LMFDB - Embedded newform 2736.2.k.o.2431.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2736,2,Mod(2431,2736)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2736.2431"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-10,0,16,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.1
Root			$-1.18614 - 1.26217i$ of defining polynomial
Character	$\chi$	$=$	2736.2431
Dual form			2736.2.k.o.2431.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37228 q^{5} -2.52434i q^{7} +2.52434i q^{11} +1.58457i q^{13} +0.372281 q^{17} +(4.00000 - 1.73205i) q^{19} +1.87953i q^{23} +0.627719 q^{25} +3.16915i q^{29} -2.74456 q^{31} +5.98844i q^{35} +1.58457i q^{37} -6.92820i q^{41} +0.644810i q^{43} +0.939764i q^{47} +0.627719 q^{49} -10.0974i q^{53} -5.98844i q^{55} +4.00000 q^{59} -0.372281 q^{61} -3.75906i q^{65} +13.4891 q^{67} +4.00000 q^{71} +13.1168 q^{73} +6.37228 q^{77} -6.74456 q^{79} +3.46410i q^{83} -0.883156 q^{85} -13.2665i q^{89} +4.00000 q^{91} +(-9.48913 + 4.10891i) q^{95} -13.2665i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} - 10 q^{17} + 16 q^{19} + 14 q^{25} + 12 q^{31} + 14 q^{49} + 16 q^{59} + 10 q^{61} + 8 q^{67} + 16 q^{71} + 18 q^{73} + 14 q^{77} - 4 q^{79} - 38 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) $ 4 * q + 2 * q^5 - 10 * q^17 + 16 * q^19 + 14 * q^25 + 12 * q^31 + 14 * q^49 + 16 * q^59 + 10 * q^61 + 8 * q^67 + 16 * q^71 + 18 * q^73 + 14 * q^77 - 4 * q^79 - 38 * q^85 + 16 * q^91 + 8 * q^95

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
$5$	−2.37228			−1.06092			−0.530458	−	0.847711i	$-0.677980\pi$
$5$	−2.37228			−1.06092			−0.530458	+	0.847711i	$0.677980\pi$
$6$		0			0
$7$		−	2.52434i		−	0.954110i	−0.878873	−	0.477055i	$-0.841704\pi$
$7$		−	2.52434i		−	0.954110i	0.878873	−	0.477055i	$-0.158296\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.52434i			0.761116i	0.924757	+	0.380558i	$0.124268\pi$
$11$			2.52434i			0.761116i	−0.924757	+	0.380558i	$0.875732\pi$
$12$		0			0
$13$			1.58457i			0.439482i	0.975558	+	0.219741i	$0.0705212\pi$
$13$			1.58457i			0.439482i	−0.975558	+	0.219741i	$0.929479\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.372281			0.0902915			0.0451457	−	0.998980i	$-0.485625\pi$
$17$	0.372281			0.0902915			0.0451457	+	0.998980i	$0.485625\pi$
$18$		0			0
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.87953i			0.391909i	0.980613	+	0.195954i	$0.0627804\pi$
$23$			1.87953i			0.391909i	−0.980613	+	0.195954i	$0.937220\pi$
$24$		0			0
$25$	0.627719			0.125544
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.16915i			0.588496i	0.955729	+	0.294248i	$0.0950692\pi$
$29$			3.16915i			0.588496i	−0.955729	+	0.294248i	$0.904931\pi$
$30$		0			0
$31$	−2.74456			−0.492938			−0.246469	−	0.969151i	$-0.579270\pi$
$31$	−2.74456			−0.492938			−0.246469	+	0.969151i	$0.579270\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			5.98844i			1.01223i
$36$		0			0
$37$			1.58457i			0.260502i	0.991481	+	0.130251i	$0.0415784\pi$
$37$			1.58457i			0.260502i	−0.991481	+	0.130251i	$0.958422\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$			0.644810i			0.0983326i	0.998791	+	0.0491663i	$0.0156564\pi$
$43$			0.644810i			0.0983326i	−0.998791	+	0.0491663i	$0.984344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			0.939764i			0.137079i	0.997648	+	0.0685393i	$0.0218339\pi$
$47$			0.939764i			0.137079i	−0.997648	+	0.0685393i	$0.978166\pi$
$48$		0			0
$49$	0.627719			0.0896741
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	10.0974i		−	1.38698i	−0.720467	−	0.693489i	$-0.756073\pi$
$53$		−	10.0974i		−	1.38698i	0.720467	−	0.693489i	$-0.243927\pi$
$54$		0			0
$55$		−	5.98844i		−	0.807481i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	−0.372281			−0.0476657			−0.0238329	−	0.999716i	$-0.507587\pi$
$61$	−0.372281			−0.0476657			−0.0238329	+	0.999716i	$0.507587\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	3.75906i		−	0.466253i
$66$		0			0
$67$	13.4891			1.64796			0.823979	−	0.566620i	$-0.191749\pi$
$67$	13.4891			1.64796			0.823979	+	0.566620i	$0.191749\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$	13.1168			1.53521			0.767605	−	0.640923i	$-0.221448\pi$
$73$	13.1168			1.53521			0.767605	+	0.640923i	$0.221448\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.37228			0.726189
$78$		0			0
$79$	−6.74456			−0.758823			−0.379411	−	0.925228i	$-0.623873\pi$
$79$	−6.74456			−0.758823			−0.379411	+	0.925228i	$0.623873\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$	−0.883156			−0.0957917
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	13.2665i		−	1.40625i	−0.711068	−	0.703123i	$-0.751788\pi$
$89$		−	13.2665i		−	1.40625i	0.711068	−	0.703123i	$-0.248212\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.48913	+	4.10891i	−0.973564	+	0.421565i
$96$		0			0
$97$		−	13.2665i		−	1.34701i	−0.739183	−	0.673504i	$-0.764788\pi$
$97$		−	13.2665i		−	1.34701i	0.739183	−	0.673504i	$-0.235212\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.25544			0.323928			0.161964	−	0.986797i	$-0.448217\pi$
$101$	3.25544			0.323928			0.161964	+	0.986797i	$0.448217\pi$
$102$		0			0
$103$	−1.25544			−0.123702			−0.0618510	−	0.998085i	$-0.519700\pi$
$103$	−1.25544			−0.123702			−0.0618510	+	0.998085i	$0.519700\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.4891			1.30404			0.652021	−	0.758200i	$-0.273921\pi$
$107$	13.4891			1.30404			0.652021	+	0.758200i	$0.273921\pi$
$108$		0			0
$109$		−	11.6819i		−	1.11893i	−0.828855	−	0.559463i	$-0.811007\pi$
$109$		−	11.6819i		−	1.11893i	0.828855	−	0.559463i	$-0.188993\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	17.0256i		−	1.60163i	−0.598912	−	0.800815i	$-0.704400\pi$
$113$		−	17.0256i		−	1.60163i	0.598912	−	0.800815i	$-0.295600\pi$
$114$		0			0
$115$		−	4.45877i		−	0.415782i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	0.939764i		−	0.0861480i
$120$		0			0
$121$	4.62772			0.420702
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.3723			0.927725
$126$		0			0
$127$	1.25544			0.111402			0.0557010	−	0.998447i	$-0.482261\pi$
$127$	1.25544			0.111402			0.0557010	+	0.998447i	$0.482261\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	7.57301i		−	0.661657i	−0.943691	−	0.330829i	$-0.892672\pi$
$131$		−	7.57301i		−	0.661657i	0.943691	−	0.330829i	$-0.107328\pi$
$132$		0			0
$133$	−4.37228	−	10.0974i	−0.379125	−	0.875551i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.1168			−1.12065			−0.560324	−	0.828274i	$-0.689323\pi$
$137$	−13.1168			−1.12065			−0.560324	+	0.828274i	$0.689323\pi$
$138$		0			0
$139$			17.6704i			1.49878i	0.662128	+	0.749390i	$0.269653\pi$
$139$			17.6704i			1.49878i	−0.662128	+	0.749390i	$0.730347\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.00000			−0.334497
$144$		0			0
$145$		−	7.51811i		−	0.624345i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.1168			1.23842			0.619210	−	0.785225i	$-0.287453\pi$
$149$	15.1168			1.23842			0.619210	+	0.785225i	$0.287453\pi$
$150$		0			0
$151$	2.74456			0.223349			0.111675	−	0.993745i	$-0.464379\pi$
$151$	2.74456			0.223349			0.111675	+	0.993745i	$0.464379\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.51087			0.522966
$156$		0			0
$157$	−15.4891			−1.23617			−0.618083	−	0.786113i	$-0.712091\pi$
$157$	−15.4891			−1.23617			−0.618083	+	0.786113i	$0.712091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.74456			0.373924
$162$		0			0
$163$			19.8997i			1.55867i	0.626608	+	0.779334i	$0.284443\pi$
$163$			19.8997i			1.55867i	−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.48913			0.424761			0.212381	−	0.977187i	$-0.431878\pi$
$167$	5.48913			0.424761			0.212381	+	0.977187i	$0.431878\pi$
$168$		0			0
$169$	10.4891			0.806856
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.33830i			0.481892i	0.970539	+	0.240946i	$0.0774576\pi$
$173$			6.33830i			0.481892i	−0.970539	+	0.240946i	$0.922542\pi$
$174$		0			0
$175$		−	1.58457i		−	0.119783i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.4891			1.60617			0.803086	−	0.595863i	$-0.203190\pi$
$179$	21.4891			1.60617			0.803086	+	0.595863i	$0.203190\pi$
$180$		0			0
$181$		−	25.5383i		−	1.89825i	−0.314901	−	0.949125i	$-0.601971\pi$
$181$		−	25.5383i		−	1.89825i	0.314901	−	0.949125i	$-0.398029\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	3.75906i		−	0.276371i
$186$		0			0
$187$			0.939764i			0.0687223i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	26.1831i		−	1.89455i	−0.320428	−	0.947273i	$-0.603827\pi$
$191$		−	26.1831i		−	1.89455i	0.320428	−	0.947273i	$-0.396173\pi$
$192$		0			0
$193$			13.8564i			0.997406i	0.866773	+	0.498703i	$0.166190\pi$
$193$			13.8564i			0.997406i	−0.866773	+	0.498703i	$0.833810\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.7446			1.19300			0.596500	−	0.802613i	$-0.296557\pi$
$197$	16.7446			1.19300			0.596500	+	0.802613i	$0.296557\pi$
$198$		0			0
$199$			18.2603i			1.29444i	0.762305	+	0.647218i	$0.224068\pi$
$199$			18.2603i			1.29444i	−0.762305	+	0.647218i	$0.775932\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.00000			0.561490
$204$		0			0
$205$			16.4356i			1.14792i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.37228	+	10.0974i	0.302437	+	0.698448i
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.52967i		−	0.104323i
$216$		0			0
$217$			6.92820i			0.470317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			0.589907i			0.0396815i
$222$		0			0
$223$	−10.7446			−0.719509			−0.359755	−	0.933047i	$-0.617140\pi$
$223$	−10.7446			−0.719509			−0.359755	+	0.933047i	$0.617140\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	−6.88316			−0.454852			−0.227426	−	0.973795i	$-0.573031\pi$
$229$	−6.88316			−0.454852			−0.227426	+	0.973795i	$0.573031\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.1168			−0.859313			−0.429657	−	0.902992i	$-0.641365\pi$
$233$	−13.1168			−0.859313			−0.429657	+	0.902992i	$0.641365\pi$
$234$		0			0
$235$		−	2.22938i		−	0.145429i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	6.57835i		−	0.425518i	−0.977105	−	0.212759i	$-0.931755\pi$
$239$		−	6.57835i		−	0.425518i	0.977105	−	0.212759i	$-0.0682449\pi$
$240$		0			0
$241$		−	3.75906i		−	0.242142i	−0.992644	−	0.121071i	$-0.961367\pi$
$241$		−	3.75906i		−	0.242142i	0.992644	−	0.121071i	$-0.0386329\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.48913			−0.0951367
$246$		0			0
$247$	2.74456	+	6.33830i	0.174632	+	0.403296i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			9.45254i			0.596639i	0.954466	+	0.298320i	$0.0964261\pi$
$251$			9.45254i			0.596639i	−0.954466	+	0.298320i	$0.903574\pi$
$252$		0			0
$253$	−4.74456			−0.298288
$254$		0			0
$255$		0			0
$256$		0			0
$257$			17.0256i			1.06202i	0.847364	+	0.531012i	$0.178188\pi$
$257$			17.0256i			1.06202i	−0.847364	+	0.531012i	$0.821812\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$		0			0
$261$		0			0
$262$		0			0
$263$			14.7962i			0.912371i	0.889885	+	0.456185i	$0.150785\pi$
$263$			14.7962i			0.912371i	−0.889885	+	0.456185i	$0.849215\pi$
$264$		0			0
$265$			23.9538i			1.47147i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	23.9538i		−	1.46049i	−0.683187	−	0.730243i	$-0.739407\pi$
$269$		−	23.9538i		−	1.46049i	0.683187	−	0.730243i	$-0.260593\pi$
$270$		0			0
$271$			0.294954i			0.0179172i	0.999960	+	0.00895858i	$0.00285164\pi$
$271$			0.294954i			0.0179172i	−0.999960	+	0.00895858i	$0.997148\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1.58457i			0.0955534i
$276$		0			0
$277$	−8.37228			−0.503042			−0.251521	−	0.967852i	$-0.580931\pi$
$277$	−8.37228			−0.503042			−0.251521	+	0.967852i	$0.580931\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.16915i			0.189056i	0.995522	+	0.0945278i	$0.0301341\pi$
$281$			3.16915i			0.189056i	−0.995522	+	0.0945278i	$0.969866\pi$
$282$		0			0
$283$		−	29.0573i		−	1.72728i	−0.504110	−	0.863640i	$-0.668179\pi$
$283$		−	29.0573i		−	1.72728i	0.504110	−	0.863640i	$-0.331821\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−17.4891			−1.03235
$288$		0			0
$289$	−16.8614			−0.991847
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	13.8564i		−	0.809500i	−0.914427	−	0.404750i	$-0.867359\pi$
$293$		−	13.8564i		−	0.809500i	0.914427	−	0.404750i	$-0.132641\pi$
$294$		0			0
$295$	−9.48913			−0.552478
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.97825			−0.172237
$300$		0			0
$301$	1.62772			0.0938201
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.883156			0.0505694
$306$		0			0
$307$	5.48913			0.313281			0.156640	−	0.987656i	$-0.449934\pi$
$307$	5.48913			0.313281			0.156640	+	0.987656i	$0.449934\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	2.22938i		−	0.126417i	−0.998000	−	0.0632084i	$-0.979867\pi$
$311$		−	2.22938i		−	0.126417i	0.998000	−	0.0632084i	$-0.0201333\pi$
$312$		0			0
$313$	−3.48913			−0.197217			−0.0986085	−	0.995126i	$-0.531439\pi$
$313$	−3.48913			−0.197217			−0.0986085	+	0.995126i	$0.531439\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			23.9538i			1.34538i	0.739926	+	0.672689i	$0.234861\pi$
$317$			23.9538i			1.34538i	−0.739926	+	0.672689i	$0.765139\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.48913	−	0.644810i	0.0828571	−	0.0358782i
$324$		0			0
$325$			0.994667i			0.0551742i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.37228			0.130788
$330$		0			0
$331$	−13.4891			−0.741429			−0.370715	−	0.928747i	$-0.620887\pi$
$331$	−13.4891			−0.741429			−0.370715	+	0.928747i	$0.620887\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−32.0000			−1.74835
$336$		0			0
$337$			20.7846i			1.13221i	0.824333	+	0.566105i	$0.191550\pi$
$337$			20.7846i			1.13221i	−0.824333	+	0.566105i	$0.808450\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	6.92820i		−	0.375183i
$342$		0			0
$343$		−	19.2549i		−	1.03967i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			16.3807i			0.879364i	0.898153	+	0.439682i	$0.144909\pi$
$347$			16.3807i			0.879364i	−0.898153	+	0.439682i	$0.855091\pi$
$348$		0			0
$349$	14.6060			0.781840			0.390920	−	0.920425i	$-0.372157\pi$
$349$	14.6060			0.781840			0.390920	+	0.920425i	$0.372157\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.510875			0.0271911			0.0135956	−	0.999908i	$-0.495672\pi$
$353$	0.510875			0.0271911			0.0135956	+	0.999908i	$0.495672\pi$
$354$		0			0
$355$	−9.48913			−0.503630
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	29.3523i		−	1.54915i	−0.632479	−	0.774577i	$-0.717963\pi$
$359$		−	29.3523i		−	1.54915i	0.632479	−	0.774577i	$-0.282037\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−31.1168			−1.62873
$366$		0			0
$367$			12.9715i			0.677109i	0.940947	+	0.338555i	$0.109938\pi$
$367$			12.9715i			0.677109i	−0.940947	+	0.338555i	$0.890062\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−25.4891			−1.32333
$372$		0			0
$373$		−	8.51278i		−	0.440775i	−0.975412	−	0.220387i	$-0.929268\pi$
$373$		−	8.51278i		−	0.440775i	0.975412	−	0.220387i	$-0.0707322\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.02175			−0.258633
$378$		0			0
$379$	−18.9783			−0.974847			−0.487424	−	0.873166i	$-0.662063\pi$
$379$	−18.9783			−0.974847			−0.487424	+	0.873166i	$0.662063\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.9783			0.765353			0.382676	−	0.923882i	$-0.375002\pi$
$383$	14.9783			0.765353			0.382676	+	0.923882i	$0.375002\pi$
$384$		0			0
$385$	−15.1168			−0.770426
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.1168			0.563646			0.281823	−	0.959466i	$-0.409061\pi$
$389$	11.1168			0.563646			0.281823	+	0.959466i	$0.409061\pi$
$390$		0			0
$391$			0.699713i			0.0353860i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.0000			0.805047
$396$		0			0
$397$	29.1168			1.46133			0.730666	−	0.682735i	$-0.239210\pi$
$397$	29.1168			1.46133			0.730666	+	0.682735i	$0.239210\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			10.6873i			0.533696i	0.963739	+	0.266848i	$0.0859822\pi$
$401$			10.6873i			0.533696i	−0.963739	+	0.266848i	$0.914018\pi$
$402$		0			0
$403$		−	4.34896i		−	0.216637i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.00000			−0.198273
$408$		0			0
$409$		−	6.92820i		−	0.342578i	−0.985221	−	0.171289i	$-0.945207\pi$
$409$		−	6.92820i		−	0.342578i	0.985221	−	0.171289i	$-0.0547931\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	10.0974i		−	0.496858i
$414$		0			0
$415$		−	8.21782i		−	0.403397i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	13.5615i		−	0.662520i	−0.943539	−	0.331260i	$-0.892526\pi$
$419$		−	13.5615i		−	0.662520i	0.943539	−	0.331260i	$-0.107474\pi$
$420$		0			0
$421$			26.1282i			1.27341i	0.771106	+	0.636706i	$0.219704\pi$
$421$			26.1282i			1.27341i	−0.771106	+	0.636706i	$0.780296\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.233688			0.0113355
$426$		0			0
$427$			0.939764i			0.0454784i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−41.4891			−1.99846			−0.999230	−	0.0392245i	$-0.987511\pi$
$431$	−41.4891			−1.99846			−0.999230	+	0.0392245i	$0.987511\pi$
$432$		0			0
$433$		−	23.3639i		−	1.12279i	−0.827546	−	0.561397i	$-0.810264\pi$
$433$		−	23.3639i		−	1.12279i	0.827546	−	0.561397i	$-0.189736\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.25544	+	7.51811i	0.155729	+	0.359640i
$438$		0			0
$439$	17.7228			0.845864			0.422932	−	0.906161i	$-0.361001\pi$
$439$	17.7228			0.845864			0.422932	+	0.906161i	$0.361001\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	3.81396i		−	0.181207i	−0.995887	−	0.0906033i	$-0.971120\pi$
$443$		−	3.81396i		−	0.181207i	0.995887	−	0.0906033i	$-0.0288795\pi$
$444$		0			0
$445$			31.4719i			1.49191i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			30.8820i			1.45741i	0.684828	+	0.728705i	$0.259877\pi$
$449$			30.8820i			1.45741i	−0.684828	+	0.728705i	$0.740123\pi$
$450$		0			0
$451$	17.4891			0.823531
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−9.48913			−0.444857
$456$		0			0
$457$	9.11684			0.426468			0.213234	−	0.977001i	$-0.431600\pi$
$457$	9.11684			0.426468			0.213234	+	0.977001i	$0.431600\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−38.3723			−1.78718			−0.893588	−	0.448889i	$-0.851820\pi$
$461$	−38.3723			−1.78718			−0.893588	+	0.448889i	$0.851820\pi$
$462$		0			0
$463$		−	29.6472i		−	1.37782i	−0.724845	−	0.688912i	$-0.758089\pi$
$463$		−	29.6472i		−	1.37782i	0.724845	−	0.688912i	$-0.241911\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			12.0318i			0.556764i	0.960470	+	0.278382i	$0.0897982\pi$
$467$			12.0318i			0.556764i	−0.960470	+	0.278382i	$0.910202\pi$
$468$		0			0
$469$		−	34.0511i		−	1.57233i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.62772			−0.0748426
$474$		0			0
$475$	2.51087	−	1.08724i	0.115207	−	0.0498860i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			11.3870i			0.520284i	0.965570	+	0.260142i	$0.0837694\pi$
$479$			11.3870i			0.520284i	−0.965570	+	0.260142i	$0.916231\pi$
$480$		0			0
$481$	−2.51087			−0.114486
$482$		0			0
$483$		0			0
$484$		0			0
$485$			31.4719i			1.42906i
$486$		0			0
$487$	40.2337			1.82316			0.911581	−	0.411120i	$-0.134862\pi$
$487$	40.2337			1.82316			0.911581	+	0.411120i	$0.134862\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			34.3461i			1.55002i	0.631951	+	0.775008i	$0.282254\pi$
$491$			34.3461i			1.55002i	−0.631951	+	0.775008i	$0.717746\pi$
$492$		0			0
$493$			1.17981i			0.0531362i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	10.0974i		−	0.452928i
$498$		0			0
$499$			14.5012i			0.649164i	0.945858	+	0.324582i	$0.105224\pi$
$499$			14.5012i			0.649164i	−0.945858	+	0.324582i	$0.894776\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			35.9306i			1.60207i	0.598619	+	0.801034i	$0.295716\pi$
$503$			35.9306i			1.60207i	−0.598619	+	0.801034i	$0.704284\pi$
$504$		0			0
$505$	−7.72281			−0.343661
$506$		0			0
$507$		0			0
$508$		0			0
$509$			13.2665i			0.588027i	0.955801	+	0.294014i	$0.0949911\pi$
$509$			13.2665i			0.588027i	−0.955801	+	0.294014i	$0.905009\pi$
$510$		0			0
$511$		−	33.1113i		−	1.46476i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.97825			0.131237
$516$		0			0
$517$	−2.37228			−0.104333
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	17.0256i		−	0.745903i	−0.927851	−	0.372952i	$-0.878346\pi$
$521$		−	17.0256i		−	0.745903i	0.927851	−	0.372952i	$-0.121654\pi$
$522$		0			0
$523$	32.0000			1.39926			0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000			1.39926			0.699631	+	0.714504i	$0.253348\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.02175			−0.0445081
$528$		0			0
$529$	19.4674			0.846408
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.9783			0.475521
$534$		0			0
$535$	−32.0000			−1.38348
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1.58457i			0.0682524i
$540$		0			0
$541$	13.1168			0.563937			0.281969	−	0.959424i	$-0.409013\pi$
$541$	13.1168			0.563937			0.281969	+	0.959424i	$0.409013\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			27.7128i			1.18709i
$546$		0			0
$547$	−6.51087			−0.278385			−0.139192	−	0.990265i	$-0.544451\pi$
$547$	−6.51087			−0.278385			−0.139192	+	0.990265i	$0.544451\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.48913	+	12.6766i	0.233845	+	0.540041i
$552$		0			0
$553$			17.0256i			0.724000i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−39.8614			−1.68898			−0.844491	−	0.535570i	$-0.820097\pi$
$557$	−39.8614			−1.68898			−0.844491	+	0.535570i	$0.820097\pi$
$558$		0			0
$559$	−1.02175			−0.0432154
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22.9783			−0.968418			−0.484209	−	0.874952i	$-0.660893\pi$
$563$	−22.9783			−0.968418			−0.484209	+	0.874952i	$0.660893\pi$
$564$		0			0
$565$			40.3894i			1.69920i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	0.589907i		−	0.0247302i	−0.999924	−	0.0123651i	$-0.996064\pi$
$569$		−	0.589907i		−	0.0247302i	0.999924	−	0.0123651i	$-0.00393603\pi$
$570$		0			0
$571$		−	23.6588i		−	0.990090i	−0.868867	−	0.495045i	$-0.835152\pi$
$571$		−	23.6588i		−	0.990090i	0.868867	−	0.495045i	$-0.164848\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.17981i			0.0492017i
$576$		0			0
$577$	2.60597			0.108488			0.0542440	−	0.998528i	$-0.482725\pi$
$577$	2.60597			0.108488			0.0542440	+	0.998528i	$0.482725\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	8.74456			0.362786
$582$		0			0
$583$	25.4891			1.05565
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	24.5986i		−	1.01529i	−0.861566	−	0.507646i	$-0.830516\pi$
$587$		−	24.5986i		−	1.01529i	0.861566	−	0.507646i	$-0.169484\pi$
$588$		0			0
$589$	−10.9783	+	4.75372i	−0.452351	+	0.195874i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.510875			0.0209791			0.0104896	−	0.999945i	$-0.496661\pi$
$593$	0.510875			0.0209791			0.0104896	+	0.999945i	$0.496661\pi$
$594$		0			0
$595$			2.22938i			0.0913958i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	24.4674			0.999710			0.499855	−	0.866109i	$-0.333387\pi$
$599$	24.4674			0.999710			0.499855	+	0.866109i	$0.333387\pi$
$600$		0			0
$601$		−	33.4612i		−	1.36491i	−0.730927	−	0.682455i	$-0.760912\pi$
$601$		−	33.4612i		−	1.36491i	0.730927	−	0.682455i	$-0.239088\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.9783			−0.446329
$606$		0			0
$607$	−29.2554			−1.18744			−0.593721	−	0.804671i	$-0.702342\pi$
$607$	−29.2554			−1.18744			−0.593721	+	0.804671i	$0.702342\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.48913			−0.0602436
$612$		0			0
$613$	−0.372281			−0.0150363			−0.00751815	−	0.999972i	$-0.502393\pi$
$613$	−0.372281			−0.0150363			−0.00751815	+	0.999972i	$0.502393\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−44.0951			−1.77520			−0.887601	−	0.460613i	$-0.847629\pi$
$617$	−44.0951			−1.77520			−0.887601	+	0.460613i	$0.847629\pi$
$618$		0			0
$619$			10.3923i			0.417702i	0.977947	+	0.208851i	$0.0669724\pi$
$619$			10.3923i			0.417702i	−0.977947	+	0.208851i	$0.933028\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−33.4891			−1.34171
$624$		0			0
$625$	−27.7446			−1.10978
$626$		0			0
$627$		0			0
$628$		0			0
$629$			0.589907i			0.0235211i
$630$		0			0
$631$			20.2496i			0.806124i	0.915173	+	0.403062i	$0.132054\pi$
$631$			20.2496i			0.806124i	−0.915173	+	0.403062i	$0.867946\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.97825			−0.118188
$636$		0			0
$637$			0.994667i			0.0394101i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	13.2665i		−	0.523995i	−0.965069	−	0.261998i	$-0.915619\pi$
$641$		−	13.2665i		−	0.523995i	0.965069	−	0.261998i	$-0.0843813\pi$
$642$		0			0
$643$		−	2.52434i		−	0.0995502i	−0.998760	−	0.0497751i	$-0.984150\pi$
$643$		−	2.52434i		−	0.0995502i	0.998760	−	0.0497751i	$-0.0158505\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	13.5065i		−	0.530997i	−0.964111	−	0.265499i	$-0.914463\pi$
$647$		−	13.5065i		−	0.530997i	0.964111	−	0.265499i	$-0.0855366\pi$
$648$		0			0
$649$			10.0974i			0.396356i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.88316			−0.191093			−0.0955463	−	0.995425i	$-0.530460\pi$
$653$	−4.88316			−0.191093			−0.0955463	+	0.995425i	$0.530460\pi$
$654$		0			0
$655$			17.9653i			0.701963i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13.4891			0.525462			0.262731	−	0.964869i	$-0.415377\pi$
$659$	13.4891			0.525462			0.262731	+	0.964869i	$0.415377\pi$
$660$		0			0
$661$			12.8617i			0.500264i	0.968212	+	0.250132i	$0.0804740\pi$
$661$			12.8617i			0.500264i	−0.968212	+	0.250132i	$0.919526\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10.3723	+	23.9538i	0.402220	+	0.928887i
$666$		0			0
$667$	−5.95650			−0.230637
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	0.939764i		−	0.0362792i
$672$		0			0
$673$			40.9793i			1.57964i	0.613341	+	0.789818i	$0.289825\pi$
$673$			40.9793i			1.57964i	−0.613341	+	0.789818i	$0.710175\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			47.9075i			1.84124i	0.390465	+	0.920618i	$0.372314\pi$
$677$			47.9075i			1.84124i	−0.390465	+	0.920618i	$0.627686\pi$
$678$		0			0
$679$	−33.4891			−1.28519
$680$		0			0
$681$		0			0
$682$		0			0
$683$	38.9783			1.49146			0.745731	−	0.666248i	$-0.232101\pi$
$683$	38.9783			1.49146			0.745731	+	0.666248i	$0.232101\pi$
$684$		0			0
$685$	31.1168			1.18891
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.0000			0.609551
$690$		0			0
$691$		−	1.93443i		−	0.0735892i	−0.999323	−	0.0367946i	$-0.988285\pi$
$691$		−	1.93443i		−	0.0735892i	0.999323	−	0.0367946i	$-0.0117147\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	41.9191i		−	1.59008i
$696$		0			0
$697$		−	2.57924i		−	0.0976957i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	19.7228			0.744920			0.372460	−	0.928048i	$-0.378514\pi$
$701$	19.7228			0.744920			0.372460	+	0.928048i	$0.378514\pi$
$702$		0			0
$703$	2.74456	+	6.33830i	0.103513	+	0.239053i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	8.21782i		−	0.309063i
$708$		0			0
$709$	−47.9565			−1.80104			−0.900522	−	0.434810i	$-0.856815\pi$
$709$	−47.9565			−1.80104			−0.900522	+	0.434810i	$0.856815\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	5.15848i		−	0.193187i
$714$		0			0
$715$	9.48913			0.354873
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	29.3523i		−	1.09466i	−0.836918	−	0.547328i	$-0.815645\pi$
$719$		−	29.3523i		−	1.09466i	0.836918	−	0.547328i	$-0.184355\pi$
$720$		0			0
$721$			3.16915i			0.118025i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			1.98933i			0.0738820i
$726$		0			0
$727$			8.16292i			0.302746i	0.988477	+	0.151373i	$0.0483695\pi$
$727$			8.16292i			0.302746i	−0.988477	+	0.151373i	$0.951631\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			0.240051i			0.00887860i
$732$		0			0
$733$	24.9783			0.922593			0.461296	−	0.887246i	$-0.347384\pi$
$733$	24.9783			0.922593			0.461296	+	0.887246i	$0.347384\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			34.0511i			1.25429i
$738$		0			0
$739$		−	5.10358i		−	0.187738i	−0.995585	−	0.0938691i	$-0.970076\pi$
$739$		−	5.10358i		−	0.187738i	0.995585	−	0.0938691i	$-0.0299235\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.9783			−0.402753			−0.201376	−	0.979514i	$-0.564541\pi$
$743$	−10.9783			−0.402753			−0.201376	+	0.979514i	$0.564541\pi$
$744$		0			0
$745$	−35.8614			−1.31386
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	34.0511i		−	1.24420i
$750$		0			0
$751$	−41.7228			−1.52249			−0.761244	−	0.648466i	$-0.775411\pi$
$751$	−41.7228			−1.52249			−0.761244	+	0.648466i	$0.775411\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.51087			−0.236955
$756$		0			0
$757$	−17.8614			−0.649184			−0.324592	−	0.945854i	$-0.605227\pi$
$757$	−17.8614			−0.649184			−0.324592	+	0.945854i	$0.605227\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	21.8614			0.792475			0.396238	−	0.918148i	$-0.370316\pi$
$761$	21.8614			0.792475			0.396238	+	0.918148i	$0.370316\pi$
$762$		0			0
$763$	−29.4891			−1.06758
$764$		0			0
$765$		0			0
$766$		0			0
$767$			6.33830i			0.228863i
$768$		0			0
$769$	−37.8614			−1.36532			−0.682659	−	0.730737i	$-0.739176\pi$
$769$	−37.8614			−1.36532			−0.682659	+	0.730737i	$0.739176\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			39.7995i			1.43149i	0.698363	+	0.715744i	$0.253912\pi$
$773$			39.7995i			1.43149i	−0.698363	+	0.715744i	$0.746088\pi$
$774$		0			0
$775$	−1.72281			−0.0618853
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.0000	−	27.7128i	−0.429945	−	0.992915i
$780$		0			0
$781$			10.0974i			0.361312i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	36.7446			1.31147
$786$		0			0
$787$	−34.9783			−1.24684			−0.623420	−	0.781887i	$-0.714257\pi$
$787$	−34.9783			−1.24684			−0.623420	+	0.781887i	$0.714257\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−42.9783			−1.52813
$792$		0			0
$793$		−	0.589907i		−	0.0209482i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			3.16915i			0.112257i	0.998424	+	0.0561285i	$0.0178756\pi$
$797$			3.16915i			0.112257i	−0.998424	+	0.0561285i	$0.982124\pi$
$798$		0			0
$799$			0.349857i			0.0123770i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			33.1113i			1.16847i
$804$		0			0
$805$	−11.2554			−0.396702
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14.8832			0.523264			0.261632	−	0.965168i	$-0.415739\pi$
$809$	14.8832			0.523264			0.261632	+	0.965168i	$0.415739\pi$
$810$		0			0
$811$	36.4674			1.28054			0.640271	−	0.768149i	$-0.278822\pi$
$811$	36.4674			1.28054			0.640271	+	0.768149i	$0.278822\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	47.2078i		−	1.65362i
$816$		0			0
$817$	1.11684	+	2.57924i	0.0390734	+	0.0902362i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.3723			−0.780798			−0.390399	−	0.920646i	$-0.627663\pi$
$821$	−22.3723			−0.780798			−0.390399	+	0.920646i	$0.627663\pi$
$822$		0			0
$823$		−	6.28339i		−	0.219025i	−0.993985	−	0.109513i	$-0.965071\pi$
$823$		−	6.28339i		−	0.219025i	0.993985	−	0.109513i	$-0.0349290\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−21.4891			−0.747250			−0.373625	−	0.927580i	$-0.621885\pi$
$827$	−21.4891			−0.747250			−0.373625	+	0.927580i	$0.621885\pi$
$828$		0			0
$829$		−	50.0820i		−	1.73942i	−0.493563	−	0.869710i	$-0.664306\pi$
$829$		−	50.0820i		−	1.73942i	0.493563	−	0.869710i	$-0.335694\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.233688			0.00809681
$834$		0			0
$835$	−13.0217			−0.450636
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	18.9565			0.653672
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−24.8832			−0.856007
$846$		0			0
$847$		−	11.6819i		−	0.401396i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.97825			−0.102093
$852$		0			0
$853$	22.0000			0.753266			0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000			0.753266			0.376633	+	0.926363i	$0.377082\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	5.74839i		−	0.196361i	−0.995169	−	0.0981806i	$-0.968698\pi$
$857$		−	5.74839i		−	0.196361i	0.995169	−	0.0981806i	$-0.0313023\pi$
$858$		0			0
$859$			35.8757i			1.22406i	0.790833	+	0.612032i	$0.209648\pi$
$859$			35.8757i			1.22406i	−0.790833	+	0.612032i	$0.790352\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.4674			−0.424394			−0.212197	−	0.977227i	$-0.568062\pi$
$863$	−12.4674			−0.424394			−0.212197	+	0.977227i	$0.568062\pi$
$864$		0			0
$865$		−	15.0362i		−	0.511247i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	17.0256i		−	0.577552i
$870$		0			0
$871$			21.3745i			0.724248i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	26.1831i		−	0.885152i
$876$		0			0
$877$			53.2511i			1.79816i	0.437781	+	0.899082i	$0.355765\pi$
$877$			53.2511i			1.79816i	−0.437781	+	0.899082i	$0.644235\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6.60597			−0.222561			−0.111280	−	0.993789i	$-0.535495\pi$
$881$	−6.60597			−0.222561			−0.111280	+	0.993789i	$0.535495\pi$
$882$		0			0
$883$		−	9.45254i		−	0.318103i	−0.987270	−	0.159052i	$-0.949156\pi$
$883$		−	9.45254i		−	0.318103i	0.987270	−	0.159052i	$-0.0508436\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.0000			−0.940148			−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000			−0.940148			−0.470074	+	0.882627i	$0.655773\pi$
$888$		0			0
$889$		−	3.16915i		−	0.106290i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.62772	+	3.75906i	0.0544695	+	0.125792i
$894$		0			0
$895$	−50.9783			−1.70401
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	8.69793i		−	0.290092i
$900$		0			0
$901$		−	3.75906i		−	0.125232i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			60.5841i			2.01388i
$906$		0			0
$907$	20.4674			0.679608			0.339804	−	0.940496i	$-0.389639\pi$
$907$	20.4674			0.679608			0.339804	+	0.940496i	$0.389639\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2.51087			−0.0831890			−0.0415945	−	0.999135i	$-0.513244\pi$
$911$	−2.51087			−0.0831890			−0.0415945	+	0.999135i	$0.513244\pi$
$912$		0			0
$913$	−8.74456			−0.289403
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−19.1168			−0.631294
$918$		0			0
$919$			24.8386i			0.819350i	0.912231	+	0.409675i	$0.134358\pi$
$919$			24.8386i			0.819350i	−0.912231	+	0.409675i	$0.865642\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			6.33830i			0.208628i
$924$		0			0
$925$			0.994667i			0.0327044i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	32.5109			1.06665			0.533324	−	0.845911i	$-0.320943\pi$
$929$	32.5109			1.06665			0.533324	+	0.845911i	$0.320943\pi$
$930$		0			0
$931$	2.51087	−	1.08724i	0.0822906	−	0.0356329i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	2.22938i		−	0.0729087i
$936$		0			0
$937$	11.6277			0.379861			0.189931	−	0.981798i	$-0.439174\pi$
$937$	11.6277			0.379861			0.189931	+	0.981798i	$0.439174\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			17.0256i			0.555017i	0.960723	+	0.277509i	$0.0895087\pi$
$941$			17.0256i			0.555017i	−0.960723	+	0.277509i	$0.910491\pi$
$942$		0			0
$943$	13.0217			0.424047
$944$		0			0
$945$		0			0
$946$		0			0
$947$			26.8280i			0.871791i	0.899997	+	0.435896i	$0.143568\pi$
$947$			26.8280i			0.871791i	−0.899997	+	0.435896i	$0.856432\pi$
$948$		0			0
$949$			20.7846i			0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	24.5437i		−	0.795047i	−0.917592	−	0.397524i	$-0.869870\pi$
$953$		−	24.5437i		−	0.795047i	0.917592	−	0.397524i	$-0.130130\pi$
$954$		0			0
$955$			62.1138i			2.00995i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			33.1113i			1.06922i
$960$		0			0
$961$	−23.4674			−0.757012
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	32.8713i		−	1.05816i
$966$		0			0
$967$		−	36.9253i		−	1.18744i	−0.804673	−	0.593719i	$-0.797659\pi$
$967$		−	36.9253i		−	1.18744i	0.804673	−	0.593719i	$-0.202341\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.4674			0.785195			0.392598	−	0.919710i	$-0.371576\pi$
$971$	24.4674			0.785195			0.392598	+	0.919710i	$0.371576\pi$
$972$		0			0
$973$	44.6060			1.43000
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	33.4612i		−	1.07052i	−0.844688	−	0.535259i	$-0.820214\pi$
$977$		−	33.4612i		−	1.07052i	0.844688	−	0.535259i	$-0.179786\pi$
$978$		0			0
$979$	33.4891			1.07032
$980$		0			0
$981$		0			0
$982$		0			0
$983$	50.9783			1.62595			0.812977	−	0.582296i	$-0.197846\pi$
$983$	50.9783			1.62595			0.812977	+	0.582296i	$0.197846\pi$
$984$		0			0
$985$	−39.7228			−1.26567
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.21194			−0.0385374
$990$		0			0
$991$	−2.74456			−0.0871839			−0.0435920	−	0.999049i	$-0.513880\pi$
$991$	−2.74456			−0.0871839			−0.0435920	+	0.999049i	$0.513880\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	43.3185i		−	1.37329i
$996$		0			0
$997$	−13.3940			−0.424193			−0.212097	−	0.977249i	$-0.568029\pi$
$997$	−13.3940			−0.424193			−0.212097	+	0.977249i	$0.568029\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.k.o.2431.1		4
3.2	odd	2		912.2.k.h.607.3	yes	4
4.3	odd	2		2736.2.k.n.2431.2		4
12.11	even	2		912.2.k.g.607.4	yes	4
19.18	odd	2		2736.2.k.n.2431.1		4
24.5	odd	2		3648.2.k.g.2431.1		4
24.11	even	2		3648.2.k.h.2431.2		4
57.56	even	2		912.2.k.g.607.3	✓	4
76.75	even	2	inner	2736.2.k.o.2431.2		4
228.227	odd	2		912.2.k.h.607.4	yes	4
456.227	odd	2		3648.2.k.g.2431.2		4
456.341	even	2		3648.2.k.h.2431.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.k.g.607.3	✓	4	57.56	even	2
912.2.k.g.607.4	yes	4	12.11	even	2
912.2.k.h.607.3	yes	4	3.2	odd	2
912.2.k.h.607.4	yes	4	228.227	odd	2
2736.2.k.n.2431.1		4	19.18	odd	2
2736.2.k.n.2431.2		4	4.3	odd	2
2736.2.k.o.2431.1		4	1.1	even	1	trivial
2736.2.k.o.2431.2		4	76.75	even	2	inner
3648.2.k.g.2431.1		4	24.5	odd	2
3648.2.k.g.2431.2		4	456.227	odd	2
3648.2.k.h.2431.1		4	456.341	even	2
3648.2.k.h.2431.2		4	24.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.37228 q^{5} -2.52434i q^{7} +2.52434i q^{11} +1.58457i q^{13} +0.372281 q^{17} +(4.00000 - 1.73205i) q^{19} +1.87953i q^{23} +0.627719 q^{25} +3.16915i q^{29} -2.74456 q^{31} +5.98844i q^{35} +1.58457i q^{37} -6.92820i q^{41} +0.644810i q^{43} +0.939764i q^{47} +0.627719 q^{49} -10.0974i q^{53} -5.98844i q^{55} +4.00000 q^{59} -0.372281 q^{61} -3.75906i q^{65} +13.4891 q^{67} +4.00000 q^{71} +13.1168 q^{73} +6.37228 q^{77} -6.74456 q^{79} +3.46410i q^{83} -0.883156 q^{85} -13.2665i q^{89} +4.00000 q^{91} +(-9.48913 + 4.10891i) q^{95} -13.2665i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - 10 q^{17} + 16 q^{19} + 14 q^{25} + 12 q^{31} + 14 q^{49} + 16 q^{59} + 10 q^{61} + 8 q^{67} + 16 q^{71} + 18 q^{73} + 14 q^{77} - 4 q^{79} - 38 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) \) 4 * q + 2 * q^5 - 10 * q^17 + 16 * q^19 + 14 * q^25 + 12 * q^31 + 14 * q^49 + 16 * q^59 + 10 * q^61 + 8 * q^67 + 16 * q^71 + 18 * q^73 + 14 * q^77 - 4 * q^79 - 38 * q^85 + 16 * q^91 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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