LMFDB - Embedded newform 1620.4.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1620,4,Mod(1,1620)] 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("1620.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1620, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$1620 = 2^{2} \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1620.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,15,0,15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.560145.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 90x + 192$ x^3 - x^2 - 90*x + 192
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2\cdot 3$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.19760$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +33.6841 q^{7} -9.49850 q^{11} +39.6841 q^{13} +106.365 q^{17} -96.5507 q^{19} +40.0552 q^{23} +25.0000 q^{25} +252.417 q^{29} +48.8727 q^{31} +168.420 q^{35} -136.476 q^{37} +139.104 q^{41} -202.359 q^{43} +300.049 q^{47} +791.617 q^{49} -344.142 q^{53} -47.4925 q^{55} -60.9339 q^{59} -122.122 q^{61} +198.420 q^{65} -744.667 q^{67} +436.971 q^{71} +586.353 q^{73} -319.948 q^{77} +473.780 q^{79} -998.864 q^{83} +531.826 q^{85} +204.463 q^{89} +1336.72 q^{91} -482.754 q^{95} -1632.78 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 15 q^{5} + 15 q^{7} + 24 q^{11} + 33 q^{13} + 42 q^{17} + 21 q^{19} - 33 q^{23} + 75 q^{25} + 222 q^{29} + 132 q^{31} + 75 q^{35} + 174 q^{37} - 99 q^{41} - 120 q^{43} + 537 q^{47} + 492 q^{49} + 267 q^{53}+ \cdots - 2556 q^{97}+O(q^{100})$ 3 * q + 15 * q^5 + 15 * q^7 + 24 * q^11 + 33 * q^13 + 42 * q^17 + 21 * q^19 - 33 * q^23 + 75 * q^25 + 222 * q^29 + 132 * q^31 + 75 * q^35 + 174 * q^37 - 99 * q^41 - 120 * q^43 + 537 * q^47 + 492 * q^49 + 267 * q^53 + 120 * q^55 - 225 * q^59 - 480 * q^61 + 165 * q^65 + 12 * q^67 + 570 * q^71 + 1062 * q^73 + 312 * q^77 - 1026 * q^79 + 702 * q^83 + 210 * q^85 + 1140 * q^89 + 1611 * q^91 + 105 * q^95 - 2556 * 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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	33.6841	1.81877	0.909385	−	0.415956i	$-0.136553\pi$
$7$	33.6841	1.81877	0.909385	+	0.415956i	$0.136553\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−9.49850	−0.260355	−0.130178	−	0.991491i	$-0.541555\pi$
$11$	−9.49850	−0.260355	−0.130178	+	0.991491i	$0.541555\pi$
$12$	0	0
$13$	39.6841	0.846645	0.423322	−	0.905979i	$-0.360864\pi$
$13$	39.6841	0.846645	0.423322	+	0.905979i	$0.360864\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	106.365	1.51749	0.758745	−	0.651387i	$-0.225813\pi$
$17$	106.365	1.51749	0.758745	+	0.651387i	$0.225813\pi$
$18$	0	0
$19$	−96.5507	−1.16580	−0.582902	−	0.812543i	$-0.698083\pi$
$19$	−96.5507	−1.16580	−0.582902	+	0.812543i	$0.698083\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	40.0552	0.363135	0.181567	−	0.983379i	$-0.441883\pi$
$23$	40.0552	0.363135	0.181567	+	0.983379i	$0.441883\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	252.417	1.61630	0.808151	−	0.588976i	$-0.200469\pi$
$29$	252.417	1.61630	0.808151	+	0.588976i	$0.200469\pi$
$30$	0	0
$31$	48.8727	0.283154	0.141577	−	0.989927i	$-0.454783\pi$
$31$	48.8727	0.283154	0.141577	+	0.989927i	$0.454783\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	168.420	0.813378
$36$	0	0
$37$	−136.476	−0.606391	−0.303195	−	0.952928i	$-0.598053\pi$
$37$	−136.476	−0.606391	−0.303195	+	0.952928i	$0.598053\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	139.104	0.529865	0.264933	−	0.964267i	$-0.414650\pi$
$41$	139.104	0.529865	0.264933	+	0.964267i	$0.414650\pi$
$42$	0	0
$43$	−202.359	−0.717662	−0.358831	−	0.933402i	$-0.616825\pi$
$43$	−202.359	−0.717662	−0.358831	+	0.933402i	$0.616825\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	300.049	0.931206	0.465603	−	0.884994i	$-0.345837\pi$
$47$	300.049	0.931206	0.465603	+	0.884994i	$0.345837\pi$
$48$	0	0
$49$	791.617	2.30792
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−344.142	−0.891915	−0.445958	−	0.895054i	$-0.647137\pi$
$53$	−344.142	−0.891915	−0.445958	+	0.895054i	$0.647137\pi$
$54$	0	0
$55$	−47.4925	−0.116434
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−60.9339	−0.134456	−0.0672281	−	0.997738i	$-0.521416\pi$
$59$	−60.9339	−0.134456	−0.0672281	+	0.997738i	$0.521416\pi$
$60$	0	0
$61$	−122.122	−0.256331	−0.128165	−	0.991753i	$-0.540909\pi$
$61$	−122.122	−0.256331	−0.128165	+	0.991753i	$0.540909\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	198.420	0.378631
$66$	0	0
$67$	−744.667	−1.35784	−0.678922	−	0.734211i	$-0.737552\pi$
$67$	−744.667	−1.35784	−0.678922	+	0.734211i	$0.737552\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	436.971	0.730408	0.365204	−	0.930928i	$-0.380999\pi$
$71$	436.971	0.730408	0.365204	+	0.930928i	$0.380999\pi$
$72$	0	0
$73$	586.353	0.940102	0.470051	−	0.882639i	$-0.344236\pi$
$73$	586.353	0.940102	0.470051	+	0.882639i	$0.344236\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−319.948	−0.473526
$78$	0	0
$79$	473.780	0.674740	0.337370	−	0.941372i	$-0.390463\pi$
$79$	473.780	0.674740	0.337370	+	0.941372i	$0.390463\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−998.864	−1.32096	−0.660479	−	0.750844i	$-0.729647\pi$
$83$	−998.864	−1.32096	−0.660479	+	0.750844i	$0.729647\pi$
$84$	0	0
$85$	531.826	0.678642
$86$	0	0
$87$	0	0
$88$	0	0
$89$	204.463	0.243517	0.121759	−	0.992560i	$-0.461147\pi$
$89$	204.463	0.243517	0.121759	+	0.992560i	$0.461147\pi$
$90$	0	0
$91$	1336.72	1.53985
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−482.754	−0.521363
$96$	0	0
$97$	−1632.78	−1.70911	−0.854557	−	0.519358i	$-0.826171\pi$
$97$	−1632.78	−1.70911	−0.854557	+	0.519358i	$0.826171\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1199.80	−1.18203	−0.591015	−	0.806661i	$-0.701273\pi$
$101$	−1199.80	−1.18203	−0.591015	+	0.806661i	$0.701273\pi$
$102$	0	0
$103$	−1609.66	−1.53985	−0.769924	−	0.638136i	$-0.779706\pi$
$103$	−1609.66	−1.53985	−0.769924	+	0.638136i	$0.779706\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−206.586	−0.186649	−0.0933245	−	0.995636i	$-0.529749\pi$
$107$	−206.586	−0.186649	−0.0933245	+	0.995636i	$0.529749\pi$
$108$	0	0
$109$	1604.44	1.40989	0.704944	−	0.709263i	$-0.250972\pi$
$109$	1604.44	1.40989	0.704944	+	0.709263i	$0.250972\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1895.03	1.57760	0.788802	−	0.614647i	$-0.210702\pi$
$113$	1895.03	1.57760	0.788802	+	0.614647i	$0.210702\pi$
$114$	0	0
$115$	200.276	0.162399
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3582.81	2.75997
$120$	0	0
$121$	−1240.78	−0.932215
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	946.537	0.661351	0.330675	−	0.943745i	$-0.392723\pi$
$127$	946.537	0.661351	0.330675	+	0.943745i	$0.392723\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1684.10	1.12321	0.561606	−	0.827405i	$-0.310184\pi$
$131$	1684.10	1.12321	0.561606	+	0.827405i	$0.310184\pi$
$132$	0	0
$133$	−3252.22	−2.12033
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−414.313	−0.258374	−0.129187	−	0.991620i	$-0.541237\pi$
$137$	−414.313	−0.258374	−0.129187	+	0.991620i	$0.541237\pi$
$138$	0	0
$139$	−317.941	−0.194010	−0.0970049	−	0.995284i	$-0.530926\pi$
$139$	−317.941	−0.194010	−0.0970049	+	0.995284i	$0.530926\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−376.939	−0.220428
$144$	0	0
$145$	1262.09	0.722832
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3037.54	−1.67010	−0.835050	−	0.550174i	$-0.814561\pi$
$149$	−3037.54	−1.67010	−0.835050	+	0.550174i	$0.814561\pi$
$150$	0	0
$151$	−2120.80	−1.14297	−0.571485	−	0.820613i	$-0.693632\pi$
$151$	−2120.80	−1.14297	−0.571485	+	0.820613i	$0.693632\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	244.363	0.126631
$156$	0	0
$157$	2508.34	1.27508	0.637539	−	0.770418i	$-0.279952\pi$
$157$	2508.34	1.27508	0.637539	+	0.770418i	$0.279952\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1349.22	0.660458
$162$	0	0
$163$	−1834.04	−0.881308	−0.440654	−	0.897677i	$-0.645253\pi$
$163$	−1834.04	−0.881308	−0.440654	+	0.897677i	$0.645253\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3188.63	1.47751	0.738753	−	0.673976i	$-0.235415\pi$
$167$	3188.63	1.47751	0.738753	+	0.673976i	$0.235415\pi$
$168$	0	0
$169$	−622.174	−0.283192
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−224.949	−0.0988587	−0.0494293	−	0.998778i	$-0.515740\pi$
$173$	−224.949	−0.0988587	−0.0494293	+	0.998778i	$0.515740\pi$
$174$	0	0
$175$	842.102	0.363754
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3815.09	1.59303	0.796517	−	0.604616i	$-0.206674\pi$
$179$	3815.09	1.59303	0.796517	+	0.604616i	$0.206674\pi$
$180$	0	0
$181$	−3356.82	−1.37851	−0.689256	−	0.724518i	$-0.742062\pi$
$181$	−3356.82	−1.37851	−0.689256	+	0.724518i	$0.742062\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−682.378	−0.271186
$186$	0	0
$187$	−1010.31	−0.395086
$188$	0	0
$189$	0	0
$190$	0	0
$191$	831.167	0.314875	0.157438	−	0.987529i	$-0.449677\pi$
$191$	831.167	0.314875	0.157438	+	0.987529i	$0.449677\pi$
$192$	0	0
$193$	3038.43	1.13322	0.566609	−	0.823987i	$-0.308255\pi$
$193$	3038.43	1.13322	0.566609	+	0.823987i	$0.308255\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1288.06	0.465839	0.232919	−	0.972496i	$-0.425172\pi$
$197$	1288.06	0.465839	0.232919	+	0.972496i	$0.425172\pi$
$198$	0	0
$199$	5069.77	1.80596	0.902981	−	0.429680i	$-0.141374\pi$
$199$	5069.77	1.80596	0.902981	+	0.429680i	$0.141374\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8502.45	2.93968
$204$	0	0
$205$	695.522	0.236963
$206$	0	0
$207$	0	0
$208$	0	0
$209$	917.087	0.303523
$210$	0	0
$211$	2523.63	0.823384	0.411692	−	0.911323i	$-0.364938\pi$
$211$	2523.63	0.823384	0.411692	+	0.911323i	$0.364938\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1011.80	−0.320948
$216$	0	0
$217$	1646.23	0.514993
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4221.00	1.28478
$222$	0	0
$223$	−2405.71	−0.722412	−0.361206	−	0.932486i	$-0.617635\pi$
$223$	−2405.71	−0.722412	−0.361206	+	0.932486i	$0.617635\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5388.29	1.57548	0.787739	−	0.616009i	$-0.211252\pi$
$227$	5388.29	1.57548	0.787739	+	0.616009i	$0.211252\pi$
$228$	0	0
$229$	4714.12	1.36034	0.680171	−	0.733054i	$-0.261906\pi$
$229$	4714.12	1.36034	0.680171	+	0.733054i	$0.261906\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5065.99	1.42439	0.712197	−	0.701979i	$-0.247700\pi$
$233$	5065.99	1.42439	0.712197	+	0.701979i	$0.247700\pi$
$234$	0	0
$235$	1500.25	0.416448
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6637.92	−1.79653	−0.898267	−	0.439451i	$-0.855173\pi$
$239$	−6637.92	−1.79653	−0.898267	+	0.439451i	$0.855173\pi$
$240$	0	0
$241$	−5058.82	−1.35215	−0.676073	−	0.736834i	$-0.736320\pi$
$241$	−5058.82	−1.35215	−0.676073	+	0.736834i	$0.736320\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3958.09	1.03213
$246$	0	0
$247$	−3831.53	−0.987021
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2428.65	0.610737	0.305368	−	0.952234i	$-0.401220\pi$
$251$	2428.65	0.610737	0.305368	+	0.952234i	$0.401220\pi$
$252$	0	0
$253$	−380.465	−0.0945439
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3720.97	0.903142	0.451571	−	0.892235i	$-0.350864\pi$
$257$	3720.97	0.903142	0.451571	+	0.892235i	$0.350864\pi$
$258$	0	0
$259$	−4597.06	−1.10289
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−217.013	−0.0508807	−0.0254403	−	0.999676i	$-0.508099\pi$
$263$	−217.013	−0.0508807	−0.0254403	+	0.999676i	$0.508099\pi$
$264$	0	0
$265$	−1720.71	−0.398877
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−766.464	−0.173725	−0.0868627	−	0.996220i	$-0.527684\pi$
$269$	−766.464	−0.173725	−0.0868627	+	0.996220i	$0.527684\pi$
$270$	0	0
$271$	4030.90	0.903541	0.451770	−	0.892134i	$-0.350793\pi$
$271$	4030.90	0.903541	0.451770	+	0.892134i	$0.350793\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−237.463	−0.0520710
$276$	0	0
$277$	−1182.89	−0.256582	−0.128291	−	0.991737i	$-0.540949\pi$
$277$	−1182.89	−0.256582	−0.128291	+	0.991737i	$0.540949\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8060.82	1.71127	0.855637	−	0.517576i	$-0.173166\pi$
$281$	8060.82	1.71127	0.855637	+	0.517576i	$0.173166\pi$
$282$	0	0
$283$	−820.131	−0.172268	−0.0861338	−	0.996284i	$-0.527451\pi$
$283$	−820.131	−0.172268	−0.0861338	+	0.996284i	$0.527451\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4685.61	0.963703
$288$	0	0
$289$	6400.55	1.30278
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−409.740	−0.0816972	−0.0408486	−	0.999165i	$-0.513006\pi$
$293$	−409.740	−0.0816972	−0.0408486	+	0.999165i	$0.513006\pi$
$294$	0	0
$295$	−304.669	−0.0601307
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1589.56	0.307446
$300$	0	0
$301$	−6816.28	−1.30526
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−610.612	−0.114635
$306$	0	0
$307$	4200.96	0.780982	0.390491	−	0.920607i	$-0.372305\pi$
$307$	4200.96	0.780982	0.390491	+	0.920607i	$0.372305\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8443.12	−1.53944	−0.769719	−	0.638382i	$-0.779604\pi$
$311$	−8443.12	−1.53944	−0.769719	+	0.638382i	$0.779604\pi$
$312$	0	0
$313$	134.035	0.0242048	0.0121024	−	0.999927i	$-0.496148\pi$
$313$	134.035	0.0242048	0.0121024	+	0.999927i	$0.496148\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9580.61	−1.69748	−0.848740	−	0.528811i	$-0.822638\pi$
$317$	−9580.61	−1.69748	−0.848740	+	0.528811i	$0.822638\pi$
$318$	0	0
$319$	−2397.59	−0.420812
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−10269.6	−1.76910
$324$	0	0
$325$	992.102	0.169329
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10106.9	1.69365
$330$	0	0
$331$	−8759.64	−1.45460	−0.727301	−	0.686318i	$-0.759226\pi$
$331$	−8759.64	−1.45460	−0.727301	+	0.686318i	$0.759226\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3723.33	−0.607246
$336$	0	0
$337$	5558.66	0.898514	0.449257	−	0.893403i	$-0.351689\pi$
$337$	5558.66	0.898514	0.449257	+	0.893403i	$0.351689\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−464.217	−0.0737207
$342$	0	0
$343$	15111.3	2.37881
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10687.1	−1.65335	−0.826675	−	0.562679i	$-0.809771\pi$
$347$	−10687.1	−1.65335	−0.826675	+	0.562679i	$0.809771\pi$
$348$	0	0
$349$	3939.46	0.604225	0.302113	−	0.953272i	$-0.402308\pi$
$349$	3939.46	0.604225	0.302113	+	0.953272i	$0.402308\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5297.90	0.798806	0.399403	−	0.916775i	$-0.369217\pi$
$353$	5297.90	0.798806	0.399403	+	0.916775i	$0.369217\pi$
$354$	0	0
$355$	2184.86	0.326648
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1103.06	−0.162165	−0.0810827	−	0.996707i	$-0.525838\pi$
$359$	−1103.06	−0.162165	−0.0810827	+	0.996707i	$0.525838\pi$
$360$	0	0
$361$	2463.05	0.359097
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2931.77	0.420426
$366$	0	0
$367$	−7315.72	−1.04054	−0.520269	−	0.854002i	$-0.674168\pi$
$367$	−7315.72	−1.04054	−0.520269	+	0.854002i	$0.674168\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11592.1	−1.62219
$372$	0	0
$373$	−1885.80	−0.261778	−0.130889	−	0.991397i	$-0.541783\pi$
$373$	−1885.80	−0.261778	−0.130889	+	0.991397i	$0.541783\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10017.0	1.36843
$378$	0	0
$379$	−5371.24	−0.727973	−0.363987	−	0.931404i	$-0.618585\pi$
$379$	−5371.24	−0.727973	−0.363987	+	0.931404i	$0.618585\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	72.2612	0.00964067	0.00482033	−	0.999988i	$-0.498466\pi$
$383$	72.2612	0.00964067	0.00482033	+	0.999988i	$0.498466\pi$
$384$	0	0
$385$	−1599.74	−0.211767
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1793.28	0.233735	0.116867	−	0.993148i	$-0.462715\pi$
$389$	1793.28	0.233735	0.116867	+	0.993148i	$0.462715\pi$
$390$	0	0
$391$	4260.48	0.551053
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2368.90	0.301753
$396$	0	0
$397$	596.085	0.0753568	0.0376784	−	0.999290i	$-0.488004\pi$
$397$	596.085	0.0753568	0.0376784	+	0.999290i	$0.488004\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	954.797	0.118903	0.0594517	−	0.998231i	$-0.481065\pi$
$401$	954.797	0.118903	0.0594517	+	0.998231i	$0.481065\pi$
$402$	0	0
$403$	1939.47	0.239731
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1296.31	0.157877
$408$	0	0
$409$	−1167.04	−0.141091	−0.0705456	−	0.997509i	$-0.522474\pi$
$409$	−1167.04	−0.141091	−0.0705456	+	0.997509i	$0.522474\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2052.50	−0.244545
$414$	0	0
$415$	−4994.32	−0.590750
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14857.0	1.73225	0.866124	−	0.499828i	$-0.166604\pi$
$419$	14857.0	1.73225	0.866124	+	0.499828i	$0.166604\pi$
$420$	0	0
$421$	2161.07	0.250176	0.125088	−	0.992146i	$-0.460079\pi$
$421$	2161.07	0.250176	0.125088	+	0.992146i	$0.460079\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2659.13	0.303498
$426$	0	0
$427$	−4113.58	−0.466207
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6031.12	0.674035	0.337017	−	0.941498i	$-0.390582\pi$
$431$	6031.12	0.674035	0.337017	+	0.941498i	$0.390582\pi$
$432$	0	0
$433$	7477.24	0.829869	0.414934	−	0.909851i	$-0.363804\pi$
$433$	7477.24	0.829869	0.414934	+	0.909851i	$0.363804\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3867.36	−0.423343
$438$	0	0
$439$	−6516.94	−0.708511	−0.354256	−	0.935149i	$-0.615266\pi$
$439$	−6516.94	−0.708511	−0.354256	+	0.935149i	$0.615266\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7912.72	0.848633	0.424317	−	0.905514i	$-0.360514\pi$
$443$	7912.72	0.848633	0.424317	+	0.905514i	$0.360514\pi$
$444$	0	0
$445$	1022.32	0.108904
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8686.04	0.912961	0.456480	−	0.889733i	$-0.349110\pi$
$449$	8686.04	0.912961	0.456480	+	0.889733i	$0.349110\pi$
$450$	0	0
$451$	−1321.28	−0.137953
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6683.61	0.688643
$456$	0	0
$457$	−6787.39	−0.694750	−0.347375	−	0.937726i	$-0.612927\pi$
$457$	−6787.39	−0.694750	−0.347375	+	0.937726i	$0.612927\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15130.9	1.52866	0.764332	−	0.644823i	$-0.223069\pi$
$461$	15130.9	1.52866	0.764332	+	0.644823i	$0.223069\pi$
$462$	0	0
$463$	12316.7	1.23629	0.618147	−	0.786062i	$-0.287883\pi$
$463$	12316.7	1.23629	0.618147	+	0.786062i	$0.287883\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16423.4	−1.62737	−0.813687	−	0.581303i	$-0.802543\pi$
$467$	−16423.4	−1.62737	−0.813687	+	0.581303i	$0.802543\pi$
$468$	0	0
$469$	−25083.4	−2.46960
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1922.11	0.186847
$474$	0	0
$475$	−2413.77	−0.233161
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11761.8	−1.12195	−0.560973	−	0.827834i	$-0.689573\pi$
$479$	−11761.8	−1.12195	−0.560973	+	0.827834i	$0.689573\pi$
$480$	0	0
$481$	−5415.91	−0.513398
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8163.92	−0.764339
$486$	0	0
$487$	−18620.0	−1.73255	−0.866277	−	0.499564i	$-0.833494\pi$
$487$	−18620.0	−1.73255	−0.866277	+	0.499564i	$0.833494\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10022.2	−0.921171	−0.460586	−	0.887615i	$-0.652361\pi$
$491$	−10022.2	−0.921171	−0.460586	+	0.887615i	$0.652361\pi$
$492$	0	0
$493$	26848.4	2.45272
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14719.0	1.32844
$498$	0	0
$499$	−3843.64	−0.344820	−0.172410	−	0.985025i	$-0.555155\pi$
$499$	−3843.64	−0.344820	−0.172410	+	0.985025i	$0.555155\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6782.43	−0.601220	−0.300610	−	0.953747i	$-0.597190\pi$
$503$	−6782.43	−0.601220	−0.300610	+	0.953747i	$0.597190\pi$
$504$	0	0
$505$	−5999.02	−0.528620
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17019.1	1.48204	0.741018	−	0.671485i	$-0.234343\pi$
$509$	17019.1	1.48204	0.741018	+	0.671485i	$0.234343\pi$
$510$	0	0
$511$	19750.8	1.70983
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8048.29	−0.688641
$516$	0	0
$517$	−2850.02	−0.242444
$518$	0	0
$519$	0	0
$520$	0	0
$521$	19278.7	1.62114	0.810572	−	0.585639i	$-0.199156\pi$
$521$	19278.7	1.62114	0.810572	+	0.585639i	$0.199156\pi$
$522$	0	0
$523$	8074.65	0.675105	0.337552	−	0.941307i	$-0.390401\pi$
$523$	8074.65	0.675105	0.337552	+	0.941307i	$0.390401\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5198.35	0.429684
$528$	0	0
$529$	−10562.6	−0.868133
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5520.23	0.448608
$534$	0	0
$535$	−1032.93	−0.0834719
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7519.18	−0.600879
$540$	0	0
$541$	−18260.8	−1.45119	−0.725594	−	0.688123i	$-0.758435\pi$
$541$	−18260.8	−1.45119	−0.725594	+	0.688123i	$0.758435\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8022.21	0.630521
$546$	0	0
$547$	10707.4	0.836958	0.418479	−	0.908226i	$-0.362563\pi$
$547$	10707.4	0.836958	0.418479	+	0.908226i	$0.362563\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24371.1	−1.88429
$552$	0	0
$553$	15958.8	1.22720
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17348.5	1.31971	0.659857	−	0.751392i	$-0.270617\pi$
$557$	17348.5	1.31971	0.659857	+	0.751392i	$0.270617\pi$
$558$	0	0
$559$	−8030.44	−0.607605
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14947.1	−1.11891	−0.559455	−	0.828860i	$-0.688990\pi$
$563$	−14947.1	−1.11891	−0.559455	+	0.828860i	$0.688990\pi$
$564$	0	0
$565$	9475.14	0.705526
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26723.0	−1.96887	−0.984435	−	0.175750i	$-0.943765\pi$
$569$	−26723.0	−1.96887	−0.984435	+	0.175750i	$0.943765\pi$
$570$	0	0
$571$	8419.93	0.617098	0.308549	−	0.951208i	$-0.400157\pi$
$571$	8419.93	0.617098	0.308549	+	0.951208i	$0.400157\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1001.38	0.0726269
$576$	0	0
$577$	2921.42	0.210780	0.105390	−	0.994431i	$-0.466391\pi$
$577$	2921.42	0.210780	0.105390	+	0.994431i	$0.466391\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−33645.8	−2.40252
$582$	0	0
$583$	3268.83	0.232215
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−64.7362	−0.00455188	−0.00227594	−	0.999997i	$-0.500724\pi$
$587$	−64.7362	−0.00455188	−0.00227594	+	0.999997i	$0.500724\pi$
$588$	0	0
$589$	−4718.69	−0.330102
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20682.2	−1.43223	−0.716117	−	0.697980i	$-0.754082\pi$
$593$	−20682.2	−1.43223	−0.716117	+	0.697980i	$0.754082\pi$
$594$	0	0
$595$	17914.1	1.23429
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20639.8	−1.40788	−0.703939	−	0.710260i	$-0.748577\pi$
$599$	−20639.8	−1.40788	−0.703939	+	0.710260i	$0.748577\pi$
$600$	0	0
$601$	−19749.2	−1.34041	−0.670205	−	0.742176i	$-0.733794\pi$
$601$	−19749.2	−1.34041	−0.670205	+	0.742176i	$0.733794\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6203.89	−0.416899
$606$	0	0
$607$	−15180.4	−1.01508	−0.507539	−	0.861629i	$-0.669445\pi$
$607$	−15180.4	−1.01508	−0.507539	+	0.861629i	$0.669445\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11907.2	0.788401
$612$	0	0
$613$	−15998.0	−1.05408	−0.527042	−	0.849839i	$-0.676699\pi$
$613$	−15998.0	−1.05408	−0.527042	+	0.849839i	$0.676699\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1827.40	0.119235	0.0596176	−	0.998221i	$-0.481012\pi$
$617$	1827.40	0.119235	0.0596176	+	0.998221i	$0.481012\pi$
$618$	0	0
$619$	−12096.6	−0.785464	−0.392732	−	0.919653i	$-0.628470\pi$
$619$	−12096.6	−0.785464	−0.392732	+	0.919653i	$0.628470\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6887.15	0.442902
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14516.3	−0.920192
$630$	0	0
$631$	20914.5	1.31948	0.659741	−	0.751493i	$-0.270666\pi$
$631$	20914.5	1.31948	0.659741	+	0.751493i	$0.270666\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4732.68	0.295765
$636$	0	0
$637$	31414.6	1.95399
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14490.4	0.892880	0.446440	−	0.894814i	$-0.352692\pi$
$641$	14490.4	0.892880	0.446440	+	0.894814i	$0.352692\pi$
$642$	0	0
$643$	11516.8	0.706345	0.353173	−	0.935558i	$-0.385103\pi$
$643$	11516.8	0.706345	0.353173	+	0.935558i	$0.385103\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18928.4	−1.15016	−0.575078	−	0.818099i	$-0.695028\pi$
$647$	−18928.4	−1.15016	−0.575078	+	0.818099i	$0.695028\pi$
$648$	0	0
$649$	578.781	0.0350064
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5144.16	−0.308279	−0.154140	−	0.988049i	$-0.549261\pi$
$653$	−5144.16	−0.308279	−0.154140	+	0.988049i	$0.549261\pi$
$654$	0	0
$655$	8420.52	0.502316
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2988.44	−0.176651	−0.0883257	−	0.996092i	$-0.528152\pi$
$659$	−2988.44	−0.176651	−0.0883257	+	0.996092i	$0.528152\pi$
$660$	0	0
$661$	−13451.0	−0.791500	−0.395750	−	0.918358i	$-0.629515\pi$
$661$	−13451.0	−0.791500	−0.395750	+	0.918358i	$0.629515\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16261.1	−0.948239
$666$	0	0
$667$	10110.6	0.586935
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1159.98	0.0667370
$672$	0	0
$673$	10085.7	0.577674	0.288837	−	0.957378i	$-0.406731\pi$
$673$	10085.7	0.577674	0.288837	+	0.957378i	$0.406731\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−20517.8	−1.16479	−0.582396	−	0.812905i	$-0.697885\pi$
$677$	−20517.8	−1.16479	−0.582396	+	0.812905i	$0.697885\pi$
$678$	0	0
$679$	−54998.8	−3.10848
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15006.6	−0.840719	−0.420359	−	0.907358i	$-0.638096\pi$
$683$	−15006.6	−0.840719	−0.420359	+	0.907358i	$0.638096\pi$
$684$	0	0
$685$	−2071.57	−0.115548
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−13657.0	−0.755136
$690$	0	0
$691$	25718.5	1.41589	0.707944	−	0.706269i	$-0.249623\pi$
$691$	25718.5	1.41589	0.707944	+	0.706269i	$0.249623\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1589.70	−0.0867639
$696$	0	0
$697$	14795.9	0.804066
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−14908.1	−0.803239	−0.401619	−	0.915807i	$-0.631552\pi$
$701$	−14908.1	−0.803239	−0.401619	+	0.915807i	$0.631552\pi$
$702$	0	0
$703$	13176.8	0.706932
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−40414.3	−2.14984
$708$	0	0
$709$	23392.0	1.23907	0.619537	−	0.784967i	$-0.287320\pi$
$709$	23392.0	1.23907	0.619537	+	0.784967i	$0.287320\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1957.61	0.102823
$714$	0	0
$715$	−1884.70	−0.0985785
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22561.1	−1.17022	−0.585110	−	0.810954i	$-0.698949\pi$
$719$	−22561.1	−1.17022	−0.585110	+	0.810954i	$0.698949\pi$
$720$	0	0
$721$	−54219.8	−2.80063
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6310.44	0.323260
$726$	0	0
$727$	−20864.7	−1.06441	−0.532206	−	0.846615i	$-0.678637\pi$
$727$	−20864.7	−1.06441	−0.532206	+	0.846615i	$0.678637\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21524.0	−1.08905
$732$	0	0
$733$	24555.8	1.23737	0.618683	−	0.785641i	$-0.287666\pi$
$733$	24555.8	1.23737	0.618683	+	0.785641i	$0.287666\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7073.22	0.353521
$738$	0	0
$739$	14842.0	0.738801	0.369400	−	0.929270i	$-0.379563\pi$
$739$	14842.0	0.738801	0.369400	+	0.929270i	$0.379563\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15518.1	0.766223	0.383111	−	0.923702i	$-0.374853\pi$
$743$	15518.1	0.766223	0.383111	+	0.923702i	$0.374853\pi$
$744$	0	0
$745$	−15187.7	−0.746891
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6958.66	−0.339471
$750$	0	0
$751$	−39293.6	−1.90925	−0.954624	−	0.297814i	$-0.903742\pi$
$751$	−39293.6	−1.90925	−0.954624	+	0.297814i	$0.903742\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10604.0	−0.511152
$756$	0	0
$757$	−30307.7	−1.45516	−0.727578	−	0.686025i	$-0.759354\pi$
$757$	−30307.7	−1.45516	−0.727578	+	0.686025i	$0.759354\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−27358.3	−1.30320	−0.651601	−	0.758562i	$-0.725902\pi$
$761$	−27358.3	−1.30320	−0.651601	+	0.758562i	$0.725902\pi$
$762$	0	0
$763$	54044.2	2.56426
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2418.11	−0.113837
$768$	0	0
$769$	17282.2	0.810419	0.405209	−	0.914224i	$-0.367199\pi$
$769$	17282.2	0.810419	0.405209	+	0.914224i	$0.367199\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−30621.8	−1.42482	−0.712411	−	0.701762i	$-0.752397\pi$
$773$	−30621.8	−1.42482	−0.712411	+	0.701762i	$0.752397\pi$
$774$	0	0
$775$	1221.82	0.0566309
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13430.6	−0.617719
$780$	0	0
$781$	−4150.57	−0.190165
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12541.7	0.570232
$786$	0	0
$787$	−32786.5	−1.48502	−0.742512	−	0.669833i	$-0.766366\pi$
$787$	−32786.5	−1.48502	−0.742512	+	0.669833i	$0.766366\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	63832.3	2.86930
$792$	0	0
$793$	−4846.32	−0.217021
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13692.1	−0.608529	−0.304265	−	0.952588i	$-0.598411\pi$
$797$	−13692.1	−0.608529	−0.304265	+	0.952588i	$0.598411\pi$
$798$	0	0
$799$	31914.8	1.41310
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5569.48	−0.244760
$804$	0	0
$805$	6746.12	0.295366
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39429.7	1.71357	0.856783	−	0.515677i	$-0.172459\pi$
$809$	39429.7	1.71357	0.856783	+	0.515677i	$0.172459\pi$
$810$	0	0
$811$	−10397.9	−0.450209	−0.225105	−	0.974335i	$-0.572272\pi$
$811$	−10397.9	−0.450209	−0.225105	+	0.974335i	$0.572272\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9170.20	−0.394133
$816$	0	0
$817$	19537.9	0.836653
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2913.08	0.123833	0.0619166	−	0.998081i	$-0.480279\pi$
$821$	2913.08	0.123833	0.0619166	+	0.998081i	$0.480279\pi$
$822$	0	0
$823$	−16934.6	−0.717257	−0.358628	−	0.933480i	$-0.616755\pi$
$823$	−16934.6	−0.717257	−0.358628	+	0.933480i	$0.616755\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1191.52	−0.0501006	−0.0250503	−	0.999686i	$-0.507975\pi$
$827$	−1191.52	−0.0501006	−0.0250503	+	0.999686i	$0.507975\pi$
$828$	0	0
$829$	−5792.37	−0.242675	−0.121337	−	0.992611i	$-0.538718\pi$
$829$	−5792.37	−0.242675	−0.121337	+	0.992611i	$0.538718\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	84200.5	3.50225
$834$	0	0
$835$	15943.1	0.660761
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2620.21	−0.107818	−0.0539091	−	0.998546i	$-0.517168\pi$
$839$	−2620.21	−0.107818	−0.0539091	+	0.998546i	$0.517168\pi$
$840$	0	0
$841$	39325.6	1.61243
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3110.87	−0.126647
$846$	0	0
$847$	−41794.5	−1.69548
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5466.57	−0.220201
$852$	0	0
$853$	1485.11	0.0596121	0.0298060	−	0.999556i	$-0.490511\pi$
$853$	1485.11	0.0596121	0.0298060	+	0.999556i	$0.490511\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13999.8	−0.558023	−0.279011	−	0.960288i	$-0.590007\pi$
$857$	−13999.8	−0.558023	−0.279011	+	0.960288i	$0.590007\pi$
$858$	0	0
$859$	46006.2	1.82737	0.913686	−	0.406421i	$-0.133223\pi$
$859$	46006.2	1.82737	0.913686	+	0.406421i	$0.133223\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37731.8	−1.48830	−0.744152	−	0.668010i	$-0.767146\pi$
$863$	−37731.8	−1.48830	−0.744152	+	0.668010i	$0.767146\pi$
$864$	0	0
$865$	−1124.74	−0.0442109
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4500.20	−0.175672
$870$	0	0
$871$	−29551.4	−1.14961
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4210.51	0.162676
$876$	0	0
$877$	40443.7	1.55723	0.778614	−	0.627504i	$-0.215923\pi$
$877$	40443.7	1.55723	0.778614	+	0.627504i	$0.215923\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−35957.5	−1.37507	−0.687535	−	0.726151i	$-0.741307\pi$
$881$	−35957.5	−1.37507	−0.687535	+	0.726151i	$0.741307\pi$
$882$	0	0
$883$	4208.35	0.160387	0.0801937	−	0.996779i	$-0.474446\pi$
$883$	4208.35	0.160387	0.0801937	+	0.996779i	$0.474446\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6615.52	0.250426	0.125213	−	0.992130i	$-0.460039\pi$
$887$	6615.52	0.250426	0.125213	+	0.992130i	$0.460039\pi$
$888$	0	0
$889$	31883.2	1.20284
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−28970.0	−1.08560
$894$	0	0
$895$	19075.4	0.712426
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12336.3	0.457663
$900$	0	0
$901$	−36604.7	−1.35347
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16784.1	−0.616489
$906$	0	0
$907$	−36352.8	−1.33084	−0.665421	−	0.746468i	$-0.731748\pi$
$907$	−36352.8	−1.33084	−0.665421	+	0.746468i	$0.731748\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2245.96	0.0816817	0.0408409	−	0.999166i	$-0.486996\pi$
$911$	2245.96	0.0816817	0.0408409	+	0.999166i	$0.486996\pi$
$912$	0	0
$913$	9487.71	0.343918
$914$	0	0
$915$	0	0
$916$	0	0
$917$	56727.5	2.04286
$918$	0	0
$919$	−13665.8	−0.490527	−0.245263	−	0.969456i	$-0.578874\pi$
$919$	−13665.8	−0.490527	−0.245263	+	0.969456i	$0.578874\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	17340.8	0.618396
$924$	0	0
$925$	−3411.89	−0.121278
$926$	0	0
$927$	0	0
$928$	0	0
$929$	43509.5	1.53660	0.768300	−	0.640091i	$-0.221103\pi$
$929$	43509.5	1.53660	0.768300	+	0.640091i	$0.221103\pi$
$930$	0	0
$931$	−76431.3	−2.69058
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5051.55	−0.176688
$936$	0	0
$937$	−6545.89	−0.228223	−0.114111	−	0.993468i	$-0.536402\pi$
$937$	−6545.89	−0.228223	−0.114111	+	0.993468i	$0.536402\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−237.479	−0.00822697	−0.00411349	−	0.999992i	$-0.501309\pi$
$941$	−237.479	−0.00822697	−0.00411349	+	0.999992i	$0.501309\pi$
$942$	0	0
$943$	5571.86	0.192412
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34862.6	−1.19629	−0.598143	−	0.801390i	$-0.704094\pi$
$947$	−34862.6	−1.19629	−0.598143	+	0.801390i	$0.704094\pi$
$948$	0	0
$949$	23268.9	0.795933
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−30830.7	−1.04796	−0.523979	−	0.851731i	$-0.675553\pi$
$953$	−30830.7	−1.04796	−0.523979	+	0.851731i	$0.675553\pi$
$954$	0	0
$955$	4155.84	0.140816
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13955.8	−0.469922
$960$	0	0
$961$	−27402.5	−0.919824
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15192.1	0.506790
$966$	0	0
$967$	−16892.7	−0.561772	−0.280886	−	0.959741i	$-0.590628\pi$
$967$	−16892.7	−0.561772	−0.280886	+	0.959741i	$0.590628\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3347.26	0.110627	0.0553135	−	0.998469i	$-0.482384\pi$
$971$	3347.26	0.110627	0.0553135	+	0.998469i	$0.482384\pi$
$972$	0	0
$973$	−10709.5	−0.352859
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21331.5	−0.698522	−0.349261	−	0.937025i	$-0.613567\pi$
$977$	−21331.5	−0.698522	−0.349261	+	0.937025i	$0.613567\pi$
$978$	0	0
$979$	−1942.09	−0.0634010
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−7531.13	−0.244360	−0.122180	−	0.992508i	$-0.538988\pi$
$983$	−7531.13	−0.244360	−0.122180	+	0.992508i	$0.538988\pi$
$984$	0	0
$985$	6440.28	0.208329
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8105.55	−0.260608
$990$	0	0
$991$	−26751.6	−0.857510	−0.428755	−	0.903421i	$-0.641048\pi$
$991$	−26751.6	−0.857510	−0.428755	+	0.903421i	$0.641048\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25348.9	0.807651
$996$	0	0
$997$	2838.54	0.0901680	0.0450840	−	0.998983i	$-0.485644\pi$
$997$	2838.54	0.0901680	0.0450840	+	0.998983i	$0.485644\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.f.1.3	yes	3
3.2	odd	2		1620.4.a.d.1.3	✓	3
9.2	odd	6		1620.4.i.u.1081.1		6
9.4	even	3		1620.4.i.s.541.1		6
9.5	odd	6		1620.4.i.u.541.1		6
9.7	even	3		1620.4.i.s.1081.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.d.1.3	✓	3	3.2	odd	2
1620.4.a.f.1.3	yes	3	1.1	even	1	trivial
1620.4.i.s.541.1		6	9.4	even	3
1620.4.i.s.1081.1		6	9.7	even	3
1620.4.i.u.541.1		6	9.5	odd	6
1620.4.i.u.1081.1		6	9.2	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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1620.4.a.f.1.3

Level	$1620$
Weight	$4$
Character	1620.1
Self dual	yes
Analytic conductor	$95.583$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$