LMFDB - Embedded newform 1764.4.a.u.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,4,Mod(1,1764)] 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("1764.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,108,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{385}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 96 $ x^2 - x - 96
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 252)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.31071$ of defining polynomial
Character	$\chi$	$=$	1764.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+19.6214 q^{5} +19.6214 q^{11} +54.0000 q^{13} +39.2428 q^{17} -2.00000 q^{19} +196.214 q^{23} +260.000 q^{25} +98.1071 q^{29} -201.000 q^{31} -202.000 q^{37} -470.914 q^{41} -244.000 q^{43} +235.457 q^{47} +647.507 q^{53} +385.000 q^{55} +608.264 q^{59} +588.000 q^{61} +1059.56 q^{65} -302.000 q^{67} -156.971 q^{71} +446.000 q^{73} -267.000 q^{79} -725.992 q^{83} +770.000 q^{85} +117.729 q^{89} -39.2428 q^{95} -595.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 108 q^{13} - 4 q^{19} + 520 q^{25} - 402 q^{31} - 404 q^{37} - 488 q^{43} + 770 q^{55} + 1176 q^{61} - 604 q^{67} + 892 q^{73} - 534 q^{79} + 1540 q^{85} - 1190 q^{97}+O(q^{100}) $ 2 * q + 108 * q^13 - 4 * q^19 + 520 * q^25 - 402 * q^31 - 404 * q^37 - 488 * q^43 + 770 * q^55 + 1176 * q^61 - 604 * q^67 + 892 * q^73 - 534 * q^79 + 1540 * q^85 - 1190 * q^97

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
$5$	19.6214	1.75499	0.877496	−	0.479583i	$-0.159212\pi$
$5$	19.6214	1.75499	0.877496	+	0.479583i	$0.159212\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	19.6214	0.537825	0.268913	−	0.963165i	$-0.413336\pi$
$11$	19.6214	0.537825	0.268913	+	0.963165i	$0.413336\pi$
$12$	0	0
$13$	54.0000	1.15207	0.576035	−	0.817425i	$-0.304599\pi$
$13$	54.0000	1.15207	0.576035	+	0.817425i	$0.304599\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	39.2428	0.559870	0.279935	−	0.960019i	$-0.409687\pi$
$17$	39.2428	0.559870	0.279935	+	0.960019i	$0.409687\pi$
$18$	0	0
$19$	−2.00000	−0.0241490	−0.0120745	−	0.999927i	$-0.503844\pi$
$19$	−2.00000	−0.0241490	−0.0120745	+	0.999927i	$0.503844\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	196.214	1.77885	0.889424	−	0.457084i	$-0.151106\pi$
$23$	196.214	1.77885	0.889424	+	0.457084i	$0.151106\pi$
$24$	0	0
$25$	260.000	2.08000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	98.1071	0.628208	0.314104	−	0.949389i	$-0.398296\pi$
$29$	98.1071	0.628208	0.314104	+	0.949389i	$0.398296\pi$
$30$	0	0
$31$	−201.000	−1.16454	−0.582269	−	0.812996i	$-0.697835\pi$
$31$	−201.000	−1.16454	−0.582269	+	0.812996i	$0.697835\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−202.000	−0.897530	−0.448765	−	0.893650i	$-0.648136\pi$
$37$	−202.000	−0.897530	−0.448765	+	0.893650i	$0.648136\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−470.914	−1.79377	−0.896883	−	0.442268i	$-0.854174\pi$
$41$	−470.914	−1.79377	−0.896883	+	0.442268i	$0.854174\pi$
$42$	0	0
$43$	−244.000	−0.865341	−0.432670	−	0.901552i	$-0.642429\pi$
$43$	−244.000	−0.865341	−0.432670	+	0.901552i	$0.642429\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	235.457	0.730743	0.365372	−	0.930862i	$-0.380942\pi$
$47$	235.457	0.730743	0.365372	+	0.930862i	$0.380942\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	647.507	1.67815	0.839074	−	0.544017i	$-0.183097\pi$
$53$	647.507	1.67815	0.839074	+	0.544017i	$0.183097\pi$
$54$	0	0
$55$	385.000	0.943880
$56$	0	0
$57$	0	0
$58$	0	0
$59$	608.264	1.34219	0.671095	−	0.741371i	$-0.265824\pi$
$59$	608.264	1.34219	0.671095	+	0.741371i	$0.265824\pi$
$60$	0	0
$61$	588.000	1.23419	0.617096	−	0.786888i	$-0.288309\pi$
$61$	588.000	1.23419	0.617096	+	0.786888i	$0.288309\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1059.56	2.02187
$66$	0	0
$67$	−302.000	−0.550674	−0.275337	−	0.961348i	$-0.588789\pi$
$67$	−302.000	−0.550674	−0.275337	+	0.961348i	$0.588789\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−156.971	−0.262381	−0.131191	−	0.991357i	$-0.541880\pi$
$71$	−156.971	−0.262381	−0.131191	+	0.991357i	$0.541880\pi$
$72$	0	0
$73$	446.000	0.715073	0.357537	−	0.933899i	$-0.383617\pi$
$73$	446.000	0.715073	0.357537	+	0.933899i	$0.383617\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−267.000	−0.380251	−0.190126	−	0.981760i	$-0.560890\pi$
$79$	−267.000	−0.380251	−0.190126	+	0.981760i	$0.560890\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−725.992	−0.960097	−0.480048	−	0.877242i	$-0.659381\pi$
$83$	−725.992	−0.960097	−0.480048	+	0.877242i	$0.659381\pi$
$84$	0	0
$85$	770.000	0.982567
$86$	0	0
$87$	0	0
$88$	0	0
$89$	117.729	0.140216	0.0701078	−	0.997539i	$-0.477666\pi$
$89$	117.729	0.140216	0.0701078	+	0.997539i	$0.477666\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−39.2428	−0.0423814
$96$	0	0
$97$	−595.000	−0.622815	−0.311408	−	0.950276i	$-0.600800\pi$
$97$	−595.000	−0.622815	−0.311408	+	0.950276i	$0.600800\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1216.53	−1.19851	−0.599253	−	0.800560i	$-0.704536\pi$
$101$	−1216.53	−1.19851	−0.599253	+	0.800560i	$0.704536\pi$
$102$	0	0
$103$	−1664.00	−1.59183	−0.795916	−	0.605406i	$-0.793011\pi$
$103$	−1664.00	−1.59183	−0.795916	+	0.605406i	$0.793011\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1589.33	−1.43595	−0.717976	−	0.696068i	$-0.754931\pi$
$107$	−1589.33	−1.43595	−0.717976	+	0.696068i	$0.754931\pi$
$108$	0	0
$109$	358.000	0.314589	0.157294	−	0.987552i	$-0.449723\pi$
$109$	358.000	0.314589	0.157294	+	0.987552i	$0.449723\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1098.80	0.914746	0.457373	−	0.889275i	$-0.348790\pi$
$113$	1098.80	0.914746	0.457373	+	0.889275i	$0.348790\pi$
$114$	0	0
$115$	3850.00	3.12186
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−946.000	−0.710744
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2648.89	1.89539
$126$	0	0
$127$	2127.00	1.48615	0.743074	−	0.669210i	$-0.233367\pi$
$127$	2127.00	1.48615	0.743074	+	0.669210i	$0.233367\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1510.85	−1.00766	−0.503830	−	0.863803i	$-0.668076\pi$
$131$	−1510.85	−1.00766	−0.503830	+	0.863803i	$0.668076\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2550.78	−1.59072	−0.795358	−	0.606139i	$-0.792717\pi$
$137$	−2550.78	−1.59072	−0.795358	+	0.606139i	$0.792717\pi$
$138$	0	0
$139$	2438.00	1.48769	0.743843	−	0.668354i	$-0.233001\pi$
$139$	2438.00	1.48769	0.743843	+	0.668354i	$0.233001\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1059.56	0.619612
$144$	0	0
$145$	1925.00	1.10250
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1530.47	−0.841483	−0.420742	−	0.907180i	$-0.638230\pi$
$149$	−1530.47	−0.841483	−0.420742	+	0.907180i	$0.638230\pi$
$150$	0	0
$151$	−947.000	−0.510369	−0.255185	−	0.966892i	$-0.582136\pi$
$151$	−947.000	−0.510369	−0.255185	+	0.966892i	$0.582136\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3943.90	−2.04376
$156$	0	0
$157$	1540.00	0.782837	0.391418	−	0.920213i	$-0.371985\pi$
$157$	1540.00	0.782837	0.391418	+	0.920213i	$0.371985\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1090.00	−0.523775	−0.261888	−	0.965098i	$-0.584345\pi$
$163$	−1090.00	−0.523775	−0.261888	+	0.965098i	$0.584345\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2079.87	0.963744	0.481872	−	0.876242i	$-0.339957\pi$
$167$	2079.87	0.963744	0.481872	+	0.876242i	$0.339957\pi$
$168$	0	0
$169$	719.000	0.327264
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1844.41	0.810567	0.405284	−	0.914191i	$-0.367173\pi$
$173$	1844.41	0.810567	0.405284	+	0.914191i	$0.367173\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1177.29	−0.491589	−0.245794	−	0.969322i	$-0.579049\pi$
$179$	−1177.29	−0.491589	−0.245794	+	0.969322i	$0.579049\pi$
$180$	0	0
$181$	2004.00	0.822962	0.411481	−	0.911418i	$-0.365012\pi$
$181$	2004.00	0.822962	0.411481	+	0.911418i	$0.365012\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3963.53	−1.57516
$186$	0	0
$187$	770.000	0.301112
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4473.68	−1.69479	−0.847394	−	0.530965i	$-0.821829\pi$
$191$	−4473.68	−1.69479	−0.847394	+	0.530965i	$0.821829\pi$
$192$	0	0
$193$	4649.00	1.73390	0.866949	−	0.498397i	$-0.166078\pi$
$193$	4649.00	1.73390	0.866949	+	0.498397i	$0.166078\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4198.98	1.51860	0.759302	−	0.650738i	$-0.225540\pi$
$197$	4198.98	1.51860	0.759302	+	0.650738i	$0.225540\pi$
$198$	0	0
$199$	616.000	0.219433	0.109716	−	0.993963i	$-0.465006\pi$
$199$	616.000	0.219433	0.109716	+	0.993963i	$0.465006\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9240.00	−3.14805
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−39.2428	−0.0129880
$210$	0	0
$211$	−3208.00	−1.04667	−0.523336	−	0.852126i	$-0.675313\pi$
$211$	−3208.00	−1.04667	−0.523336	+	0.852126i	$0.675313\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4787.63	−1.51867
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2119.11	0.645009
$222$	0	0
$223$	2065.00	0.620101	0.310051	−	0.950720i	$-0.399654\pi$
$223$	2065.00	0.620101	0.310051	+	0.950720i	$0.399654\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1824.79	0.533549	0.266775	−	0.963759i	$-0.414042\pi$
$227$	1824.79	0.533549	0.266775	+	0.963759i	$0.414042\pi$
$228$	0	0
$229$	3908.00	1.12772	0.563860	−	0.825870i	$-0.309316\pi$
$229$	3908.00	1.12772	0.563860	+	0.825870i	$0.309316\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4434.44	1.24682	0.623412	−	0.781894i	$-0.285746\pi$
$233$	4434.44	1.24682	0.623412	+	0.781894i	$0.285746\pi$
$234$	0	0
$235$	4620.00	1.28245
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1530.47	−0.414217	−0.207109	−	0.978318i	$-0.566405\pi$
$239$	−1530.47	−0.414217	−0.207109	+	0.978318i	$0.566405\pi$
$240$	0	0
$241$	3913.00	1.04589	0.522943	−	0.852368i	$-0.324834\pi$
$241$	3913.00	1.04589	0.522943	+	0.852368i	$0.324834\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−108.000	−0.0278214
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4650.28	−1.16941	−0.584707	−	0.811245i	$-0.698790\pi$
$251$	−4650.28	−1.16941	−0.584707	+	0.811245i	$0.698790\pi$
$252$	0	0
$253$	3850.00	0.956709
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2001.38	0.485770	0.242885	−	0.970055i	$-0.421906\pi$
$257$	2001.38	0.485770	0.242885	+	0.970055i	$0.421906\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−470.914	−0.110410	−0.0552049	−	0.998475i	$-0.517581\pi$
$263$	−470.914	−0.110410	−0.0552049	+	0.998475i	$0.517581\pi$
$264$	0	0
$265$	12705.0	2.94514
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2021.01	0.458078	0.229039	−	0.973417i	$-0.426442\pi$
$269$	2021.01	0.458078	0.229039	+	0.973417i	$0.426442\pi$
$270$	0	0
$271$	−1197.00	−0.268312	−0.134156	−	0.990960i	$-0.542832\pi$
$271$	−1197.00	−0.268312	−0.134156	+	0.990960i	$0.542832\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5101.57	1.11868
$276$	0	0
$277$	1532.00	0.332307	0.166153	−	0.986100i	$-0.446865\pi$
$277$	1532.00	0.332307	0.166153	+	0.986100i	$0.446865\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−510.157	−0.108304	−0.0541520	−	0.998533i	$-0.517246\pi$
$281$	−510.157	−0.108304	−0.0541520	+	0.998533i	$0.517246\pi$
$282$	0	0
$283$	6284.00	1.31995	0.659974	−	0.751289i	$-0.270567\pi$
$283$	6284.00	1.31995	0.659974	+	0.751289i	$0.270567\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3373.00	−0.686546
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3708.45	−0.739419	−0.369710	−	0.929147i	$-0.620543\pi$
$293$	−3708.45	−0.739419	−0.369710	+	0.929147i	$0.620543\pi$
$294$	0	0
$295$	11935.0	2.35553
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10595.6	2.04936
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11537.4	2.16600
$306$	0	0
$307$	−1762.00	−0.327566	−0.163783	−	0.986496i	$-0.552370\pi$
$307$	−1762.00	−0.327566	−0.163783	+	0.986496i	$0.552370\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6671.28	1.21638	0.608189	−	0.793792i	$-0.291896\pi$
$311$	6671.28	1.21638	0.608189	+	0.793792i	$0.291896\pi$
$312$	0	0
$313$	−1321.00	−0.238554	−0.119277	−	0.992861i	$-0.538058\pi$
$313$	−1321.00	−0.238554	−0.119277	+	0.992861i	$0.538058\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6298.47	1.11595	0.557977	−	0.829856i	$-0.311578\pi$
$317$	6298.47	1.11595	0.557977	+	0.829856i	$0.311578\pi$
$318$	0	0
$319$	1925.00	0.337866
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−78.4857	−0.0135203
$324$	0	0
$325$	14040.0	2.39630
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9736.00	−1.61673	−0.808367	−	0.588679i	$-0.799648\pi$
$331$	−9736.00	−1.61673	−0.808367	+	0.588679i	$0.799648\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5925.67	−0.966429
$336$	0	0
$337$	−6747.00	−1.09060	−0.545300	−	0.838241i	$-0.683584\pi$
$337$	−6747.00	−1.09060	−0.545300	+	0.838241i	$0.683584\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3943.90	−0.626318
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2119.11	−0.327838	−0.163919	−	0.986474i	$-0.552414\pi$
$347$	−2119.11	−0.327838	−0.163919	+	0.986474i	$0.552414\pi$
$348$	0	0
$349$	−9646.00	−1.47948	−0.739740	−	0.672893i	$-0.765052\pi$
$349$	−9646.00	−1.47948	−0.739740	+	0.672893i	$0.765052\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−470.914	−0.0710035	−0.0355017	−	0.999370i	$-0.511303\pi$
$353$	−470.914	−0.0710035	−0.0355017	+	0.999370i	$0.511303\pi$
$354$	0	0
$355$	−3080.00	−0.460477
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6396.58	0.940386	0.470193	−	0.882564i	$-0.344184\pi$
$359$	6396.58	0.940386	0.470193	+	0.882564i	$0.344184\pi$
$360$	0	0
$361$	−6855.00	−0.999417
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8751.15	1.25495
$366$	0	0
$367$	6757.00	0.961070	0.480535	−	0.876976i	$-0.340443\pi$
$367$	6757.00	0.961070	0.480535	+	0.876976i	$0.340443\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8644.00	1.19992	0.599959	−	0.800031i	$-0.295184\pi$
$373$	8644.00	1.19992	0.599959	+	0.800031i	$0.295184\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5297.78	0.723739
$378$	0	0
$379$	7372.00	0.999140	0.499570	−	0.866273i	$-0.333491\pi$
$379$	7372.00	0.999140	0.499570	+	0.866273i	$0.333491\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7495.38	−0.999990	−0.499995	−	0.866028i	$-0.666665\pi$
$383$	−7495.38	−0.999990	−0.499995	+	0.866028i	$0.666665\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13617.3	−1.77487	−0.887433	−	0.460937i	$-0.847513\pi$
$389$	−13617.3	−1.77487	−0.887433	+	0.460937i	$0.847513\pi$
$390$	0	0
$391$	7700.00	0.995923
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5238.92	−0.667338
$396$	0	0
$397$	4632.00	0.585575	0.292788	−	0.956177i	$-0.405417\pi$
$397$	4632.00	0.585575	0.292788	+	0.956177i	$0.405417\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5180.05	−0.645086	−0.322543	−	0.946555i	$-0.604538\pi$
$401$	−5180.05	−0.645086	−0.322543	+	0.946555i	$0.604538\pi$
$402$	0	0
$403$	−10854.0	−1.34163
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3963.53	−0.482714
$408$	0	0
$409$	4573.00	0.552861	0.276431	−	0.961034i	$-0.410848\pi$
$409$	4573.00	0.552861	0.276431	+	0.961034i	$0.410848\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−14245.0	−1.68496
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2590.03	0.301984	0.150992	−	0.988535i	$-0.451753\pi$
$419$	2590.03	0.301984	0.150992	+	0.988535i	$0.451753\pi$
$420$	0	0
$421$	6096.00	0.705703	0.352851	−	0.935679i	$-0.385212\pi$
$421$	6096.00	0.705703	0.352851	+	0.935679i	$0.385212\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	10203.1	1.16453
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16952.9	1.89465	0.947323	−	0.320279i	$-0.103777\pi$
$431$	16952.9	1.89465	0.947323	+	0.320279i	$0.103777\pi$
$432$	0	0
$433$	5206.00	0.577793	0.288897	−	0.957360i	$-0.406712\pi$
$433$	5206.00	0.577793	0.288897	+	0.957360i	$0.406712\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−392.428	−0.0429574
$438$	0	0
$439$	12411.0	1.34930	0.674652	−	0.738136i	$-0.264294\pi$
$439$	12411.0	1.34930	0.674652	+	0.738136i	$0.264294\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13401.4	−1.43729	−0.718647	−	0.695375i	$-0.755238\pi$
$443$	−13401.4	−1.43729	−0.718647	+	0.695375i	$0.755238\pi$
$444$	0	0
$445$	2310.00	0.246078
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4669.90	−0.490838	−0.245419	−	0.969417i	$-0.578925\pi$
$449$	−4669.90	−0.490838	−0.245419	+	0.969417i	$0.578925\pi$
$450$	0	0
$451$	−9240.00	−0.964733
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14827.0	1.51768	0.758838	−	0.651280i	$-0.225767\pi$
$457$	14827.0	1.51768	0.758838	+	0.651280i	$0.225767\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1451.98	−0.146693	−0.0733467	−	0.997307i	$-0.523368\pi$
$461$	−1451.98	−0.146693	−0.0733467	+	0.997307i	$0.523368\pi$
$462$	0	0
$463$	−2276.00	−0.228455	−0.114228	−	0.993455i	$-0.536439\pi$
$463$	−2276.00	−0.228455	−0.114228	+	0.993455i	$0.536439\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5572.48	−0.552171	−0.276085	−	0.961133i	$-0.589037\pi$
$467$	−5572.48	−0.552171	−0.276085	+	0.961133i	$0.589037\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4787.63	−0.465402
$474$	0	0
$475$	−520.000	−0.0502300
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11145.0	−1.06310	−0.531552	−	0.847026i	$-0.678391\pi$
$479$	−11145.0	−1.06310	−0.531552	+	0.847026i	$0.678391\pi$
$480$	0	0
$481$	−10908.0	−1.03402
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11674.7	−1.09304
$486$	0	0
$487$	18467.0	1.71832	0.859158	−	0.511711i	$-0.170988\pi$
$487$	18467.0	1.71832	0.859158	+	0.511711i	$0.170988\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3826.18	−0.351676	−0.175838	−	0.984419i	$-0.556263\pi$
$491$	−3826.18	−0.351676	−0.175838	+	0.984419i	$0.556263\pi$
$492$	0	0
$493$	3850.00	0.351715
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13922.0	−1.24897	−0.624483	−	0.781038i	$-0.714690\pi$
$499$	−13922.0	−1.24897	−0.624483	+	0.781038i	$0.714690\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20837.9	1.84715	0.923577	−	0.383414i	$-0.125252\pi$
$503$	20837.9	1.84715	0.923577	+	0.383414i	$0.125252\pi$
$504$	0	0
$505$	−23870.0	−2.10337
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14068.6	−1.22510	−0.612552	−	0.790430i	$-0.709857\pi$
$509$	−14068.6	−1.22510	−0.612552	+	0.790430i	$0.709857\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−32650.0	−2.79366
$516$	0	0
$517$	4620.00	0.393012
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12714.7	1.06917	0.534587	−	0.845113i	$-0.320467\pi$
$521$	12714.7	1.06917	0.534587	+	0.845113i	$0.320467\pi$
$522$	0	0
$523$	−16466.0	−1.37669	−0.688344	−	0.725384i	$-0.741662\pi$
$523$	−16466.0	−1.37669	−0.688344	+	0.725384i	$0.741662\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7887.81	−0.651989
$528$	0	0
$529$	26333.0	2.16430
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−25429.4	−2.06654
$534$	0	0
$535$	−31185.0	−2.52008
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	384.000	0.0305165	0.0152583	−	0.999884i	$-0.495143\pi$
$541$	384.000	0.0305165	0.0152583	+	0.999884i	$0.495143\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7024.47	0.552101
$546$	0	0
$547$	10110.0	0.790260	0.395130	−	0.918625i	$-0.370699\pi$
$547$	10110.0	0.790260	0.395130	+	0.918625i	$0.370699\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−196.214	−0.0151706
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1628.58	0.123887	0.0619435	−	0.998080i	$-0.480270\pi$
$557$	1628.58	0.123887	0.0619435	+	0.998080i	$0.480270\pi$
$558$	0	0
$559$	−13176.0	−0.996933
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13676.1	−1.02377	−0.511883	−	0.859055i	$-0.671052\pi$
$563$	−13676.1	−1.02377	−0.511883	+	0.859055i	$0.671052\pi$
$564$	0	0
$565$	21560.0	1.60537
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14951.5	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$569$	−14951.5	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$570$	0	0
$571$	−10536.0	−0.772185	−0.386093	−	0.922460i	$-0.626176\pi$
$571$	−10536.0	−0.772185	−0.386093	+	0.922460i	$0.626176\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	51015.7	3.70000
$576$	0	0
$577$	−2287.00	−0.165007	−0.0825035	−	0.996591i	$-0.526292\pi$
$577$	−2287.00	−0.165007	−0.0825035	+	0.996591i	$0.526292\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	12705.0	0.902551
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3904.66	−0.274553	−0.137277	−	0.990533i	$-0.543835\pi$
$587$	−3904.66	−0.274553	−0.137277	+	0.990533i	$0.543835\pi$
$588$	0	0
$589$	402.000	0.0281224
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15657.9	−1.08430	−0.542152	−	0.840280i	$-0.682390\pi$
$593$	−15657.9	−1.08430	−0.542152	+	0.840280i	$0.682390\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21544.3	1.46958	0.734789	−	0.678296i	$-0.237281\pi$
$599$	21544.3	1.46958	0.734789	+	0.678296i	$0.237281\pi$
$600$	0	0
$601$	17089.0	1.15986	0.579929	−	0.814667i	$-0.303080\pi$
$601$	17089.0	1.15986	0.579929	+	0.814667i	$0.303080\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−18561.9	−1.24735
$606$	0	0
$607$	−4947.00	−0.330795	−0.165397	−	0.986227i	$-0.552891\pi$
$607$	−4947.00	−0.330795	−0.165397	+	0.986227i	$0.552891\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12714.7	0.841867
$612$	0	0
$613$	−18030.0	−1.18797	−0.593984	−	0.804477i	$-0.702446\pi$
$613$	−18030.0	−1.18797	−0.593984	+	0.804477i	$0.702446\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16403.5	−1.07031	−0.535154	−	0.844754i	$-0.679746\pi$
$617$	−16403.5	−1.07031	−0.535154	+	0.844754i	$0.679746\pi$
$618$	0	0
$619$	−26308.0	−1.70825	−0.854126	−	0.520066i	$-0.825907\pi$
$619$	−26308.0	−1.70825	−0.854126	+	0.520066i	$0.825907\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19475.0	1.24640
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7927.05	−0.502500
$630$	0	0
$631$	−10813.0	−0.682185	−0.341092	−	0.940030i	$-0.610797\pi$
$631$	−10813.0	−0.682185	−0.341092	+	0.940030i	$0.610797\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	41734.8	2.60818
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12636.2	−0.778627	−0.389313	−	0.921105i	$-0.627288\pi$
$641$	−12636.2	−0.778627	−0.389313	+	0.921105i	$0.627288\pi$
$642$	0	0
$643$	−11174.0	−0.685318	−0.342659	−	0.939460i	$-0.611328\pi$
$643$	−11174.0	−0.685318	−0.342659	+	0.939460i	$0.611328\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19542.9	1.18750	0.593750	−	0.804650i	$-0.297647\pi$
$647$	19542.9	1.18750	0.593750	+	0.804650i	$0.297647\pi$
$648$	0	0
$649$	11935.0	0.721864
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14657.2	0.878377	0.439189	−	0.898395i	$-0.355266\pi$
$653$	14657.2	0.878377	0.439189	+	0.898395i	$0.355266\pi$
$654$	0	0
$655$	−29645.0	−1.76844
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9810.71	−0.579926	−0.289963	−	0.957038i	$-0.593643\pi$
$659$	−9810.71	−0.579926	−0.289963	+	0.957038i	$0.593643\pi$
$660$	0	0
$661$	8024.00	0.472159	0.236080	−	0.971734i	$-0.424137\pi$
$661$	8024.00	0.472159	0.236080	+	0.971734i	$0.424137\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19250.0	1.11749
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11537.4	0.663780
$672$	0	0
$673$	−29791.0	−1.70633	−0.853164	−	0.521643i	$-0.825319\pi$
$673$	−29791.0	−1.70633	−0.853164	+	0.521643i	$0.825319\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8888.50	0.504598	0.252299	−	0.967649i	$-0.418813\pi$
$677$	8888.50	0.504598	0.252299	+	0.967649i	$0.418813\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5003.46	0.280311	0.140155	−	0.990130i	$-0.455240\pi$
$683$	5003.46	0.280311	0.140155	+	0.990130i	$0.455240\pi$
$684$	0	0
$685$	−50050.0	−2.79170
$686$	0	0
$687$	0	0
$688$	0	0
$689$	34965.4	1.93334
$690$	0	0
$691$	−27176.0	−1.49613	−0.748064	−	0.663627i	$-0.769016\pi$
$691$	−27176.0	−1.49613	−0.748064	+	0.663627i	$0.769016\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	47837.0	2.61088
$696$	0	0
$697$	−18480.0	−1.00428
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9320.17	−0.502166	−0.251083	−	0.967966i	$-0.580787\pi$
$701$	−9320.17	−0.502166	−0.251083	+	0.967966i	$0.580787\pi$
$702$	0	0
$703$	404.000	0.0216745
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5434.00	−0.287839	−0.143920	−	0.989589i	$-0.545971\pi$
$709$	−5434.00	−0.287839	−0.143920	+	0.989589i	$0.545971\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−39439.0	−2.07153
$714$	0	0
$715$	20790.0	1.08742
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18012.5	−0.934286	−0.467143	−	0.884182i	$-0.654717\pi$
$719$	−18012.5	−0.934286	−0.467143	+	0.884182i	$0.654717\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	25507.8	1.30667
$726$	0	0
$727$	−29353.0	−1.49744	−0.748722	−	0.662884i	$-0.769332\pi$
$727$	−29353.0	−1.49744	−0.748722	+	0.662884i	$0.769332\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9575.25	−0.484478
$732$	0	0
$733$	11436.0	0.576260	0.288130	−	0.957591i	$-0.406967\pi$
$733$	11436.0	0.576260	0.288130	+	0.957591i	$0.406967\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5925.67	−0.296167
$738$	0	0
$739$	−78.0000	−0.00388265	−0.00194132	−	0.999998i	$-0.500618\pi$
$739$	−78.0000	−0.00388265	−0.00194132	+	0.999998i	$0.500618\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17306.1	0.854507	0.427254	−	0.904132i	$-0.359481\pi$
$743$	17306.1	0.854507	0.427254	+	0.904132i	$0.359481\pi$
$744$	0	0
$745$	−30030.0	−1.47680
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26391.0	1.28232	0.641159	−	0.767408i	$-0.278454\pi$
$751$	26391.0	1.28232	0.641159	+	0.767408i	$0.278454\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18581.5	−0.895695
$756$	0	0
$757$	−660.000	−0.0316884	−0.0158442	−	0.999874i	$-0.505044\pi$
$757$	−660.000	−0.0316884	−0.0158442	+	0.999874i	$0.505044\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5140.81	−0.244881	−0.122440	−	0.992476i	$-0.539072\pi$
$761$	−5140.81	−0.244881	−0.122440	+	0.992476i	$0.539072\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32846.3	1.54630
$768$	0	0
$769$	−8773.00	−0.411395	−0.205697	−	0.978616i	$-0.565946\pi$
$769$	−8773.00	−0.411395	−0.205697	+	0.978616i	$0.565946\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	11027.2	0.513095	0.256547	−	0.966532i	$-0.417415\pi$
$773$	11027.2	0.513095	0.256547	+	0.966532i	$0.417415\pi$
$774$	0	0
$775$	−52260.0	−2.42224
$776$	0	0
$777$	0	0
$778$	0	0
$779$	941.828	0.0433177
$780$	0	0
$781$	−3080.00	−0.141115
$782$	0	0
$783$	0	0
$784$	0	0
$785$	30217.0	1.37387
$786$	0	0
$787$	5880.00	0.266327	0.133164	−	0.991094i	$-0.457486\pi$
$787$	5880.00	0.266327	0.133164	+	0.991094i	$0.457486\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	31752.0	1.42187
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22898.2	−1.01769	−0.508843	−	0.860859i	$-0.669927\pi$
$797$	−22898.2	−1.01769	−0.508843	+	0.860859i	$0.669927\pi$
$798$	0	0
$799$	9240.00	0.409121
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8751.15	0.384585
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3217.91	0.139846	0.0699232	−	0.997552i	$-0.477725\pi$
$809$	3217.91	0.139846	0.0699232	+	0.997552i	$0.477725\pi$
$810$	0	0
$811$	6062.00	0.262473	0.131237	−	0.991351i	$-0.458105\pi$
$811$	6062.00	0.262473	0.131237	+	0.991351i	$0.458105\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−21387.3	−0.919222
$816$	0	0
$817$	488.000	0.0208971
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6808.63	−0.289431	−0.144716	−	0.989473i	$-0.546227\pi$
$821$	−6808.63	−0.289431	−0.144716	+	0.989473i	$0.546227\pi$
$822$	0	0
$823$	−32240.0	−1.36551	−0.682756	−	0.730647i	$-0.739219\pi$
$823$	−32240.0	−1.36551	−0.682756	+	0.730647i	$0.739219\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28313.7	−1.19052	−0.595262	−	0.803531i	$-0.702952\pi$
$827$	−28313.7	−1.19052	−0.595262	+	0.803531i	$0.702952\pi$
$828$	0	0
$829$	−32002.0	−1.34074	−0.670371	−	0.742026i	$-0.733865\pi$
$829$	−32002.0	−1.34074	−0.670371	+	0.742026i	$0.733865\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	40810.0	1.69136
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5376.27	−0.221227	−0.110613	−	0.993864i	$-0.535282\pi$
$839$	−5376.27	−0.221227	−0.110613	+	0.993864i	$0.535282\pi$
$840$	0	0
$841$	−14764.0	−0.605355
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14107.8	0.574347
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−39635.3	−1.59657
$852$	0	0
$853$	19966.0	0.801434	0.400717	−	0.916202i	$-0.368761\pi$
$853$	19966.0	0.801434	0.400717	+	0.916202i	$0.368761\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13892.0	0.553723	0.276861	−	0.960910i	$-0.410706\pi$
$857$	13892.0	0.553723	0.276861	+	0.960910i	$0.410706\pi$
$858$	0	0
$859$	33376.0	1.32570	0.662849	−	0.748753i	$-0.269347\pi$
$859$	33376.0	1.32570	0.662849	+	0.748753i	$0.269347\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8986.61	0.354470	0.177235	−	0.984169i	$-0.443285\pi$
$863$	8986.61	0.354470	0.177235	+	0.984169i	$0.443285\pi$
$864$	0	0
$865$	36190.0	1.42254
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5238.92	−0.204509
$870$	0	0
$871$	−16308.0	−0.634415
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4520.00	−0.174036	−0.0870180	−	0.996207i	$-0.527734\pi$
$877$	−4520.00	−0.174036	−0.0870180	+	0.996207i	$0.527734\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	37555.4	1.43618	0.718089	−	0.695951i	$-0.245017\pi$
$881$	37555.4	1.43618	0.718089	+	0.695951i	$0.245017\pi$
$882$	0	0
$883$	−7636.00	−0.291021	−0.145511	−	0.989357i	$-0.546483\pi$
$883$	−7636.00	−0.291021	−0.145511	+	0.989357i	$0.546483\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24369.8	−0.922500	−0.461250	−	0.887270i	$-0.652599\pi$
$887$	−24369.8	−0.922500	−0.461250	+	0.887270i	$0.652599\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−470.914	−0.0176467
$894$	0	0
$895$	−23100.0	−0.862735
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19719.5	−0.731572
$900$	0	0
$901$	25410.0	0.939545
$902$	0	0
$903$	0	0
$904$	0	0
$905$	39321.3	1.44429
$906$	0	0
$907$	−12102.0	−0.443043	−0.221522	−	0.975155i	$-0.571102\pi$
$907$	−12102.0	−0.443043	−0.221522	+	0.975155i	$0.571102\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5180.05	0.188390	0.0941948	−	0.995554i	$-0.469972\pi$
$911$	5180.05	0.188390	0.0941948	+	0.995554i	$0.469972\pi$
$912$	0	0
$913$	−14245.0	−0.516364
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−33788.0	−1.21280	−0.606400	−	0.795160i	$-0.707387\pi$
$919$	−33788.0	−1.21280	−0.606400	+	0.795160i	$0.707387\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8476.45	−0.302281
$924$	0	0
$925$	−52520.0	−1.86686
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1412.74	0.0498930	0.0249465	−	0.999689i	$-0.492058\pi$
$929$	1412.74	0.0498930	0.0249465	+	0.999689i	$0.492058\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	15108.5	0.528450
$936$	0	0
$937$	−17603.0	−0.613730	−0.306865	−	0.951753i	$-0.599280\pi$
$937$	−17603.0	−0.613730	−0.306865	+	0.951753i	$0.599280\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19091.6	0.661392	0.330696	−	0.943737i	$-0.392717\pi$
$941$	19091.6	0.661392	0.330696	+	0.943737i	$0.392717\pi$
$942$	0	0
$943$	−92400.0	−3.19084
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−55646.3	−1.90946	−0.954732	−	0.297466i	$-0.903858\pi$
$947$	−55646.3	−1.90946	−0.954732	+	0.297466i	$0.903858\pi$
$948$	0	0
$949$	24084.0	0.823814
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17345.3	−0.589581	−0.294790	−	0.955562i	$-0.595250\pi$
$953$	−17345.3	−0.589581	−0.294790	+	0.955562i	$0.595250\pi$
$954$	0	0
$955$	−87780.0	−2.97434
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	10610.0	0.356148
$962$	0	0
$963$	0	0
$964$	0	0
$965$	91220.0	3.04298
$966$	0	0
$967$	−4747.00	−0.157863	−0.0789313	−	0.996880i	$-0.525151\pi$
$967$	−4747.00	−0.157863	−0.0789313	+	0.996880i	$0.525151\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32905.1	1.08751	0.543757	−	0.839243i	$-0.317001\pi$
$971$	32905.1	1.08751	0.543757	+	0.839243i	$0.317001\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20955.7	−0.686214	−0.343107	−	0.939296i	$-0.611479\pi$
$977$	−20955.7	−0.686214	−0.343107	+	0.939296i	$0.611479\pi$
$978$	0	0
$979$	2310.00	0.0754116
$980$	0	0
$981$	0	0
$982$	0	0
$983$	33905.8	1.10013	0.550065	−	0.835122i	$-0.314603\pi$
$983$	33905.8	1.10013	0.550065	+	0.835122i	$0.314603\pi$
$984$	0	0
$985$	82390.0	2.66514
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−47876.3	−1.53931
$990$	0	0
$991$	13469.0	0.431743	0.215871	−	0.976422i	$-0.430741\pi$
$991$	13469.0	0.431743	0.215871	+	0.976422i	$0.430741\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12086.8	0.385103
$996$	0	0
$997$	−14282.0	−0.453677	−0.226838	−	0.973932i	$-0.572839\pi$
$997$	−14282.0	−0.453677	−0.226838	+	0.973932i	$0.572839\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.a.u.1.2		2
3.2	odd	2	inner	1764.4.a.u.1.1		2
7.2	even	3		1764.4.k.x.361.1		4
7.3	odd	6		252.4.k.e.37.2	yes	4
7.4	even	3		1764.4.k.x.1549.1		4
7.5	odd	6		252.4.k.e.109.2	yes	4
7.6	odd	2		1764.4.a.r.1.1		2
21.2	odd	6		1764.4.k.x.361.2		4
21.5	even	6		252.4.k.e.109.1	yes	4
21.11	odd	6		1764.4.k.x.1549.2		4
21.17	even	6		252.4.k.e.37.1	✓	4
21.20	even	2		1764.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.k.e.37.1	✓	4	21.17	even	6
252.4.k.e.37.2	yes	4	7.3	odd	6
252.4.k.e.109.1	yes	4	21.5	even	6
252.4.k.e.109.2	yes	4	7.5	odd	6
1764.4.a.r.1.1		2	7.6	odd	2
1764.4.a.r.1.2		2	21.20	even	2
1764.4.a.u.1.1		2	3.2	odd	2	inner
1764.4.a.u.1.2		2	1.1	even	1	trivial
1764.4.k.x.361.1		4	7.2	even	3
1764.4.k.x.361.2		4	21.2	odd	6
1764.4.k.x.1549.1		4	7.4	even	3
1764.4.k.x.1549.2		4	21.11	odd	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)

\(f(q)\)	\(=\)	\(q+19.6214 q^{5} +19.6214 q^{11} +54.0000 q^{13} +39.2428 q^{17} -2.00000 q^{19} +196.214 q^{23} +260.000 q^{25} +98.1071 q^{29} -201.000 q^{31} -202.000 q^{37} -470.914 q^{41} -244.000 q^{43} +235.457 q^{47} +647.507 q^{53} +385.000 q^{55} +608.264 q^{59} +588.000 q^{61} +1059.56 q^{65} -302.000 q^{67} -156.971 q^{71} +446.000 q^{73} -267.000 q^{79} -725.992 q^{83} +770.000 q^{85} +117.729 q^{89} -39.2428 q^{95} -595.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 108 q^{13} - 4 q^{19} + 520 q^{25} - 402 q^{31} - 404 q^{37} - 488 q^{43} + 770 q^{55} + 1176 q^{61} - 604 q^{67} + 892 q^{73} - 534 q^{79} + 1540 q^{85} - 1190 q^{97}+O(q^{100}) \) 2 * q + 108 * q^13 - 4 * q^19 + 520 * q^25 - 402 * q^31 - 404 * q^37 - 488 * q^43 + 770 * q^55 + 1176 * q^61 - 604 * q^67 + 892 * q^73 - 534 * q^79 + 1540 * q^85 - 1190 * q^97

Introduction

L-functions

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