LMFDB - Embedded newform 1620.4.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 167x^{5} + 940x^{4} + 1153x^{3} - 13196x^{2} + 16629x + 714 $ x^7 - 2x^6 - 167x^5 + 940x^4 + 1153x^3 - 13196x^2 + 16629x + 714
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.01973$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} +24.6035 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +24.6035 q^{7} +62.6277 q^{11} +59.6522 q^{13} +85.2021 q^{17} -75.1746 q^{19} +97.5551 q^{23} +25.0000 q^{25} -159.251 q^{29} +318.510 q^{31} -123.017 q^{35} +393.612 q^{37} -85.0813 q^{41} -40.7635 q^{43} +90.2908 q^{47} +262.331 q^{49} -691.499 q^{53} -313.139 q^{55} -283.519 q^{59} -432.156 q^{61} -298.261 q^{65} +308.360 q^{67} -302.675 q^{71} -457.563 q^{73} +1540.86 q^{77} -841.695 q^{79} +788.180 q^{83} -426.010 q^{85} -1119.35 q^{89} +1467.65 q^{91} +375.873 q^{95} +1197.70 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) $ 7 * q - 35 * q^5 + 8 * q^7	$ 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) $ 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	24.6035	1.32846	0.664232	−	0.747527i	$-0.268759\pi$
$7$	24.6035	1.32846	0.664232	+	0.747527i	$0.268759\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	62.6277	1.71663	0.858316	−	0.513121i	$-0.171511\pi$
$11$	62.6277	1.71663	0.858316	+	0.513121i	$0.171511\pi$
$12$	0	0
$13$	59.6522	1.27266	0.636329	−	0.771418i	$-0.280452\pi$
$13$	59.6522	1.27266	0.636329	+	0.771418i	$0.280452\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	85.2021	1.21556	0.607780	−	0.794105i	$-0.292060\pi$
$17$	85.2021	1.21556	0.607780	+	0.794105i	$0.292060\pi$
$18$	0	0
$19$	−75.1746	−0.907697	−0.453849	−	0.891079i	$-0.649949\pi$
$19$	−75.1746	−0.907697	−0.453849	+	0.891079i	$0.649949\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	97.5551	0.884420	0.442210	−	0.896912i	$-0.354195\pi$
$23$	97.5551	0.884420	0.442210	+	0.896912i	$0.354195\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−159.251	−1.01973	−0.509865	−	0.860254i	$-0.670305\pi$
$29$	−159.251	−1.01973	−0.509865	+	0.860254i	$0.670305\pi$
$30$	0	0
$31$	318.510	1.84536	0.922680	−	0.385568i	$-0.125994\pi$
$31$	318.510	1.84536	0.922680	+	0.385568i	$0.125994\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−123.017	−0.594107
$36$	0	0
$37$	393.612	1.74890	0.874451	−	0.485114i	$-0.161222\pi$
$37$	393.612	1.74890	0.874451	+	0.485114i	$0.161222\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−85.0813	−0.324085	−0.162042	−	0.986784i	$-0.551808\pi$
$41$	−85.0813	−0.324085	−0.162042	+	0.986784i	$0.551808\pi$
$42$	0	0
$43$	−40.7635	−0.144567	−0.0722835	−	0.997384i	$-0.523029\pi$
$43$	−40.7635	−0.144567	−0.0722835	+	0.997384i	$0.523029\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	90.2908	0.280218	0.140109	−	0.990136i	$-0.455255\pi$
$47$	90.2908	0.280218	0.140109	+	0.990136i	$0.455255\pi$
$48$	0	0
$49$	262.331	0.764814
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−691.499	−1.79216	−0.896081	−	0.443890i	$-0.853598\pi$
$53$	−691.499	−1.79216	−0.896081	+	0.443890i	$0.853598\pi$
$54$	0	0
$55$	−313.139	−0.767702
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−283.519	−0.625610	−0.312805	−	0.949817i	$-0.601269\pi$
$59$	−283.519	−0.625610	−0.312805	+	0.949817i	$0.601269\pi$
$60$	0	0
$61$	−432.156	−0.907081	−0.453540	−	0.891236i	$-0.649839\pi$
$61$	−432.156	−0.907081	−0.453540	+	0.891236i	$0.649839\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−298.261	−0.569150
$66$	0	0
$67$	308.360	0.562272	0.281136	−	0.959668i	$-0.409289\pi$
$67$	308.360	0.562272	0.281136	+	0.959668i	$0.409289\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−302.675	−0.505928	−0.252964	−	0.967476i	$-0.581405\pi$
$71$	−302.675	−0.505928	−0.252964	+	0.967476i	$0.581405\pi$
$72$	0	0
$73$	−457.563	−0.733613	−0.366806	−	0.930297i	$-0.619549\pi$
$73$	−457.563	−0.733613	−0.366806	+	0.930297i	$0.619549\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1540.86	2.28048
$78$	0	0
$79$	−841.695	−1.19871	−0.599355	−	0.800483i	$-0.704576\pi$
$79$	−841.695	−1.19871	−0.599355	+	0.800483i	$0.704576\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	788.180	1.04234	0.521169	−	0.853454i	$-0.325496\pi$
$83$	788.180	1.04234	0.521169	+	0.853454i	$0.325496\pi$
$84$	0	0
$85$	−426.010	−0.543615
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1119.35	−1.33316	−0.666578	−	0.745436i	$-0.732241\pi$
$89$	−1119.35	−1.33316	−0.666578	+	0.745436i	$0.732241\pi$
$90$	0	0
$91$	1467.65	1.69068
$92$	0	0
$93$	0	0
$94$	0	0
$95$	375.873	0.405934
$96$	0	0
$97$	1197.70	1.25369	0.626846	−	0.779144i	$-0.284346\pi$
$97$	1197.70	1.25369	0.626846	+	0.779144i	$0.284346\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1417.36	−1.39636	−0.698180	−	0.715922i	$-0.746006\pi$
$101$	−1417.36	−1.39636	−0.698180	+	0.715922i	$0.746006\pi$
$102$	0	0
$103$	415.294	0.397283	0.198641	−	0.980072i	$-0.436347\pi$
$103$	415.294	0.397283	0.198641	+	0.980072i	$0.436347\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−766.240	−0.692292	−0.346146	−	0.938181i	$-0.612510\pi$
$107$	−766.240	−0.692292	−0.346146	+	0.938181i	$0.612510\pi$
$108$	0	0
$109$	349.214	0.306868	0.153434	−	0.988159i	$-0.450967\pi$
$109$	349.214	0.306868	0.153434	+	0.988159i	$0.450967\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1119.94	0.932345	0.466173	−	0.884694i	$-0.345633\pi$
$113$	1119.94	0.932345	0.466173	+	0.884694i	$0.345633\pi$
$114$	0	0
$115$	−487.776	−0.395525
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2096.27	1.61483
$120$	0	0
$121$	2591.23	1.94683
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−871.649	−0.609027	−0.304513	−	0.952508i	$-0.598494\pi$
$127$	−871.649	−0.609027	−0.304513	+	0.952508i	$0.598494\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	120.965	0.0806776	0.0403388	−	0.999186i	$-0.487156\pi$
$131$	120.965	0.0806776	0.0403388	+	0.999186i	$0.487156\pi$
$132$	0	0
$133$	−1849.56	−1.20584
$134$	0	0
$135$	0	0
$136$	0	0
$137$	454.704	0.283562	0.141781	−	0.989898i	$-0.454717\pi$
$137$	454.704	0.283562	0.141781	+	0.989898i	$0.454717\pi$
$138$	0	0
$139$	2340.61	1.42826	0.714130	−	0.700013i	$-0.246822\pi$
$139$	2340.61	1.42826	0.714130	+	0.700013i	$0.246822\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3735.88	2.18469
$144$	0	0
$145$	796.256	0.456037
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2168.37	−1.19221	−0.596106	−	0.802906i	$-0.703286\pi$
$149$	−2168.37	−1.19221	−0.596106	+	0.802906i	$0.703286\pi$
$150$	0	0
$151$	2689.02	1.44920	0.724601	−	0.689169i	$-0.242024\pi$
$151$	2689.02	1.44920	0.724601	+	0.689169i	$0.242024\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1592.55	−0.825270
$156$	0	0
$157$	−2716.43	−1.38086	−0.690430	−	0.723399i	$-0.742579\pi$
$157$	−2716.43	−1.38086	−0.690430	+	0.723399i	$0.742579\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2400.20	1.17492
$162$	0	0
$163$	289.025	0.138885	0.0694423	−	0.997586i	$-0.477878\pi$
$163$	289.025	0.138885	0.0694423	+	0.997586i	$0.477878\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	358.558	0.166144	0.0830721	−	0.996544i	$-0.473527\pi$
$167$	358.558	0.166144	0.0830721	+	0.996544i	$0.473527\pi$
$168$	0	0
$169$	1361.39	0.619658
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1350.34	0.593435	0.296718	−	0.954965i	$-0.404108\pi$
$173$	1350.34	0.593435	0.296718	+	0.954965i	$0.404108\pi$
$174$	0	0
$175$	615.087	0.265693
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1192.29	−0.497854	−0.248927	−	0.968522i	$-0.580078\pi$
$179$	−1192.29	−0.497854	−0.248927	+	0.968522i	$0.580078\pi$
$180$	0	0
$181$	−1785.62	−0.733283	−0.366641	−	0.930362i	$-0.619492\pi$
$181$	−1785.62	−0.733283	−0.366641	+	0.930362i	$0.619492\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1968.06	−0.782133
$186$	0	0
$187$	5336.01	2.08667
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1572.85	0.595851	0.297925	−	0.954589i	$-0.403705\pi$
$191$	1572.85	0.595851	0.297925	+	0.954589i	$0.403705\pi$
$192$	0	0
$193$	298.079	0.111172	0.0555860	−	0.998454i	$-0.482297\pi$
$193$	298.079	0.111172	0.0555860	+	0.998454i	$0.482297\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−881.991	−0.318981	−0.159490	−	0.987199i	$-0.550985\pi$
$197$	−881.991	−0.318981	−0.159490	+	0.987199i	$0.550985\pi$
$198$	0	0
$199$	2647.66	0.943155	0.471578	−	0.881825i	$-0.343685\pi$
$199$	2647.66	0.943155	0.471578	+	0.881825i	$0.343685\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3918.13	−1.35467
$204$	0	0
$205$	425.407	0.144935
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4708.01	−1.55818
$210$	0	0
$211$	3664.57	1.19564	0.597819	−	0.801631i	$-0.296034\pi$
$211$	3664.57	1.19564	0.597819	+	0.801631i	$0.296034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	203.818	0.0646523
$216$	0	0
$217$	7836.46	2.45149
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5082.49	1.54699
$222$	0	0
$223$	3976.40	1.19408	0.597039	−	0.802212i	$-0.296344\pi$
$223$	3976.40	1.19408	0.597039	+	0.802212i	$0.296344\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−443.547	−0.129688	−0.0648442	−	0.997895i	$-0.520655\pi$
$227$	−443.547	−0.129688	−0.0648442	+	0.997895i	$0.520655\pi$
$228$	0	0
$229$	−695.966	−0.200833	−0.100416	−	0.994946i	$-0.532017\pi$
$229$	−695.966	−0.200833	−0.100416	+	0.994946i	$0.532017\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1469.06	−0.413052	−0.206526	−	0.978441i	$-0.566216\pi$
$233$	−1469.06	−0.413052	−0.206526	+	0.978441i	$0.566216\pi$
$234$	0	0
$235$	−451.454	−0.125317
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3589.80	−0.971569	−0.485784	−	0.874079i	$-0.661466\pi$
$239$	−3589.80	−0.971569	−0.485784	+	0.874079i	$0.661466\pi$
$240$	0	0
$241$	−3900.63	−1.04258	−0.521290	−	0.853380i	$-0.674549\pi$
$241$	−3900.63	−1.04258	−0.521290	+	0.853380i	$0.674549\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1311.66	−0.342035
$246$	0	0
$247$	−4484.33	−1.15519
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5079.37	1.27732	0.638659	−	0.769490i	$-0.279489\pi$
$251$	5079.37	1.27732	0.638659	+	0.769490i	$0.279489\pi$
$252$	0	0
$253$	6109.65	1.51822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3147.36	−0.763917	−0.381959	−	0.924179i	$-0.624750\pi$
$257$	−3147.36	−0.763917	−0.381959	+	0.924179i	$0.624750\pi$
$258$	0	0
$259$	9684.22	2.32335
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−297.262	−0.0696957	−0.0348479	−	0.999393i	$-0.511095\pi$
$263$	−297.262	−0.0696957	−0.0348479	+	0.999393i	$0.511095\pi$
$264$	0	0
$265$	3457.49	0.801480
$266$	0	0
$267$	0	0
$268$	0	0
$269$	98.5534	0.0223379	0.0111690	−	0.999938i	$-0.496445\pi$
$269$	98.5534	0.0223379	0.0111690	+	0.999938i	$0.496445\pi$
$270$	0	0
$271$	2602.14	0.583280	0.291640	−	0.956528i	$-0.405799\pi$
$271$	2602.14	0.583280	0.291640	+	0.956528i	$0.405799\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1565.69	0.343327
$276$	0	0
$277$	6368.64	1.38142	0.690712	−	0.723130i	$-0.257297\pi$
$277$	6368.64	1.38142	0.690712	+	0.723130i	$0.257297\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1875.56	−0.398173	−0.199086	−	0.979982i	$-0.563797\pi$
$281$	−1875.56	−0.398173	−0.199086	+	0.979982i	$0.563797\pi$
$282$	0	0
$283$	−2069.70	−0.434738	−0.217369	−	0.976089i	$-0.569748\pi$
$283$	−2069.70	−0.434738	−0.217369	+	0.976089i	$0.569748\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2093.30	−0.430534
$288$	0	0
$289$	2346.39	0.477588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2288.50	0.456298	0.228149	−	0.973626i	$-0.426733\pi$
$293$	2288.50	0.456298	0.228149	+	0.973626i	$0.426733\pi$
$294$	0	0
$295$	1417.59	0.279781
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5819.38	1.12556
$300$	0	0
$301$	−1002.92	−0.192052
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2160.78	0.405659
$306$	0	0
$307$	−8234.22	−1.53079	−0.765394	−	0.643563i	$-0.777456\pi$
$307$	−8234.22	−1.53079	−0.765394	+	0.643563i	$0.777456\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5957.67	−1.08626	−0.543132	−	0.839647i	$-0.682762\pi$
$311$	−5957.67	−1.08626	−0.543132	+	0.839647i	$0.682762\pi$
$312$	0	0
$313$	5132.65	0.926884	0.463442	−	0.886127i	$-0.346614\pi$
$313$	5132.65	0.926884	0.463442	+	0.886127i	$0.346614\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7685.78	1.36176	0.680878	−	0.732397i	$-0.261599\pi$
$317$	7685.78	1.36176	0.680878	+	0.732397i	$0.261599\pi$
$318$	0	0
$319$	−9973.53	−1.75050
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6405.03	−1.10336
$324$	0	0
$325$	1491.31	0.254532
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2221.47	0.372260
$330$	0	0
$331$	−11581.2	−1.92314	−0.961571	−	0.274556i	$-0.911469\pi$
$331$	−11581.2	−1.92314	−0.961571	+	0.274556i	$0.911469\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1541.80	−0.251456
$336$	0	0
$337$	6679.30	1.07966	0.539828	−	0.841775i	$-0.318489\pi$
$337$	6679.30	1.07966	0.539828	+	0.841775i	$0.318489\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19947.6	3.16780
$342$	0	0
$343$	−1984.73	−0.312436
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5100.24	−0.789035	−0.394518	−	0.918888i	$-0.629088\pi$
$347$	−5100.24	−0.789035	−0.394518	+	0.918888i	$0.629088\pi$
$348$	0	0
$349$	−2392.38	−0.366938	−0.183469	−	0.983026i	$-0.558733\pi$
$349$	−2392.38	−0.366938	−0.183469	+	0.983026i	$0.558733\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3758.39	−0.566683	−0.283341	−	0.959019i	$-0.591443\pi$
$353$	−3758.39	−0.566683	−0.283341	+	0.959019i	$0.591443\pi$
$354$	0	0
$355$	1513.37	0.226258
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−796.747	−0.117133	−0.0585664	−	0.998284i	$-0.518653\pi$
$359$	−796.747	−0.117133	−0.0585664	+	0.998284i	$0.518653\pi$
$360$	0	0
$361$	−1207.77	−0.176086
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2287.82	0.328082
$366$	0	0
$367$	−11297.1	−1.60682	−0.803408	−	0.595429i	$-0.796982\pi$
$367$	−11297.1	−1.60682	−0.803408	+	0.595429i	$0.796982\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17013.3	−2.38082
$372$	0	0
$373$	9953.17	1.38165	0.690825	−	0.723022i	$-0.257248\pi$
$373$	9953.17	1.38165	0.690825	+	0.723022i	$0.257248\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9499.68	−1.29777
$378$	0	0
$379$	−5649.80	−0.765727	−0.382864	−	0.923805i	$-0.625062\pi$
$379$	−5649.80	−0.765727	−0.382864	+	0.923805i	$0.625062\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	518.964	0.0692371	0.0346186	−	0.999401i	$-0.488978\pi$
$383$	518.964	0.0692371	0.0346186	+	0.999401i	$0.488978\pi$
$384$	0	0
$385$	−7704.30	−1.01986
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2323.03	−0.302783	−0.151391	−	0.988474i	$-0.548375\pi$
$389$	−2323.03	−0.302783	−0.151391	+	0.988474i	$0.548375\pi$
$390$	0	0
$391$	8311.90	1.07507
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4208.48	0.536080
$396$	0	0
$397$	4598.47	0.581336	0.290668	−	0.956824i	$-0.406122\pi$
$397$	4598.47	0.581336	0.290668	+	0.956824i	$0.406122\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9429.04	1.17422	0.587112	−	0.809506i	$-0.300265\pi$
$401$	9429.04	1.17422	0.587112	+	0.809506i	$0.300265\pi$
$402$	0	0
$403$	18999.8	2.34851
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24651.0	3.00222
$408$	0	0
$409$	5964.60	0.721101	0.360551	−	0.932740i	$-0.382589\pi$
$409$	5964.60	0.721101	0.360551	+	0.932740i	$0.382589\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6975.54	−0.831099
$414$	0	0
$415$	−3940.90	−0.466147
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9949.14	1.16002	0.580009	−	0.814610i	$-0.303049\pi$
$419$	9949.14	1.16002	0.580009	+	0.814610i	$0.303049\pi$
$420$	0	0
$421$	−3192.17	−0.369541	−0.184771	−	0.982782i	$-0.559154\pi$
$421$	−3192.17	−0.369541	−0.184771	+	0.982782i	$0.559154\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2130.05	0.243112
$426$	0	0
$427$	−10632.5	−1.20502
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8301.53	−0.927774	−0.463887	−	0.885894i	$-0.653546\pi$
$431$	−8301.53	−0.927774	−0.463887	+	0.885894i	$0.653546\pi$
$432$	0	0
$433$	4712.24	0.522993	0.261496	−	0.965204i	$-0.415784\pi$
$433$	4712.24	0.522993	0.261496	+	0.965204i	$0.415784\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7333.67	−0.802785
$438$	0	0
$439$	2633.89	0.286352	0.143176	−	0.989697i	$-0.454268\pi$
$439$	2633.89	0.286352	0.143176	+	0.989697i	$0.454268\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9341.37	−1.00186	−0.500928	−	0.865489i	$-0.667008\pi$
$443$	−9341.37	−1.00186	−0.500928	+	0.865489i	$0.667008\pi$
$444$	0	0
$445$	5596.75	0.596205
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16365.3	−1.72010	−0.860051	−	0.510208i	$-0.829568\pi$
$449$	−16365.3	−1.72010	−0.860051	+	0.510208i	$0.829568\pi$
$450$	0	0
$451$	−5328.45	−0.556334
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7338.26	−0.756095
$456$	0	0
$457$	10661.5	1.09130	0.545648	−	0.838014i	$-0.316283\pi$
$457$	10661.5	1.09130	0.545648	+	0.838014i	$0.316283\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2344.89	−0.236903	−0.118452	−	0.992960i	$-0.537793\pi$
$461$	−2344.89	−0.236903	−0.118452	+	0.992960i	$0.537793\pi$
$462$	0	0
$463$	3729.73	0.374374	0.187187	−	0.982324i	$-0.440063\pi$
$463$	3729.73	0.374374	0.187187	+	0.982324i	$0.440063\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4100.10	0.406274	0.203137	−	0.979150i	$-0.434886\pi$
$467$	4100.10	0.406274	0.203137	+	0.979150i	$0.434886\pi$
$468$	0	0
$469$	7586.74	0.746957
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2552.93	−0.248168
$474$	0	0
$475$	−1879.37	−0.181539
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5316.49	0.507133	0.253566	−	0.967318i	$-0.418396\pi$
$479$	5316.49	0.507133	0.253566	+	0.967318i	$0.418396\pi$
$480$	0	0
$481$	23479.8	2.22575
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5988.50	−0.560668
$486$	0	0
$487$	3958.43	0.368324	0.184162	−	0.982896i	$-0.441043\pi$
$487$	3958.43	0.368324	0.184162	+	0.982896i	$0.441043\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4051.71	0.372405	0.186203	−	0.982511i	$-0.440382\pi$
$491$	4051.71	0.372405	0.186203	+	0.982511i	$0.440382\pi$
$492$	0	0
$493$	−13568.5	−1.23954
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7446.85	−0.672106
$498$	0	0
$499$	−9102.91	−0.816638	−0.408319	−	0.912839i	$-0.633885\pi$
$499$	−9102.91	−0.816638	−0.408319	+	0.912839i	$0.633885\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4980.47	−0.441487	−0.220744	−	0.975332i	$-0.570848\pi$
$503$	−4980.47	−0.441487	−0.220744	+	0.975332i	$0.570848\pi$
$504$	0	0
$505$	7086.79	0.624471
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16384.8	1.42680	0.713402	−	0.700755i	$-0.247154\pi$
$509$	16384.8	1.42680	0.713402	+	0.700755i	$0.247154\pi$
$510$	0	0
$511$	−11257.7	−0.974578
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2076.47	−0.177670
$516$	0	0
$517$	5654.70	0.481032
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4814.03	−0.404811	−0.202406	−	0.979302i	$-0.564876\pi$
$521$	−4814.03	−0.404811	−0.202406	+	0.979302i	$0.564876\pi$
$522$	0	0
$523$	−815.794	−0.0682069	−0.0341034	−	0.999418i	$-0.510858\pi$
$523$	−815.794	−0.0682069	−0.0341034	+	0.999418i	$0.510858\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27137.7	2.24315
$528$	0	0
$529$	−2649.99	−0.217802
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5075.29	−0.412449
$534$	0	0
$535$	3831.20	0.309602
$536$	0	0
$537$	0	0
$538$	0	0
$539$	16429.2	1.31290
$540$	0	0
$541$	3305.78	0.262711	0.131355	−	0.991335i	$-0.458067\pi$
$541$	3305.78	0.262711	0.131355	+	0.991335i	$0.458067\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1746.07	−0.137235
$546$	0	0
$547$	−6784.31	−0.530304	−0.265152	−	0.964207i	$-0.585422\pi$
$547$	−6784.31	−0.530304	−0.265152	+	0.964207i	$0.585422\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11971.6	0.925606
$552$	0	0
$553$	−20708.6	−1.59244
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5708.87	−0.434278	−0.217139	−	0.976141i	$-0.569672\pi$
$557$	−5708.87	−0.434278	−0.217139	+	0.976141i	$0.569672\pi$
$558$	0	0
$559$	−2431.64	−0.183984
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22005.0	−1.64725	−0.823625	−	0.567134i	$-0.808052\pi$
$563$	−22005.0	−1.64725	−0.823625	+	0.567134i	$0.808052\pi$
$564$	0	0
$565$	−5599.69	−0.416957
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21931.0	1.61581	0.807906	−	0.589312i	$-0.200601\pi$
$569$	21931.0	1.61581	0.807906	+	0.589312i	$0.200601\pi$
$570$	0	0
$571$	9973.02	0.730925	0.365462	−	0.930826i	$-0.380911\pi$
$571$	9973.02	0.730925	0.365462	+	0.930826i	$0.380911\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2438.88	0.176884
$576$	0	0
$577$	−2721.94	−0.196388	−0.0981941	−	0.995167i	$-0.531307\pi$
$577$	−2721.94	−0.196388	−0.0981941	+	0.995167i	$0.531307\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	19392.0	1.38471
$582$	0	0
$583$	−43307.0	−3.07649
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26169.0	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$587$	26169.0	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$588$	0	0
$589$	−23943.9	−1.67503
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19409.1	−1.34408	−0.672038	−	0.740516i	$-0.734581\pi$
$593$	−19409.1	−1.34408	−0.672038	+	0.740516i	$0.734581\pi$
$594$	0	0
$595$	−10481.3	−0.722173
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15512.3	1.05812	0.529062	−	0.848583i	$-0.322544\pi$
$599$	15512.3	1.05812	0.529062	+	0.848583i	$0.322544\pi$
$600$	0	0
$601$	23446.5	1.59135	0.795675	−	0.605724i	$-0.207116\pi$
$601$	23446.5	1.59135	0.795675	+	0.605724i	$0.207116\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−12956.1	−0.870648
$606$	0	0
$607$	−16731.4	−1.11879	−0.559394	−	0.828902i	$-0.688966\pi$
$607$	−16731.4	−1.11879	−0.559394	+	0.828902i	$0.688966\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5386.05	0.356622
$612$	0	0
$613$	−5457.03	−0.359555	−0.179777	−	0.983707i	$-0.557538\pi$
$613$	−5457.03	−0.359555	−0.179777	+	0.983707i	$0.557538\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4950.96	0.323044	0.161522	−	0.986869i	$-0.448360\pi$
$617$	4950.96	0.323044	0.161522	+	0.986869i	$0.448360\pi$
$618$	0	0
$619$	6489.61	0.421389	0.210694	−	0.977552i	$-0.432428\pi$
$619$	6489.61	0.421389	0.210694	+	0.977552i	$0.432428\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−27539.9	−1.77105
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	33536.5	2.12590
$630$	0	0
$631$	−16613.1	−1.04811	−0.524054	−	0.851685i	$-0.675581\pi$
$631$	−16613.1	−1.04811	−0.524054	+	0.851685i	$0.675581\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4358.25	0.272365
$636$	0	0
$637$	15648.6	0.973346
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18780.6	−1.15724	−0.578618	−	0.815599i	$-0.696408\pi$
$641$	−18780.6	−1.15724	−0.578618	+	0.815599i	$0.696408\pi$
$642$	0	0
$643$	−10080.8	−0.618268	−0.309134	−	0.951018i	$-0.600039\pi$
$643$	−10080.8	−0.618268	−0.309134	+	0.951018i	$0.600039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14010.0	−0.851296	−0.425648	−	0.904889i	$-0.639954\pi$
$647$	−14010.0	−0.851296	−0.425648	+	0.904889i	$0.639954\pi$
$648$	0	0
$649$	−17756.1	−1.07394
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21666.8	−1.29845	−0.649226	−	0.760596i	$-0.724907\pi$
$653$	−21666.8	−1.29845	−0.649226	+	0.760596i	$0.724907\pi$
$654$	0	0
$655$	−604.825	−0.0360801
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15395.5	0.910049	0.455025	−	0.890479i	$-0.349630\pi$
$659$	15395.5	0.910049	0.455025	+	0.890479i	$0.349630\pi$
$660$	0	0
$661$	−19663.9	−1.15709	−0.578544	−	0.815651i	$-0.696379\pi$
$661$	−19663.9	−1.15709	−0.578544	+	0.815651i	$0.696379\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9247.79	0.539269
$666$	0	0
$667$	−15535.8	−0.901870
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−27064.9	−1.55712
$672$	0	0
$673$	1142.23	0.0654229	0.0327114	−	0.999465i	$-0.489586\pi$
$673$	1142.23	0.0654229	0.0327114	+	0.999465i	$0.489586\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21429.3	−1.21654	−0.608269	−	0.793731i	$-0.708136\pi$
$677$	−21429.3	−1.21654	−0.608269	+	0.793731i	$0.708136\pi$
$678$	0	0
$679$	29467.6	1.66548
$680$	0	0
$681$	0	0
$682$	0	0
$683$	246.980	0.0138366	0.00691832	−	0.999976i	$-0.497798\pi$
$683$	246.980	0.0138366	0.00691832	+	0.999976i	$0.497798\pi$
$684$	0	0
$685$	−2273.52	−0.126813
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−41249.4	−2.28081
$690$	0	0
$691$	−30236.1	−1.66459	−0.832297	−	0.554329i	$-0.812975\pi$
$691$	−30236.1	−1.66459	−0.832297	+	0.554329i	$0.812975\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11703.1	−0.638737
$696$	0	0
$697$	−7249.10	−0.393945
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30606.5	1.64906	0.824530	−	0.565818i	$-0.191440\pi$
$701$	30606.5	1.64906	0.824530	+	0.565818i	$0.191440\pi$
$702$	0	0
$703$	−29589.6	−1.58747
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−34871.9	−1.85501
$708$	0	0
$709$	−12296.3	−0.651338	−0.325669	−	0.945484i	$-0.605590\pi$
$709$	−12296.3	−0.651338	−0.325669	+	0.945484i	$0.605590\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	31072.3	1.63207
$714$	0	0
$715$	−18679.4	−0.977021
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32946.1	−1.70888	−0.854438	−	0.519553i	$-0.826098\pi$
$719$	−32946.1	−1.70888	−0.854438	+	0.519553i	$0.826098\pi$
$720$	0	0
$721$	10217.7	0.527775
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3981.28	−0.203946
$726$	0	0
$727$	11429.3	0.583066	0.291533	−	0.956561i	$-0.405835\pi$
$727$	11429.3	0.583066	0.291533	+	0.956561i	$0.405835\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3473.14	−0.175730
$732$	0	0
$733$	−22923.6	−1.15512	−0.577560	−	0.816348i	$-0.695995\pi$
$733$	−22923.6	−1.15512	−0.577560	+	0.816348i	$0.695995\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19311.9	0.965215
$738$	0	0
$739$	−4769.73	−0.237426	−0.118713	−	0.992929i	$-0.537877\pi$
$739$	−4769.73	−0.237426	−0.118713	+	0.992929i	$0.537877\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14027.9	−0.692643	−0.346322	−	0.938116i	$-0.612569\pi$
$743$	−14027.9	−0.692643	−0.346322	+	0.938116i	$0.612569\pi$
$744$	0	0
$745$	10841.8	0.533174
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−18852.2	−0.919684
$750$	0	0
$751$	−28193.1	−1.36988	−0.684941	−	0.728599i	$-0.740172\pi$
$751$	−28193.1	−1.36988	−0.684941	+	0.728599i	$0.740172\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−13445.1	−0.648102
$756$	0	0
$757$	−13350.5	−0.640992	−0.320496	−	0.947250i	$-0.603850\pi$
$757$	−13350.5	−0.640992	−0.320496	+	0.947250i	$0.603850\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−485.851	−0.0231434	−0.0115717	−	0.999933i	$-0.503683\pi$
$761$	−485.851	−0.0231434	−0.0115717	+	0.999933i	$0.503683\pi$
$762$	0	0
$763$	8591.87	0.407663
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16912.5	−0.796187
$768$	0	0
$769$	14135.9	0.662877	0.331438	−	0.943477i	$-0.392466\pi$
$769$	14135.9	0.662877	0.331438	+	0.943477i	$0.392466\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25340.8	−1.17910	−0.589552	−	0.807731i	$-0.700696\pi$
$773$	−25340.8	−1.17910	−0.589552	+	0.807731i	$0.700696\pi$
$774$	0	0
$775$	7962.76	0.369072
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6395.96	0.294171
$780$	0	0
$781$	−18955.8	−0.868492
$782$	0	0
$783$	0	0
$784$	0	0
$785$	13582.2	0.617539
$786$	0	0
$787$	−22200.9	−1.00556	−0.502780	−	0.864414i	$-0.667689\pi$
$787$	−22200.9	−1.00556	−0.502780	+	0.864414i	$0.667689\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	27554.4	1.23859
$792$	0	0
$793$	−25779.1	−1.15440
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31871.8	1.41651	0.708255	−	0.705956i	$-0.249483\pi$
$797$	31871.8	1.41651	0.708255	+	0.705956i	$0.249483\pi$
$798$	0	0
$799$	7692.96	0.340623
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−28656.1	−1.25934
$804$	0	0
$805$	−12001.0	−0.525440
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29707.7	1.29106	0.645529	−	0.763736i	$-0.276637\pi$
$809$	29707.7	1.29106	0.645529	+	0.763736i	$0.276637\pi$
$810$	0	0
$811$	34903.2	1.51124	0.755621	−	0.655009i	$-0.227335\pi$
$811$	34903.2	1.51124	0.755621	+	0.655009i	$0.227335\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1445.12	−0.0621110
$816$	0	0
$817$	3064.38	0.131223
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42164.3	1.79238	0.896191	−	0.443669i	$-0.146323\pi$
$821$	42164.3	1.79238	0.896191	+	0.443669i	$0.146323\pi$
$822$	0	0
$823$	−4253.01	−0.180134	−0.0900671	−	0.995936i	$-0.528708\pi$
$823$	−4253.01	−0.180134	−0.0900671	+	0.995936i	$0.528708\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31806.5	1.33739	0.668694	−	0.743538i	$-0.266854\pi$
$827$	31806.5	1.33739	0.668694	+	0.743538i	$0.266854\pi$
$828$	0	0
$829$	32483.0	1.36089	0.680447	−	0.732797i	$-0.261785\pi$
$829$	32483.0	1.36089	0.680447	+	0.732797i	$0.261785\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	22351.2	0.929678
$834$	0	0
$835$	−1792.79	−0.0743019
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18744.1	−0.771296	−0.385648	−	0.922646i	$-0.626022\pi$
$839$	−18744.1	−0.771296	−0.385648	+	0.922646i	$0.626022\pi$
$840$	0	0
$841$	971.913	0.0398505
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6806.94	−0.277119
$846$	0	0
$847$	63753.2	2.58629
$848$	0	0
$849$	0	0
$850$	0	0
$851$	38398.9	1.54676
$852$	0	0
$853$	30128.1	1.20934	0.604670	−	0.796476i	$-0.293305\pi$
$853$	30128.1	1.20934	0.604670	+	0.796476i	$0.293305\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18398.5	0.733348	0.366674	−	0.930349i	$-0.380496\pi$
$857$	18398.5	0.733348	0.366674	+	0.930349i	$0.380496\pi$
$858$	0	0
$859$	−18261.4	−0.725344	−0.362672	−	0.931917i	$-0.618135\pi$
$859$	−18261.4	−0.725344	−0.362672	+	0.931917i	$0.618135\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44247.0	1.74529	0.872645	−	0.488355i	$-0.162403\pi$
$863$	44247.0	1.74529	0.872645	+	0.488355i	$0.162403\pi$
$864$	0	0
$865$	−6751.69	−0.265392
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−52713.4	−2.05775
$870$	0	0
$871$	18394.4	0.715580
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3075.43	−0.118821
$876$	0	0
$877$	−21813.8	−0.839908	−0.419954	−	0.907545i	$-0.637954\pi$
$877$	−21813.8	−0.839908	−0.419954	+	0.907545i	$0.637954\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10425.7	0.398694	0.199347	−	0.979929i	$-0.436118\pi$
$881$	10425.7	0.398694	0.199347	+	0.979929i	$0.436118\pi$
$882$	0	0
$883$	3527.13	0.134425	0.0672125	−	0.997739i	$-0.478589\pi$
$883$	3527.13	0.134425	0.0672125	+	0.997739i	$0.478589\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28746.7	1.08819	0.544093	−	0.839025i	$-0.316874\pi$
$887$	28746.7	1.08819	0.544093	+	0.839025i	$0.316874\pi$
$888$	0	0
$889$	−21445.6	−0.809069
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6787.58	−0.254353
$894$	0	0
$895$	5961.44	0.222647
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−50723.1	−1.88177
$900$	0	0
$901$	−58917.1	−2.17848
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8928.11	0.327934
$906$	0	0
$907$	−32306.6	−1.18271	−0.591357	−	0.806410i	$-0.701408\pi$
$907$	−32306.6	−1.18271	−0.591357	+	0.806410i	$0.701408\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	47387.0	1.72338	0.861691	−	0.507433i	$-0.169406\pi$
$911$	47387.0	1.72338	0.861691	+	0.507433i	$0.169406\pi$
$912$	0	0
$913$	49361.9	1.78931
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2976.16	0.107177
$918$	0	0
$919$	−25628.3	−0.919914	−0.459957	−	0.887941i	$-0.652135\pi$
$919$	−25628.3	−0.919914	−0.459957	+	0.887941i	$0.652135\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18055.2	−0.643873
$924$	0	0
$925$	9840.29	0.349780
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36544.6	1.29062	0.645312	−	0.763919i	$-0.276727\pi$
$929$	36544.6	1.29062	0.645312	+	0.763919i	$0.276727\pi$
$930$	0	0
$931$	−19720.7	−0.694219
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−26680.0	−0.933188
$936$	0	0
$937$	−46590.4	−1.62438	−0.812189	−	0.583394i	$-0.801724\pi$
$937$	−46590.4	−1.62438	−0.812189	+	0.583394i	$0.801724\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13500.3	0.467691	0.233846	−	0.972274i	$-0.424869\pi$
$941$	13500.3	0.467691	0.233846	+	0.972274i	$0.424869\pi$
$942$	0	0
$943$	−8300.12	−0.286627
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35855.6	−1.23036	−0.615179	−	0.788387i	$-0.710916\pi$
$947$	−35855.6	−1.23036	−0.615179	+	0.788387i	$0.710916\pi$
$948$	0	0
$949$	−27294.7	−0.933638
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−44408.4	−1.50948	−0.754738	−	0.656027i	$-0.772236\pi$
$953$	−44408.4	−1.50948	−0.754738	+	0.656027i	$0.772236\pi$
$954$	0	0
$955$	−7864.25	−0.266473
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11187.3	0.376701
$960$	0	0
$961$	71657.8	2.40535
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1490.39	−0.0497176
$966$	0	0
$967$	−3625.68	−0.120573	−0.0602865	−	0.998181i	$-0.519201\pi$
$967$	−3625.68	−0.120573	−0.0602865	+	0.998181i	$0.519201\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7066.70	0.233554	0.116777	−	0.993158i	$-0.462744\pi$
$971$	7066.70	0.233554	0.116777	+	0.993158i	$0.462744\pi$
$972$	0	0
$973$	57587.2	1.89739
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12594.5	−0.412419	−0.206209	−	0.978508i	$-0.566113\pi$
$977$	−12594.5	−0.412419	−0.206209	+	0.978508i	$0.566113\pi$
$978$	0	0
$979$	−70102.3	−2.28854
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17290.7	−0.561025	−0.280513	−	0.959850i	$-0.590504\pi$
$983$	−17290.7	−0.561025	−0.280513	+	0.959850i	$0.590504\pi$
$984$	0	0
$985$	4409.95	0.142653
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3976.69	−0.127858
$990$	0	0
$991$	−355.437	−0.0113934	−0.00569669	−	0.999984i	$-0.501813\pi$
$991$	−355.437	−0.0113934	−0.00569669	+	0.999984i	$0.501813\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−13238.3	−0.421792
$996$	0	0
$997$	−18017.9	−0.572349	−0.286175	−	0.958177i	$-0.592384\pi$
$997$	−18017.9	−0.572349	−0.286175	+	0.958177i	$0.592384\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.k.1.6		7
3.2	odd	2		1620.4.a.l.1.6		7
9.2	odd	6		180.4.i.c.121.5	yes	14
9.4	even	3		540.4.i.c.181.2		14
9.5	odd	6		180.4.i.c.61.5	✓	14
9.7	even	3		540.4.i.c.361.2		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.c.61.5	✓	14	9.5	odd	6
180.4.i.c.121.5	yes	14	9.2	odd	6
540.4.i.c.181.2		14	9.4	even	3
540.4.i.c.361.2		14	9.7	even	3
1620.4.a.k.1.6		7	1.1	even	1	trivial
1620.4.a.l.1.6		7	3.2	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +24.6035 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +24.6035 q^{7} +62.6277 q^{11} +59.6522 q^{13} +85.2021 q^{17} -75.1746 q^{19} +97.5551 q^{23} +25.0000 q^{25} -159.251 q^{29} +318.510 q^{31} -123.017 q^{35} +393.612 q^{37} -85.0813 q^{41} -40.7635 q^{43} +90.2908 q^{47} +262.331 q^{49} -691.499 q^{53} -313.139 q^{55} -283.519 q^{59} -432.156 q^{61} -298.261 q^{65} +308.360 q^{67} -302.675 q^{71} -457.563 q^{73} +1540.86 q^{77} -841.695 q^{79} +788.180 q^{83} -426.010 q^{85} -1119.35 q^{89} +1467.65 q^{91} +375.873 q^{95} +1197.70 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) \) 7 * q - 35 * q^5 + 8 * q^7	\( 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) \) 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * q^97

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