LMFDB - Embedded newform 592.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,4,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("592.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	592.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:	$34.9291307234$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1264493.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 22x^{2} - 29x - 4 $ x^4 - 22x^2 - 29x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.25916$ of defining polynomial
Character	$\chi$	$=$	592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.43262 q^{3} -21.9509 q^{5} +5.22823 q^{7} +14.3786 q^{9} +O(q^{10})$	$q-6.43262 q^{3} -21.9509 q^{5} +5.22823 q^{7} +14.3786 q^{9} +63.3844 q^{11} -25.4136 q^{13} +141.202 q^{15} +15.9174 q^{17} -99.7015 q^{19} -33.6312 q^{21} +73.4096 q^{23} +356.844 q^{25} +81.1885 q^{27} +159.290 q^{29} +274.803 q^{31} -407.728 q^{33} -114.765 q^{35} -37.0000 q^{37} +163.476 q^{39} -354.038 q^{41} -414.930 q^{43} -315.625 q^{45} +110.851 q^{47} -315.666 q^{49} -102.391 q^{51} -379.369 q^{53} -1391.35 q^{55} +641.342 q^{57} -331.573 q^{59} +521.791 q^{61} +75.1748 q^{63} +557.852 q^{65} +114.559 q^{67} -472.216 q^{69} +219.212 q^{71} -88.8324 q^{73} -2295.44 q^{75} +331.388 q^{77} +628.287 q^{79} -910.478 q^{81} -998.754 q^{83} -349.403 q^{85} -1024.65 q^{87} -70.3152 q^{89} -132.868 q^{91} -1767.71 q^{93} +2188.54 q^{95} -407.700 q^{97} +911.381 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 19 q^{5} + 6 q^{7} - 7 q^{9}+O(q^{10}) $ 4 * q + q^3 - 19 * q^5 + 6 * q^7 - 7 * q^9	$ 4 q + q^{3} - 19 q^{5} + 6 q^{7} - 7 q^{9} + 77 q^{11} - 115 q^{13} + 172 q^{15} - 180 q^{17} - 4 q^{19} - 208 q^{21} + 189 q^{23} + 19 q^{25} + 94 q^{27} - 99 q^{29} + 251 q^{31} - 434 q^{33} - 124 q^{35} - 148 q^{37} - 351 q^{39} - 497 q^{41} - 694 q^{43} - 122 q^{45} + 50 q^{47} - 198 q^{49} - 1270 q^{51} - 290 q^{53} - 1621 q^{55} + 330 q^{57} - 586 q^{59} + 991 q^{61} - 798 q^{63} - 94 q^{65} - 823 q^{67} + 89 q^{69} + 96 q^{71} + 121 q^{73} - 3026 q^{75} - 314 q^{77} - 475 q^{79} - 456 q^{81} - 820 q^{83} - 750 q^{85} - 687 q^{87} - 880 q^{89} - 626 q^{91} - 430 q^{93} + 1538 q^{95} - 904 q^{97} - 202 q^{99}+O(q^{100}) $ 4 * q + q^3 - 19 * q^5 + 6 * q^7 - 7 * q^9 + 77 * q^11 - 115 * q^13 + 172 * q^15 - 180 * q^17 - 4 * q^19 - 208 * q^21 + 189 * q^23 + 19 * q^25 + 94 * q^27 - 99 * q^29 + 251 * q^31 - 434 * q^33 - 124 * q^35 - 148 * q^37 - 351 * q^39 - 497 * q^41 - 694 * q^43 - 122 * q^45 + 50 * q^47 - 198 * q^49 - 1270 * q^51 - 290 * q^53 - 1621 * q^55 + 330 * q^57 - 586 * q^59 + 991 * q^61 - 798 * q^63 - 94 * q^65 - 823 * q^67 + 89 * q^69 + 96 * q^71 + 121 * q^73 - 3026 * q^75 - 314 * q^77 - 475 * q^79 - 456 * q^81 - 820 * q^83 - 750 * q^85 - 687 * q^87 - 880 * q^89 - 626 * q^91 - 430 * q^93 + 1538 * q^95 - 904 * q^97 - 202 * q^99

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
$3$	−6.43262	−1.23796	−0.618979	−	0.785407i	$-0.712454\pi$
$3$	−6.43262	−1.23796	−0.618979	+	0.785407i	$0.712454\pi$
$4$	0	0
$5$	−21.9509	−1.96335	−0.981676	−	0.190556i	$-0.938971\pi$
$5$	−21.9509	−1.96335	−0.981676	+	0.190556i	$0.938971\pi$
$6$	0	0
$7$	5.22823	0.282298	0.141149	−	0.989988i	$-0.454920\pi$
$7$	5.22823	0.282298	0.141149	+	0.989988i	$0.454920\pi$
$8$	0	0
$9$	14.3786	0.532542
$10$	0	0
$11$	63.3844	1.73737	0.868687	−	0.495361i	$-0.164964\pi$
$11$	63.3844	1.73737	0.868687	+	0.495361i	$0.164964\pi$
$12$	0	0
$13$	−25.4136	−0.542189	−0.271095	−	0.962553i	$-0.587386\pi$
$13$	−25.4136	−0.542189	−0.271095	+	0.962553i	$0.587386\pi$
$14$	0	0
$15$	141.202	2.43055
$16$	0	0
$17$	15.9174	0.227091	0.113545	−	0.993533i	$-0.463779\pi$
$17$	15.9174	0.227091	0.113545	+	0.993533i	$0.463779\pi$
$18$	0	0
$19$	−99.7015	−1.20385	−0.601923	−	0.798554i	$-0.705599\pi$
$19$	−99.7015	−1.20385	−0.601923	+	0.798554i	$0.705599\pi$
$20$	0	0
$21$	−33.6312	−0.349473
$22$	0	0
$23$	73.4096	0.665520	0.332760	−	0.943012i	$-0.392020\pi$
$23$	73.4096	0.665520	0.332760	+	0.943012i	$0.392020\pi$
$24$	0	0
$25$	356.844	2.85475
$26$	0	0
$27$	81.1885	0.578694
$28$	0	0
$29$	159.290	1.01998	0.509989	−	0.860181i	$-0.329649\pi$
$29$	159.290	1.01998	0.509989	+	0.860181i	$0.329649\pi$
$30$	0	0
$31$	274.803	1.59213	0.796067	−	0.605209i	$-0.206910\pi$
$31$	274.803	1.59213	0.796067	+	0.605209i	$0.206910\pi$
$32$	0	0
$33$	−407.728	−2.15080
$34$	0	0
$35$	−114.765	−0.554250
$36$	0	0
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	0	0
$39$	163.476	0.671208
$40$	0	0
$41$	−354.038	−1.34857	−0.674287	−	0.738470i	$-0.735549\pi$
$41$	−354.038	−1.34857	−0.674287	+	0.738470i	$0.735549\pi$
$42$	0	0
$43$	−414.930	−1.47154	−0.735770	−	0.677232i	$-0.763180\pi$
$43$	−414.930	−1.47154	−0.735770	+	0.677232i	$0.763180\pi$
$44$	0	0
$45$	−315.625	−1.04557
$46$	0	0
$47$	110.851	0.344027	0.172014	−	0.985095i	$-0.444973\pi$
$47$	110.851	0.344027	0.172014	+	0.985095i	$0.444973\pi$
$48$	0	0
$49$	−315.666	−0.920308
$50$	0	0
$51$	−102.391	−0.281129
$52$	0	0
$53$	−379.369	−0.983213	−0.491607	−	0.870817i	$-0.663590\pi$
$53$	−379.369	−0.983213	−0.491607	+	0.870817i	$0.663590\pi$
$54$	0	0
$55$	−1391.35	−3.41108
$56$	0	0
$57$	641.342	1.49031
$58$	0	0
$59$	−331.573	−0.731646	−0.365823	−	0.930685i	$-0.619212\pi$
$59$	−331.573	−0.731646	−0.365823	+	0.930685i	$0.619212\pi$
$60$	0	0
$61$	521.791	1.09522	0.547611	−	0.836733i	$-0.315537\pi$
$61$	521.791	1.09522	0.547611	+	0.836733i	$0.315537\pi$
$62$	0	0
$63$	75.1748	0.150335
$64$	0	0
$65$	557.852	1.06451
$66$	0	0
$67$	114.559	0.208889	0.104444	−	0.994531i	$-0.466694\pi$
$67$	114.559	0.208889	0.104444	+	0.994531i	$0.466694\pi$
$68$	0	0
$69$	−472.216	−0.823887
$70$	0	0
$71$	219.212	0.366418	0.183209	−	0.983074i	$-0.441352\pi$
$71$	219.212	0.366418	0.183209	+	0.983074i	$0.441352\pi$
$72$	0	0
$73$	−88.8324	−0.142425	−0.0712127	−	0.997461i	$-0.522687\pi$
$73$	−88.8324	−0.142425	−0.0712127	+	0.997461i	$0.522687\pi$
$74$	0	0
$75$	−2295.44	−3.53407
$76$	0	0
$77$	331.388	0.490457
$78$	0	0
$79$	628.287	0.894782	0.447391	−	0.894339i	$-0.352353\pi$
$79$	628.287	0.894782	0.447391	+	0.894339i	$0.352353\pi$
$80$	0	0
$81$	−910.478	−1.24894
$82$	0	0
$83$	−998.754	−1.32081	−0.660407	−	0.750908i	$-0.729616\pi$
$83$	−998.754	−1.32081	−0.660407	+	0.750908i	$0.729616\pi$
$84$	0	0
$85$	−349.403	−0.445860
$86$	0	0
$87$	−1024.65	−1.26269
$88$	0	0
$89$	−70.3152	−0.0837461	−0.0418730	−	0.999123i	$-0.513332\pi$
$89$	−70.3152	−0.0837461	−0.0418730	+	0.999123i	$0.513332\pi$
$90$	0	0
$91$	−132.868	−0.153059
$92$	0	0
$93$	−1767.71	−1.97100
$94$	0	0
$95$	2188.54	2.36358
$96$	0	0
$97$	−407.700	−0.426760	−0.213380	−	0.976969i	$-0.568447\pi$
$97$	−407.700	−0.426760	−0.213380	+	0.976969i	$0.568447\pi$
$98$	0	0
$99$	911.381	0.925225
$100$	0	0
$101$	−262.531	−0.258641	−0.129321	−	0.991603i	$-0.541280\pi$
$101$	−262.531	−0.258641	−0.129321	+	0.991603i	$0.541280\pi$
$102$	0	0
$103$	836.599	0.800316	0.400158	−	0.916446i	$-0.368955\pi$
$103$	836.599	0.800316	0.400158	+	0.916446i	$0.368955\pi$
$104$	0	0
$105$	738.237	0.686139
$106$	0	0
$107$	−403.695	−0.364735	−0.182368	−	0.983230i	$-0.558376\pi$
$107$	−403.695	−0.364735	−0.182368	+	0.983230i	$0.558376\pi$
$108$	0	0
$109$	−69.3904	−0.0609761	−0.0304881	−	0.999535i	$-0.509706\pi$
$109$	−69.3904	−0.0609761	−0.0304881	+	0.999535i	$0.509706\pi$
$110$	0	0
$111$	238.007	0.203519
$112$	0	0
$113$	1290.95	1.07471	0.537355	−	0.843356i	$-0.319423\pi$
$113$	1290.95	1.07471	0.537355	+	0.843356i	$0.319423\pi$
$114$	0	0
$115$	−1611.41	−1.30665
$116$	0	0
$117$	−365.413	−0.288739
$118$	0	0
$119$	83.2200	0.0641073
$120$	0	0
$121$	2686.58	2.01847
$122$	0	0
$123$	2277.39	1.66948
$124$	0	0
$125$	−5089.20	−3.64154
$126$	0	0
$127$	−110.325	−0.0770847	−0.0385423	−	0.999257i	$-0.512271\pi$
$127$	−110.325	−0.0770847	−0.0385423	+	0.999257i	$0.512271\pi$
$128$	0	0
$129$	2669.09	1.82171
$130$	0	0
$131$	−909.741	−0.606752	−0.303376	−	0.952871i	$-0.598114\pi$
$131$	−909.741	−0.606752	−0.303376	+	0.952871i	$0.598114\pi$
$132$	0	0
$133$	−521.262	−0.339843
$134$	0	0
$135$	−1782.16	−1.13618
$136$	0	0
$137$	−2285.57	−1.42533	−0.712663	−	0.701507i	$-0.752511\pi$
$137$	−2285.57	−1.42533	−0.712663	+	0.701507i	$0.752511\pi$
$138$	0	0
$139$	2689.27	1.64102	0.820508	−	0.571635i	$-0.193690\pi$
$139$	2689.27	1.64102	0.820508	+	0.571635i	$0.193690\pi$
$140$	0	0
$141$	−713.063	−0.425892
$142$	0	0
$143$	−1610.83	−0.941986
$144$	0	0
$145$	−3496.56	−2.00258
$146$	0	0
$147$	2030.56	1.13930
$148$	0	0
$149$	−3089.90	−1.69889	−0.849443	−	0.527680i	$-0.823062\pi$
$149$	−3089.90	−1.69889	−0.849443	+	0.527680i	$0.823062\pi$
$150$	0	0
$151$	−264.413	−0.142501	−0.0712503	−	0.997458i	$-0.522699\pi$
$151$	−264.413	−0.142501	−0.0712503	+	0.997458i	$0.522699\pi$
$152$	0	0
$153$	228.871	0.120935
$154$	0	0
$155$	−6032.19	−3.12592
$156$	0	0
$157$	3349.65	1.70275	0.851374	−	0.524559i	$-0.175770\pi$
$157$	3349.65	1.70275	0.851374	+	0.524559i	$0.175770\pi$
$158$	0	0
$159$	2440.34	1.21718
$160$	0	0
$161$	383.802	0.187875
$162$	0	0
$163$	2998.38	1.44080	0.720402	−	0.693557i	$-0.243957\pi$
$163$	2998.38	1.44080	0.720402	+	0.693557i	$0.243957\pi$
$164$	0	0
$165$	8950.02	4.22278
$166$	0	0
$167$	−365.683	−0.169446	−0.0847229	−	0.996405i	$-0.527000\pi$
$167$	−365.683	−0.169446	−0.0847229	+	0.996405i	$0.527000\pi$
$168$	0	0
$169$	−1551.15	−0.706031
$170$	0	0
$171$	−1433.57	−0.641099
$172$	0	0
$173$	−4338.20	−1.90651	−0.953257	−	0.302160i	$-0.902292\pi$
$173$	−4338.20	−1.90651	−0.953257	+	0.302160i	$0.902292\pi$
$174$	0	0
$175$	1865.66	0.805891
$176$	0	0
$177$	2132.88	0.905747
$178$	0	0
$179$	2913.79	1.21669	0.608343	−	0.793674i	$-0.291835\pi$
$179$	2913.79	1.21669	0.608343	+	0.793674i	$0.291835\pi$
$180$	0	0
$181$	−256.845	−0.105476	−0.0527379	−	0.998608i	$-0.516795\pi$
$181$	−256.845	−0.105476	−0.0527379	+	0.998608i	$0.516795\pi$
$182$	0	0
$183$	−3356.49	−1.35584
$184$	0	0
$185$	812.185	0.322773
$186$	0	0
$187$	1008.92	0.394542
$188$	0	0
$189$	424.472	0.163364
$190$	0	0
$191$	1154.76	0.437463	0.218731	−	0.975785i	$-0.429808\pi$
$191$	1154.76	0.437463	0.218731	+	0.975785i	$0.429808\pi$
$192$	0	0
$193$	1607.12	0.599393	0.299696	−	0.954035i	$-0.403115\pi$
$193$	1607.12	0.599393	0.299696	+	0.954035i	$0.403115\pi$
$194$	0	0
$195$	−3588.45	−1.31782
$196$	0	0
$197$	494.233	0.178744	0.0893721	−	0.995998i	$-0.471514\pi$
$197$	494.233	0.178744	0.0893721	+	0.995998i	$0.471514\pi$
$198$	0	0
$199$	828.008	0.294954	0.147477	−	0.989065i	$-0.452885\pi$
$199$	828.008	0.294954	0.147477	+	0.989065i	$0.452885\pi$
$200$	0	0
$201$	−736.912	−0.258596
$202$	0	0
$203$	832.804	0.287938
$204$	0	0
$205$	7771.48	2.64772
$206$	0	0
$207$	1055.53	0.354417
$208$	0	0
$209$	−6319.52	−2.09153
$210$	0	0
$211$	−2955.87	−0.964410	−0.482205	−	0.876058i	$-0.660164\pi$
$211$	−2955.87	−0.964410	−0.482205	+	0.876058i	$0.660164\pi$
$212$	0	0
$213$	−1410.11	−0.453610
$214$	0	0
$215$	9108.11	2.88915
$216$	0	0
$217$	1436.73	0.449456
$218$	0	0
$219$	571.426	0.176317
$220$	0	0
$221$	−404.519	−0.123126
$222$	0	0
$223$	−4409.23	−1.32405	−0.662026	−	0.749481i	$-0.730303\pi$
$223$	−4409.23	−1.32405	−0.662026	+	0.749481i	$0.730303\pi$
$224$	0	0
$225$	5130.93	1.52028
$226$	0	0
$227$	−2480.08	−0.725148	−0.362574	−	0.931955i	$-0.618102\pi$
$227$	−2480.08	−0.725148	−0.362574	+	0.931955i	$0.618102\pi$
$228$	0	0
$229$	3613.65	1.04278	0.521390	−	0.853318i	$-0.325414\pi$
$229$	3613.65	1.04278	0.521390	+	0.853318i	$0.325414\pi$
$230$	0	0
$231$	−2131.70	−0.607166
$232$	0	0
$233$	−2040.68	−0.573774	−0.286887	−	0.957964i	$-0.592620\pi$
$233$	−2040.68	−0.573774	−0.286887	+	0.957964i	$0.592620\pi$
$234$	0	0
$235$	−2433.29	−0.675447
$236$	0	0
$237$	−4041.53	−1.10770
$238$	0	0
$239$	−3374.09	−0.913187	−0.456593	−	0.889675i	$-0.650931\pi$
$239$	−3374.09	−0.913187	−0.456593	+	0.889675i	$0.650931\pi$
$240$	0	0
$241$	−3220.60	−0.860818	−0.430409	−	0.902634i	$-0.641631\pi$
$241$	−3220.60	−0.860818	−0.430409	+	0.902634i	$0.641631\pi$
$242$	0	0
$243$	3664.67	0.967444
$244$	0	0
$245$	6929.16	1.80689
$246$	0	0
$247$	2533.77	0.652713
$248$	0	0
$249$	6424.61	1.63511
$250$	0	0
$251$	−1396.90	−0.351281	−0.175641	−	0.984454i	$-0.556200\pi$
$251$	−1396.90	−0.351281	−0.175641	+	0.984454i	$0.556200\pi$
$252$	0	0
$253$	4653.03	1.15626
$254$	0	0
$255$	2247.58	0.551956
$256$	0	0
$257$	−3878.18	−0.941301	−0.470650	−	0.882320i	$-0.655981\pi$
$257$	−3878.18	−0.941301	−0.470650	+	0.882320i	$0.655981\pi$
$258$	0	0
$259$	−193.444	−0.0464095
$260$	0	0
$261$	2290.37	0.543181
$262$	0	0
$263$	−4945.43	−1.15950	−0.579749	−	0.814795i	$-0.696850\pi$
$263$	−4945.43	−1.15950	−0.579749	+	0.814795i	$0.696850\pi$
$264$	0	0
$265$	8327.50	1.93039
$266$	0	0
$267$	452.311	0.103674
$268$	0	0
$269$	577.744	0.130950	0.0654752	−	0.997854i	$-0.479144\pi$
$269$	577.744	0.130950	0.0654752	+	0.997854i	$0.479144\pi$
$270$	0	0
$271$	1677.77	0.376079	0.188039	−	0.982161i	$-0.439787\pi$
$271$	1677.77	0.376079	0.188039	+	0.982161i	$0.439787\pi$
$272$	0	0
$273$	854.690	0.189481
$274$	0	0
$275$	22618.4	4.95978
$276$	0	0
$277$	6113.02	1.32598	0.662989	−	0.748630i	$-0.269288\pi$
$277$	6113.02	1.32598	0.662989	+	0.748630i	$0.269288\pi$
$278$	0	0
$279$	3951.29	0.847878
$280$	0	0
$281$	5243.02	1.11307	0.556535	−	0.830824i	$-0.312131\pi$
$281$	5243.02	1.11307	0.556535	+	0.830824i	$0.312131\pi$
$282$	0	0
$283$	−4944.32	−1.03855	−0.519274	−	0.854608i	$-0.673798\pi$
$283$	−4944.32	−1.03855	−0.519274	+	0.854608i	$0.673798\pi$
$284$	0	0
$285$	−14078.1	−2.92601
$286$	0	0
$287$	−1850.99	−0.380699
$288$	0	0
$289$	−4659.64	−0.948430
$290$	0	0
$291$	2622.58	0.528311
$292$	0	0
$293$	−7919.01	−1.57895	−0.789477	−	0.613780i	$-0.789648\pi$
$293$	−7919.01	−1.57895	−0.789477	+	0.613780i	$0.789648\pi$
$294$	0	0
$295$	7278.34	1.43648
$296$	0	0
$297$	5146.09	1.00541
$298$	0	0
$299$	−1865.60	−0.360838
$300$	0	0
$301$	−2169.35	−0.415413
$302$	0	0
$303$	1688.76	0.320187
$304$	0	0
$305$	−11453.8	−2.15031
$306$	0	0
$307$	−9949.94	−1.84975	−0.924875	−	0.380272i	$-0.875830\pi$
$307$	−9949.94	−1.84975	−0.924875	+	0.380272i	$0.875830\pi$
$308$	0	0
$309$	−5381.53	−0.990759
$310$	0	0
$311$	2268.46	0.413610	0.206805	−	0.978382i	$-0.433693\pi$
$311$	2268.46	0.413610	0.206805	+	0.978382i	$0.433693\pi$
$312$	0	0
$313$	−9555.19	−1.72553	−0.862766	−	0.505604i	$-0.831270\pi$
$313$	−9555.19	−1.72553	−0.862766	+	0.505604i	$0.831270\pi$
$314$	0	0
$315$	−1650.16	−0.295161
$316$	0	0
$317$	−4491.78	−0.795848	−0.397924	−	0.917418i	$-0.630269\pi$
$317$	−4491.78	−0.795848	−0.397924	+	0.917418i	$0.630269\pi$
$318$	0	0
$319$	10096.5	1.77209
$320$	0	0
$321$	2596.82	0.451527
$322$	0	0
$323$	−1586.99	−0.273383
$324$	0	0
$325$	−9068.69	−1.54782
$326$	0	0
$327$	446.362	0.0754859
$328$	0	0
$329$	579.555	0.0971182
$330$	0	0
$331$	1867.66	0.310139	0.155069	−	0.987904i	$-0.450440\pi$
$331$	1867.66	0.310139	0.155069	+	0.987904i	$0.450440\pi$
$332$	0	0
$333$	−532.009	−0.0875493
$334$	0	0
$335$	−2514.67	−0.410122
$336$	0	0
$337$	−4481.95	−0.724473	−0.362236	−	0.932086i	$-0.617987\pi$
$337$	−4481.95	−0.724473	−0.362236	+	0.932086i	$0.617987\pi$
$338$	0	0
$339$	−8304.19	−1.33045
$340$	0	0
$341$	17418.2	2.76613
$342$	0	0
$343$	−3443.66	−0.542099
$344$	0	0
$345$	10365.6	1.61758
$346$	0	0
$347$	−2011.24	−0.311150	−0.155575	−	0.987824i	$-0.549723\pi$
$347$	−2011.24	−0.311150	−0.155575	+	0.987824i	$0.549723\pi$
$348$	0	0
$349$	2043.66	0.313451	0.156726	−	0.987642i	$-0.449906\pi$
$349$	2043.66	0.313451	0.156726	+	0.987642i	$0.449906\pi$
$350$	0	0
$351$	−2063.29	−0.313762
$352$	0	0
$353$	−1972.58	−0.297421	−0.148711	−	0.988881i	$-0.547512\pi$
$353$	−1972.58	−0.297421	−0.148711	+	0.988881i	$0.547512\pi$
$354$	0	0
$355$	−4811.91	−0.719408
$356$	0	0
$357$	−535.323	−0.0793622
$358$	0	0
$359$	−12863.6	−1.89113	−0.945567	−	0.325428i	$-0.894492\pi$
$359$	−12863.6	−1.89113	−0.945567	+	0.325428i	$0.894492\pi$
$360$	0	0
$361$	3081.38	0.449247
$362$	0	0
$363$	−17281.8	−2.49878
$364$	0	0
$365$	1949.96	0.279631
$366$	0	0
$367$	7161.72	1.01863	0.509317	−	0.860579i	$-0.329898\pi$
$367$	7161.72	1.01863	0.509317	+	0.860579i	$0.329898\pi$
$368$	0	0
$369$	−5090.59	−0.718172
$370$	0	0
$371$	−1983.43	−0.277559
$372$	0	0
$373$	−4977.48	−0.690949	−0.345475	−	0.938428i	$-0.612282\pi$
$373$	−4977.48	−0.690949	−0.345475	+	0.938428i	$0.612282\pi$
$374$	0	0
$375$	32736.9	4.50807
$376$	0	0
$377$	−4048.13	−0.553022
$378$	0	0
$379$	−2536.18	−0.343733	−0.171867	−	0.985120i	$-0.554980\pi$
$379$	−2536.18	−0.343733	−0.171867	+	0.985120i	$0.554980\pi$
$380$	0	0
$381$	709.679	0.0954277
$382$	0	0
$383$	5864.69	0.782432	0.391216	−	0.920299i	$-0.372054\pi$
$383$	5864.69	0.782432	0.391216	+	0.920299i	$0.372054\pi$
$384$	0	0
$385$	−7274.29	−0.962940
$386$	0	0
$387$	−5966.12	−0.783657
$388$	0	0
$389$	11438.9	1.49094	0.745469	−	0.666541i	$-0.232226\pi$
$389$	11438.9	1.49094	0.745469	+	0.666541i	$0.232226\pi$
$390$	0	0
$391$	1168.49	0.151134
$392$	0	0
$393$	5852.02	0.751134
$394$	0	0
$395$	−13791.5	−1.75677
$396$	0	0
$397$	4131.61	0.522316	0.261158	−	0.965296i	$-0.415896\pi$
$397$	4131.61	0.522316	0.261158	+	0.965296i	$0.415896\pi$
$398$	0	0
$399$	3353.08	0.420712
$400$	0	0
$401$	13571.2	1.69005	0.845026	−	0.534725i	$-0.179585\pi$
$401$	13571.2	1.69005	0.845026	+	0.534725i	$0.179585\pi$
$402$	0	0
$403$	−6983.74	−0.863238
$404$	0	0
$405$	19985.9	2.45211
$406$	0	0
$407$	−2345.22	−0.285623
$408$	0	0
$409$	−8474.21	−1.02451	−0.512253	−	0.858835i	$-0.671189\pi$
$409$	−8474.21	−1.02451	−0.512253	+	0.858835i	$0.671189\pi$
$410$	0	0
$411$	14702.2	1.76449
$412$	0	0
$413$	−1733.54	−0.206542
$414$	0	0
$415$	21923.6	2.59322
$416$	0	0
$417$	−17299.1	−2.03151
$418$	0	0
$419$	5181.27	0.604108	0.302054	−	0.953291i	$-0.402328\pi$
$419$	5181.27	0.604108	0.302054	+	0.953291i	$0.402328\pi$
$420$	0	0
$421$	11389.2	1.31846	0.659232	−	0.751939i	$-0.270881\pi$
$421$	11389.2	1.31846	0.659232	+	0.751939i	$0.270881\pi$
$422$	0	0
$423$	1593.89	0.183209
$424$	0	0
$425$	5680.05	0.648289
$426$	0	0
$427$	2728.04	0.309179
$428$	0	0
$429$	10361.8	1.16614
$430$	0	0
$431$	−10185.3	−1.13831	−0.569154	−	0.822231i	$-0.692729\pi$
$431$	−10185.3	−1.13831	−0.569154	+	0.822231i	$0.692729\pi$
$432$	0	0
$433$	−2394.77	−0.265786	−0.132893	−	0.991130i	$-0.542427\pi$
$433$	−2394.77	−0.265786	−0.132893	+	0.991130i	$0.542427\pi$
$434$	0	0
$435$	22492.1	2.47911
$436$	0	0
$437$	−7319.05	−0.801184
$438$	0	0
$439$	−14072.5	−1.52994	−0.764971	−	0.644065i	$-0.777247\pi$
$439$	−14072.5	−1.52994	−0.764971	+	0.644065i	$0.777247\pi$
$440$	0	0
$441$	−4538.84	−0.490102
$442$	0	0
$443$	−4450.37	−0.477299	−0.238650	−	0.971106i	$-0.576705\pi$
$443$	−4450.37	−0.477299	−0.238650	+	0.971106i	$0.576705\pi$
$444$	0	0
$445$	1543.49	0.164423
$446$	0	0
$447$	19876.1	2.10315
$448$	0	0
$449$	11927.5	1.25366	0.626831	−	0.779155i	$-0.284352\pi$
$449$	11927.5	1.25366	0.626831	+	0.779155i	$0.284352\pi$
$450$	0	0
$451$	−22440.5	−2.34298
$452$	0	0
$453$	1700.87	0.176410
$454$	0	0
$455$	2916.58	0.300509
$456$	0	0
$457$	−16260.7	−1.66443	−0.832214	−	0.554454i	$-0.812927\pi$
$457$	−16260.7	−1.66443	−0.832214	+	0.554454i	$0.812927\pi$
$458$	0	0
$459$	1292.31	0.131416
$460$	0	0
$461$	−5581.99	−0.563947	−0.281973	−	0.959422i	$-0.590989\pi$
$461$	−5581.99	−0.563947	−0.281973	+	0.959422i	$0.590989\pi$
$462$	0	0
$463$	−8967.28	−0.900097	−0.450048	−	0.893004i	$-0.648593\pi$
$463$	−8967.28	−0.900097	−0.450048	+	0.893004i	$0.648593\pi$
$464$	0	0
$465$	38802.8	3.86976
$466$	0	0
$467$	−371.995	−0.0368605	−0.0184303	−	0.999830i	$-0.505867\pi$
$467$	−371.995	−0.0368605	−0.0184303	+	0.999830i	$0.505867\pi$
$468$	0	0
$469$	598.938	0.0589689
$470$	0	0
$471$	−21547.1	−2.10793
$472$	0	0
$473$	−26300.1	−2.55662
$474$	0	0
$475$	−35577.9	−3.43669
$476$	0	0
$477$	−5454.80	−0.523602
$478$	0	0
$479$	−7204.53	−0.687230	−0.343615	−	0.939111i	$-0.611652\pi$
$479$	−7204.53	−0.687230	−0.343615	+	0.939111i	$0.611652\pi$
$480$	0	0
$481$	940.303	0.0891354
$482$	0	0
$483$	−2468.86	−0.232581
$484$	0	0
$485$	8949.41	0.837880
$486$	0	0
$487$	4391.87	0.408654	0.204327	−	0.978903i	$-0.434499\pi$
$487$	4391.87	0.408654	0.204327	+	0.978903i	$0.434499\pi$
$488$	0	0
$489$	−19287.4	−1.78366
$490$	0	0
$491$	3357.81	0.308627	0.154313	−	0.988022i	$-0.450683\pi$
$491$	3357.81	0.308627	0.154313	+	0.988022i	$0.450683\pi$
$492$	0	0
$493$	2535.49	0.231628
$494$	0	0
$495$	−20005.7	−1.81654
$496$	0	0
$497$	1146.09	0.103439
$498$	0	0
$499$	12559.7	1.12676	0.563378	−	0.826199i	$-0.309502\pi$
$499$	12559.7	1.12676	0.563378	+	0.826199i	$0.309502\pi$
$500$	0	0
$501$	2352.30	0.209767
$502$	0	0
$503$	13358.4	1.18414	0.592068	−	0.805888i	$-0.298312\pi$
$503$	13358.4	1.18414	0.592068	+	0.805888i	$0.298312\pi$
$504$	0	0
$505$	5762.80	0.507804
$506$	0	0
$507$	9977.96	0.874037
$508$	0	0
$509$	−16650.8	−1.44997	−0.724985	−	0.688765i	$-0.758153\pi$
$509$	−16650.8	−1.44997	−0.724985	+	0.688765i	$0.758153\pi$
$510$	0	0
$511$	−464.436	−0.0402064
$512$	0	0
$513$	−8094.61	−0.696659
$514$	0	0
$515$	−18364.1	−1.57130
$516$	0	0
$517$	7026.23	0.597704
$518$	0	0
$519$	27906.0	2.36019
$520$	0	0
$521$	−116.315	−0.00978091	−0.00489046	−	0.999988i	$-0.501557\pi$
$521$	−116.315	−0.00978091	−0.00489046	+	0.999988i	$0.501557\pi$
$522$	0	0
$523$	15471.9	1.29357	0.646786	−	0.762672i	$-0.276113\pi$
$523$	15471.9	1.29357	0.646786	+	0.762672i	$0.276113\pi$
$524$	0	0
$525$	−12001.1	−0.997660
$526$	0	0
$527$	4374.16	0.361559
$528$	0	0
$529$	−6778.03	−0.557083
$530$	0	0
$531$	−4767.56	−0.389632
$532$	0	0
$533$	8997.39	0.731182
$534$	0	0
$535$	8861.50	0.716104
$536$	0	0
$537$	−18743.3	−1.50621
$538$	0	0
$539$	−20008.3	−1.59892
$540$	0	0
$541$	−6162.98	−0.489773	−0.244887	−	0.969552i	$-0.578751\pi$
$541$	−6162.98	−0.489773	−0.244887	+	0.969552i	$0.578751\pi$
$542$	0	0
$543$	1652.19	0.130575
$544$	0	0
$545$	1523.19	0.119718
$546$	0	0
$547$	2993.45	0.233987	0.116993	−	0.993133i	$-0.462674\pi$
$547$	2993.45	0.233987	0.116993	+	0.993133i	$0.462674\pi$
$548$	0	0
$549$	7502.64	0.583251
$550$	0	0
$551$	−15881.4	−1.22790
$552$	0	0
$553$	3284.83	0.252595
$554$	0	0
$555$	−5224.48	−0.399580
$556$	0	0
$557$	3241.58	0.246589	0.123294	−	0.992370i	$-0.460654\pi$
$557$	3241.58	0.246589	0.123294	+	0.992370i	$0.460654\pi$
$558$	0	0
$559$	10544.9	0.797853
$560$	0	0
$561$	−6489.99	−0.488427
$562$	0	0
$563$	−13156.3	−0.984852	−0.492426	−	0.870354i	$-0.663890\pi$
$563$	−13156.3	−0.984852	−0.492426	+	0.870354i	$0.663890\pi$
$564$	0	0
$565$	−28337.6	−2.11004
$566$	0	0
$567$	−4760.19	−0.352573
$568$	0	0
$569$	−2289.78	−0.168704	−0.0843519	−	0.996436i	$-0.526882\pi$
$569$	−2289.78	−0.168704	−0.0843519	+	0.996436i	$0.526882\pi$
$570$	0	0
$571$	−14654.8	−1.07405	−0.537025	−	0.843566i	$-0.680452\pi$
$571$	−14654.8	−1.07405	−0.537025	+	0.843566i	$0.680452\pi$
$572$	0	0
$573$	−7428.13	−0.541561
$574$	0	0
$575$	26195.8	1.89990
$576$	0	0
$577$	−20760.1	−1.49784	−0.748921	−	0.662659i	$-0.769428\pi$
$577$	−20760.1	−1.49784	−0.748921	+	0.662659i	$0.769428\pi$
$578$	0	0
$579$	−10338.0	−0.742024
$580$	0	0
$581$	−5221.72	−0.372863
$582$	0	0
$583$	−24046.1	−1.70821
$584$	0	0
$585$	8021.15	0.566896
$586$	0	0
$587$	22530.5	1.58421	0.792107	−	0.610383i	$-0.208984\pi$
$587$	22530.5	1.58421	0.792107	+	0.610383i	$0.208984\pi$
$588$	0	0
$589$	−27398.3	−1.91668
$590$	0	0
$591$	−3179.21	−0.221278
$592$	0	0
$593$	7249.23	0.502007	0.251004	−	0.967986i	$-0.419239\pi$
$593$	7249.23	0.502007	0.251004	+	0.967986i	$0.419239\pi$
$594$	0	0
$595$	−1826.76	−0.125865
$596$	0	0
$597$	−5326.26	−0.365141
$598$	0	0
$599$	−23792.3	−1.62292	−0.811458	−	0.584411i	$-0.801326\pi$
$599$	−23792.3	−1.62292	−0.811458	+	0.584411i	$0.801326\pi$
$600$	0	0
$601$	2515.26	0.170715	0.0853574	−	0.996350i	$-0.472797\pi$
$601$	2515.26	0.170715	0.0853574	+	0.996350i	$0.472797\pi$
$602$	0	0
$603$	1647.19	0.111242
$604$	0	0
$605$	−58973.1	−3.96297
$606$	0	0
$607$	23125.6	1.54635	0.773177	−	0.634190i	$-0.218666\pi$
$607$	23125.6	1.54635	0.773177	+	0.634190i	$0.218666\pi$
$608$	0	0
$609$	−5357.11	−0.356455
$610$	0	0
$611$	−2817.12	−0.186528
$612$	0	0
$613$	−16552.6	−1.09062	−0.545312	−	0.838233i	$-0.683589\pi$
$613$	−16552.6	−1.09062	−0.545312	+	0.838233i	$0.683589\pi$
$614$	0	0
$615$	−49991.0	−3.27777
$616$	0	0
$617$	−14612.2	−0.953425	−0.476712	−	0.879059i	$-0.658172\pi$
$617$	−14612.2	−0.953425	−0.476712	+	0.879059i	$0.658172\pi$
$618$	0	0
$619$	4970.70	0.322761	0.161381	−	0.986892i	$-0.448405\pi$
$619$	4970.70	0.322761	0.161381	+	0.986892i	$0.448405\pi$
$620$	0	0
$621$	5960.02	0.385133
$622$	0	0
$623$	−367.624	−0.0236413
$624$	0	0
$625$	67107.3	4.29486
$626$	0	0
$627$	40651.1	2.58923
$628$	0	0
$629$	−588.945	−0.0373335
$630$	0	0
$631$	−13107.6	−0.826952	−0.413476	−	0.910515i	$-0.635685\pi$
$631$	−13107.6	−0.826952	−0.413476	+	0.910515i	$0.635685\pi$
$632$	0	0
$633$	19014.0	1.19390
$634$	0	0
$635$	2421.74	0.151344
$636$	0	0
$637$	8022.20	0.498981
$638$	0	0
$639$	3151.97	0.195133
$640$	0	0
$641$	27443.4	1.69103	0.845515	−	0.533952i	$-0.179294\pi$
$641$	27443.4	1.69103	0.845515	+	0.533952i	$0.179294\pi$
$642$	0	0
$643$	−16671.2	−1.02247	−0.511234	−	0.859442i	$-0.670811\pi$
$643$	−16671.2	−1.02247	−0.511234	+	0.859442i	$0.670811\pi$
$644$	0	0
$645$	−58589.0	−3.57665
$646$	0	0
$647$	−13670.7	−0.830678	−0.415339	−	0.909667i	$-0.636337\pi$
$647$	−13670.7	−0.830678	−0.415339	+	0.909667i	$0.636337\pi$
$648$	0	0
$649$	−21016.5	−1.27114
$650$	0	0
$651$	−9241.97	−0.556408
$652$	0	0
$653$	15344.4	0.919561	0.459780	−	0.888033i	$-0.347928\pi$
$653$	15344.4	0.919561	0.459780	+	0.888033i	$0.347928\pi$
$654$	0	0
$655$	19969.7	1.19127
$656$	0	0
$657$	−1277.29	−0.0758475
$658$	0	0
$659$	−31695.3	−1.87356	−0.936778	−	0.349924i	$-0.886208\pi$
$659$	−31695.3	−1.87356	−0.936778	+	0.349924i	$0.886208\pi$
$660$	0	0
$661$	−11965.3	−0.704078	−0.352039	−	0.935985i	$-0.614512\pi$
$661$	−11965.3	−0.704078	−0.352039	+	0.935985i	$0.614512\pi$
$662$	0	0
$663$	2602.12	0.152425
$664$	0	0
$665$	11442.2	0.667232
$666$	0	0
$667$	11693.4	0.678817
$668$	0	0
$669$	28362.9	1.63912
$670$	0	0
$671$	33073.4	1.90281
$672$	0	0
$673$	−14818.6	−0.848759	−0.424379	−	0.905485i	$-0.639508\pi$
$673$	−14818.6	−0.848759	−0.424379	+	0.905485i	$0.639508\pi$
$674$	0	0
$675$	28971.6	1.65203
$676$	0	0
$677$	−4761.64	−0.270317	−0.135159	−	0.990824i	$-0.543154\pi$
$677$	−4761.64	−0.270317	−0.135159	+	0.990824i	$0.543154\pi$
$678$	0	0
$679$	−2131.55	−0.120473
$680$	0	0
$681$	15953.4	0.897703
$682$	0	0
$683$	11880.9	0.665610	0.332805	−	0.942996i	$-0.392005\pi$
$683$	11880.9	0.665610	0.332805	+	0.942996i	$0.392005\pi$
$684$	0	0
$685$	50170.5	2.79842
$686$	0	0
$687$	−23245.2	−1.29092
$688$	0	0
$689$	9641.12	0.533088
$690$	0	0
$691$	21469.7	1.18198	0.590988	−	0.806681i	$-0.298738\pi$
$691$	21469.7	1.18198	0.590988	+	0.806681i	$0.298738\pi$
$692$	0	0
$693$	4764.91	0.261189
$694$	0	0
$695$	−59032.1	−3.22189
$696$	0	0
$697$	−5635.38	−0.306249
$698$	0	0
$699$	13126.9	0.710308
$700$	0	0
$701$	−3839.35	−0.206862	−0.103431	−	0.994637i	$-0.532982\pi$
$701$	−3839.35	−0.206862	−0.103431	+	0.994637i	$0.532982\pi$
$702$	0	0
$703$	3688.95	0.197911
$704$	0	0
$705$	15652.4	0.836176
$706$	0	0
$707$	−1372.57	−0.0730139
$708$	0	0
$709$	19020.0	1.00749	0.503745	−	0.863853i	$-0.331955\pi$
$709$	19020.0	1.00749	0.503745	+	0.863853i	$0.331955\pi$
$710$	0	0
$711$	9033.90	0.476509
$712$	0	0
$713$	20173.2	1.05960
$714$	0	0
$715$	35359.2	1.84945
$716$	0	0
$717$	21704.2	1.13049
$718$	0	0
$719$	−4274.48	−0.221712	−0.110856	−	0.993836i	$-0.535359\pi$
$719$	−4274.48	−0.221712	−0.110856	+	0.993836i	$0.535359\pi$
$720$	0	0
$721$	4373.93	0.225928
$722$	0	0
$723$	20716.9	1.06566
$724$	0	0
$725$	56841.7	2.91179
$726$	0	0
$727$	3597.47	0.183525	0.0917626	−	0.995781i	$-0.470750\pi$
$727$	3597.47	0.183525	0.0917626	+	0.995781i	$0.470750\pi$
$728$	0	0
$729$	1009.46	0.0512858
$730$	0	0
$731$	−6604.62	−0.334173
$732$	0	0
$733$	−6014.23	−0.303057	−0.151528	−	0.988453i	$-0.548419\pi$
$733$	−6014.23	−0.303057	−0.151528	+	0.988453i	$0.548419\pi$
$734$	0	0
$735$	−44572.7	−2.23685
$736$	0	0
$737$	7261.22	0.362918
$738$	0	0
$739$	16249.9	0.808882	0.404441	−	0.914564i	$-0.367466\pi$
$739$	16249.9	0.808882	0.404441	+	0.914564i	$0.367466\pi$
$740$	0	0
$741$	−16298.8	−0.808032
$742$	0	0
$743$	5527.96	0.272949	0.136475	−	0.990644i	$-0.456423\pi$
$743$	5527.96	0.272949	0.136475	+	0.990644i	$0.456423\pi$
$744$	0	0
$745$	67826.2	3.33551
$746$	0	0
$747$	−14360.7	−0.703388
$748$	0	0
$749$	−2110.61	−0.102964
$750$	0	0
$751$	10340.8	0.502452	0.251226	−	0.967928i	$-0.419166\pi$
$751$	10340.8	0.502452	0.251226	+	0.967928i	$0.419166\pi$
$752$	0	0
$753$	8985.74	0.434872
$754$	0	0
$755$	5804.11	0.279779
$756$	0	0
$757$	−8559.36	−0.410958	−0.205479	−	0.978662i	$-0.565875\pi$
$757$	−8559.36	−0.410958	−0.205479	+	0.978662i	$0.565875\pi$
$758$	0	0
$759$	−29931.2	−1.43140
$760$	0	0
$761$	9656.20	0.459970	0.229985	−	0.973194i	$-0.426132\pi$
$761$	9656.20	0.459970	0.229985	+	0.973194i	$0.426132\pi$
$762$	0	0
$763$	−362.789	−0.0172134
$764$	0	0
$765$	−5023.93	−0.237439
$766$	0	0
$767$	8426.45	0.396690
$768$	0	0
$769$	13660.4	0.640581	0.320290	−	0.947319i	$-0.396220\pi$
$769$	13660.4	0.640581	0.320290	+	0.947319i	$0.396220\pi$
$770$	0	0
$771$	24946.9	1.16529
$772$	0	0
$773$	−13501.3	−0.628213	−0.314107	−	0.949388i	$-0.601705\pi$
$773$	−13501.3	−0.628213	−0.314107	+	0.949388i	$0.601705\pi$
$774$	0	0
$775$	98062.0	4.54515
$776$	0	0
$777$	1244.36	0.0574530
$778$	0	0
$779$	35298.1	1.62348
$780$	0	0
$781$	13894.6	0.636605
$782$	0	0
$783$	12932.5	0.590255
$784$	0	0
$785$	−73528.1	−3.34310
$786$	0	0
$787$	22425.6	1.01574	0.507868	−	0.861435i	$-0.330434\pi$
$787$	22425.6	1.01574	0.507868	+	0.861435i	$0.330434\pi$
$788$	0	0
$789$	31812.1	1.43541
$790$	0	0
$791$	6749.38	0.303388
$792$	0	0
$793$	−13260.6	−0.593818
$794$	0	0
$795$	−53567.7	−2.38975
$796$	0	0
$797$	31861.9	1.41607	0.708034	−	0.706178i	$-0.249582\pi$
$797$	31861.9	1.41607	0.708034	+	0.706178i	$0.249582\pi$
$798$	0	0
$799$	1764.46	0.0781255
$800$	0	0
$801$	−1011.04	−0.0445983
$802$	0	0
$803$	−5630.59	−0.247446
$804$	0	0
$805$	−8424.83	−0.368865
$806$	0	0
$807$	−3716.41	−0.162111
$808$	0	0
$809$	−31815.3	−1.38265	−0.691327	−	0.722542i	$-0.742974\pi$
$809$	−31815.3	−1.38265	−0.691327	+	0.722542i	$0.742974\pi$
$810$	0	0
$811$	6697.35	0.289982	0.144991	−	0.989433i	$-0.453685\pi$
$811$	6697.35	0.289982	0.144991	+	0.989433i	$0.453685\pi$
$812$	0	0
$813$	−10792.5	−0.465570
$814$	0	0
$815$	−65817.2	−2.82881
$816$	0	0
$817$	41369.1	1.77151
$818$	0	0
$819$	−1910.46	−0.0815103
$820$	0	0
$821$	−33398.7	−1.41976	−0.709880	−	0.704322i	$-0.751251\pi$
$821$	−33398.7	−1.41976	−0.709880	+	0.704322i	$0.751251\pi$
$822$	0	0
$823$	23636.6	1.00112	0.500558	−	0.865703i	$-0.333128\pi$
$823$	23636.6	1.00112	0.500558	+	0.865703i	$0.333128\pi$
$824$	0	0
$825$	−145495.	−6.14000
$826$	0	0
$827$	1368.22	0.0575303	0.0287652	−	0.999586i	$-0.490843\pi$
$827$	1368.22	0.0575303	0.0287652	+	0.999586i	$0.490843\pi$
$828$	0	0
$829$	−36528.3	−1.53037	−0.765187	−	0.643808i	$-0.777354\pi$
$829$	−36528.3	−1.53037	−0.765187	+	0.643808i	$0.777354\pi$
$830$	0	0
$831$	−39322.7	−1.64150
$832$	0	0
$833$	−5024.59	−0.208994
$834$	0	0
$835$	8027.10	0.332682
$836$	0	0
$837$	22310.9	0.921358
$838$	0	0
$839$	14641.4	0.602475	0.301238	−	0.953549i	$-0.402600\pi$
$839$	14641.4	0.602475	0.301238	+	0.953549i	$0.402600\pi$
$840$	0	0
$841$	984.259	0.0403567
$842$	0	0
$843$	−33726.4	−1.37793
$844$	0	0
$845$	34049.2	1.38619
$846$	0	0
$847$	14046.1	0.569810
$848$	0	0
$849$	31804.9	1.28568
$850$	0	0
$851$	−2716.16	−0.109411
$852$	0	0
$853$	9315.63	0.373929	0.186964	−	0.982367i	$-0.440135\pi$
$853$	9315.63	0.373929	0.186964	+	0.982367i	$0.440135\pi$
$854$	0	0
$855$	31468.2	1.25870
$856$	0	0
$857$	−41258.3	−1.64452	−0.822261	−	0.569110i	$-0.807288\pi$
$857$	−41258.3	−1.64452	−0.822261	+	0.569110i	$0.807288\pi$
$858$	0	0
$859$	10131.6	0.402430	0.201215	−	0.979547i	$-0.435511\pi$
$859$	10131.6	0.402430	0.201215	+	0.979547i	$0.435511\pi$
$860$	0	0
$861$	11906.7	0.471290
$862$	0	0
$863$	−6617.87	−0.261037	−0.130519	−	0.991446i	$-0.541664\pi$
$863$	−6617.87	−0.261037	−0.130519	+	0.991446i	$0.541664\pi$
$864$	0	0
$865$	95227.6	3.74316
$866$	0	0
$867$	29973.7	1.17412
$868$	0	0
$869$	39823.6	1.55457
$870$	0	0
$871$	−2911.34	−0.113257
$872$	0	0
$873$	−5862.17	−0.227267
$874$	0	0
$875$	−26607.5	−1.02800
$876$	0	0
$877$	8075.49	0.310935	0.155467	−	0.987841i	$-0.450312\pi$
$877$	8075.49	0.310935	0.155467	+	0.987841i	$0.450312\pi$
$878$	0	0
$879$	50940.0	1.95468
$880$	0	0
$881$	40691.7	1.55612	0.778058	−	0.628192i	$-0.216205\pi$
$881$	40691.7	1.55612	0.778058	+	0.628192i	$0.216205\pi$
$882$	0	0
$883$	−38479.5	−1.46652	−0.733260	−	0.679948i	$-0.762002\pi$
$883$	−38479.5	−1.46652	−0.733260	+	0.679948i	$0.762002\pi$
$884$	0	0
$885$	−46818.8	−1.77830
$886$	0	0
$887$	−27759.4	−1.05081	−0.525405	−	0.850852i	$-0.676086\pi$
$887$	−27759.4	−1.05081	−0.525405	+	0.850852i	$0.676086\pi$
$888$	0	0
$889$	−576.804	−0.0217608
$890$	0	0
$891$	−57710.1	−2.16988
$892$	0	0
$893$	−11052.0	−0.414156
$894$	0	0
$895$	−63960.4	−2.38878
$896$	0	0
$897$	12000.7	0.446703
$898$	0	0
$899$	43773.4	1.62394
$900$	0	0
$901$	−6038.58	−0.223279
$902$	0	0
$903$	13954.6	0.514264
$904$	0	0
$905$	5637.99	0.207086
$906$	0	0
$907$	25063.2	0.917540	0.458770	−	0.888555i	$-0.348290\pi$
$907$	25063.2	0.917540	0.458770	+	0.888555i	$0.348290\pi$
$908$	0	0
$909$	−3774.83	−0.137737
$910$	0	0
$911$	8517.47	0.309765	0.154883	−	0.987933i	$-0.450500\pi$
$911$	8517.47	0.309765	0.154883	+	0.987933i	$0.450500\pi$
$912$	0	0
$913$	−63305.4	−2.29475
$914$	0	0
$915$	73678.0	2.66199
$916$	0	0
$917$	−4756.34	−0.171285
$918$	0	0
$919$	10124.9	0.363429	0.181714	−	0.983351i	$-0.441835\pi$
$919$	10124.9	0.363429	0.181714	+	0.983351i	$0.441835\pi$
$920$	0	0
$921$	64004.2	2.28991
$922$	0	0
$923$	−5570.96	−0.198668
$924$	0	0
$925$	−13203.2	−0.469319
$926$	0	0
$927$	12029.2	0.426202
$928$	0	0
$929$	−21701.6	−0.766423	−0.383212	−	0.923661i	$-0.625182\pi$
$929$	−21701.6	−0.766423	−0.383212	+	0.923661i	$0.625182\pi$
$930$	0	0
$931$	31472.3	1.10791
$932$	0	0
$933$	−14592.1	−0.512032
$934$	0	0
$935$	−22146.7	−0.774625
$936$	0	0
$937$	−45485.7	−1.58586	−0.792931	−	0.609311i	$-0.791446\pi$
$937$	−45485.7	−1.58586	−0.792931	+	0.609311i	$0.791446\pi$
$938$	0	0
$939$	61464.9	2.13614
$940$	0	0
$941$	14943.1	0.517674	0.258837	−	0.965921i	$-0.416661\pi$
$941$	14943.1	0.517674	0.258837	+	0.965921i	$0.416661\pi$
$942$	0	0
$943$	−25989.8	−0.897503
$944$	0	0
$945$	−9317.57	−0.320741
$946$	0	0
$947$	−12370.8	−0.424495	−0.212247	−	0.977216i	$-0.568078\pi$
$947$	−12370.8	−0.424495	−0.212247	+	0.977216i	$0.568078\pi$
$948$	0	0
$949$	2257.55	0.0772215
$950$	0	0
$951$	28894.0	0.985227
$952$	0	0
$953$	−15807.8	−0.537317	−0.268659	−	0.963235i	$-0.586580\pi$
$953$	−15807.8	−0.537317	−0.268659	+	0.963235i	$0.586580\pi$
$954$	0	0
$955$	−25348.0	−0.858894
$956$	0	0
$957$	−64946.9	−2.19377
$958$	0	0
$959$	−11949.5	−0.402366
$960$	0	0
$961$	45725.8	1.53489
$962$	0	0
$963$	−5804.59	−0.194237
$964$	0	0
$965$	−35277.8	−1.17682
$966$	0	0
$967$	−15029.0	−0.499792	−0.249896	−	0.968273i	$-0.580396\pi$
$967$	−15029.0	−0.499792	−0.249896	+	0.968273i	$0.580396\pi$
$968$	0	0
$969$	10208.5	0.338436
$970$	0	0
$971$	13603.6	0.449599	0.224799	−	0.974405i	$-0.427827\pi$
$971$	13603.6	0.449599	0.224799	+	0.974405i	$0.427827\pi$
$972$	0	0
$973$	14060.1	0.463255
$974$	0	0
$975$	58335.5	1.91613
$976$	0	0
$977$	27442.8	0.898643	0.449321	−	0.893370i	$-0.351666\pi$
$977$	27442.8	0.898643	0.449321	+	0.893370i	$0.351666\pi$
$978$	0	0
$979$	−4456.89	−0.145498
$980$	0	0
$981$	−997.739	−0.0324723
$982$	0	0
$983$	−5490.69	−0.178154	−0.0890772	−	0.996025i	$-0.528392\pi$
$983$	−5490.69	−0.178154	−0.0890772	+	0.996025i	$0.528392\pi$
$984$	0	0
$985$	−10848.9	−0.350938
$986$	0	0
$987$	−3728.06	−0.120228
$988$	0	0
$989$	−30459.8	−0.979340
$990$	0	0
$991$	−13574.9	−0.435136	−0.217568	−	0.976045i	$-0.569812\pi$
$991$	−13574.9	−0.435136	−0.217568	+	0.976045i	$0.569812\pi$
$992$	0	0
$993$	−12014.0	−0.383939
$994$	0	0
$995$	−18175.6	−0.579099
$996$	0	0
$997$	16328.3	0.518679	0.259339	−	0.965786i	$-0.416495\pi$
$997$	16328.3	0.518679	0.259339	+	0.965786i	$0.416495\pi$
$998$	0	0
$999$	−3003.97	−0.0951367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.4.a.e.1.1		4
4.3	odd	2		148.4.a.a.1.4	✓	4
8.3	odd	2		2368.4.a.j.1.1		4
8.5	even	2		2368.4.a.i.1.4		4
12.11	even	2		1332.4.a.e.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.4.a.a.1.4	✓	4	4.3	odd	2
592.4.a.e.1.1		4	1.1	even	1	trivial
1332.4.a.e.1.4		4	12.11	even	2
2368.4.a.i.1.4		4	8.5	even	2
2368.4.a.j.1.1		4	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:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)

\(f(q)\)	\(=\)	\(q-6.43262 q^{3} -21.9509 q^{5} +5.22823 q^{7} +14.3786 q^{9} +O(q^{10})\)	\(q-6.43262 q^{3} -21.9509 q^{5} +5.22823 q^{7} +14.3786 q^{9} +63.3844 q^{11} -25.4136 q^{13} +141.202 q^{15} +15.9174 q^{17} -99.7015 q^{19} -33.6312 q^{21} +73.4096 q^{23} +356.844 q^{25} +81.1885 q^{27} +159.290 q^{29} +274.803 q^{31} -407.728 q^{33} -114.765 q^{35} -37.0000 q^{37} +163.476 q^{39} -354.038 q^{41} -414.930 q^{43} -315.625 q^{45} +110.851 q^{47} -315.666 q^{49} -102.391 q^{51} -379.369 q^{53} -1391.35 q^{55} +641.342 q^{57} -331.573 q^{59} +521.791 q^{61} +75.1748 q^{63} +557.852 q^{65} +114.559 q^{67} -472.216 q^{69} +219.212 q^{71} -88.8324 q^{73} -2295.44 q^{75} +331.388 q^{77} +628.287 q^{79} -910.478 q^{81} -998.754 q^{83} -349.403 q^{85} -1024.65 q^{87} -70.3152 q^{89} -132.868 q^{91} -1767.71 q^{93} +2188.54 q^{95} -407.700 q^{97} +911.381 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 19 q^{5} + 6 q^{7} - 7 q^{9}+O(q^{10}) \) 4 * q + q^3 - 19 * q^5 + 6 * q^7 - 7 * q^9	\( 4 q + q^{3} - 19 q^{5} + 6 q^{7} - 7 q^{9} + 77 q^{11} - 115 q^{13} + 172 q^{15} - 180 q^{17} - 4 q^{19} - 208 q^{21} + 189 q^{23} + 19 q^{25} + 94 q^{27} - 99 q^{29} + 251 q^{31} - 434 q^{33} - 124 q^{35} - 148 q^{37} - 351 q^{39} - 497 q^{41} - 694 q^{43} - 122 q^{45} + 50 q^{47} - 198 q^{49} - 1270 q^{51} - 290 q^{53} - 1621 q^{55} + 330 q^{57} - 586 q^{59} + 991 q^{61} - 798 q^{63} - 94 q^{65} - 823 q^{67} + 89 q^{69} + 96 q^{71} + 121 q^{73} - 3026 q^{75} - 314 q^{77} - 475 q^{79} - 456 q^{81} - 820 q^{83} - 750 q^{85} - 687 q^{87} - 880 q^{89} - 626 q^{91} - 430 q^{93} + 1538 q^{95} - 904 q^{97} - 202 q^{99}+O(q^{100}) \) 4 * q + q^3 - 19 * q^5 + 6 * q^7 - 7 * q^9 + 77 * q^11 - 115 * q^13 + 172 * q^15 - 180 * q^17 - 4 * q^19 - 208 * q^21 + 189 * q^23 + 19 * q^25 + 94 * q^27 - 99 * q^29 + 251 * q^31 - 434 * q^33 - 124 * q^35 - 148 * q^37 - 351 * q^39 - 497 * q^41 - 694 * q^43 - 122 * q^45 + 50 * q^47 - 198 * q^49 - 1270 * q^51 - 290 * q^53 - 1621 * q^55 + 330 * q^57 - 586 * q^59 + 991 * q^61 - 798 * q^63 - 94 * q^65 - 823 * q^67 + 89 * q^69 + 96 * q^71 + 121 * q^73 - 3026 * q^75 - 314 * q^77 - 475 * q^79 - 456 * q^81 - 820 * q^83 - 750 * q^85 - 687 * q^87 - 880 * q^89 - 626 * q^91 - 430 * q^93 + 1538 * q^95 - 904 * q^97 - 202 * q^99

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