	1.08985i
$780$		0			0
$781$			196.682i			0.251833i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−647.569			−0.824928
$786$		0			0
$787$		−	734.077i		−	0.932754i	−0.884586	−	0.466377i	$-0.845559\pi$
$787$		−	734.077i		−	0.932754i	0.884586	−	0.466377i	$-0.154441\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			278.969i			0.352680i
$792$		0			0
$793$			965.393i			1.21739i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1208.84i			1.51673i	0.651827	+	0.758367i	$0.274003\pi$
$797$			1208.84i			1.51673i	−0.651827	+	0.758367i	$0.725997\pi$
$798$		0			0
$799$	−611.054			−0.764774
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.518975			0.000646295
$804$		0			0
$805$	254.497			0.316146
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1294.97			−1.60070			−0.800349	−	0.599534i	$-0.795353\pi$
$809$	−1294.97			−1.60070			−0.800349	+	0.599534i	$0.795353\pi$
$810$		0			0
$811$			698.570i			0.861369i	0.902503	+	0.430685i	$0.141728\pi$
$811$			698.570i			0.861369i	−0.902503	+	0.430685i	$0.858272\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2441.38			−2.99556
$816$		0			0
$817$	−277.347	+	359.231i	−0.339470	+	0.439695i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−642.039			−0.782021			−0.391011	−	0.920386i	$-0.627874\pi$
$821$	−642.039			−0.782021			−0.391011	+	0.920386i	$0.627874\pi$
$822$		0			0
$823$	808.191			0.982006			0.491003	−	0.871158i	$-0.336630\pi$
$823$	808.191			0.982006			0.491003	+	0.871158i	$0.336630\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1092.17i			1.32064i	0.750983	+	0.660321i	$0.229580\pi$
$827$			1092.17i			1.32064i	−0.750983	+	0.660321i	$0.770420\pi$
$828$		0			0
$829$		−	552.397i		−	0.666341i	−0.942867	−	0.333170i	$-0.891882\pi$
$829$		−	552.397i		−	0.666341i	0.942867	−	0.333170i	$-0.108118\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−454.932			−0.546137
$834$		0			0
$835$		−	1446.03i		−	1.73177i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			308.616i			0.367838i	0.982941	+	0.183919i	$0.0588784\pi$
$839$			308.616i			0.367838i	−0.982941	+	0.183919i	$0.941122\pi$
$840$		0			0
$841$	−172.387			−0.204978
$842$		0			0
$843$		0			0
$844$		0			0
$845$	182.578			0.216069
$846$		0			0
$847$	−416.382			−0.491597
$848$		0			0
$849$		0			0
$850$		0			0
$851$			34.2839i			0.0402866i
$852$		0			0
$853$	513.281			0.601737			0.300868	−	0.953666i	$-0.402724\pi$
$853$	513.281			0.601737			0.300868	+	0.953666i	$0.402724\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	902.528i		−	1.05312i	−0.850137	−	0.526562i	$-0.823481\pi$
$857$		−	902.528i		−	1.05312i	0.850137	−	0.526562i	$-0.176519\pi$
$858$		0			0
$859$	279.345			0.325197			0.162599	−	0.986692i	$-0.448012\pi$
$859$	279.345			0.325197			0.162599	+	0.986692i	$0.448012\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	974.538i		−	1.12924i	−0.825349	−	0.564622i	$-0.809022\pi$
$863$		−	974.538i		−	1.12924i	0.825349	−	0.564622i	$-0.190978\pi$
$864$		0			0
$865$			1605.95i			1.85659i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			85.4095i			0.0982848i
$870$		0			0
$871$	1741.88			1.99987
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1057.32			−1.20836
$876$		0			0
$877$		−	1355.21i		−	1.54527i	−0.634848	−	0.772637i	$-0.718937\pi$
$877$		−	1355.21i		−	1.54527i	0.634848	−	0.772637i	$-0.281063\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1598.89			1.81486			0.907431	−	0.420200i	$-0.138040\pi$
$881$	1598.89			1.81486			0.907431	+	0.420200i	$0.138040\pi$
$882$		0			0
$883$	−298.404			−0.337944			−0.168972	−	0.985621i	$-0.554045\pi$
$883$	−298.404			−0.337944			−0.168972	+	0.985621i	$0.554045\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			277.416i			0.312757i	0.987697	+	0.156379i	$0.0499820\pi$
$887$			277.416i			0.312757i	−0.987697	+	0.156379i	$0.950018\pi$
$888$		0			0
$889$			19.2085i			0.0216069i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−503.458	+	652.100i	−0.563782	+	0.730235i
$894$		0			0
$895$		−	222.014i		−	0.248060i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	393.282			0.437467
$900$		0			0
$901$		−	922.312i		−	1.02365i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	657.136i		−	0.726117i
$906$		0			0
$907$		−	588.225i		−	0.648539i	−0.945965	−	0.324270i	$-0.894881\pi$
$907$		−	588.225i		−	0.648539i	0.945965	−	0.324270i	$-0.105119\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	456.868i		−	0.501502i	−0.968052	−	0.250751i	$-0.919323\pi$
$911$		−	456.868i		−	0.501502i	0.968052	−	0.250751i	$-0.0806775\pi$
$912$		0			0
$913$	428.228			0.469034
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−100.080			−0.109139
$918$		0			0
$919$	−3.98423			−0.00433540			−0.00216770	−	0.999998i	$-0.500690\pi$
$919$	−3.98423			−0.00433540			−0.00216770	+	0.999998i	$0.500690\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−618.501			−0.670099
$924$		0			0
$925$		−	264.889i		−	0.286366i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	742.264			0.798993			0.399496	−	0.916735i	$-0.369185\pi$
$929$	742.264			0.798993			0.399496	+	0.916735i	$0.369185\pi$
$930$		0			0
$931$	−374.826	+	485.491i	−0.402606	+	0.521472i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	548.640			0.586781
$936$		0			0
$937$	415.097			0.443006			0.221503	−	0.975160i	$-0.428904\pi$
$937$	415.097			0.443006			0.221503	+	0.975160i	$0.428904\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			883.401i			0.938790i	0.882988	+	0.469395i	$0.155528\pi$
$941$			883.401i			0.938790i	−0.882988	+	0.469395i	$0.844472\pi$
$942$		0			0
$943$		−	511.777i		−	0.542711i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	465.843			0.491915			0.245957	−	0.969281i	$-0.420898\pi$
$947$	465.843			0.491915			0.245957	+	0.969281i	$0.420898\pi$
$948$		0			0
$949$			1.63201i			0.00171972i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			479.320i			0.502959i	0.967863	+	0.251479i	$0.0809171\pi$
$953$			479.320i			0.502959i	−0.967863	+	0.251479i	$0.919083\pi$
$954$		0			0
$955$	1770.27			1.85369
$956$		0			0
$957$		0			0
$958$		0			0
$959$	273.905			0.285615
$960$		0			0
$961$	808.372			0.841178
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2006.97i			2.07976i
$966$		0			0
$967$	48.0426			0.0496821			0.0248411	−	0.999691i	$-0.492092\pi$
$967$	48.0426			0.0496821			0.0248411	+	0.999691i	$0.492092\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			954.276i			0.982776i	0.870941	+	0.491388i	$0.163510\pi$
$971$			954.276i			0.982776i	−0.870941	+	0.491388i	$0.836490\pi$
$972$		0			0
$973$	−744.727			−0.765393
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	230.509i		−	0.235935i	−0.993017	−	0.117968i	$-0.962362\pi$
$977$		−	230.509i		−	0.235935i	0.993017	−	0.117968i	$-0.0376379\pi$
$978$		0			0
$979$		−	68.7078i		−	0.0701816i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			525.483i			0.534571i	0.963617	+	0.267285i	$0.0861266\pi$
$983$			525.483i			0.534571i	−0.963617	+	0.267285i	$0.913873\pi$
$984$		0			0
$985$	1642.19			1.66720
$986$		0			0
$987$		0			0
$988$		0			0
$989$	167.186			0.169046
$990$		0			0
$991$			988.597i			0.997575i	0.866724	+	0.498787i	$0.166221\pi$
$991$			988.597i			0.997575i	−0.866724	+	0.498787i	$0.833779\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3036.58			−3.05184
$996$		0			0
$997$	307.985			0.308912			0.154456	−	0.988000i	$-0.450638\pi$
$997$	307.985			0.308912			0.154456	+	0.988000i	$0.450638\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.3.o.d.721.1	✓	20
3.2	odd	2	inner	1368.3.o.d.721.19	yes	20
4.3	odd	2		2736.3.o.q.721.1		20
12.11	even	2		2736.3.o.q.721.19		20
19.18	odd	2	inner	1368.3.o.d.721.2	yes	20
57.56	even	2	inner	1368.3.o.d.721.20	yes	20
76.75	even	2		2736.3.o.q.721.2		20
228.227	odd	2		2736.3.o.q.721.20		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.3.o.d.721.1	✓	20	1.1	even	1	trivial
1368.3.o.d.721.2	yes	20	19.18	odd	2	inner
1368.3.o.d.721.19	yes	20	3.2	odd	2	inner
1368.3.o.d.721.20	yes	20	57.56	even	2	inner
2736.3.o.q.721.1		20	4.3	odd	2
2736.3.o.q.721.2		20	76.75	even	2
2736.3.o.q.721.19		20	12.11	even	2
2736.3.o.q.721.20		20	228.227	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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1368.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.89263 q^{5} +4.08883 q^{7} +O(q^{10})\)	\(q-8.89263 q^{5} +4.08883 q^{7} -4.37788 q^{11} -13.7670i q^{13} +14.0927 q^{17} +(11.6112 - 15.0393i) q^{19} -6.99929 q^{23} +54.0788 q^{25} +31.8337i q^{29} -12.3543i q^{31} -36.3604 q^{35} -4.89820i q^{37} +73.1184i q^{41} -23.8862 q^{43} -43.3597 q^{47} -32.2815 q^{49} -65.4462i q^{53} +38.9309 q^{55} -62.3993i q^{59} -70.1235 q^{61} +122.425i q^{65} +126.526i q^{67} -44.9262i q^{71} -0.118545 q^{73} -17.9004 q^{77} -19.5093i q^{79} -97.8163 q^{83} -125.321 q^{85} +15.6943i q^{89} -56.2911i q^{91} +(-103.254 + 133.739i) q^{95} -6.98703i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 16 q^{7}+O(q^{10}) \) 20 * q - 16 * q^7	\( 20 q - 16 q^{7} - 8 q^{19} + 68 q^{25} - 128 q^{43} + 116 q^{49} + 144 q^{55} - 104 q^{61} - 88 q^{73} - 280 q^{85}+O(q^{100}) \) 20 * q - 16 * q^7 - 8 * q^19 + 68 * q^25 - 128 * q^43 + 116 * q^49 + 144 * q^55 - 104 * q^61 - 88 * q^73 - 280 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more