LMFDB - Embedded newform 3744.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.a (trivial)

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:	yes
Analytic conductor:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	3744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{5} -5.23607 q^{7} +O(q^{10})$	$q-1.23607 q^{5} -5.23607 q^{7} +4.47214 q^{11} -1.00000 q^{13} -4.47214 q^{17} +1.23607 q^{19} -2.47214 q^{23} -3.47214 q^{25} +8.47214 q^{29} -9.23607 q^{31} +6.47214 q^{35} -4.47214 q^{37} +9.23607 q^{41} -6.47214 q^{43} -0.472136 q^{47} +20.4164 q^{49} -0.472136 q^{53} -5.52786 q^{55} -6.94427 q^{59} +3.52786 q^{61} +1.23607 q^{65} +7.70820 q^{67} +10.0000 q^{71} -4.47214 q^{73} -23.4164 q^{77} -8.94427 q^{79} -2.94427 q^{83} +5.52786 q^{85} +15.7082 q^{89} +5.23607 q^{91} -1.52786 q^{95} +16.4721 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 6 * q^7	$ 2 q + 2 q^{5} - 6 q^{7} - 2 q^{13} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{29} - 14 q^{31} + 4 q^{35} + 14 q^{41} - 4 q^{43} + 8 q^{47} + 14 q^{49} + 8 q^{53} - 20 q^{55} + 4 q^{59} + 16 q^{61} - 2 q^{65} + 2 q^{67} + 20 q^{71} - 20 q^{77} + 12 q^{83} + 20 q^{85} + 18 q^{89} + 6 q^{91} - 12 q^{95} + 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 6 * q^7 - 2 * q^13 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 8 * q^29 - 14 * q^31 + 4 * q^35 + 14 * q^41 - 4 * q^43 + 8 * q^47 + 14 * q^49 + 8 * q^53 - 20 * q^55 + 4 * q^59 + 16 * q^61 - 2 * q^65 + 2 * q^67 + 20 * q^71 - 20 * q^77 + 12 * q^83 + 20 * q^85 + 18 * q^89 + 6 * q^91 - 12 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−5.23607	−1.97905	−0.989524	−	0.144370i	$-0.953885\pi$
$7$	−5.23607	−1.97905	−0.989524	+	0.144370i	$0.953885\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	1.23607	0.283573	0.141787	−	0.989897i	$-0.454715\pi$
$19$	1.23607	0.283573	0.141787	+	0.989897i	$0.454715\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.47214	−0.515476	−0.257738	−	0.966215i	$-0.582977\pi$
$23$	−2.47214	−0.515476	−0.257738	+	0.966215i	$0.582977\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	−9.23607	−1.65885	−0.829423	−	0.558620i	$-0.811331\pi$
$31$	−9.23607	−1.65885	−0.829423	+	0.558620i	$0.811331\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.47214	1.09399
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.23607	1.44243	0.721216	−	0.692711i	$-0.243584\pi$
$41$	9.23607	1.44243	0.721216	+	0.692711i	$0.243584\pi$
$42$	0	0
$43$	−6.47214	−0.986991	−0.493496	−	0.869748i	$-0.664281\pi$
$43$	−6.47214	−0.986991	−0.493496	+	0.869748i	$0.664281\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.472136	−0.0688681	−0.0344341	−	0.999407i	$-0.510963\pi$
$47$	−0.472136	−0.0688681	−0.0344341	+	0.999407i	$0.510963\pi$
$48$	0	0
$49$	20.4164	2.91663
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	−5.52786	−0.745377
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.94427	−0.904067	−0.452034	−	0.892001i	$-0.649301\pi$
$59$	−6.94427	−0.904067	−0.452034	+	0.892001i	$0.649301\pi$
$60$	0	0
$61$	3.52786	0.451697	0.225848	−	0.974162i	$-0.427485\pi$
$61$	3.52786	0.451697	0.225848	+	0.974162i	$0.427485\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.23607	0.153315
$66$	0	0
$67$	7.70820	0.941707	0.470853	−	0.882211i	$-0.343946\pi$
$67$	7.70820	0.941707	0.470853	+	0.882211i	$0.343946\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	−4.47214	−0.523424	−0.261712	−	0.965146i	$-0.584287\pi$
$73$	−4.47214	−0.523424	−0.261712	+	0.965146i	$0.584287\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−23.4164	−2.66855
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.94427	−0.323176	−0.161588	−	0.986858i	$-0.551662\pi$
$83$	−2.94427	−0.323176	−0.161588	+	0.986858i	$0.551662\pi$
$84$	0	0
$85$	5.52786	0.599581
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.7082	1.66507	0.832533	−	0.553975i	$-0.186890\pi$
$89$	15.7082	1.66507	0.832533	+	0.553975i	$0.186890\pi$
$90$	0	0
$91$	5.23607	0.548889
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.52786	−0.156756
$96$	0	0
$97$	16.4721	1.67249	0.836246	−	0.548354i	$-0.184746\pi$
$97$	16.4721	1.67249	0.836246	+	0.548354i	$0.184746\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	11.4164	1.12489	0.562446	−	0.826834i	$-0.309860\pi$
$103$	11.4164	1.12489	0.562446	+	0.826834i	$0.309860\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.47214	0.625685	0.312842	−	0.949805i	$-0.398719\pi$
$107$	6.47214	0.625685	0.312842	+	0.949805i	$0.398719\pi$
$108$	0	0
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.94427	0.653262	0.326631	−	0.945152i	$-0.394087\pi$
$113$	6.94427	0.653262	0.326631	+	0.945152i	$0.394087\pi$
$114$	0	0
$115$	3.05573	0.284948
$116$	0	0
$117$	0	0
$118$	0	0
$119$	23.4164	2.14658
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	−11.4164	−1.01304	−0.506521	−	0.862228i	$-0.669069\pi$
$127$	−11.4164	−1.01304	−0.506521	+	0.862228i	$0.669069\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.52786	−0.832453	−0.416227	−	0.909261i	$-0.636648\pi$
$131$	−9.52786	−0.832453	−0.416227	+	0.909261i	$0.636648\pi$
$132$	0	0
$133$	−6.47214	−0.561205
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.65248	−0.739231	−0.369615	−	0.929185i	$-0.620511\pi$
$137$	−8.65248	−0.739231	−0.369615	+	0.929185i	$0.620511\pi$
$138$	0	0
$139$	16.9443	1.43719	0.718597	−	0.695427i	$-0.244785\pi$
$139$	16.9443	1.43719	0.718597	+	0.695427i	$0.244785\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.47214	−0.373979
$144$	0	0
$145$	−10.4721	−0.869664
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.1803	0.834006	0.417003	−	0.908905i	$-0.363080\pi$
$149$	10.1803	0.834006	0.417003	+	0.908905i	$0.363080\pi$
$150$	0	0
$151$	14.1803	1.15398	0.576990	−	0.816751i	$-0.304227\pi$
$151$	14.1803	1.15398	0.576990	+	0.816751i	$0.304227\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.4164	0.916988
$156$	0	0
$157$	−3.52786	−0.281554	−0.140777	−	0.990041i	$-0.544960\pi$
$157$	−3.52786	−0.281554	−0.140777	+	0.990041i	$0.544960\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	12.9443	1.02015
$162$	0	0
$163$	6.76393	0.529792	0.264896	−	0.964277i	$-0.414662\pi$
$163$	6.76393	0.529792	0.264896	+	0.964277i	$0.414662\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.4721	0.965123	0.482561	−	0.875862i	$-0.339707\pi$
$167$	12.4721	0.965123	0.482561	+	0.875862i	$0.339707\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.9443	0.832078	0.416039	−	0.909347i	$-0.363418\pi$
$173$	10.9443	0.832078	0.416039	+	0.909347i	$0.363418\pi$
$174$	0	0
$175$	18.1803	1.37430
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.52786	−0.413172	−0.206586	−	0.978428i	$-0.566235\pi$
$179$	−5.52786	−0.413172	−0.206586	+	0.978428i	$0.566235\pi$
$180$	0	0
$181$	25.4164	1.88919	0.944593	−	0.328243i	$-0.106456\pi$
$181$	25.4164	1.88919	0.944593	+	0.328243i	$0.106456\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.52786	0.406417
$186$	0	0
$187$	−20.0000	−1.46254
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.41641	0.247203	0.123601	−	0.992332i	$-0.460556\pi$
$191$	3.41641	0.247203	0.123601	+	0.992332i	$0.460556\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.6525	1.18644	0.593220	−	0.805040i	$-0.297856\pi$
$197$	16.6525	1.18644	0.593220	+	0.805040i	$0.297856\pi$
$198$	0	0
$199$	−1.52786	−0.108307	−0.0541537	−	0.998533i	$-0.517246\pi$
$199$	−1.52786	−0.108307	−0.0541537	+	0.998533i	$0.517246\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−44.3607	−3.11351
$204$	0	0
$205$	−11.4164	−0.797357
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.52786	0.382370
$210$	0	0
$211$	−8.94427	−0.615749	−0.307875	−	0.951427i	$-0.599618\pi$
$211$	−8.94427	−0.615749	−0.307875	+	0.951427i	$0.599618\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	48.3607	3.28294
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.47214	0.300828
$222$	0	0
$223$	−6.18034	−0.413866	−0.206933	−	0.978355i	$-0.566348\pi$
$223$	−6.18034	−0.413866	−0.206933	+	0.978355i	$0.566348\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.41641	−0.0940103	−0.0470051	−	0.998895i	$-0.514968\pi$
$227$	−1.41641	−0.0940103	−0.0470051	+	0.998895i	$0.514968\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.4721	1.86527	0.932636	−	0.360819i	$-0.117503\pi$
$233$	28.4721	1.86527	0.932636	+	0.360819i	$0.117503\pi$
$234$	0	0
$235$	0.583592	0.0380694
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.8885	−1.54522	−0.772611	−	0.634880i	$-0.781050\pi$
$239$	−23.8885	−1.54522	−0.772611	+	0.634880i	$0.781050\pi$
$240$	0	0
$241$	−1.41641	−0.0912389	−0.0456194	−	0.998959i	$-0.514526\pi$
$241$	−1.41641	−0.0912389	−0.0456194	+	0.998959i	$0.514526\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−25.2361	−1.61227
$246$	0	0
$247$	−1.23607	−0.0786491
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	0	0
$253$	−11.0557	−0.695068
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.41641	0.0883531	0.0441765	−	0.999024i	$-0.485934\pi$
$257$	1.41641	0.0883531	0.0441765	+	0.999024i	$0.485934\pi$
$258$	0	0
$259$	23.4164	1.45502
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.0000	1.72655	0.863277	−	0.504730i	$-0.168408\pi$
$263$	28.0000	1.72655	0.863277	+	0.504730i	$0.168408\pi$
$264$	0	0
$265$	0.583592	0.0358498
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−17.4164	−1.06190	−0.530949	−	0.847404i	$-0.678164\pi$
$269$	−17.4164	−1.06190	−0.530949	+	0.847404i	$0.678164\pi$
$270$	0	0
$271$	12.6525	0.768583	0.384292	−	0.923212i	$-0.374446\pi$
$271$	12.6525	0.768583	0.384292	+	0.923212i	$0.374446\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.5279	−0.936365
$276$	0	0
$277$	1.05573	0.0634326	0.0317163	−	0.999497i	$-0.489903\pi$
$277$	1.05573	0.0634326	0.0317163	+	0.999497i	$0.489903\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.29180	0.256027	0.128014	−	0.991772i	$-0.459140\pi$
$281$	4.29180	0.256027	0.128014	+	0.991772i	$0.459140\pi$
$282$	0	0
$283$	−30.4721	−1.81138	−0.905690	−	0.423940i	$-0.860647\pi$
$283$	−30.4721	−1.81138	−0.905690	+	0.423940i	$0.860647\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−48.3607	−2.85464
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−33.5967	−1.96274	−0.981371	−	0.192120i	$-0.938464\pi$
$293$	−33.5967	−1.96274	−0.981371	+	0.192120i	$0.938464\pi$
$294$	0	0
$295$	8.58359	0.499756
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.47214	0.142967
$300$	0	0
$301$	33.8885	1.95330
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.36068	−0.249692
$306$	0	0
$307$	8.29180	0.473238	0.236619	−	0.971603i	$-0.423961\pi$
$307$	8.29180	0.473238	0.236619	+	0.971603i	$0.423961\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.4721	−1.50110	−0.750549	−	0.660815i	$-0.770211\pi$
$311$	−26.4721	−1.50110	−0.750549	+	0.660815i	$0.770211\pi$
$312$	0	0
$313$	−3.88854	−0.219793	−0.109897	−	0.993943i	$-0.535052\pi$
$313$	−3.88854	−0.219793	−0.109897	+	0.993943i	$0.535052\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.6525	0.935296	0.467648	−	0.883915i	$-0.345101\pi$
$317$	16.6525	0.935296	0.467648	+	0.883915i	$0.345101\pi$
$318$	0	0
$319$	37.8885	2.12135
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.52786	−0.307579
$324$	0	0
$325$	3.47214	0.192599
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.47214	0.136293
$330$	0	0
$331$	−13.2361	−0.727520	−0.363760	−	0.931493i	$-0.618507\pi$
$331$	−13.2361	−0.727520	−0.363760	+	0.931493i	$0.618507\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.52786	−0.520563
$336$	0	0
$337$	−22.9443	−1.24985	−0.624927	−	0.780683i	$-0.714871\pi$
$337$	−22.9443	−1.24985	−0.624927	+	0.780683i	$0.714871\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−41.3050	−2.23679
$342$	0	0
$343$	−70.2492	−3.79310
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	10.3607	0.554594	0.277297	−	0.960784i	$-0.410561\pi$
$349$	10.3607	0.554594	0.277297	+	0.960784i	$0.410561\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.2918	0.654226	0.327113	−	0.944985i	$-0.393924\pi$
$353$	12.2918	0.654226	0.327113	+	0.944985i	$0.393924\pi$
$354$	0	0
$355$	−12.3607	−0.656037
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.4164	0.919203	0.459601	−	0.888125i	$-0.347992\pi$
$359$	17.4164	0.919203	0.459601	+	0.888125i	$0.347992\pi$
$360$	0	0
$361$	−17.4721	−0.919586
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.52786	0.289342
$366$	0	0
$367$	−15.0557	−0.785903	−0.392951	−	0.919559i	$-0.628546\pi$
$367$	−15.0557	−0.785903	−0.392951	+	0.919559i	$0.628546\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.47214	0.128347
$372$	0	0
$373$	−10.9443	−0.566673	−0.283336	−	0.959021i	$-0.591441\pi$
$373$	−10.9443	−0.566673	−0.283336	+	0.959021i	$0.591441\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.47214	−0.436337
$378$	0	0
$379$	32.0689	1.64727	0.823634	−	0.567122i	$-0.191943\pi$
$379$	32.0689	1.64727	0.823634	+	0.567122i	$0.191943\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.41641	0.276766	0.138383	−	0.990379i	$-0.455810\pi$
$383$	5.41641	0.276766	0.138383	+	0.990379i	$0.455810\pi$
$384$	0	0
$385$	28.9443	1.47514
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.88854	0.197157	0.0985785	−	0.995129i	$-0.468570\pi$
$389$	3.88854	0.197157	0.0985785	+	0.995129i	$0.468570\pi$
$390$	0	0
$391$	11.0557	0.559112
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.0557	0.556274
$396$	0	0
$397$	3.52786	0.177058	0.0885292	−	0.996074i	$-0.471783\pi$
$397$	3.52786	0.177058	0.0885292	+	0.996074i	$0.471783\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.29180	−0.414073	−0.207036	−	0.978333i	$-0.566382\pi$
$401$	−8.29180	−0.414073	−0.207036	+	0.978333i	$0.566382\pi$
$402$	0	0
$403$	9.23607	0.460081
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	−22.3607	−1.10566	−0.552832	−	0.833293i	$-0.686453\pi$
$409$	−22.3607	−1.10566	−0.552832	+	0.833293i	$0.686453\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	36.3607	1.78919
$414$	0	0
$415$	3.63932	0.178647
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.4721	0.707010	0.353505	−	0.935433i	$-0.384990\pi$
$419$	14.4721	0.707010	0.353505	+	0.935433i	$0.384990\pi$
$420$	0	0
$421$	−10.9443	−0.533391	−0.266696	−	0.963781i	$-0.585932\pi$
$421$	−10.9443	−0.533391	−0.266696	+	0.963781i	$0.585932\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.5279	0.753212
$426$	0	0
$427$	−18.4721	−0.893929
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.4721	−0.600762	−0.300381	−	0.953819i	$-0.597114\pi$
$431$	−12.4721	−0.600762	−0.300381	+	0.953819i	$0.597114\pi$
$432$	0	0
$433$	1.41641	0.0680682	0.0340341	−	0.999421i	$-0.489165\pi$
$433$	1.41641	0.0680682	0.0340341	+	0.999421i	$0.489165\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.05573	−0.146175
$438$	0	0
$439$	19.4164	0.926695	0.463347	−	0.886177i	$-0.346648\pi$
$439$	19.4164	0.926695	0.463347	+	0.886177i	$0.346648\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.52786	0.452682	0.226341	−	0.974048i	$-0.427324\pi$
$443$	9.52786	0.452682	0.226341	+	0.974048i	$0.427324\pi$
$444$	0	0
$445$	−19.4164	−0.920426
$446$	0	0
$447$	0	0
$448$	0	0
$449$	36.6525	1.72974	0.864869	−	0.501998i	$-0.167402\pi$
$449$	36.6525	1.72974	0.864869	+	0.501998i	$0.167402\pi$
$450$	0	0
$451$	41.3050	1.94497
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.47214	−0.303418
$456$	0	0
$457$	27.5279	1.28770	0.643850	−	0.765152i	$-0.277336\pi$
$457$	27.5279	1.28770	0.643850	+	0.765152i	$0.277336\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.23607	−0.0575694	−0.0287847	−	0.999586i	$-0.509164\pi$
$461$	−1.23607	−0.0575694	−0.0287847	+	0.999586i	$0.509164\pi$
$462$	0	0
$463$	13.8197	0.642254	0.321127	−	0.947036i	$-0.395938\pi$
$463$	13.8197	0.642254	0.321127	+	0.947036i	$0.395938\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.9443	1.70958	0.854789	−	0.518976i	$-0.173687\pi$
$467$	36.9443	1.70958	0.854789	+	0.518976i	$0.173687\pi$
$468$	0	0
$469$	−40.3607	−1.86368
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−28.9443	−1.33086
$474$	0	0
$475$	−4.29180	−0.196921
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	4.47214	0.203912
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20.3607	−0.924531
$486$	0	0
$487$	16.6525	0.754596	0.377298	−	0.926092i	$-0.376853\pi$
$487$	16.6525	0.754596	0.377298	+	0.926092i	$0.376853\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9.88854	−0.446264	−0.223132	−	0.974788i	$-0.571628\pi$
$491$	−9.88854	−0.446264	−0.223132	+	0.974788i	$0.571628\pi$
$492$	0	0
$493$	−37.8885	−1.70641
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−52.3607	−2.34870
$498$	0	0
$499$	−29.2361	−1.30879	−0.654393	−	0.756155i	$-0.727076\pi$
$499$	−29.2361	−1.30879	−0.654393	+	0.756155i	$0.727076\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.9443	−1.11221	−0.556105	−	0.831112i	$-0.687705\pi$
$503$	−24.9443	−1.11221	−0.556105	+	0.831112i	$0.687705\pi$
$504$	0	0
$505$	12.3607	0.550043
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.1803	−1.69231	−0.846157	−	0.532934i	$-0.821089\pi$
$509$	−38.1803	−1.69231	−0.846157	+	0.532934i	$0.821089\pi$
$510$	0	0
$511$	23.4164	1.03588
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.1115	−0.621825
$516$	0	0
$517$	−2.11146	−0.0928617
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.111456	0.00488298	0.00244149	−	0.999997i	$-0.499223\pi$
$521$	0.111456	0.00488298	0.00244149	+	0.999997i	$0.499223\pi$
$522$	0	0
$523$	−2.11146	−0.0923275	−0.0461638	−	0.998934i	$-0.514700\pi$
$523$	−2.11146	−0.0923275	−0.0461638	+	0.998934i	$0.514700\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	41.3050	1.79927
$528$	0	0
$529$	−16.8885	−0.734285
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.23607	−0.400059
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	0	0
$538$	0	0
$539$	91.3050	3.93278
$540$	0	0
$541$	37.7771	1.62416	0.812082	−	0.583543i	$-0.198334\pi$
$541$	37.7771	1.62416	0.812082	+	0.583543i	$0.198334\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−18.4721	−0.791259
$546$	0	0
$547$	39.7771	1.70075	0.850373	−	0.526181i	$-0.176376\pi$
$547$	39.7771	1.70075	0.850373	+	0.526181i	$0.176376\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.4721	0.446128
$552$	0	0
$553$	46.8328	1.99153
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.0689	1.18932	0.594658	−	0.803978i	$-0.297287\pi$
$557$	28.0689	1.18932	0.594658	+	0.803978i	$0.297287\pi$
$558$	0	0
$559$	6.47214	0.273742
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.47214	−0.104188	−0.0520941	−	0.998642i	$-0.516590\pi$
$563$	−2.47214	−0.104188	−0.0520941	+	0.998642i	$0.516590\pi$
$564$	0	0
$565$	−8.58359	−0.361114
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.41641	0.394756	0.197378	−	0.980327i	$-0.436757\pi$
$569$	9.41641	0.394756	0.197378	+	0.980327i	$0.436757\pi$
$570$	0	0
$571$	−11.4164	−0.477762	−0.238881	−	0.971049i	$-0.576781\pi$
$571$	−11.4164	−0.477762	−0.238881	+	0.971049i	$0.576781\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.58359	0.357961
$576$	0	0
$577$	24.8328	1.03380	0.516902	−	0.856045i	$-0.327085\pi$
$577$	24.8328	1.03380	0.516902	+	0.856045i	$0.327085\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.4164	0.639580
$582$	0	0
$583$	−2.11146	−0.0874476
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.9443	0.947011	0.473506	−	0.880791i	$-0.342988\pi$
$587$	22.9443	0.947011	0.473506	+	0.880791i	$0.342988\pi$
$588$	0	0
$589$	−11.4164	−0.470405
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.65248	0.191054	0.0955271	−	0.995427i	$-0.469546\pi$
$593$	4.65248	0.191054	0.0955271	+	0.995427i	$0.469546\pi$
$594$	0	0
$595$	−28.9443	−1.18660
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−35.8885	−1.46392	−0.731962	−	0.681345i	$-0.761395\pi$
$601$	−35.8885	−1.46392	−0.731962	+	0.681345i	$0.761395\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.1246	−0.452280
$606$	0	0
$607$	16.9443	0.687747	0.343873	−	0.939016i	$-0.388261\pi$
$607$	16.9443	0.687747	0.343873	+	0.939016i	$0.388261\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.472136	0.0191006
$612$	0	0
$613$	0.472136	0.0190694	0.00953470	−	0.999955i	$-0.496965\pi$
$613$	0.472136	0.0190694	0.00953470	+	0.999955i	$0.496965\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.0689	1.61311	0.806556	−	0.591157i	$-0.201329\pi$
$617$	40.0689	1.61311	0.806556	+	0.591157i	$0.201329\pi$
$618$	0	0
$619$	−9.59675	−0.385726	−0.192863	−	0.981226i	$-0.561777\pi$
$619$	−9.59675	−0.385726	−0.192863	+	0.981226i	$0.561777\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−82.2492	−3.29525
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.0000	0.797452
$630$	0	0
$631$	−11.7082	−0.466096	−0.233048	−	0.972465i	$-0.574870\pi$
$631$	−11.7082	−0.466096	−0.233048	+	0.972465i	$0.574870\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.1115	0.559996
$636$	0	0
$637$	−20.4164	−0.808928
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.47214	−0.176639	−0.0883194	−	0.996092i	$-0.528150\pi$
$641$	−4.47214	−0.176639	−0.0883194	+	0.996092i	$0.528150\pi$
$642$	0	0
$643$	−42.1803	−1.66343	−0.831715	−	0.555203i	$-0.812641\pi$
$643$	−42.1803	−1.66343	−0.831715	+	0.555203i	$0.812641\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	0	0
$649$	−31.0557	−1.21904
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.4721	−0.801137	−0.400568	−	0.916267i	$-0.631187\pi$
$653$	−20.4721	−0.801137	−0.400568	+	0.916267i	$0.631187\pi$
$654$	0	0
$655$	11.7771	0.460169
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10.8328	−0.421987	−0.210993	−	0.977488i	$-0.567670\pi$
$659$	−10.8328	−0.421987	−0.210993	+	0.977488i	$0.567670\pi$
$660$	0	0
$661$	−28.4721	−1.10744	−0.553719	−	0.832704i	$-0.686792\pi$
$661$	−28.4721	−1.10744	−0.553719	+	0.832704i	$0.686792\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	−20.9443	−0.810965
$668$	0	0
$669$	0	0
$670$	0	0
$671$	15.7771	0.609068
$672$	0	0
$673$	−44.4721	−1.71427	−0.857137	−	0.515088i	$-0.827759\pi$
$673$	−44.4721	−1.71427	−0.857137	+	0.515088i	$0.827759\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.41641	0.0544370	0.0272185	−	0.999630i	$-0.491335\pi$
$677$	1.41641	0.0544370	0.0272185	+	0.999630i	$0.491335\pi$
$678$	0	0
$679$	−86.2492	−3.30994
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29.4164	−1.12559	−0.562794	−	0.826597i	$-0.690273\pi$
$683$	−29.4164	−1.12559	−0.562794	+	0.826597i	$0.690273\pi$
$684$	0	0
$685$	10.6950	0.408637
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.472136	0.0179869
$690$	0	0
$691$	−7.12461	−0.271033	−0.135517	−	0.990775i	$-0.543269\pi$
$691$	−7.12461	−0.271033	−0.135517	+	0.990775i	$0.543269\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.9443	−0.794462
$696$	0	0
$697$	−41.3050	−1.56454
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.8885	1.05334	0.526668	−	0.850071i	$-0.323441\pi$
$701$	27.8885	1.05334	0.526668	+	0.850071i	$0.323441\pi$
$702$	0	0
$703$	−5.52786	−0.208487
$704$	0	0
$705$	0	0
$706$	0	0
$707$	52.3607	1.96923
$708$	0	0
$709$	−39.8885	−1.49805	−0.749023	−	0.662544i	$-0.769477\pi$
$709$	−39.8885	−1.49805	−0.749023	+	0.662544i	$0.769477\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22.8328	0.855096
$714$	0	0
$715$	5.52786	0.206730
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.47214	−0.0921951	−0.0460976	−	0.998937i	$-0.514679\pi$
$719$	−2.47214	−0.0921951	−0.0460976	+	0.998937i	$0.514679\pi$
$720$	0	0
$721$	−59.7771	−2.22622
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−29.4164	−1.09250
$726$	0	0
$727$	−16.3607	−0.606784	−0.303392	−	0.952866i	$-0.598119\pi$
$727$	−16.3607	−0.606784	−0.303392	+	0.952866i	$0.598119\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	28.9443	1.07054
$732$	0	0
$733$	27.8885	1.03009	0.515043	−	0.857164i	$-0.327776\pi$
$733$	27.8885	1.03009	0.515043	+	0.857164i	$0.327776\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	34.4721	1.26980
$738$	0	0
$739$	13.2361	0.486897	0.243448	−	0.969914i	$-0.421721\pi$
$739$	13.2361	0.486897	0.243448	+	0.969914i	$0.421721\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.0557	0.772460	0.386230	−	0.922403i	$-0.373777\pi$
$743$	21.0557	0.772460	0.386230	+	0.922403i	$0.373777\pi$
$744$	0	0
$745$	−12.5836	−0.461027
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−33.8885	−1.23826
$750$	0	0
$751$	25.5279	0.931525	0.465762	−	0.884910i	$-0.345780\pi$
$751$	25.5279	0.931525	0.465762	+	0.884910i	$0.345780\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.5279	−0.637904
$756$	0	0
$757$	5.41641	0.196863	0.0984313	−	0.995144i	$-0.468618\pi$
$757$	5.41641	0.196863	0.0984313	+	0.995144i	$0.468618\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.81966	−0.0659626	−0.0329813	−	0.999456i	$-0.510500\pi$
$761$	−1.81966	−0.0659626	−0.0329813	+	0.999456i	$0.510500\pi$
$762$	0	0
$763$	−78.2492	−2.83281
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.94427	0.250743
$768$	0	0
$769$	14.9443	0.538904	0.269452	−	0.963014i	$-0.413157\pi$
$769$	14.9443	0.538904	0.269452	+	0.963014i	$0.413157\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21.8197	−0.784799	−0.392399	−	0.919795i	$-0.628355\pi$
$773$	−21.8197	−0.784799	−0.392399	+	0.919795i	$0.628355\pi$
$774$	0	0
$775$	32.0689	1.15195
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.4164	0.409035
$780$	0	0
$781$	44.7214	1.60026
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.36068	0.155639
$786$	0	0
$787$	−34.7639	−1.23920	−0.619600	−	0.784918i	$-0.712705\pi$
$787$	−34.7639	−1.23920	−0.619600	+	0.784918i	$0.712705\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−36.3607	−1.29284
$792$	0	0
$793$	−3.52786	−0.125278
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−41.7771	−1.47982	−0.739910	−	0.672706i	$-0.765132\pi$
$797$	−41.7771	−1.47982	−0.739910	+	0.672706i	$0.765132\pi$
$798$	0	0
$799$	2.11146	0.0746979
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	2.40325	0.0843896	0.0421948	−	0.999109i	$-0.486565\pi$
$811$	2.40325	0.0843896	0.0421948	+	0.999109i	$0.486565\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.36068	−0.292862
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.12461	0.248651	0.124325	−	0.992242i	$-0.460323\pi$
$821$	7.12461	0.248651	0.124325	+	0.992242i	$0.460323\pi$
$822$	0	0
$823$	−6.47214	−0.225604	−0.112802	−	0.993617i	$-0.535983\pi$
$823$	−6.47214	−0.225604	−0.112802	+	0.993617i	$0.535983\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37.4164	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$827$	−37.4164	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$828$	0	0
$829$	16.4721	0.572101	0.286050	−	0.958215i	$-0.407657\pi$
$829$	16.4721	0.572101	0.286050	+	0.958215i	$0.407657\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−91.3050	−3.16353
$834$	0	0
$835$	−15.4164	−0.533507
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.88854	−0.134247	−0.0671237	−	0.997745i	$-0.521382\pi$
$839$	−3.88854	−0.134247	−0.0671237	+	0.997745i	$0.521382\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.23607	−0.0425220
$846$	0	0
$847$	−47.1246	−1.61922
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.0557	0.378985
$852$	0	0
$853$	−41.4164	−1.41807	−0.709035	−	0.705173i	$-0.750869\pi$
$853$	−41.4164	−1.41807	−0.709035	+	0.705173i	$0.750869\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.4721	−0.562677	−0.281339	−	0.959609i	$-0.590778\pi$
$857$	−16.4721	−0.562677	−0.281339	+	0.959609i	$0.590778\pi$
$858$	0	0
$859$	16.9443	0.578131	0.289066	−	0.957309i	$-0.406655\pi$
$859$	16.9443	0.578131	0.289066	+	0.957309i	$0.406655\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	55.3050	1.88260	0.941301	−	0.337568i	$-0.109604\pi$
$863$	55.3050	1.88260	0.941301	+	0.337568i	$0.109604\pi$
$864$	0	0
$865$	−13.5279	−0.459961
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	−7.70820	−0.261183
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−54.8328	−1.85369
$876$	0	0
$877$	11.5279	0.389268	0.194634	−	0.980876i	$-0.437648\pi$
$877$	11.5279	0.389268	0.194634	+	0.980876i	$0.437648\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−19.5279	−0.657910	−0.328955	−	0.944346i	$-0.606697\pi$
$881$	−19.5279	−0.657910	−0.328955	+	0.944346i	$0.606697\pi$
$882$	0	0
$883$	−14.4721	−0.487026	−0.243513	−	0.969898i	$-0.578300\pi$
$883$	−14.4721	−0.487026	−0.243513	+	0.969898i	$0.578300\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.5836	0.691129	0.345565	−	0.938395i	$-0.387687\pi$
$887$	20.5836	0.691129	0.345565	+	0.938395i	$0.387687\pi$
$888$	0	0
$889$	59.7771	2.00486
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.583592	−0.0195292
$894$	0	0
$895$	6.83282	0.228396
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−78.2492	−2.60976
$900$	0	0
$901$	2.11146	0.0703428
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.4164	−1.04432
$906$	0	0
$907$	13.3050	0.441784	0.220892	−	0.975298i	$-0.429103\pi$
$907$	13.3050	0.441784	0.220892	+	0.975298i	$0.429103\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−33.5279	−1.11083	−0.555414	−	0.831574i	$-0.687440\pi$
$911$	−33.5279	−1.11083	−0.555414	+	0.831574i	$0.687440\pi$
$912$	0	0
$913$	−13.1672	−0.435770
$914$	0	0
$915$	0	0
$916$	0	0
$917$	49.8885	1.64746
$918$	0	0
$919$	5.88854	0.194245	0.0971226	−	0.995272i	$-0.469036\pi$
$919$	5.88854	0.194245	0.0971226	+	0.995272i	$0.469036\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.0000	−0.329154
$924$	0	0
$925$	15.5279	0.510553
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.6525	−0.808821	−0.404411	−	0.914577i	$-0.632523\pi$
$929$	−24.6525	−0.808821	−0.404411	+	0.914577i	$0.632523\pi$
$930$	0	0
$931$	25.2361	0.827079
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.7214	0.808475
$936$	0	0
$937$	−41.1935	−1.34573	−0.672866	−	0.739764i	$-0.734937\pi$
$937$	−41.1935	−1.34573	−0.672866	+	0.739764i	$0.734937\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	45.2361	1.47465	0.737327	−	0.675536i	$-0.236088\pi$
$941$	45.2361	1.47465	0.737327	+	0.675536i	$0.236088\pi$
$942$	0	0
$943$	−22.8328	−0.743539
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.94427	−0.225659	−0.112829	−	0.993614i	$-0.535991\pi$
$947$	−6.94427	−0.225659	−0.112829	+	0.993614i	$0.535991\pi$
$948$	0	0
$949$	4.47214	0.145172
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13.4164	−0.434600	−0.217300	−	0.976105i	$-0.569725\pi$
$953$	−13.4164	−0.434600	−0.217300	+	0.976105i	$0.569725\pi$
$954$	0	0
$955$	−4.22291	−0.136650
$956$	0	0
$957$	0	0
$958$	0	0
$959$	45.3050	1.46297
$960$	0	0
$961$	54.3050	1.75177
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.41641	0.238743
$966$	0	0
$967$	3.70820	0.119248	0.0596239	−	0.998221i	$-0.481010\pi$
$967$	3.70820	0.119248	0.0596239	+	0.998221i	$0.481010\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.8328	0.347642	0.173821	−	0.984777i	$-0.444389\pi$
$971$	10.8328	0.347642	0.173821	+	0.984777i	$0.444389\pi$
$972$	0	0
$973$	−88.7214	−2.84428
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51.4853	1.64716	0.823580	−	0.567200i	$-0.191973\pi$
$977$	51.4853	1.64716	0.823580	+	0.567200i	$0.191973\pi$
$978$	0	0
$979$	70.2492	2.24517
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.0557	0.799154	0.399577	−	0.916700i	$-0.369157\pi$
$983$	25.0557	0.799154	0.399577	+	0.916700i	$0.369157\pi$
$984$	0	0
$985$	−20.5836	−0.655848
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.88854	0.0598709
$996$	0	0
$997$	7.88854	0.249833	0.124916	−	0.992167i	$-0.460134\pi$
$997$	7.88854	0.249833	0.124916	+	0.992167i	$0.460134\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.a.v.1.1		2
3.2	odd	2		1248.2.a.m.1.2	yes	2
4.3	odd	2		3744.2.a.w.1.1		2
8.3	odd	2		7488.2.a.ch.1.2		2
8.5	even	2		7488.2.a.cg.1.2		2
12.11	even	2		1248.2.a.k.1.2	✓	2
24.5	odd	2		2496.2.a.bg.1.1		2
24.11	even	2		2496.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.a.k.1.2	✓	2	12.11	even	2
1248.2.a.m.1.2	yes	2	3.2	odd	2
2496.2.a.bg.1.1		2	24.5	odd	2
2496.2.a.bj.1.1		2	24.11	even	2
3744.2.a.v.1.1		2	1.1	even	1	trivial
3744.2.a.w.1.1		2	4.3	odd	2
7488.2.a.cg.1.2		2	8.5	even	2
7488.2.a.ch.1.2		2	8.3	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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23607 q^{5} -5.23607 q^{7} +O(q^{10})\)	\(q-1.23607 q^{5} -5.23607 q^{7} +4.47214 q^{11} -1.00000 q^{13} -4.47214 q^{17} +1.23607 q^{19} -2.47214 q^{23} -3.47214 q^{25} +8.47214 q^{29} -9.23607 q^{31} +6.47214 q^{35} -4.47214 q^{37} +9.23607 q^{41} -6.47214 q^{43} -0.472136 q^{47} +20.4164 q^{49} -0.472136 q^{53} -5.52786 q^{55} -6.94427 q^{59} +3.52786 q^{61} +1.23607 q^{65} +7.70820 q^{67} +10.0000 q^{71} -4.47214 q^{73} -23.4164 q^{77} -8.94427 q^{79} -2.94427 q^{83} +5.52786 q^{85} +15.7082 q^{89} +5.23607 q^{91} -1.52786 q^{95} +16.4721 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 6 * q^7	\( 2 q + 2 q^{5} - 6 q^{7} - 2 q^{13} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 8 q^{29} - 14 q^{31} + 4 q^{35} + 14 q^{41} - 4 q^{43} + 8 q^{47} + 14 q^{49} + 8 q^{53} - 20 q^{55} + 4 q^{59} + 16 q^{61} - 2 q^{65} + 2 q^{67} + 20 q^{71} - 20 q^{77} + 12 q^{83} + 20 q^{85} + 18 q^{89} + 6 q^{91} - 12 q^{95} + 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 6 * q^7 - 2 * q^13 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 8 * q^29 - 14 * q^31 + 4 * q^35 + 14 * q^41 - 4 * q^43 + 8 * q^47 + 14 * q^49 + 8 * q^53 - 20 * q^55 + 4 * q^59 + 16 * q^61 - 2 * q^65 + 2 * q^67 + 20 * q^71 - 20 * q^77 + 12 * q^83 + 20 * q^85 + 18 * q^89 + 6 * q^91 - 12 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more