LMFDB - Embedded newform 2736.3.o.r.721.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, 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("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (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:	$74.5506003290$
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} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + \cdots + 36100 $ x^20 - 4x^19 + 80x^18 - 152x^17 + 4326x^16 - 10096x^15 + 70116x^14 - 93436x^13 + 597327x^12 - 837564x^11 + 2838686x^10 - 2898276x^9 + 7652658x^8 - 6988600x^7 + 12843126x^6 - 6054124x^5 + 7434909x^4 - 621764x^3 + 3099194x^2 - 329080*x + 36100
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{37} $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.18
Root			$-0.330374 + 0.572225i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.r.721.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.79518 q^{5} -9.11667 q^{7} +O(q^{10})$	$q+7.79518 q^{5} -9.11667 q^{7} +18.2215 q^{11} +1.61213i q^{13} +10.5905 q^{17} +(-3.02719 + 18.7573i) q^{19} -33.7879 q^{23} +35.7648 q^{25} +53.8674i q^{29} +54.3737i q^{31} -71.0661 q^{35} -20.2750i q^{37} -13.2653i q^{41} -45.1340 q^{43} -28.6535 q^{47} +34.1137 q^{49} +8.24155i q^{53} +142.040 q^{55} -73.0524i q^{59} +71.0212 q^{61} +12.5668i q^{65} -85.4984i q^{67} +113.658i q^{71} -68.1028 q^{73} -166.120 q^{77} +99.4367i q^{79} +133.816 q^{83} +82.5544 q^{85} +89.2673i q^{89} -14.6973i q^{91} +(-23.5975 + 146.216i) q^{95} +85.7792i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{7}+O(q^{10}) $ 20 * q + 16 * q^7	$ 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) $ 20 * q + 16 * q^7 + 16 * q^11 - 32 * q^17 - 40 * q^19 + 64 * q^23 + 68 * q^25 - 208 * q^35 - 64 * q^43 + 48 * q^47 + 20 * q^49 + 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 + 224 * q^83 - 136 * q^85 - 320 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	7.79518			1.55904			0.779518	−	0.626380i	$-0.215464\pi$
$5$	7.79518			1.55904			0.779518	+	0.626380i	$0.215464\pi$
$6$		0			0
$7$	−9.11667			−1.30238			−0.651191	−	0.758914i	$-0.725730\pi$
$7$	−9.11667			−1.30238			−0.651191	+	0.758914i	$0.725730\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18.2215			1.65650			0.828251	−	0.560357i	$-0.189336\pi$
$11$	18.2215			1.65650			0.828251	+	0.560357i	$0.189336\pi$
$12$		0			0
$13$			1.61213i			0.124010i	0.998076	+	0.0620050i	$0.0197495\pi$
$13$			1.61213i			0.124010i	−0.998076	+	0.0620050i	$0.980251\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	10.5905			0.622968			0.311484	−	0.950251i	$-0.399174\pi$
$17$	10.5905			0.622968			0.311484	+	0.950251i	$0.399174\pi$
$18$		0			0
$19$	−3.02719	+	18.7573i	−0.159326	+	0.987226i
$19$	−3.02719	+	18.7573i	−0.159326	+	0.987226i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−33.7879			−1.46904			−0.734519	−	0.678589i	$-0.762592\pi$
$23$	−33.7879			−1.46904			−0.734519	+	0.678589i	$0.762592\pi$
$24$		0			0
$25$	35.7648			1.43059
$26$		0			0
$27$		0			0
$28$		0			0
$29$			53.8674i			1.85750i	0.370712	+	0.928748i	$0.379114\pi$
$29$			53.8674i			1.85750i	−0.370712	+	0.928748i	$0.620886\pi$
$30$		0			0
$31$			54.3737i			1.75399i	0.480499	+	0.876995i	$0.340456\pi$
$31$			54.3737i			1.75399i	−0.480499	+	0.876995i	$0.659544\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−71.0661			−2.03046
$36$		0			0
$37$		−	20.2750i		−	0.547974i	−0.961733	−	0.273987i	$-0.911657\pi$
$37$		−	20.2750i		−	0.547974i	0.961733	−	0.273987i	$-0.0883425\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	13.2653i		−	0.323544i	−0.986828	−	0.161772i	$-0.948279\pi$
$41$		−	13.2653i		−	0.323544i	0.986828	−	0.161772i	$-0.0517209\pi$
$42$		0			0
$43$	−45.1340			−1.04963			−0.524814	−	0.851217i	$-0.675865\pi$
$43$	−45.1340			−1.04963			−0.524814	+	0.851217i	$0.675865\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−28.6535			−0.609649			−0.304824	−	0.952409i	$-0.598598\pi$
$47$	−28.6535			−0.609649			−0.304824	+	0.952409i	$0.598598\pi$
$48$		0			0
$49$	34.1137			0.696199
$50$		0			0
$51$		0			0
$52$		0			0
$53$			8.24155i			0.155501i	0.996973	+	0.0777505i	$0.0247738\pi$
$53$			8.24155i			0.155501i	−0.996973	+	0.0777505i	$0.975226\pi$
$54$		0			0
$55$	142.040			2.58255
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	73.0524i		−	1.23818i	−0.785322	−	0.619088i	$-0.787502\pi$
$59$		−	73.0524i		−	1.23818i	0.785322	−	0.619088i	$-0.212498\pi$
$60$		0			0
$61$	71.0212			1.16428			0.582141	−	0.813088i	$-0.302215\pi$
$61$	71.0212			1.16428			0.582141	+	0.813088i	$0.302215\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			12.5668i			0.193336i
$66$		0			0
$67$		−	85.4984i		−	1.27610i	−0.769997	−	0.638048i	$-0.779742\pi$
$67$		−	85.4984i		−	1.27610i	0.769997	−	0.638048i	$-0.220258\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			113.658i			1.60082i	0.599453	+	0.800410i	$0.295385\pi$
$71$			113.658i			1.60082i	−0.599453	+	0.800410i	$0.704615\pi$
$72$		0			0
$73$	−68.1028			−0.932916			−0.466458	−	0.884543i	$-0.654470\pi$
$73$	−68.1028			−0.932916			−0.466458	+	0.884543i	$0.654470\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−166.120			−2.15740
$78$		0			0
$79$			99.4367i			1.25869i	0.777125	+	0.629346i	$0.216677\pi$
$79$			99.4367i			1.25869i	−0.777125	+	0.629346i	$0.783323\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	133.816			1.61224			0.806118	−	0.591755i	$-0.201565\pi$
$83$	133.816			1.61224			0.806118	+	0.591755i	$0.201565\pi$
$84$		0			0
$85$	82.5544			0.971229
$86$		0			0
$87$		0			0
$88$		0			0
$89$			89.2673i			1.00300i	0.865157	+	0.501502i	$0.167219\pi$
$89$			89.2673i			1.00300i	−0.865157	+	0.501502i	$0.832781\pi$
$90$		0			0
$91$		−	14.6973i		−	0.161508i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−23.5975	+	146.216i	−0.248395	+	1.53912i
$96$		0			0
$97$			85.7792i			0.884322i	0.896936	+	0.442161i	$0.145788\pi$
$97$			85.7792i			0.884322i	−0.896936	+	0.442161i	$0.854212\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	51.0106			0.505056			0.252528	−	0.967590i	$-0.418738\pi$
$101$	51.0106			0.505056			0.252528	+	0.967590i	$0.418738\pi$
$102$		0			0
$103$			108.423i			1.05265i	0.850284	+	0.526325i	$0.176430\pi$
$103$			108.423i			1.05265i	−0.850284	+	0.526325i	$0.823570\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	135.501i		−	1.26637i	−0.774001	−	0.633184i	$-0.781748\pi$
$107$		−	135.501i		−	1.26637i	0.774001	−	0.633184i	$-0.218252\pi$
$108$		0			0
$109$		−	21.5235i		−	0.197464i	−0.995114	−	0.0987318i	$-0.968521\pi$
$109$		−	21.5235i		−	0.197464i	0.995114	−	0.0987318i	$-0.0314786\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.6752i		−	0.112170i	−0.998426	−	0.0560850i	$-0.982138\pi$
$113$		−	12.6752i		−	0.112170i	0.998426	−	0.0560850i	$-0.0178618\pi$
$114$		0			0
$115$	−263.382			−2.29028
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−96.5497			−0.811342
$120$		0			0
$121$	211.024			1.74400
$122$		0			0
$123$		0			0
$124$		0			0
$125$	83.9133			0.671306
$126$		0			0
$127$		−	59.5553i		−	0.468939i	−0.972123	−	0.234470i	$-0.924665\pi$
$127$		−	59.5553i		−	0.468939i	0.972123	−	0.234470i	$-0.0753353\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−123.518			−0.942888			−0.471444	−	0.881896i	$-0.656267\pi$
$131$	−123.518			−0.942888			−0.471444	+	0.881896i	$0.656267\pi$
$132$		0			0
$133$	27.5979	−	171.004i	0.207503	−	1.28575i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−145.804			−1.06427			−0.532133	−	0.846661i	$-0.678609\pi$
$137$	−145.804			−1.06427			−0.532133	+	0.846661i	$0.678609\pi$
$138$		0			0
$139$	7.81824			0.0562463			0.0281232	−	0.999604i	$-0.491047\pi$
$139$	7.81824			0.0562463			0.0281232	+	0.999604i	$0.491047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			29.3755i			0.205423i
$144$		0			0
$145$			419.906i			2.89590i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	220.138			1.47743			0.738717	−	0.674016i	$-0.235432\pi$
$149$	220.138			1.47743			0.738717	+	0.674016i	$0.235432\pi$
$150$		0			0
$151$		−	118.430i		−	0.784308i	−0.919900	−	0.392154i	$-0.871730\pi$
$151$		−	118.430i		−	0.784308i	0.919900	−	0.392154i	$-0.128270\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			423.853i			2.73453i
$156$		0			0
$157$	97.9568			0.623929			0.311964	−	0.950094i	$-0.399013\pi$
$157$	97.9568			0.623929			0.311964	+	0.950094i	$0.399013\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	308.033			1.91325
$162$		0			0
$163$	−86.9716			−0.533568			−0.266784	−	0.963756i	$-0.585961\pi$
$163$	−86.9716			−0.533568			−0.266784	+	0.963756i	$0.585961\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			226.279i			1.35496i	0.735541	+	0.677481i	$0.236928\pi$
$167$			226.279i			1.35496i	−0.735541	+	0.677481i	$0.763072\pi$
$168$		0			0
$169$	166.401			0.984622
$170$		0			0
$171$		0			0
$172$		0			0
$173$			34.1937i			0.197652i	0.995105	+	0.0988258i	$0.0315087\pi$
$173$			34.1937i			0.197652i	−0.995105	+	0.0988258i	$0.968491\pi$
$174$		0			0
$175$	−326.056			−1.86318
$176$		0			0
$177$		0			0
$178$		0			0
$179$			302.014i			1.68723i	0.536951	+	0.843613i	$0.319576\pi$
$179$			302.014i			1.68723i	−0.536951	+	0.843613i	$0.680424\pi$
$180$		0			0
$181$		−	36.1103i		−	0.199504i	−0.995012	−	0.0997522i	$-0.968195\pi$
$181$		−	36.1103i		−	0.199504i	0.995012	−	0.0997522i	$-0.0318050\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	158.047i		−	0.854311i
$186$		0			0
$187$	192.974			1.03195
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−68.1599			−0.356858			−0.178429	−	0.983953i	$-0.557101\pi$
$191$	−68.1599			−0.356858			−0.178429	+	0.983953i	$0.557101\pi$
$192$		0			0
$193$			171.070i			0.886372i	0.896430	+	0.443186i	$0.146152\pi$
$193$			171.070i			0.886372i	−0.896430	+	0.443186i	$0.853848\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−71.7597			−0.364263			−0.182131	−	0.983274i	$-0.558300\pi$
$197$	−71.7597			−0.364263			−0.182131	+	0.983274i	$0.558300\pi$
$198$		0			0
$199$	266.869			1.34105			0.670525	−	0.741887i	$-0.266069\pi$
$199$	266.869			1.34105			0.670525	+	0.741887i	$0.266069\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	491.091i		−	2.41917i
$204$		0			0
$205$		−	103.405i		−	0.504416i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−55.1601	+	341.787i	−0.263924	+	1.63534i
$210$		0			0
$211$			1.77156i			0.00839602i	0.999991	+	0.00419801i	$0.00133627\pi$
$211$			1.77156i			0.00839602i	−0.999991	+	0.00419801i	$0.998664\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−351.827			−1.63641
$216$		0			0
$217$		−	495.707i		−	2.28437i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			17.0732i			0.0772542i
$222$		0			0
$223$		−	42.4239i		−	0.190242i	−0.995466	−	0.0951210i	$-0.969676\pi$
$223$		−	42.4239i		−	0.190242i	0.995466	−	0.0951210i	$-0.0303238\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	147.370i		−	0.649205i	−0.945851	−	0.324602i	$-0.894769\pi$
$227$		−	147.370i		−	0.649205i	0.945851	−	0.324602i	$-0.105231\pi$
$228$		0			0
$229$	−216.495			−0.945391			−0.472696	−	0.881226i	$-0.656719\pi$
$229$	−216.495			−0.945391			−0.472696	+	0.881226i	$0.656719\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−312.649			−1.34184			−0.670920	−	0.741530i	$-0.734101\pi$
$233$	−312.649			−1.34184			−0.670920	+	0.741530i	$0.734101\pi$
$234$		0			0
$235$	−223.359			−0.950464
$236$		0			0
$237$		0			0
$238$		0			0
$239$	243.162			1.01741			0.508707	−	0.860940i	$-0.330124\pi$
$239$	243.162			1.01741			0.508707	+	0.860940i	$0.330124\pi$
$240$		0			0
$241$			58.2062i			0.241519i	0.992682	+	0.120760i	$0.0385331\pi$
$241$			58.2062i			0.241519i	−0.992682	+	0.120760i	$0.961467\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	265.923			1.08540
$246$		0			0
$247$	−30.2392	−	4.88023i	−0.122426	−	0.0197580i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	211.136			0.841181			0.420590	−	0.907251i	$-0.361823\pi$
$251$	211.136			0.841181			0.420590	+	0.907251i	$0.361823\pi$
$252$		0			0
$253$	−615.666			−2.43346
$254$		0			0
$255$		0			0
$256$		0			0
$257$			435.639i			1.69509i	0.530721	+	0.847547i	$0.321921\pi$
$257$			435.639i			1.69509i	−0.530721	+	0.847547i	$0.678079\pi$
$258$		0			0
$259$			184.841i			0.713671i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	158.502			0.602670			0.301335	−	0.953518i	$-0.402568\pi$
$263$	158.502			0.602670			0.301335	+	0.953518i	$0.402568\pi$
$264$		0			0
$265$			64.2444i			0.242432i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	39.3715i		−	0.146362i	−0.997319	−	0.0731811i	$-0.976685\pi$
$269$		−	39.3715i		−	0.146362i	0.997319	−	0.0731811i	$-0.0233151\pi$
$270$		0			0
$271$	−152.209			−0.561658			−0.280829	−	0.959758i	$-0.590609\pi$
$271$	−152.209			−0.561658			−0.280829	+	0.959758i	$0.590609\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	651.689			2.36978
$276$		0			0
$277$	−6.52424			−0.0235532			−0.0117766	−	0.999931i	$-0.503749\pi$
$277$	−6.52424			−0.0235532			−0.0117766	+	0.999931i	$0.503749\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	527.885i		−	1.87859i	−0.343106	−	0.939297i	$-0.611479\pi$
$281$		−	527.885i		−	1.87859i	0.343106	−	0.939297i	$-0.388521\pi$
$282$		0			0
$283$	40.5272			0.143205			0.0716027	−	0.997433i	$-0.477189\pi$
$283$	40.5272			0.143205			0.0716027	+	0.997433i	$0.477189\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			120.935i			0.421378i
$288$		0			0
$289$	−176.842			−0.611911
$290$		0			0
$291$		0			0
$292$		0			0
$293$			92.3726i			0.315265i	0.987498	+	0.157632i	$0.0503861\pi$
$293$			92.3726i			0.315265i	−0.987498	+	0.157632i	$0.949614\pi$
$294$		0			0
$295$		−	569.456i		−	1.93036i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	54.4704i		−	0.182175i
$300$		0			0
$301$	411.472			1.36702
$302$		0			0
$303$		0			0
$304$		0			0
$305$	553.623			1.81516
$306$		0			0
$307$			91.0319i			0.296521i	0.988948	+	0.148260i	$0.0473674\pi$
$307$			91.0319i			0.296521i	−0.988948	+	0.148260i	$0.952633\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−72.4384			−0.232921			−0.116460	−	0.993195i	$-0.537155\pi$
$311$	−72.4384			−0.232921			−0.116460	+	0.993195i	$0.537155\pi$
$312$		0			0
$313$	131.269			0.419391			0.209695	−	0.977767i	$-0.432753\pi$
$313$	131.269			0.419391			0.209695	+	0.977767i	$0.432753\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	153.554i		−	0.484398i	−0.970227	−	0.242199i	$-0.922131\pi$
$317$		−	153.554i		−	0.484398i	0.970227	−	0.242199i	$-0.0778686\pi$
$318$		0			0
$319$			981.546i			3.07695i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−32.0593	+	198.648i	−0.0992549	+	0.615010i
$324$		0			0
$325$			57.6575i			0.177408i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	261.225			0.793996
$330$		0			0
$331$			5.50727i			0.0166383i	0.999965	+	0.00831913i	$0.00264809\pi$
$331$			5.50727i			0.0166383i	−0.999965	+	0.00831913i	$0.997352\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	666.475i		−	1.98948i
$336$		0			0
$337$			211.812i			0.628521i	0.949337	+	0.314261i	$0.101757\pi$
$337$			211.812i			0.628521i	−0.949337	+	0.314261i	$0.898243\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			990.772i			2.90549i
$342$		0			0
$343$	135.713			0.395665
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−75.2470			−0.216850			−0.108425	−	0.994105i	$-0.534581\pi$
$347$	−75.2470			−0.216850			−0.108425	+	0.994105i	$0.534581\pi$
$348$		0			0
$349$	−304.278			−0.871856			−0.435928	−	0.899982i	$-0.643580\pi$
$349$	−304.278			−0.871856			−0.435928	+	0.899982i	$0.643580\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−182.048			−0.515716			−0.257858	−	0.966183i	$-0.583017\pi$
$353$	−182.048			−0.515716			−0.257858	+	0.966183i	$0.583017\pi$
$354$		0			0
$355$			885.986i			2.49574i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	339.783			0.946471			0.473236	−	0.880936i	$-0.343086\pi$
$359$	339.783			0.946471			0.473236	+	0.880936i	$0.343086\pi$
$360$		0			0
$361$	−342.672	−	113.564i	−0.949231	−	0.314581i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−530.874			−1.45445
$366$		0			0
$367$	467.305			1.27331			0.636656	−	0.771148i	$-0.280317\pi$
$367$	467.305			1.27331			0.636656	+	0.771148i	$0.280317\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	75.1356i		−	0.202522i
$372$		0			0
$373$			634.590i			1.70131i	0.525722	+	0.850656i	$0.323795\pi$
$373$			634.590i			1.70131i	−0.525722	+	0.850656i	$0.676205\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−86.8412			−0.230348
$378$		0			0
$379$		−	173.785i		−	0.458536i	−0.973363	−	0.229268i	$-0.926367\pi$
$379$		−	173.785i		−	0.458536i	0.973363	−	0.229268i	$-0.0736331\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			427.629i			1.11653i	0.829664	+	0.558263i	$0.188532\pi$
$383$			427.629i			1.11653i	−0.829664	+	0.558263i	$0.811468\pi$
$384$		0			0
$385$	−1294.93			−3.36346
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−349.951			−0.899618			−0.449809	−	0.893125i	$-0.648508\pi$
$389$	−349.951			−0.899618			−0.449809	+	0.893125i	$0.648508\pi$
$390$		0			0
$391$	−357.829			−0.915163
$392$		0			0
$393$		0			0
$394$		0			0
$395$			775.127i			1.96235i
$396$		0			0
$397$	654.552			1.64875			0.824373	−	0.566047i	$-0.191528\pi$
$397$	654.552			1.64875			0.824373	+	0.566047i	$0.191528\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			500.325i			1.24769i	0.781547	+	0.623846i	$0.214431\pi$
$401$			500.325i			1.24769i	−0.781547	+	0.623846i	$0.785569\pi$
$402$		0			0
$403$	−87.6575			−0.217512
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	369.442i		−	0.907720i
$408$		0			0
$409$			360.037i			0.880287i	0.897927	+	0.440143i	$0.145072\pi$
$409$			360.037i			0.880287i	−0.897927	+	0.440143i	$0.854928\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			665.995i			1.61258i
$414$		0			0
$415$	1043.12			2.51353
$416$		0			0
$417$		0			0
$418$		0			0
$419$	303.613			0.724613			0.362307	−	0.932059i	$-0.381989\pi$
$419$	303.613			0.724613			0.362307	+	0.932059i	$0.381989\pi$
$420$		0			0
$421$		−	534.380i		−	1.26931i	−0.772795	−	0.634656i	$-0.781142\pi$
$421$		−	534.380i		−	1.26931i	0.772795	−	0.634656i	$-0.218858\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	378.765			0.891212
$426$		0			0
$427$	−647.477			−1.51634
$428$		0			0
$429$		0			0
$430$		0			0
$431$			450.553i			1.04537i	0.852527	+	0.522683i	$0.175069\pi$
$431$			450.553i			1.04537i	−0.852527	+	0.522683i	$0.824931\pi$
$432$		0			0
$433$		−	367.589i		−	0.848935i	−0.905443	−	0.424467i	$-0.860461\pi$
$433$		−	367.589i		−	0.848935i	0.905443	−	0.424467i	$-0.139539\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	102.282	−	633.769i	0.234056	−	1.45027i
$438$		0			0
$439$		−	91.3152i		−	0.208007i	−0.994577	−	0.104004i	$-0.966835\pi$
$439$		−	91.3152i		−	0.208007i	0.994577	−	0.104004i	$-0.0331654\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−257.932			−0.582240			−0.291120	−	0.956687i	$-0.594028\pi$
$443$	−257.932			−0.582240			−0.291120	+	0.956687i	$0.594028\pi$
$444$		0			0
$445$			695.854i			1.56372i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	265.025i		−	0.590257i	−0.955458	−	0.295128i	$-0.904638\pi$
$449$		−	265.025i		−	0.590257i	0.955458	−	0.295128i	$-0.0953624\pi$
$450$		0			0
$451$		−	241.714i		−	0.535951i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	114.568i		−	0.251797i
$456$		0			0
$457$	2.70629			0.00592187			0.00296093	−	0.999996i	$-0.499058\pi$
$457$	2.70629			0.00592187			0.00296093	+	0.999996i	$0.499058\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	676.701			1.46790			0.733949	−	0.679205i	$-0.237675\pi$
$461$	676.701			1.46790			0.733949	+	0.679205i	$0.237675\pi$
$462$		0			0
$463$	790.741			1.70786			0.853932	−	0.520384i	$-0.174211\pi$
$463$	790.741			1.70786			0.853932	+	0.520384i	$0.174211\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	425.853			0.911890			0.455945	−	0.890008i	$-0.349301\pi$
$467$	425.853			0.911890			0.455945	+	0.890008i	$0.349301\pi$
$468$		0			0
$469$			779.461i			1.66196i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−822.410			−1.73871
$474$		0			0
$475$	−108.267	+	670.850i	−0.227930	+	1.41232i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−882.086			−1.84152			−0.920758	−	0.390134i	$-0.872429\pi$
$479$	−882.086			−1.84152			−0.920758	+	0.390134i	$0.872429\pi$
$480$		0			0
$481$	32.6860			0.0679543
$482$		0			0
$483$		0			0
$484$		0			0
$485$			668.664i			1.37869i
$486$		0			0
$487$		−	392.908i		−	0.806793i	−0.915025	−	0.403396i	$-0.867830\pi$
$487$		−	392.908i		−	0.806793i	0.915025	−	0.403396i	$-0.132170\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	351.132			0.715137			0.357569	−	0.933887i	$-0.383606\pi$
$491$	351.132			0.715137			0.357569	+	0.933887i	$0.383606\pi$
$492$		0			0
$493$			570.480i			1.15716i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	1036.18i		−	2.08488i
$498$		0			0
$499$	258.991			0.519020			0.259510	−	0.965740i	$-0.416439\pi$
$499$	258.991			0.519020			0.259510	+	0.965740i	$0.416439\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	437.058			0.868902			0.434451	−	0.900695i	$-0.356942\pi$
$503$	437.058			0.868902			0.434451	+	0.900695i	$0.356942\pi$
$504$		0			0
$505$	397.637			0.787399
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	207.944i		−	0.408534i	−0.978915	−	0.204267i	$-0.934519\pi$
$509$		−	207.944i		−	0.408534i	0.978915	−	0.204267i	$-0.0654810\pi$
$510$		0			0
$511$	620.871			1.21501
$512$		0			0
$513$		0			0
$514$		0			0
$515$			845.176i			1.64112i
$516$		0			0
$517$	−522.111			−1.00988
$518$		0			0
$519$		0			0
$520$		0			0
$521$			316.477i			0.607441i	0.952761	+	0.303720i	$0.0982289\pi$
$521$			316.477i			0.607441i	−0.952761	+	0.303720i	$0.901771\pi$
$522$		0			0
$523$			389.576i			0.744887i	0.928055	+	0.372443i	$0.121480\pi$
$523$			389.576i			0.744887i	−0.928055	+	0.372443i	$0.878520\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			575.842i			1.09268i
$528$		0			0
$529$	612.619			1.15807
$530$		0			0
$531$		0			0
$532$		0			0
$533$	21.3854			0.0401227
$534$		0			0
$535$		−	1056.26i		−	1.97431i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	621.604			1.15325
$540$		0			0
$541$	588.194			1.08723			0.543617	−	0.839333i	$-0.317054\pi$
$541$	588.194			1.08723			0.543617	+	0.839333i	$0.317054\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	167.780i		−	0.307853i
$546$		0			0
$547$		−	907.139i		−	1.65839i	−0.558960	−	0.829195i	$-0.688800\pi$
$547$		−	907.139i		−	1.65839i	0.558960	−	0.829195i	$-0.311200\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1010.41	−	163.067i	−1.83377	−	0.295947i
$552$		0			0
$553$		−	906.532i		−	1.63930i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−129.282			−0.232105			−0.116052	−	0.993243i	$-0.537024\pi$
$557$	−129.282			−0.232105			−0.116052	+	0.993243i	$0.537024\pi$
$558$		0			0
$559$		−	72.7619i		−	0.130164i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			323.099i			0.573888i	0.957947	+	0.286944i	$0.0926393\pi$
$563$			323.099i			0.573888i	−0.957947	+	0.286944i	$0.907361\pi$
$564$		0			0
$565$		−	98.8054i		−	0.174877i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			48.7239i			0.0856307i	0.999083	+	0.0428154i	$0.0136327\pi$
$569$			48.7239i			0.0856307i	−0.999083	+	0.0428154i	$0.986367\pi$
$570$		0			0
$571$	986.929			1.72842			0.864211	−	0.503129i	$-0.167818\pi$
$571$	986.929			1.72842			0.864211	+	0.503129i	$0.167818\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1208.41			−2.10159
$576$		0			0
$577$	248.909			0.431385			0.215693	−	0.976461i	$-0.430799\pi$
$577$	248.909			0.431385			0.215693	+	0.976461i	$0.430799\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1219.95			−2.09975
$582$		0			0
$583$			150.174i			0.257588i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−32.9707			−0.0561681			−0.0280840	−	0.999606i	$-0.508941\pi$
$587$	−32.9707			−0.0561681			−0.0280840	+	0.999606i	$0.508941\pi$
$588$		0			0
$589$	−1019.90	−	164.600i	−1.73159	−	0.279456i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−943.621			−1.59127			−0.795633	−	0.605779i	$-0.792862\pi$
$593$	−943.621			−1.59127			−0.795633	+	0.605779i	$0.792862\pi$
$594$		0			0
$595$	−752.622			−1.26491
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	550.122i		−	0.918400i	−0.888333	−	0.459200i	$-0.848136\pi$
$599$		−	550.122i		−	0.918400i	0.888333	−	0.459200i	$-0.151864\pi$
$600$		0			0
$601$		−	123.366i		−	0.205268i	−0.994719	−	0.102634i	$-0.967273\pi$
$601$		−	123.366i		−	0.205268i	0.994719	−	0.102634i	$-0.0327270\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1644.97			2.71896
$606$		0			0
$607$		−	617.674i		−	1.01759i	−0.860889	−	0.508793i	$-0.830092\pi$
$607$		−	617.674i		−	1.01759i	0.860889	−	0.508793i	$-0.169908\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	46.1932i		−	0.0756026i
$612$		0			0
$613$	255.678			0.417092			0.208546	−	0.978013i	$-0.433127\pi$
$613$	255.678			0.417092			0.208546	+	0.978013i	$0.433127\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−849.542			−1.37689			−0.688445	−	0.725288i	$-0.741707\pi$
$617$	−849.542			−1.37689			−0.688445	+	0.725288i	$0.741707\pi$
$618$		0			0
$619$	−729.780			−1.17897			−0.589483	−	0.807781i	$-0.700668\pi$
$619$	−729.780			−1.17897			−0.589483	+	0.807781i	$0.700668\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	813.821i		−	1.30629i
$624$		0			0
$625$	−240.001			−0.384001
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	214.722i		−	0.341370i
$630$		0			0
$631$	−816.085			−1.29332			−0.646660	−	0.762778i	$-0.723835\pi$
$631$	−816.085			−1.29332			−0.646660	+	0.762778i	$0.723835\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	464.244i		−	0.731093i
$636$		0			0
$637$			54.9958i			0.0863356i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	36.6936i		−	0.0572443i	−0.999590	−	0.0286221i	$-0.990888\pi$
$641$		−	36.6936i		−	0.0572443i	0.999590	−	0.0286221i	$-0.00911196\pi$
$642$		0			0
$643$	862.977			1.34211			0.671055	−	0.741407i	$-0.265841\pi$
$643$	862.977			1.34211			0.671055	+	0.741407i	$0.265841\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−327.176			−0.505681			−0.252841	−	0.967508i	$-0.581365\pi$
$647$	−327.176			−0.505681			−0.252841	+	0.967508i	$0.581365\pi$
$648$		0			0
$649$		−	1331.13i		−	2.05104i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	31.3418			0.0479966			0.0239983	−	0.999712i	$-0.492360\pi$
$653$	31.3418			0.0479966			0.0239983	+	0.999712i	$0.492360\pi$
$654$		0			0
$655$	−962.847			−1.46999
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1062.89i		−	1.61289i	−0.591312	−	0.806443i	$-0.701390\pi$
$659$		−	1062.89i		−	1.61289i	0.591312	−	0.806443i	$-0.298610\pi$
$660$		0			0
$661$		−	556.662i		−	0.842151i	−0.907026	−	0.421075i	$-0.861653\pi$
$661$		−	556.662i		−	0.842151i	0.907026	−	0.421075i	$-0.138347\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	215.131	−	1333.01i	0.323505	−	2.00452i
$666$		0			0
$667$		−	1820.06i		−	2.72873i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1294.11			1.92864
$672$		0			0
$673$		−	426.796i		−	0.634170i	−0.948397	−	0.317085i	$-0.897296\pi$
$673$		−	426.796i		−	0.634170i	0.948397	−	0.317085i	$-0.102704\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1141.62i		−	1.68629i	−0.537684	−	0.843146i	$-0.680701\pi$
$677$		−	1141.62i		−	1.68629i	0.537684	−	0.843146i	$-0.319299\pi$
$678$		0			0
$679$		−	782.021i		−	1.15173i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	599.266i		−	0.877403i	−0.898633	−	0.438701i	$-0.855439\pi$
$683$		−	599.266i		−	0.877403i	0.898633	−	0.438701i	$-0.144561\pi$
$684$		0			0
$685$	−1136.57			−1.65923
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−13.2865			−0.0192837
$690$		0			0
$691$	565.025			0.817692			0.408846	−	0.912603i	$-0.365931\pi$
$691$	565.025			0.817692			0.408846	+	0.912603i	$0.365931\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	60.9446			0.0876900
$696$		0			0
$697$		−	140.486i		−	0.201557i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−975.154			−1.39109			−0.695545	−	0.718483i	$-0.744837\pi$
$701$	−975.154			−1.39109			−0.695545	+	0.718483i	$0.744837\pi$
$702$		0			0
$703$	380.305	+	61.3764i	0.540974	+	0.0873064i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−465.047			−0.657775
$708$		0			0
$709$	−101.197			−0.142732			−0.0713660	−	0.997450i	$-0.522736\pi$
$709$	−101.197			−0.142732			−0.0713660	+	0.997450i	$0.522736\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1837.17i		−	2.57668i
$714$		0			0
$715$			228.987i			0.320262i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	678.538			0.943725			0.471862	−	0.881672i	$-0.343582\pi$
$719$	678.538			0.943725			0.471862	+	0.881672i	$0.343582\pi$
$720$		0			0
$721$		−	988.456i		−	1.37095i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			1926.55i			2.65732i
$726$		0			0
$727$	606.487			0.834232			0.417116	−	0.908853i	$-0.363041\pi$
$727$	606.487			0.834232			0.417116	+	0.908853i	$0.363041\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−477.989			−0.653884
$732$		0			0
$733$	−27.4763			−0.0374848			−0.0187424	−	0.999824i	$-0.505966\pi$
$733$	−27.4763			−0.0374848			−0.0187424	+	0.999824i	$0.505966\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1557.91i		−	2.11386i
$738$		0			0
$739$	327.283			0.442873			0.221437	−	0.975175i	$-0.428925\pi$
$739$	327.283			0.442873			0.221437	+	0.975175i	$0.428925\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	589.829i		−	0.793848i	−0.917851	−	0.396924i	$-0.870078\pi$
$743$		−	589.829i		−	0.793848i	0.917851	−	0.396924i	$-0.129922\pi$
$744$		0			0
$745$	1716.01			2.30337
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1235.32i			1.64930i
$750$		0			0
$751$			1380.49i			1.83821i	0.394016	+	0.919104i	$0.371086\pi$
$751$			1380.49i			1.83821i	−0.394016	+	0.919104i	$0.628914\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	923.186i		−	1.22276i
$756$		0			0
$757$	947.977			1.25228			0.626140	−	0.779710i	$-0.284634\pi$
$757$	947.977			1.25228			0.626140	+	0.779710i	$0.284634\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−246.939			−0.324492			−0.162246	−	0.986750i	$-0.551874\pi$
$761$	−246.939			−0.324492			−0.162246	+	0.986750i	$0.551874\pi$
$762$		0			0
$763$			196.223i			0.257173i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	117.770			0.153546
$768$		0			0
$769$	−412.025			−0.535794			−0.267897	−	0.963448i	$-0.586329\pi$
$769$	−412.025			−0.535794			−0.267897	+	0.963448i	$0.586329\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			593.063i			0.767222i	0.923495	+	0.383611i	$0.125320\pi$
$773$			593.063i			0.767222i	−0.923495	+	0.383611i	$0.874680\pi$
$774$		0			0
$775$			1944.66i			2.50924i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	248.821	+	40.1566i	0.319411	+	0.0515489i
$780$		0			0
$781$			2071.03i			2.65176i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	763.590			0.972727
$786$		0			0
$787$		−	1319.49i		−	1.67661i	−0.545200	−	0.838306i	$-0.683546\pi$
$787$		−	1319.49i		−	1.67661i	0.545200	−	0.838306i	$-0.316454\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			115.556i			0.146088i
$792$		0			0
$793$			114.495i			0.144383i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	3.54319i		−	0.00444566i	−0.999998	−	0.00222283i	$-0.999292\pi$
$797$		−	3.54319i		−	0.00444566i	0.999998	−	0.00222283i	$-0.000707550\pi$
$798$		0			0
$799$	−303.453			−0.379792
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1240.94			−1.54538
$804$		0			0
$805$	2401.17			2.98282
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−257.128			−0.317835			−0.158917	−	0.987292i	$-0.550800\pi$
$809$	−257.128			−0.317835			−0.158917	+	0.987292i	$0.550800\pi$
$810$		0			0
$811$		−	1251.31i		−	1.54292i	−0.636275	−	0.771462i	$-0.719526\pi$
$811$		−	1251.31i		−	1.54292i	0.636275	−	0.771462i	$-0.280474\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−677.959			−0.831851
$816$		0			0
$817$	136.629	−	846.591i	0.167233	−	1.03622i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1557.49			−1.89706			−0.948532	−	0.316682i	$-0.897431\pi$
$821$	−1557.49			−1.89706			−0.948532	+	0.316682i	$0.897431\pi$
$822$		0			0
$823$	830.344			1.00892			0.504462	−	0.863434i	$-0.331691\pi$
$823$	830.344			1.00892			0.504462	+	0.863434i	$0.331691\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1224.43i		−	1.48057i	−0.672291	−	0.740287i	$-0.734690\pi$
$827$		−	1224.43i		−	1.48057i	0.672291	−	0.740287i	$-0.265310\pi$
$828$		0			0
$829$		−	1239.78i		−	1.49551i	−0.663973	−	0.747757i	$-0.731131\pi$
$829$		−	1239.78i		−	1.49551i	0.663973	−	0.747757i	$-0.268869\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	361.280			0.433709
$834$		0			0
$835$			1763.88i			2.11243i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	943.241i		−	1.12424i	−0.827054	−	0.562122i	$-0.809985\pi$
$839$		−	943.241i		−	1.12424i	0.827054	−	0.562122i	$-0.190015\pi$
$840$		0			0
$841$	−2060.69			−2.45029
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1297.13			1.53506
$846$		0			0
$847$	−1923.84			−2.27136
$848$		0			0
$849$		0			0
$850$		0			0
$851$			685.050i			0.804994i
$852$		0			0
$853$	530.339			0.621733			0.310867	−	0.950454i	$-0.399381\pi$
$853$	530.339			0.621733			0.310867	+	0.950454i	$0.399381\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	431.790i		−	0.503839i	−0.967748	−	0.251920i	$-0.918938\pi$
$857$		−	431.790i		−	0.503839i	0.967748	−	0.251920i	$-0.0810619\pi$
$858$		0			0
$859$	−426.502			−0.496510			−0.248255	−	0.968695i	$-0.579857\pi$
$859$	−426.502			−0.496510			−0.248255	+	0.968695i	$0.579857\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			586.564i			0.679680i	0.940483	+	0.339840i	$0.110373\pi$
$863$			586.564i			0.679680i	−0.940483	+	0.339840i	$0.889627\pi$
$864$		0			0
$865$			266.546i			0.308146i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1811.89i			2.08503i
$870$		0			0
$871$	137.834			0.158249
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−765.010			−0.874297
$876$		0			0
$877$			415.783i			0.474096i	0.971498	+	0.237048i	$0.0761799\pi$
$877$			415.783i			0.474096i	−0.971498	+	0.237048i	$0.923820\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	208.821			0.237027			0.118513	−	0.992952i	$-0.462187\pi$
$881$	208.821			0.237027			0.118513	+	0.992952i	$0.462187\pi$
$882$		0			0
$883$	684.488			0.775185			0.387592	−	0.921831i	$-0.373307\pi$
$883$	684.488			0.775185			0.387592	+	0.921831i	$0.373307\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			143.535i			0.161821i	0.996721	+	0.0809105i	$0.0257828\pi$
$887$			143.535i			0.161821i	−0.996721	+	0.0809105i	$0.974217\pi$
$888$		0			0
$889$			542.946i			0.610738i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	86.7396	−	537.462i	0.0971328	−	0.601861i
$894$		0			0
$895$			2354.25i			2.63045i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2928.97			−3.25803
$900$		0			0
$901$			87.2818i			0.0968721i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	281.486i		−	0.311034i
$906$		0			0
$907$		−	471.912i		−	0.520300i	−0.965568	−	0.260150i	$-0.916228\pi$
$907$		−	471.912i		−	0.520300i	0.965568	−	0.260150i	$-0.0837720\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1432.48i			1.57242i	0.617957	+	0.786212i	$0.287961\pi$
$911$			1432.48i			1.57242i	−0.617957	+	0.786212i	$0.712039\pi$
$912$		0			0
$913$	2438.32			2.67067
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1126.08			1.22800
$918$		0			0
$919$	−1793.99			−1.95211			−0.976055	−	0.217522i	$-0.930203\pi$
$919$	−1793.99			−1.95211			−0.976055	+	0.217522i	$0.930203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−183.232			−0.198518
$924$		0			0
$925$		−	725.132i		−	0.783926i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1285.15			1.38337			0.691686	−	0.722198i	$-0.256868\pi$
$929$	1285.15			1.38337			0.691686	+	0.722198i	$0.256868\pi$
$930$		0			0
$931$	−103.269	+	639.881i	−0.110922	+	0.687305i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1504.27			1.60884
$936$		0			0
$937$	−1097.35			−1.17113			−0.585564	−	0.810626i	$-0.699127\pi$
$937$	−1097.35			−1.17113			−0.585564	+	0.810626i	$0.699127\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	700.411i		−	0.744326i	−0.928167	−	0.372163i	$-0.878616\pi$
$941$		−	700.411i		−	0.744326i	0.928167	−	0.372163i	$-0.121384\pi$
$942$		0			0
$943$			448.206i			0.475298i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	148.645			0.156964			0.0784820	−	0.996916i	$-0.474993\pi$
$947$	148.645			0.156964			0.0784820	+	0.996916i	$0.474993\pi$
$948$		0			0
$949$		−	109.791i		−	0.115691i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	161.856i		−	0.169838i	−0.996388	−	0.0849192i	$-0.972937\pi$
$953$		−	161.856i		−	0.169838i	0.996388	−	0.0849192i	$-0.0270632\pi$
$954$		0			0
$955$	−531.318			−0.556354
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1329.25			1.38608
$960$		0			0
$961$	−1995.50			−2.07648
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1333.52i			1.38188i
$966$		0			0
$967$	−1614.01			−1.66909			−0.834543	−	0.550943i	$-0.814268\pi$
$967$	−1614.01			−1.66909			−0.834543	+	0.550943i	$0.814268\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	932.963i		−	0.960827i	−0.877042	−	0.480414i	$-0.840487\pi$
$971$		−	932.963i		−	0.960827i	0.877042	−	0.480414i	$-0.159513\pi$
$972$		0			0
$973$	−71.2764			−0.0732542
$974$		0			0
$975$		0			0
$976$		0			0
$977$			279.278i			0.285852i	0.989733	+	0.142926i	$0.0456511\pi$
$977$			279.278i			0.285852i	−0.989733	+	0.142926i	$0.954349\pi$
$978$		0			0
$979$			1626.59i			1.66148i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	789.785i		−	0.803444i	−0.915762	−	0.401722i	$-0.868412\pi$
$983$		−	789.785i		−	0.803444i	0.915762	−	0.401722i	$-0.131588\pi$
$984$		0			0
$985$	−559.380			−0.567898
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1524.98			1.54194
$990$		0			0
$991$			475.481i			0.479799i	0.970798	+	0.239900i	$0.0771145\pi$
$991$			475.481i			0.479799i	−0.970798	+	0.239900i	$0.922885\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2080.29			2.09075
$996$		0			0
$997$	467.526			0.468932			0.234466	−	0.972124i	$-0.424666\pi$
$997$	467.526			0.468932			0.234466	+	0.972124i	$0.424666\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.r.721.18		20
3.2	odd	2		912.3.o.e.721.12		20
4.3	odd	2		1368.3.o.c.721.18		20
12.11	even	2		456.3.o.a.265.2	✓	20
19.18	odd	2	inner	2736.3.o.r.721.17		20
57.56	even	2		912.3.o.e.721.2		20
76.75	even	2		1368.3.o.c.721.17		20
228.227	odd	2		456.3.o.a.265.12	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.3.o.a.265.2	✓	20	12.11	even	2
456.3.o.a.265.12	yes	20	228.227	odd	2
912.3.o.e.721.2		20	57.56	even	2
912.3.o.e.721.12		20	3.2	odd	2
1368.3.o.c.721.17		20	76.75	even	2
1368.3.o.c.721.18		20	4.3	odd	2
2736.3.o.r.721.17		20	19.18	odd	2	inner
2736.3.o.r.721.18		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.79518 q^{5} -9.11667 q^{7} +O(q^{10})\)	\(q+7.79518 q^{5} -9.11667 q^{7} +18.2215 q^{11} +1.61213i q^{13} +10.5905 q^{17} +(-3.02719 + 18.7573i) q^{19} -33.7879 q^{23} +35.7648 q^{25} +53.8674i q^{29} +54.3737i q^{31} -71.0661 q^{35} -20.2750i q^{37} -13.2653i q^{41} -45.1340 q^{43} -28.6535 q^{47} +34.1137 q^{49} +8.24155i q^{53} +142.040 q^{55} -73.0524i q^{59} +71.0212 q^{61} +12.5668i q^{65} -85.4984i q^{67} +113.658i q^{71} -68.1028 q^{73} -166.120 q^{77} +99.4367i q^{79} +133.816 q^{83} +82.5544 q^{85} +89.2673i q^{89} -14.6973i q^{91} +(-23.5975 + 146.216i) q^{95} +85.7792i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7}+O(q^{10}) \) 20 * q + 16 * q^7	\( 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) \) 20 * q + 16 * q^7 + 16 * q^11 - 32 * q^17 - 40 * q^19 + 64 * q^23 + 68 * q^25 - 208 * q^35 - 64 * q^43 + 48 * q^47 + 20 * q^49 + 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 + 224 * q^83 - 136 * q^85 - 320 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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