LMFDB - Embedded newform 600.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(1,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	600.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:	$35.4011460034$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.72015$ of defining polynomial
Character	$\chi$	$=$	600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -21.8806 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -21.8806 q^{7} +9.00000 q^{9} +54.6418 q^{11} -82.7612 q^{13} +100.403 q^{17} -84.1194 q^{19} -65.6418 q^{21} +0.880613 q^{23} +27.0000 q^{27} -99.1194 q^{29} -78.9255 q^{31} +163.926 q^{33} -390.806 q^{37} -248.284 q^{39} +104.642 q^{41} -241.403 q^{43} -512.567 q^{47} +135.761 q^{49} +301.209 q^{51} -284.642 q^{53} -252.358 q^{57} -709.045 q^{59} +470.522 q^{61} -196.926 q^{63} +667.583 q^{67} +2.64184 q^{69} -51.5378 q^{71} -371.284 q^{73} -1195.60 q^{77} -79.3428 q^{79} +81.0000 q^{81} +682.180 q^{83} -297.358 q^{87} +628.478 q^{89} +1810.87 q^{91} -236.777 q^{93} -1519.09 q^{97} +491.777 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} - 16 q^{11} - 82 q^{13} - 8 q^{17} - 210 q^{19} - 6 q^{21} - 40 q^{23} + 54 q^{27} - 240 q^{29} + 218 q^{31} - 48 q^{33} - 364 q^{37} - 246 q^{39} + 84 q^{41} - 274 q^{43} - 524 q^{47} + 188 q^{49} - 24 q^{51} - 444 q^{53} - 630 q^{57} - 1084 q^{59} + 774 q^{61} - 18 q^{63} - 210 q^{67} - 120 q^{69} + 1108 q^{71} - 492 q^{73} - 2600 q^{77} - 1328 q^{79} + 162 q^{81} + 28 q^{83} - 720 q^{87} + 1424 q^{89} + 1826 q^{91} + 654 q^{93} - 2370 q^{97} - 144 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9 - 16 * q^11 - 82 * q^13 - 8 * q^17 - 210 * q^19 - 6 * q^21 - 40 * q^23 + 54 * q^27 - 240 * q^29 + 218 * q^31 - 48 * q^33 - 364 * q^37 - 246 * q^39 + 84 * q^41 - 274 * q^43 - 524 * q^47 + 188 * q^49 - 24 * q^51 - 444 * q^53 - 630 * q^57 - 1084 * q^59 + 774 * q^61 - 18 * q^63 - 210 * q^67 - 120 * q^69 + 1108 * q^71 - 492 * q^73 - 2600 * q^77 - 1328 * q^79 + 162 * q^81 + 28 * q^83 - 720 * q^87 + 1424 * q^89 + 1826 * q^91 + 654 * q^93 - 2370 * q^97 - 144 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−21.8806	−1.18144	−0.590721	−	0.806876i	$-0.701157\pi$
$7$	−21.8806	−1.18144	−0.590721	+	0.806876i	$0.701157\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	54.6418	1.49774	0.748870	−	0.662717i	$-0.230597\pi$
$11$	54.6418	1.49774	0.748870	+	0.662717i	$0.230597\pi$
$12$	0	0
$13$	−82.7612	−1.76568	−0.882840	−	0.469674i	$-0.844371\pi$
$13$	−82.7612	−1.76568	−0.882840	+	0.469674i	$0.844371\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	100.403	1.43243	0.716215	−	0.697879i	$-0.245873\pi$
$17$	100.403	1.43243	0.716215	+	0.697879i	$0.245873\pi$
$18$	0	0
$19$	−84.1194	−1.01570	−0.507850	−	0.861445i	$-0.669560\pi$
$19$	−84.1194	−1.01570	−0.507850	+	0.861445i	$0.669560\pi$
$20$	0	0
$21$	−65.6418	−0.682106
$22$	0	0
$23$	0.880613	0.00798350	0.00399175	−	0.999992i	$-0.498729\pi$
$23$	0.880613	0.00798350	0.00399175	+	0.999992i	$0.498729\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−99.1194	−0.634690	−0.317345	−	0.948310i	$-0.602791\pi$
$29$	−99.1194	−0.634690	−0.317345	+	0.948310i	$0.602791\pi$
$30$	0	0
$31$	−78.9255	−0.457272	−0.228636	−	0.973512i	$-0.573427\pi$
$31$	−78.9255	−0.457272	−0.228636	+	0.973512i	$0.573427\pi$
$32$	0	0
$33$	163.926	0.864720
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−390.806	−1.73644	−0.868218	−	0.496183i	$-0.834735\pi$
$37$	−390.806	−1.73644	−0.868218	+	0.496183i	$0.834735\pi$
$38$	0	0
$39$	−248.284	−1.01942
$40$	0	0
$41$	104.642	0.398593	0.199296	−	0.979939i	$-0.436134\pi$
$41$	104.642	0.398593	0.199296	+	0.979939i	$0.436134\pi$
$42$	0	0
$43$	−241.403	−0.856131	−0.428065	−	0.903748i	$-0.640805\pi$
$43$	−241.403	−0.856131	−0.428065	+	0.903748i	$0.640805\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−512.567	−1.59076	−0.795379	−	0.606112i	$-0.792728\pi$
$47$	−512.567	−1.59076	−0.795379	+	0.606112i	$0.792728\pi$
$48$	0	0
$49$	135.761	0.395805
$50$	0	0
$51$	301.209	0.827014
$52$	0	0
$53$	−284.642	−0.737709	−0.368854	−	0.929487i	$-0.620250\pi$
$53$	−284.642	−0.737709	−0.368854	+	0.929487i	$0.620250\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−252.358	−0.586415
$58$	0	0
$59$	−709.045	−1.56457	−0.782286	−	0.622919i	$-0.785947\pi$
$59$	−709.045	−1.56457	−0.782286	+	0.622919i	$0.785947\pi$
$60$	0	0
$61$	470.522	0.987610	0.493805	−	0.869573i	$-0.335606\pi$
$61$	470.522	0.987610	0.493805	+	0.869573i	$0.335606\pi$
$62$	0	0
$63$	−196.926	−0.393814
$64$	0	0
$65$	0	0
$66$	0	0
$67$	667.583	1.21729	0.608643	−	0.793444i	$-0.291714\pi$
$67$	667.583	1.21729	0.608643	+	0.793444i	$0.291714\pi$
$68$	0	0
$69$	2.64184	0.00460928
$70$	0	0
$71$	−51.5378	−0.0861466	−0.0430733	−	0.999072i	$-0.513715\pi$
$71$	−51.5378	−0.0861466	−0.0430733	+	0.999072i	$0.513715\pi$
$72$	0	0
$73$	−371.284	−0.595280	−0.297640	−	0.954678i	$-0.596200\pi$
$73$	−371.284	−0.595280	−0.297640	+	0.954678i	$0.596200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1195.60	−1.76949
$78$	0	0
$79$	−79.3428	−0.112997	−0.0564985	−	0.998403i	$-0.517994\pi$
$79$	−79.3428	−0.112997	−0.0564985	+	0.998403i	$0.517994\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	682.180	0.902156	0.451078	−	0.892485i	$-0.351040\pi$
$83$	682.180	0.902156	0.451078	+	0.892485i	$0.351040\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−297.358	−0.366438
$88$	0	0
$89$	628.478	0.748522	0.374261	−	0.927323i	$-0.377896\pi$
$89$	628.478	0.748522	0.374261	+	0.927323i	$0.377896\pi$
$90$	0	0
$91$	1810.87	2.08605
$92$	0	0
$93$	−236.777	−0.264006
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1519.09	−1.59011	−0.795053	−	0.606541i	$-0.792557\pi$
$97$	−1519.09	−1.59011	−0.795053	+	0.606541i	$0.792557\pi$
$98$	0	0
$99$	491.777	0.499246
$100$	0	0
$101$	977.777	0.963291	0.481646	−	0.876366i	$-0.340039\pi$
$101$	977.777	0.963291	0.481646	+	0.876366i	$0.340039\pi$
$102$	0	0
$103$	−759.194	−0.726268	−0.363134	−	0.931737i	$-0.618293\pi$
$103$	−759.194	−0.726268	−0.363134	+	0.931737i	$0.618293\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−945.448	−0.854205	−0.427102	−	0.904203i	$-0.640466\pi$
$107$	−945.448	−0.854205	−0.427102	+	0.904203i	$0.640466\pi$
$108$	0	0
$109$	−688.314	−0.604849	−0.302425	−	0.953173i	$-0.597796\pi$
$109$	−688.314	−0.604849	−0.302425	+	0.953173i	$0.597796\pi$
$110$	0	0
$111$	−1172.42	−1.00253
$112$	0	0
$113$	350.865	0.292094	0.146047	−	0.989278i	$-0.453345\pi$
$113$	350.865	0.292094	0.146047	+	0.989278i	$0.453345\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−744.851	−0.588560
$118$	0	0
$119$	−2196.88	−1.69233
$120$	0	0
$121$	1654.73	1.24322
$122$	0	0
$123$	313.926	0.230128
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1588.21	−1.10969	−0.554846	−	0.831953i	$-0.687223\pi$
$127$	−1588.21	−1.10969	−0.554846	+	0.831953i	$0.687223\pi$
$128$	0	0
$129$	−724.209	−0.494287
$130$	0	0
$131$	−2156.02	−1.43795	−0.718977	−	0.695034i	$-0.755389\pi$
$131$	−2156.02	−1.43795	−0.718977	+	0.695034i	$0.755389\pi$
$132$	0	0
$133$	1840.58	1.19999
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−316.716	−0.197510	−0.0987551	−	0.995112i	$-0.531486\pi$
$137$	−316.716	−0.197510	−0.0987551	+	0.995112i	$0.531486\pi$
$138$	0	0
$139$	−2624.87	−1.60171	−0.800857	−	0.598855i	$-0.795623\pi$
$139$	−2624.87	−1.60171	−0.800857	+	0.598855i	$0.795623\pi$
$140$	0	0
$141$	−1537.70	−0.918425
$142$	0	0
$143$	−4522.23	−2.64453
$144$	0	0
$145$	0	0
$146$	0	0
$147$	407.284	0.228518
$148$	0	0
$149$	2782.87	1.53008	0.765038	−	0.643985i	$-0.222720\pi$
$149$	2782.87	1.53008	0.765038	+	0.643985i	$0.222720\pi$
$150$	0	0
$151$	1227.25	0.661407	0.330704	−	0.943735i	$-0.392714\pi$
$151$	1227.25	0.661407	0.330704	+	0.943735i	$0.392714\pi$
$152$	0	0
$153$	903.628	0.477477
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3187.33	1.62023	0.810117	−	0.586268i	$-0.199404\pi$
$157$	3187.33	1.62023	0.810117	+	0.586268i	$0.199404\pi$
$158$	0	0
$159$	−853.926	−0.425916
$160$	0	0
$161$	−19.2684	−0.00943204
$162$	0	0
$163$	−2481.64	−1.19250	−0.596249	−	0.802800i	$-0.703343\pi$
$163$	−2481.64	−1.19250	−0.596249	+	0.802800i	$0.703343\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1904.55	0.882508	0.441254	−	0.897382i	$-0.354534\pi$
$167$	1904.55	0.882508	0.441254	+	0.897382i	$0.354534\pi$
$168$	0	0
$169$	4652.42	2.11762
$170$	0	0
$171$	−757.074	−0.338567
$172$	0	0
$173$	3788.88	1.66511	0.832553	−	0.553946i	$-0.186879\pi$
$173$	3788.88	1.66511	0.832553	+	0.553946i	$0.186879\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2127.13	−0.903306
$178$	0	0
$179$	−1390.63	−0.580672	−0.290336	−	0.956925i	$-0.593767\pi$
$179$	−1390.63	−0.580672	−0.290336	+	0.956925i	$0.593767\pi$
$180$	0	0
$181$	2512.55	1.03180	0.515902	−	0.856647i	$-0.327457\pi$
$181$	2512.55	1.03180	0.515902	+	0.856647i	$0.327457\pi$
$182$	0	0
$183$	1411.57	0.570197
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5486.21	2.14541
$188$	0	0
$189$	−590.777	−0.227369
$190$	0	0
$191$	2872.82	1.08833	0.544163	−	0.838980i	$-0.316847\pi$
$191$	2872.82	1.08833	0.544163	+	0.838980i	$0.316847\pi$
$192$	0	0
$193$	−4130.79	−1.54063	−0.770314	−	0.637665i	$-0.779900\pi$
$193$	−4130.79	−1.54063	−0.770314	+	0.637665i	$0.779900\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2243.52	−0.811393	−0.405697	−	0.914008i	$-0.632971\pi$
$197$	−2243.52	−0.811393	−0.405697	+	0.914008i	$0.632971\pi$
$198$	0	0
$199$	1111.28	0.395864	0.197932	−	0.980216i	$-0.436578\pi$
$199$	1111.28	0.395864	0.197932	+	0.980216i	$0.436578\pi$
$200$	0	0
$201$	2002.75	0.702801
$202$	0	0
$203$	2168.79	0.749849
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.92552	0.00266117
$208$	0	0
$209$	−4596.44	−1.52125
$210$	0	0
$211$	−4546.63	−1.48343	−0.741713	−	0.670717i	$-0.765986\pi$
$211$	−4546.63	−1.48343	−0.741713	+	0.670717i	$0.765986\pi$
$212$	0	0
$213$	−154.613	−0.0497368
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1726.94	0.540241
$218$	0	0
$219$	−1113.85	−0.343685
$220$	0	0
$221$	−8309.48	−2.52921
$222$	0	0
$223$	−3012.93	−0.904755	−0.452378	−	0.891826i	$-0.649424\pi$
$223$	−3012.93	−0.904755	−0.452378	+	0.891826i	$0.649424\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1515.27	−0.443049	−0.221524	−	0.975155i	$-0.571103\pi$
$227$	−1515.27	−0.443049	−0.221524	+	0.975155i	$0.571103\pi$
$228$	0	0
$229$	2539.98	0.732956	0.366478	−	0.930427i	$-0.380564\pi$
$229$	2539.98	0.732956	0.366478	+	0.930427i	$0.380564\pi$
$230$	0	0
$231$	−3586.79	−1.02162
$232$	0	0
$233$	4772.91	1.34199	0.670996	−	0.741461i	$-0.265867\pi$
$233$	4772.91	1.34199	0.670996	+	0.741461i	$0.265867\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−238.029	−0.0652389
$238$	0	0
$239$	244.344	0.0661309	0.0330655	−	0.999453i	$-0.489473\pi$
$239$	244.344	0.0661309	0.0330655	+	0.999453i	$0.489473\pi$
$240$	0	0
$241$	5573.18	1.48963	0.744813	−	0.667273i	$-0.232538\pi$
$241$	5573.18	1.48963	0.744813	+	0.667273i	$0.232538\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6961.82	1.79340
$248$	0	0
$249$	2046.54	0.520860
$250$	0	0
$251$	−5786.78	−1.45521	−0.727606	−	0.685995i	$-0.759367\pi$
$251$	−5786.78	−1.45521	−0.727606	+	0.685995i	$0.759367\pi$
$252$	0	0
$253$	48.1183	0.0119572
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7780.75	1.88852	0.944260	−	0.329200i	$-0.106779\pi$
$257$	7780.75	1.88852	0.944260	+	0.329200i	$0.106779\pi$
$258$	0	0
$259$	8551.08	2.05150
$260$	0	0
$261$	−892.074	−0.211563
$262$	0	0
$263$	5122.61	1.20104	0.600521	−	0.799609i	$-0.294960\pi$
$263$	5122.61	1.20104	0.600521	+	0.799609i	$0.294960\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1885.43	0.432160
$268$	0	0
$269$	7125.12	1.61497	0.807484	−	0.589890i	$-0.200829\pi$
$269$	7125.12	1.61497	0.807484	+	0.589890i	$0.200829\pi$
$270$	0	0
$271$	1761.58	0.394865	0.197433	−	0.980316i	$-0.436740\pi$
$271$	1761.58	0.394865	0.197433	+	0.980316i	$0.436740\pi$
$272$	0	0
$273$	5432.60	1.20438
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5810.41	−1.26034	−0.630169	−	0.776458i	$-0.717014\pi$
$277$	−5810.41	−1.26034	−0.630169	+	0.776458i	$0.717014\pi$
$278$	0	0
$279$	−710.330	−0.152424
$280$	0	0
$281$	−4126.19	−0.875972	−0.437986	−	0.898982i	$-0.644308\pi$
$281$	−4126.19	−0.875972	−0.437986	+	0.898982i	$0.644308\pi$
$282$	0	0
$283$	−718.330	−0.150884	−0.0754422	−	0.997150i	$-0.524037\pi$
$283$	−718.330	−0.150884	−0.0754422	+	0.997150i	$0.524037\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2289.63	−0.470914
$288$	0	0
$289$	5167.78	1.05186
$290$	0	0
$291$	−4557.27	−0.918048
$292$	0	0
$293$	3756.21	0.748942	0.374471	−	0.927239i	$-0.377824\pi$
$293$	3756.21	0.748942	0.374471	+	0.927239i	$0.377824\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1475.33	0.288240
$298$	0	0
$299$	−72.8806	−0.0140963
$300$	0	0
$301$	5282.05	1.01147
$302$	0	0
$303$	2933.33	0.556156
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7599.52	1.41279	0.706397	−	0.707816i	$-0.250319\pi$
$307$	7599.52	1.41279	0.706397	+	0.707816i	$0.250319\pi$
$308$	0	0
$309$	−2277.58	−0.419311
$310$	0	0
$311$	2850.03	0.519648	0.259824	−	0.965656i	$-0.416336\pi$
$311$	2850.03	0.519648	0.259824	+	0.965656i	$0.416336\pi$
$312$	0	0
$313$	−799.684	−0.144411	−0.0722057	−	0.997390i	$-0.523004\pi$
$313$	−799.684	−0.144411	−0.0722057	+	0.997390i	$0.523004\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1003.79	0.177850	0.0889250	−	0.996038i	$-0.471657\pi$
$317$	1003.79	0.177850	0.0889250	+	0.996038i	$0.471657\pi$
$318$	0	0
$319$	−5416.07	−0.950600
$320$	0	0
$321$	−2836.34	−0.493175
$322$	0	0
$323$	−8445.84	−1.45492
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2064.94	−0.349210
$328$	0	0
$329$	11215.3	1.87939
$330$	0	0
$331$	8367.68	1.38951	0.694757	−	0.719245i	$-0.255512\pi$
$331$	8367.68	1.38951	0.694757	+	0.719245i	$0.255512\pi$
$332$	0	0
$333$	−3517.26	−0.578812
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5034.79	0.813835	0.406918	−	0.913465i	$-0.366604\pi$
$337$	5034.79	0.813835	0.406918	+	0.913465i	$0.366604\pi$
$338$	0	0
$339$	1052.60	0.168641
$340$	0	0
$341$	−4312.64	−0.684875
$342$	0	0
$343$	4534.51	0.713821
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2105.53	0.325737	0.162868	−	0.986648i	$-0.447925\pi$
$347$	2105.53	0.325737	0.162868	+	0.986648i	$0.447925\pi$
$348$	0	0
$349$	−8813.47	−1.35179	−0.675894	−	0.736999i	$-0.736242\pi$
$349$	−8813.47	−1.35179	−0.675894	+	0.736999i	$0.736242\pi$
$350$	0	0
$351$	−2234.55	−0.339805
$352$	0	0
$353$	−9348.98	−1.40962	−0.704810	−	0.709396i	$-0.748968\pi$
$353$	−9348.98	−1.40962	−0.704810	+	0.709396i	$0.748968\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6590.64	−0.977069
$358$	0	0
$359$	−9233.33	−1.35743	−0.678714	−	0.734403i	$-0.737462\pi$
$359$	−9233.33	−1.35743	−0.678714	+	0.734403i	$0.737462\pi$
$360$	0	0
$361$	217.071	0.0316477
$362$	0	0
$363$	4964.19	0.717775
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6996.60	0.995148	0.497574	−	0.867421i	$-0.334224\pi$
$367$	6996.60	0.995148	0.497574	+	0.867421i	$0.334224\pi$
$368$	0	0
$369$	941.777	0.132864
$370$	0	0
$371$	6228.14	0.871560
$372$	0	0
$373$	−2945.48	−0.408877	−0.204438	−	0.978879i	$-0.565537\pi$
$373$	−2945.48	−0.408877	−0.204438	+	0.978879i	$0.565537\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8203.24	1.12066
$378$	0	0
$379$	8499.83	1.15200	0.575998	−	0.817451i	$-0.304613\pi$
$379$	8499.83	1.15200	0.575998	+	0.817451i	$0.304613\pi$
$380$	0	0
$381$	−4764.63	−0.640681
$382$	0	0
$383$	6426.01	0.857320	0.428660	−	0.903466i	$-0.358986\pi$
$383$	6426.01	0.857320	0.428660	+	0.903466i	$0.358986\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2172.63	−0.285377
$388$	0	0
$389$	−675.805	−0.0880840	−0.0440420	−	0.999030i	$-0.514024\pi$
$389$	−675.805	−0.0880840	−0.0440420	+	0.999030i	$0.514024\pi$
$390$	0	0
$391$	88.4162	0.0114358
$392$	0	0
$393$	−6468.05	−0.830203
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9331.43	1.17967	0.589837	−	0.807522i	$-0.299192\pi$
$397$	9331.43	1.17967	0.589837	+	0.807522i	$0.299192\pi$
$398$	0	0
$399$	5521.75	0.692815
$400$	0	0
$401$	−427.213	−0.0532021	−0.0266010	−	0.999646i	$-0.508468\pi$
$401$	−427.213	−0.0532021	−0.0266010	+	0.999646i	$0.508468\pi$
$402$	0	0
$403$	6531.97	0.807396
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21354.4	−2.60073
$408$	0	0
$409$	1243.18	0.150297	0.0751484	−	0.997172i	$-0.476057\pi$
$409$	1243.18	0.150297	0.0751484	+	0.997172i	$0.476057\pi$
$410$	0	0
$411$	−950.149	−0.114033
$412$	0	0
$413$	15514.3	1.84845
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7874.60	−0.924750
$418$	0	0
$419$	−3802.17	−0.443313	−0.221656	−	0.975125i	$-0.571146\pi$
$419$	−3802.17	−0.443313	−0.221656	+	0.975125i	$0.571146\pi$
$420$	0	0
$421$	−12785.6	−1.48012	−0.740059	−	0.672542i	$-0.765203\pi$
$421$	−12785.6	−1.48012	−0.740059	+	0.672542i	$0.765203\pi$
$422$	0	0
$423$	−4613.11	−0.530253
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10295.3	−1.16680
$428$	0	0
$429$	−13566.7	−1.52682
$430$	0	0
$431$	−9588.11	−1.07156	−0.535781	−	0.844357i	$-0.679983\pi$
$431$	−9588.11	−1.07156	−0.535781	+	0.844357i	$0.679983\pi$
$432$	0	0
$433$	−8493.83	−0.942697	−0.471348	−	0.881947i	$-0.656233\pi$
$433$	−8493.83	−0.942697	−0.471348	+	0.881947i	$0.656233\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−74.0766	−0.00810885
$438$	0	0
$439$	5167.40	0.561792	0.280896	−	0.959738i	$-0.409368\pi$
$439$	5167.40	0.561792	0.280896	+	0.959738i	$0.409368\pi$
$440$	0	0
$441$	1221.85	0.131935
$442$	0	0
$443$	−8192.86	−0.878679	−0.439339	−	0.898321i	$-0.644787\pi$
$443$	−8192.86	−0.878679	−0.439339	+	0.898321i	$0.644787\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8348.60	0.883390
$448$	0	0
$449$	−18252.3	−1.91844	−0.959218	−	0.282666i	$-0.908781\pi$
$449$	−18252.3	−1.91844	−0.959218	+	0.282666i	$0.908781\pi$
$450$	0	0
$451$	5717.82	0.596988
$452$	0	0
$453$	3681.76	0.381864
$454$	0	0
$455$	0	0
$456$	0	0
$457$	330.123	0.0337910	0.0168955	−	0.999857i	$-0.494622\pi$
$457$	330.123	0.0337910	0.0168955	+	0.999857i	$0.494622\pi$
$458$	0	0
$459$	2710.88	0.275671
$460$	0	0
$461$	−4773.18	−0.482232	−0.241116	−	0.970496i	$-0.577513\pi$
$461$	−4773.18	−0.482232	−0.241116	+	0.970496i	$0.577513\pi$
$462$	0	0
$463$	−7860.46	−0.788999	−0.394499	−	0.918896i	$-0.629082\pi$
$463$	−7860.46	−0.788999	−0.394499	+	0.918896i	$0.629082\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10520.8	−1.04250	−0.521249	−	0.853404i	$-0.674534\pi$
$467$	−10520.8	−1.04250	−0.521249	+	0.853404i	$0.674534\pi$
$468$	0	0
$469$	−14607.1	−1.43815
$470$	0	0
$471$	9561.99	0.935442
$472$	0	0
$473$	−13190.7	−1.28226
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2561.78	−0.245903
$478$	0	0
$479$	−3013.64	−0.287467	−0.143734	−	0.989616i	$-0.545911\pi$
$479$	−3013.64	−0.287467	−0.143734	+	0.989616i	$0.545911\pi$
$480$	0	0
$481$	32343.6	3.06599
$482$	0	0
$483$	−57.8051	−0.00544559
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7048.81	−0.655876	−0.327938	−	0.944699i	$-0.606354\pi$
$487$	−7048.81	−0.655876	−0.327938	+	0.944699i	$0.606354\pi$
$488$	0	0
$489$	−7444.93	−0.688489
$490$	0	0
$491$	6338.68	0.582608	0.291304	−	0.956630i	$-0.405911\pi$
$491$	6338.68	0.582608	0.291304	+	0.956630i	$0.405911\pi$
$492$	0	0
$493$	−9951.89	−0.909149
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1127.68	0.101777
$498$	0	0
$499$	−5402.66	−0.484682	−0.242341	−	0.970191i	$-0.577915\pi$
$499$	−5402.66	−0.484682	−0.242341	+	0.970191i	$0.577915\pi$
$500$	0	0
$501$	5713.66	0.509516
$502$	0	0
$503$	7770.85	0.688837	0.344419	−	0.938816i	$-0.388076\pi$
$503$	7770.85	0.688837	0.344419	+	0.938816i	$0.388076\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13957.3	1.22261
$508$	0	0
$509$	729.019	0.0634837	0.0317418	−	0.999496i	$-0.489895\pi$
$509$	729.019	0.0634837	0.0317418	+	0.999496i	$0.489895\pi$
$510$	0	0
$511$	8123.91	0.703289
$512$	0	0
$513$	−2271.22	−0.195472
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−28007.6	−2.38254
$518$	0	0
$519$	11366.6	0.961349
$520$	0	0
$521$	14783.8	1.24317	0.621585	−	0.783346i	$-0.286489\pi$
$521$	14783.8	1.24317	0.621585	+	0.783346i	$0.286489\pi$
$522$	0	0
$523$	−395.434	−0.0330614	−0.0165307	−	0.999863i	$-0.505262\pi$
$523$	−395.434	−0.0330614	−0.0165307	+	0.999863i	$0.505262\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7924.36	−0.655011
$528$	0	0
$529$	−12166.2	−0.999936
$530$	0	0
$531$	−6381.40	−0.521524
$532$	0	0
$533$	−8660.29	−0.703787
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4171.88	−0.335251
$538$	0	0
$539$	7418.24	0.592813
$540$	0	0
$541$	−16507.9	−1.31188	−0.655942	−	0.754811i	$-0.727728\pi$
$541$	−16507.9	−1.31188	−0.655942	+	0.754811i	$0.727728\pi$
$542$	0	0
$543$	7537.66	0.595713
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9242.35	−0.722440	−0.361220	−	0.932481i	$-0.617640\pi$
$547$	−9242.35	−0.722440	−0.361220	+	0.932481i	$0.617640\pi$
$548$	0	0
$549$	4234.70	0.329203
$550$	0	0
$551$	8337.86	0.644655
$552$	0	0
$553$	1736.07	0.133499
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10449.5	0.794898	0.397449	−	0.917624i	$-0.369896\pi$
$557$	10449.5	0.794898	0.397449	+	0.917624i	$0.369896\pi$
$558$	0	0
$559$	19978.8	1.51165
$560$	0	0
$561$	16458.6	1.23865
$562$	0	0
$563$	13670.1	1.02331	0.511656	−	0.859191i	$-0.329032\pi$
$563$	13670.1	1.02331	0.511656	+	0.859191i	$0.329032\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1772.33	−0.131271
$568$	0	0
$569$	13616.7	1.00323	0.501617	−	0.865090i	$-0.332739\pi$
$569$	13616.7	1.00323	0.501617	+	0.865090i	$0.332739\pi$
$570$	0	0
$571$	−11833.4	−0.867270	−0.433635	−	0.901089i	$-0.642769\pi$
$571$	−11833.4	−0.867270	−0.433635	+	0.901089i	$0.642769\pi$
$572$	0	0
$573$	8618.47	0.628345
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7131.30	−0.514523	−0.257262	−	0.966342i	$-0.582820\pi$
$577$	−7131.30	−0.514523	−0.257262	+	0.966342i	$0.582820\pi$
$578$	0	0
$579$	−12392.4	−0.889482
$580$	0	0
$581$	−14926.5	−1.06584
$582$	0	0
$583$	−15553.4	−1.10490
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25214.3	−1.77292	−0.886461	−	0.462804i	$-0.846843\pi$
$587$	−25214.3	−1.77292	−0.886461	+	0.462804i	$0.846843\pi$
$588$	0	0
$589$	6639.17	0.464452
$590$	0	0
$591$	−6730.57	−0.468458
$592$	0	0
$593$	−4218.04	−0.292098	−0.146049	−	0.989277i	$-0.546656\pi$
$593$	−4218.04	−0.292098	−0.146049	+	0.989277i	$0.546656\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3333.85	0.228552
$598$	0	0
$599$	5956.57	0.406308	0.203154	−	0.979147i	$-0.434881\pi$
$599$	5956.57	0.406308	0.203154	+	0.979147i	$0.434881\pi$
$600$	0	0
$601$	−5182.29	−0.351731	−0.175865	−	0.984414i	$-0.556272\pi$
$601$	−5182.29	−0.351731	−0.175865	+	0.984414i	$0.556272\pi$
$602$	0	0
$603$	6008.24	0.405762
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10142.0	−0.678171	−0.339086	−	0.940755i	$-0.610118\pi$
$607$	−10142.0	−0.678171	−0.339086	+	0.940755i	$0.610118\pi$
$608$	0	0
$609$	6506.38	0.432926
$610$	0	0
$611$	42420.7	2.80877
$612$	0	0
$613$	4022.95	0.265066	0.132533	−	0.991179i	$-0.457689\pi$
$613$	4022.95	0.265066	0.132533	+	0.991179i	$0.457689\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27074.4	1.76657	0.883285	−	0.468837i	$-0.155327\pi$
$617$	27074.4	1.76657	0.883285	+	0.468837i	$0.155327\pi$
$618$	0	0
$619$	−17245.7	−1.11981	−0.559905	−	0.828557i	$-0.689162\pi$
$619$	−17245.7	−1.11981	−0.559905	+	0.828557i	$0.689162\pi$
$620$	0	0
$621$	23.7766	0.00153643
$622$	0	0
$623$	−13751.5	−0.884336
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−13789.3	−0.878297
$628$	0	0
$629$	−39238.1	−2.48732
$630$	0	0
$631$	−14254.4	−0.899302	−0.449651	−	0.893204i	$-0.648452\pi$
$631$	−14254.4	−0.899302	−0.449651	+	0.893204i	$0.648452\pi$
$632$	0	0
$633$	−13639.9	−0.856456
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−11235.8	−0.698865
$638$	0	0
$639$	−463.840	−0.0287155
$640$	0	0
$641$	2611.61	0.160924	0.0804621	−	0.996758i	$-0.474360\pi$
$641$	2611.61	0.160924	0.0804621	+	0.996758i	$0.474360\pi$
$642$	0	0
$643$	27414.4	1.68136	0.840682	−	0.541529i	$-0.182154\pi$
$643$	27414.4	1.68136	0.840682	+	0.541529i	$0.182154\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12820.8	−0.779041	−0.389521	−	0.921018i	$-0.627359\pi$
$647$	−12820.8	−0.779041	−0.389521	+	0.921018i	$0.627359\pi$
$648$	0	0
$649$	−38743.5	−2.34332
$650$	0	0
$651$	5180.82	0.311908
$652$	0	0
$653$	15786.4	0.946049	0.473025	−	0.881049i	$-0.343162\pi$
$653$	15786.4	0.946049	0.473025	+	0.881049i	$0.343162\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3341.55	−0.198427
$658$	0	0
$659$	4822.61	0.285072	0.142536	−	0.989790i	$-0.454474\pi$
$659$	4822.61	0.285072	0.142536	+	0.989790i	$0.454474\pi$
$660$	0	0
$661$	−23624.9	−1.39017	−0.695084	−	0.718929i	$-0.744633\pi$
$661$	−23624.9	−1.39017	−0.695084	+	0.718929i	$0.744633\pi$
$662$	0	0
$663$	−24928.4	−1.46024
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−87.2858	−0.00506705
$668$	0	0
$669$	−9038.78	−0.522361
$670$	0	0
$671$	25710.2	1.47918
$672$	0	0
$673$	8903.65	0.509971	0.254985	−	0.966945i	$-0.417929\pi$
$673$	8903.65	0.509971	0.254985	+	0.966945i	$0.417929\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9034.56	0.512890	0.256445	−	0.966559i	$-0.417449\pi$
$677$	9034.56	0.512890	0.256445	+	0.966559i	$0.417449\pi$
$678$	0	0
$679$	33238.6	1.87862
$680$	0	0
$681$	−4545.81	−0.255794
$682$	0	0
$683$	−28676.8	−1.60657	−0.803284	−	0.595596i	$-0.796916\pi$
$683$	−28676.8	−1.60657	−0.803284	+	0.595596i	$0.796916\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7619.95	0.423172
$688$	0	0
$689$	23557.3	1.30256
$690$	0	0
$691$	−20653.1	−1.13702	−0.568509	−	0.822677i	$-0.692480\pi$
$691$	−20653.1	−1.13702	−0.568509	+	0.822677i	$0.692480\pi$
$692$	0	0
$693$	−10760.4	−0.589831
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10506.4	0.570957
$698$	0	0
$699$	14318.7	0.774799
$700$	0	0
$701$	−11000.1	−0.592678	−0.296339	−	0.955083i	$-0.595766\pi$
$701$	−11000.1	−0.592678	−0.296339	+	0.955083i	$0.595766\pi$
$702$	0	0
$703$	32874.4	1.76370
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21394.4	−1.13807
$708$	0	0
$709$	−372.560	−0.0197345	−0.00986725	−	0.999951i	$-0.503141\pi$
$709$	−372.560	−0.0197345	−0.00986725	+	0.999951i	$0.503141\pi$
$710$	0	0
$711$	−714.086	−0.0376657
$712$	0	0
$713$	−69.5028	−0.00365063
$714$	0	0
$715$	0	0
$716$	0	0
$717$	733.032	0.0381807
$718$	0	0
$719$	−7207.32	−0.373835	−0.186918	−	0.982376i	$-0.559850\pi$
$719$	−7207.32	−0.373835	−0.186918	+	0.982376i	$0.559850\pi$
$720$	0	0
$721$	16611.6	0.858043
$722$	0	0
$723$	16719.5	0.860036
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23423.6	1.19496	0.597479	−	0.801885i	$-0.296169\pi$
$727$	23423.6	1.19496	0.597479	+	0.801885i	$0.296169\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−24237.6	−1.22635
$732$	0	0
$733$	20368.0	1.02634	0.513172	−	0.858286i	$-0.328470\pi$
$733$	20368.0	1.02634	0.513172	+	0.858286i	$0.328470\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36477.9	1.82318
$738$	0	0
$739$	2603.64	0.129603	0.0648014	−	0.997898i	$-0.479359\pi$
$739$	2603.64	0.129603	0.0648014	+	0.997898i	$0.479359\pi$
$740$	0	0
$741$	20885.5	1.03542
$742$	0	0
$743$	−8627.83	−0.426009	−0.213004	−	0.977051i	$-0.568325\pi$
$743$	−8627.83	−0.426009	−0.213004	+	0.977051i	$0.568325\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6139.62	0.300719
$748$	0	0
$749$	20687.0	1.00919
$750$	0	0
$751$	−18735.5	−0.910343	−0.455172	−	0.890404i	$-0.650422\pi$
$751$	−18735.5	−0.910343	−0.455172	+	0.890404i	$0.650422\pi$
$752$	0	0
$753$	−17360.3	−0.840167
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10787.3	−0.517926	−0.258963	−	0.965887i	$-0.583381\pi$
$757$	−10787.3	−0.517926	−0.258963	+	0.965887i	$0.583381\pi$
$758$	0	0
$759$	144.355	0.00690349
$760$	0	0
$761$	−2175.95	−0.103651	−0.0518253	−	0.998656i	$-0.516504\pi$
$761$	−2175.95	−0.103651	−0.0518253	+	0.998656i	$0.516504\pi$
$762$	0	0
$763$	15060.7	0.714594
$764$	0	0
$765$	0	0
$766$	0	0
$767$	58681.4	2.76253
$768$	0	0
$769$	18705.8	0.877176	0.438588	−	0.898688i	$-0.355479\pi$
$769$	18705.8	0.877176	0.438588	+	0.898688i	$0.355479\pi$
$770$	0	0
$771$	23342.2	1.09034
$772$	0	0
$773$	−8243.60	−0.383573	−0.191786	−	0.981437i	$-0.561428\pi$
$773$	−8243.60	−0.383573	−0.191786	+	0.981437i	$0.561428\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	25653.2	1.18443
$778$	0	0
$779$	−8802.41	−0.404851
$780$	0	0
$781$	−2816.12	−0.129025
$782$	0	0
$783$	−2676.22	−0.122146
$784$	0	0
$785$	0	0
$786$	0	0
$787$	29442.1	1.33354	0.666770	−	0.745263i	$-0.267676\pi$
$787$	29442.1	1.33354	0.666770	+	0.745263i	$0.267676\pi$
$788$	0	0
$789$	15367.8	0.693422
$790$	0	0
$791$	−7677.15	−0.345092
$792$	0	0
$793$	−38941.0	−1.74380
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42045.4	−1.86866	−0.934332	−	0.356403i	$-0.884003\pi$
$797$	−42045.4	−1.86866	−0.934332	+	0.356403i	$0.884003\pi$
$798$	0	0
$799$	−51463.3	−2.27865
$800$	0	0
$801$	5656.30	0.249507
$802$	0	0
$803$	−20287.6	−0.891575
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21375.4	0.932402
$808$	0	0
$809$	43757.6	1.90165	0.950825	−	0.309727i	$-0.100238\pi$
$809$	43757.6	1.90165	0.950825	+	0.309727i	$0.100238\pi$
$810$	0	0
$811$	3273.00	0.141715	0.0708574	−	0.997486i	$-0.477426\pi$
$811$	3273.00	0.141715	0.0708574	+	0.997486i	$0.477426\pi$
$812$	0	0
$813$	5284.74	0.227976
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20306.7	0.869572
$818$	0	0
$819$	16297.8	0.695349
$820$	0	0
$821$	−40442.6	−1.71919	−0.859595	−	0.510976i	$-0.829284\pi$
$821$	−40442.6	−1.71919	−0.859595	+	0.510976i	$0.829284\pi$
$822$	0	0
$823$	27236.8	1.15360	0.576802	−	0.816884i	$-0.304300\pi$
$823$	27236.8	1.15360	0.576802	+	0.816884i	$0.304300\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33344.5	1.40206	0.701028	−	0.713133i	$-0.252725\pi$
$827$	33344.5	1.40206	0.701028	+	0.713133i	$0.252725\pi$
$828$	0	0
$829$	−32085.0	−1.34422	−0.672110	−	0.740452i	$-0.734612\pi$
$829$	−32085.0	−1.34422	−0.672110	+	0.740452i	$0.734612\pi$
$830$	0	0
$831$	−17431.2	−0.727656
$832$	0	0
$833$	13630.8	0.566964
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2130.99	−0.0880021
$838$	0	0
$839$	−7072.24	−0.291014	−0.145507	−	0.989357i	$-0.546481\pi$
$839$	−7072.24	−0.291014	−0.145507	+	0.989357i	$0.546481\pi$
$840$	0	0
$841$	−14564.3	−0.597169
$842$	0	0
$843$	−12378.6	−0.505743
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−36206.5	−1.46880
$848$	0	0
$849$	−2154.99	−0.0871131
$850$	0	0
$851$	−344.149	−0.0138628
$852$	0	0
$853$	10225.0	0.410431	0.205215	−	0.978717i	$-0.434211\pi$
$853$	10225.0	0.410431	0.205215	+	0.978717i	$0.434211\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9783.87	−0.389977	−0.194989	−	0.980805i	$-0.562467\pi$
$857$	−9783.87	−0.389977	−0.194989	+	0.980805i	$0.562467\pi$
$858$	0	0
$859$	−22931.8	−0.910853	−0.455427	−	0.890273i	$-0.650513\pi$
$859$	−22931.8	−0.910853	−0.455427	+	0.890273i	$0.650513\pi$
$860$	0	0
$861$	−6868.88	−0.271883
$862$	0	0
$863$	23506.0	0.927175	0.463588	−	0.886051i	$-0.346562\pi$
$863$	23506.0	0.927175	0.463588	+	0.886051i	$0.346562\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15503.3	0.607290
$868$	0	0
$869$	−4335.44	−0.169240
$870$	0	0
$871$	−55250.0	−2.14934
$872$	0	0
$873$	−13671.8	−0.530035
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34672.0	−1.33499	−0.667497	−	0.744613i	$-0.732634\pi$
$877$	−34672.0	−1.33499	−0.667497	+	0.744613i	$0.732634\pi$
$878$	0	0
$879$	11268.6	0.432402
$880$	0	0
$881$	−2920.92	−0.111701	−0.0558504	−	0.998439i	$-0.517787\pi$
$881$	−2920.92	−0.111701	−0.0558504	+	0.998439i	$0.517787\pi$
$882$	0	0
$883$	−7123.74	−0.271498	−0.135749	−	0.990743i	$-0.543344\pi$
$883$	−7123.74	−0.271498	−0.135749	+	0.990743i	$0.543344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7923.42	0.299935	0.149968	−	0.988691i	$-0.452083\pi$
$887$	7923.42	0.299935	0.149968	+	0.988691i	$0.452083\pi$
$888$	0	0
$889$	34751.0	1.31104
$890$	0	0
$891$	4425.99	0.166415
$892$	0	0
$893$	43116.9	1.61573
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−218.642	−0.00813850
$898$	0	0
$899$	7823.05	0.290226
$900$	0	0
$901$	−28578.9	−1.05672
$902$	0	0
$903$	15846.1	0.583972
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−19771.7	−0.723825	−0.361912	−	0.932212i	$-0.617876\pi$
$907$	−19771.7	−0.723825	−0.361912	+	0.932212i	$0.617876\pi$
$908$	0	0
$909$	8799.99	0.321097
$910$	0	0
$911$	38222.7	1.39009	0.695046	−	0.718966i	$-0.255384\pi$
$911$	38222.7	1.39009	0.695046	+	0.718966i	$0.255384\pi$
$912$	0	0
$913$	37275.5	1.35119
$914$	0	0
$915$	0	0
$916$	0	0
$917$	47174.9	1.69886
$918$	0	0
$919$	39237.3	1.40840	0.704199	−	0.710003i	$-0.251306\pi$
$919$	39237.3	1.40840	0.704199	+	0.710003i	$0.251306\pi$
$920$	0	0
$921$	22798.6	0.815677
$922$	0	0
$923$	4265.33	0.152107
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6832.74	−0.242089
$928$	0	0
$929$	20024.5	0.707194	0.353597	−	0.935398i	$-0.384958\pi$
$929$	20024.5	0.707194	0.353597	+	0.935398i	$0.384958\pi$
$930$	0	0
$931$	−11420.2	−0.402020
$932$	0	0
$933$	8550.09	0.300019
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8717.10	0.303922	0.151961	−	0.988386i	$-0.451441\pi$
$937$	8717.10	0.303922	0.151961	+	0.988386i	$0.451441\pi$
$938$	0	0
$939$	−2399.05	−0.0833760
$940$	0	0
$941$	39276.2	1.36065	0.680323	−	0.732913i	$-0.261840\pi$
$941$	39276.2	1.36065	0.680323	+	0.732913i	$0.261840\pi$
$942$	0	0
$943$	92.1490	0.00318217
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20080.8	0.689060	0.344530	−	0.938775i	$-0.388038\pi$
$947$	20080.8	0.689060	0.344530	+	0.938775i	$0.388038\pi$
$948$	0	0
$949$	30727.9	1.05107
$950$	0	0
$951$	3011.37	0.102682
$952$	0	0
$953$	−38549.9	−1.31034	−0.655170	−	0.755481i	$-0.727403\pi$
$953$	−38549.9	−1.31034	−0.655170	+	0.755481i	$0.727403\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−16248.2	−0.548829
$958$	0	0
$959$	6929.95	0.233347
$960$	0	0
$961$	−23561.8	−0.790902
$962$	0	0
$963$	−8509.03	−0.284735
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24108.2	0.801724	0.400862	−	0.916138i	$-0.368711\pi$
$967$	24108.2	0.801724	0.400862	+	0.916138i	$0.368711\pi$
$968$	0	0
$969$	−25337.5	−0.839999
$970$	0	0
$971$	−41760.1	−1.38017	−0.690084	−	0.723729i	$-0.742427\pi$
$971$	−41760.1	−1.38017	−0.690084	+	0.723729i	$0.742427\pi$
$972$	0	0
$973$	57433.7	1.89233
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4532.52	−0.148422	−0.0742109	−	0.997243i	$-0.523644\pi$
$977$	−4532.52	−0.148422	−0.0742109	+	0.997243i	$0.523644\pi$
$978$	0	0
$979$	34341.2	1.12109
$980$	0	0
$981$	−6194.83	−0.201616
$982$	0	0
$983$	48822.8	1.58414	0.792068	−	0.610432i	$-0.209004\pi$
$983$	48822.8	1.58414	0.792068	+	0.610432i	$0.209004\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	33645.9	1.08507
$988$	0	0
$989$	−212.583	−0.00683492
$990$	0	0
$991$	−26937.6	−0.863473	−0.431737	−	0.902000i	$-0.642099\pi$
$991$	−26937.6	−0.863473	−0.431737	+	0.902000i	$0.642099\pi$
$992$	0	0
$993$	25103.0	0.802236
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9191.03	0.291959	0.145979	−	0.989288i	$-0.453367\pi$
$997$	9191.03	0.291959	0.145979	+	0.989288i	$0.453367\pi$
$998$	0	0
$999$	−10551.8	−0.334177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.4.a.u.1.1	yes	2
3.2	odd	2		1800.4.a.bm.1.1		2
4.3	odd	2		1200.4.a.bp.1.2		2
5.2	odd	4		600.4.f.j.49.1		4
5.3	odd	4		600.4.f.j.49.4		4
5.4	even	2		600.4.a.s.1.2	✓	2
15.2	even	4		1800.4.f.z.649.1		4
15.8	even	4		1800.4.f.z.649.4		4
15.14	odd	2		1800.4.a.bo.1.2		2
20.3	even	4		1200.4.f.x.49.1		4
20.7	even	4		1200.4.f.x.49.4		4
20.19	odd	2		1200.4.a.br.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.s.1.2	✓	2	5.4	even	2
600.4.a.u.1.1	yes	2	1.1	even	1	trivial
600.4.f.j.49.1		4	5.2	odd	4
600.4.f.j.49.4		4	5.3	odd	4
1200.4.a.bp.1.2		2	4.3	odd	2
1200.4.a.br.1.1		2	20.19	odd	2
1200.4.f.x.49.1		4	20.3	even	4
1200.4.f.x.49.4		4	20.7	even	4
1800.4.a.bm.1.1		2	3.2	odd	2
1800.4.a.bo.1.2		2	15.14	odd	2
1800.4.f.z.649.1		4	15.2	even	4
1800.4.f.z.649.4		4	15.8	even	4

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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -21.8806 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -21.8806 q^{7} +9.00000 q^{9} +54.6418 q^{11} -82.7612 q^{13} +100.403 q^{17} -84.1194 q^{19} -65.6418 q^{21} +0.880613 q^{23} +27.0000 q^{27} -99.1194 q^{29} -78.9255 q^{31} +163.926 q^{33} -390.806 q^{37} -248.284 q^{39} +104.642 q^{41} -241.403 q^{43} -512.567 q^{47} +135.761 q^{49} +301.209 q^{51} -284.642 q^{53} -252.358 q^{57} -709.045 q^{59} +470.522 q^{61} -196.926 q^{63} +667.583 q^{67} +2.64184 q^{69} -51.5378 q^{71} -371.284 q^{73} -1195.60 q^{77} -79.3428 q^{79} +81.0000 q^{81} +682.180 q^{83} -297.358 q^{87} +628.478 q^{89} +1810.87 q^{91} -236.777 q^{93} -1519.09 q^{97} +491.777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} - 16 q^{11} - 82 q^{13} - 8 q^{17} - 210 q^{19} - 6 q^{21} - 40 q^{23} + 54 q^{27} - 240 q^{29} + 218 q^{31} - 48 q^{33} - 364 q^{37} - 246 q^{39} + 84 q^{41} - 274 q^{43} - 524 q^{47} + 188 q^{49} - 24 q^{51} - 444 q^{53} - 630 q^{57} - 1084 q^{59} + 774 q^{61} - 18 q^{63} - 210 q^{67} - 120 q^{69} + 1108 q^{71} - 492 q^{73} - 2600 q^{77} - 1328 q^{79} + 162 q^{81} + 28 q^{83} - 720 q^{87} + 1424 q^{89} + 1826 q^{91} + 654 q^{93} - 2370 q^{97} - 144 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9 - 16 * q^11 - 82 * q^13 - 8 * q^17 - 210 * q^19 - 6 * q^21 - 40 * q^23 + 54 * q^27 - 240 * q^29 + 218 * q^31 - 48 * q^33 - 364 * q^37 - 246 * q^39 + 84 * q^41 - 274 * q^43 - 524 * q^47 + 188 * q^49 - 24 * q^51 - 444 * q^53 - 630 * q^57 - 1084 * q^59 + 774 * q^61 - 18 * q^63 - 210 * q^67 - 120 * q^69 + 1108 * q^71 - 492 * q^73 - 2600 * q^77 - 1328 * q^79 + 162 * q^81 + 28 * q^83 - 720 * q^87 + 1424 * q^89 + 1826 * q^91 + 654 * q^93 - 2370 * q^97 - 144 * q^99

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