LMFDB - Embedded newform 648.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(1,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 32 $ x^2 - x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.17891$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.3578 q^{5} -14.3578 q^{7} +O(q^{10})$	$q-13.3578 q^{5} -14.3578 q^{7} -39.0735 q^{11} -76.7891 q^{13} +62.4313 q^{17} -39.7891 q^{19} +128.505 q^{23} +53.4313 q^{25} -64.9360 q^{29} -9.13747 q^{31} +191.789 q^{35} +319.505 q^{37} -17.5782 q^{41} -450.945 q^{43} +581.597 q^{47} -136.853 q^{49} -329.450 q^{53} +521.936 q^{55} +241.853 q^{59} +497.524 q^{61} +1025.73 q^{65} -578.064 q^{67} +660.927 q^{71} +696.559 q^{73} +561.009 q^{77} +730.268 q^{79} -1090.48 q^{83} -833.945 q^{85} +317.588 q^{89} +1102.52 q^{91} +531.495 q^{95} -1485.65 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 - 6 * q^7	$ 2 q - 4 q^{5} - 6 q^{7} - 10 q^{11} - 40 q^{13} + 34 q^{17} + 34 q^{19} + 98 q^{23} + 16 q^{25} + 120 q^{29} - 200 q^{31} + 270 q^{35} + 480 q^{37} + 192 q^{41} - 334 q^{43} + 300 q^{47} - 410 q^{49} + 68 q^{53} + 794 q^{55} + 620 q^{59} + 200 q^{61} + 1370 q^{65} - 1406 q^{67} + 1390 q^{71} + 1802 q^{73} + 804 q^{77} - 334 q^{79} - 500 q^{83} - 1100 q^{85} + 90 q^{89} + 1410 q^{91} + 1222 q^{95} - 200 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 6 * q^7 - 10 * q^11 - 40 * q^13 + 34 * q^17 + 34 * q^19 + 98 * q^23 + 16 * q^25 + 120 * q^29 - 200 * q^31 + 270 * q^35 + 480 * q^37 + 192 * q^41 - 334 * q^43 + 300 * q^47 - 410 * q^49 + 68 * q^53 + 794 * q^55 + 620 * q^59 + 200 * q^61 + 1370 * q^65 - 1406 * q^67 + 1390 * q^71 + 1802 * q^73 + 804 * q^77 - 334 * q^79 - 500 * q^83 - 1100 * q^85 + 90 * q^89 + 1410 * q^91 + 1222 * q^95 - 200 * 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
$5$	−13.3578	−1.19476	−0.597380	−	0.801959i	$-0.703791\pi$
$5$	−13.3578	−1.19476	−0.597380	+	0.801959i	$0.703791\pi$
$6$	0	0
$7$	−14.3578	−0.775249	−0.387625	−	0.921817i	$-0.626704\pi$
$7$	−14.3578	−0.775249	−0.387625	+	0.921817i	$0.626704\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−39.0735	−1.07101	−0.535504	−	0.844533i	$-0.679878\pi$
$11$	−39.0735	−1.07101	−0.535504	+	0.844533i	$0.679878\pi$
$12$	0	0
$13$	−76.7891	−1.63827	−0.819133	−	0.573604i	$-0.805545\pi$
$13$	−76.7891	−1.63827	−0.819133	+	0.573604i	$0.805545\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	62.4313	0.890694	0.445347	−	0.895358i	$-0.353080\pi$
$17$	62.4313	0.890694	0.445347	+	0.895358i	$0.353080\pi$
$18$	0	0
$19$	−39.7891	−0.480434	−0.240217	−	0.970719i	$-0.577219\pi$
$19$	−39.7891	−0.480434	−0.240217	+	0.970719i	$0.577219\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	128.505	1.16500	0.582502	−	0.812829i	$-0.302074\pi$
$23$	128.505	1.16500	0.582502	+	0.812829i	$0.302074\pi$
$24$	0	0
$25$	53.4313	0.427450
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−64.9360	−0.415804	−0.207902	−	0.978150i	$-0.566663\pi$
$29$	−64.9360	−0.415804	−0.207902	+	0.978150i	$0.566663\pi$
$30$	0	0
$31$	−9.13747	−0.0529399	−0.0264700	−	0.999650i	$-0.508427\pi$
$31$	−9.13747	−0.0529399	−0.0264700	+	0.999650i	$0.508427\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	191.789	0.926236
$36$	0	0
$37$	319.505	1.41963	0.709814	−	0.704389i	$-0.248779\pi$
$37$	319.505	1.41963	0.709814	+	0.704389i	$0.248779\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−17.5782	−0.0669573	−0.0334786	−	0.999439i	$-0.510659\pi$
$41$	−17.5782	−0.0669573	−0.0334786	+	0.999439i	$0.510659\pi$
$42$	0	0
$43$	−450.945	−1.59927	−0.799634	−	0.600488i	$-0.794973\pi$
$43$	−450.945	−1.59927	−0.799634	+	0.600488i	$0.794973\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	581.597	1.80499	0.902496	−	0.430698i	$-0.141732\pi$
$47$	581.597	1.80499	0.902496	+	0.430698i	$0.141732\pi$
$48$	0	0
$49$	−136.853	−0.398989
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−329.450	−0.853839	−0.426919	−	0.904290i	$-0.640401\pi$
$53$	−329.450	−0.853839	−0.426919	+	0.904290i	$0.640401\pi$
$54$	0	0
$55$	521.936	1.27960
$56$	0	0
$57$	0	0
$58$	0	0
$59$	241.853	0.533671	0.266836	−	0.963742i	$-0.414022\pi$
$59$	241.853	0.533671	0.266836	+	0.963742i	$0.414022\pi$
$60$	0	0
$61$	497.524	1.04428	0.522142	−	0.852858i	$-0.325133\pi$
$61$	497.524	1.04428	0.522142	+	0.852858i	$0.325133\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1025.73	1.95733
$66$	0	0
$67$	−578.064	−1.05406	−0.527028	−	0.849848i	$-0.676694\pi$
$67$	−578.064	−1.05406	−0.527028	+	0.849848i	$0.676694\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	660.927	1.10475	0.552377	−	0.833594i	$-0.313721\pi$
$71$	660.927	1.10475	0.552377	+	0.833594i	$0.313721\pi$
$72$	0	0
$73$	696.559	1.11680	0.558398	−	0.829573i	$-0.311416\pi$
$73$	696.559	1.11680	0.558398	+	0.829573i	$0.311416\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	561.009	0.830298
$78$	0	0
$79$	730.268	1.04002	0.520010	−	0.854160i	$-0.325928\pi$
$79$	730.268	1.04002	0.520010	+	0.854160i	$0.325928\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1090.48	−1.44212	−0.721058	−	0.692875i	$-0.756344\pi$
$83$	−1090.48	−1.44212	−0.721058	+	0.692875i	$0.756344\pi$
$84$	0	0
$85$	−833.945	−1.06417
$86$	0	0
$87$	0	0
$88$	0	0
$89$	317.588	0.378250	0.189125	−	0.981953i	$-0.439435\pi$
$89$	317.588	0.378250	0.189125	+	0.981953i	$0.439435\pi$
$90$	0	0
$91$	1102.52	1.27006
$92$	0	0
$93$	0	0
$94$	0	0
$95$	531.495	0.574003
$96$	0	0
$97$	−1485.65	−1.55511	−0.777553	−	0.628817i	$-0.783539\pi$
$97$	−1485.65	−1.55511	−0.777553	+	0.628817i	$0.783539\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1003.27	0.988402	0.494201	−	0.869348i	$-0.335461\pi$
$101$	1003.27	0.988402	0.494201	+	0.869348i	$0.335461\pi$
$102$	0	0
$103$	−1717.60	−1.64311	−0.821553	−	0.570133i	$-0.806892\pi$
$103$	−1717.60	−1.64311	−0.821553	+	0.570133i	$0.806892\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1006.27	0.909161	0.454581	−	0.890706i	$-0.349789\pi$
$107$	1006.27	0.909161	0.454581	+	0.890706i	$0.349789\pi$
$108$	0	0
$109$	1724.94	1.51577	0.757885	−	0.652388i	$-0.226233\pi$
$109$	1724.94	1.51577	0.757885	+	0.652388i	$0.226233\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	627.019	0.521991	0.260995	−	0.965340i	$-0.415949\pi$
$113$	627.019	0.521991	0.260995	+	0.965340i	$0.415949\pi$
$114$	0	0
$115$	−1716.54	−1.39190
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−896.377	−0.690510
$120$	0	0
$121$	195.735	0.147058
$122$	0	0
$123$	0	0
$124$	0	0
$125$	956.002	0.684059
$126$	0	0
$127$	−1990.05	−1.39046	−0.695230	−	0.718788i	$-0.744697\pi$
$127$	−1990.05	−1.39046	−0.695230	+	0.718788i	$0.744697\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−156.414	−0.104321	−0.0521603	−	0.998639i	$-0.516611\pi$
$131$	−156.414	−0.104321	−0.0521603	+	0.998639i	$0.516611\pi$
$132$	0	0
$133$	571.284	0.372456
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2119.31	1.32164	0.660822	−	0.750543i	$-0.270208\pi$
$137$	2119.31	1.32164	0.660822	+	0.750543i	$0.270208\pi$
$138$	0	0
$139$	−2686.77	−1.63949	−0.819745	−	0.572729i	$-0.805885\pi$
$139$	−2686.77	−1.63949	−0.819745	+	0.572729i	$0.805885\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3000.41	1.75460
$144$	0	0
$145$	867.403	0.496785
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1551.56	0.853080	0.426540	−	0.904469i	$-0.359732\pi$
$149$	1551.56	0.853080	0.426540	+	0.904469i	$0.359732\pi$
$150$	0	0
$151$	−18.1092	−0.00975962	−0.00487981	−	0.999988i	$-0.501553\pi$
$151$	−18.1092	−0.00975962	−0.00487981	+	0.999988i	$0.501553\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	122.057	0.0632505
$156$	0	0
$157$	3741.51	1.90194	0.950971	−	0.309280i	$-0.100088\pi$
$157$	3741.51	1.90194	0.950971	+	0.309280i	$0.100088\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1845.05	−0.903168
$162$	0	0
$163$	2608.90	1.25365	0.626826	−	0.779160i	$-0.284354\pi$
$163$	2608.90	1.25365	0.626826	+	0.779160i	$0.284354\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1562.89	0.724193	0.362097	−	0.932141i	$-0.382061\pi$
$167$	1562.89	0.724193	0.362097	+	0.932141i	$0.382061\pi$
$168$	0	0
$169$	3699.56	1.68392
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1502.77	0.660425	0.330212	−	0.943907i	$-0.392880\pi$
$173$	1502.77	0.660425	0.330212	+	0.943907i	$0.392880\pi$
$174$	0	0
$175$	−767.156	−0.331380
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4126.50	−1.72307	−0.861534	−	0.507700i	$-0.830496\pi$
$179$	−4126.50	−1.72307	−0.861534	+	0.507700i	$0.830496\pi$
$180$	0	0
$181$	1582.33	0.649800	0.324900	−	0.945748i	$-0.394669\pi$
$181$	1582.33	0.649800	0.324900	+	0.945748i	$0.394669\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4267.89	−1.69611
$186$	0	0
$187$	−2439.40	−0.953941
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2575.88	−0.975833	−0.487916	−	0.872890i	$-0.662243\pi$
$191$	−2575.88	−0.975833	−0.487916	+	0.872890i	$0.662243\pi$
$192$	0	0
$193$	−1395.30	−0.520392	−0.260196	−	0.965556i	$-0.583787\pi$
$193$	−1395.30	−0.520392	−0.260196	+	0.965556i	$0.583787\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5047.62	−1.82552	−0.912761	−	0.408494i	$-0.866054\pi$
$197$	−5047.62	−1.82552	−0.912761	+	0.408494i	$0.866054\pi$
$198$	0	0
$199$	−2441.47	−0.869704	−0.434852	−	0.900502i	$-0.643199\pi$
$199$	−2441.47	−0.869704	−0.434852	+	0.900502i	$0.643199\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	932.339	0.322352
$204$	0	0
$205$	234.806	0.0799978
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1554.70	0.514548
$210$	0	0
$211$	4266.16	1.39192	0.695959	−	0.718081i	$-0.254979\pi$
$211$	4266.16	1.39192	0.695959	+	0.718081i	$0.254979\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6023.65	1.91074
$216$	0	0
$217$	131.194	0.0410416
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4794.04	−1.45919
$222$	0	0
$223$	1751.15	0.525855	0.262927	−	0.964816i	$-0.415312\pi$
$223$	1751.15	0.525855	0.262927	+	0.964816i	$0.415312\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2200.05	−0.643269	−0.321635	−	0.946864i	$-0.604232\pi$
$227$	−2200.05	−0.643269	−0.321635	+	0.946864i	$0.604232\pi$
$228$	0	0
$229$	−1191.29	−0.343767	−0.171884	−	0.985117i	$-0.554985\pi$
$229$	−1191.29	−0.343767	−0.171884	+	0.985117i	$0.554985\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2644.52	−0.743555	−0.371778	−	0.928322i	$-0.621252\pi$
$233$	−2644.52	−0.743555	−0.371778	+	0.928322i	$0.621252\pi$
$234$	0	0
$235$	−7768.87	−2.15653
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1437.52	−0.389061	−0.194530	−	0.980896i	$-0.562318\pi$
$239$	−1437.52	−0.389061	−0.194530	+	0.980896i	$0.562318\pi$
$240$	0	0
$241$	3267.08	0.873240	0.436620	−	0.899646i	$-0.356175\pi$
$241$	3267.08	0.873240	0.436620	+	0.899646i	$0.356175\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1828.06	0.476695
$246$	0	0
$247$	3055.37	0.787078
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1871.89	0.470727	0.235364	−	0.971907i	$-0.424372\pi$
$251$	1871.89	0.470727	0.235364	+	0.971907i	$0.424372\pi$
$252$	0	0
$253$	−5021.12	−1.24773
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4690.82	1.13854	0.569271	−	0.822150i	$-0.307225\pi$
$257$	4690.82	1.13854	0.569271	+	0.822150i	$0.307225\pi$
$258$	0	0
$259$	−4587.39	−1.10057
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4593.38	−1.07696	−0.538479	−	0.842639i	$-0.681001\pi$
$263$	−4593.38	−1.07696	−0.538479	+	0.842639i	$0.681001\pi$
$264$	0	0
$265$	4400.73	1.02013
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−83.0074	−0.0188143	−0.00940716	−	0.999956i	$-0.502994\pi$
$269$	−83.0074	−0.0188143	−0.00940716	+	0.999956i	$0.502994\pi$
$270$	0	0
$271$	−3464.08	−0.776487	−0.388244	−	0.921557i	$-0.626918\pi$
$271$	−3464.08	−0.776487	−0.388244	+	0.921557i	$0.626918\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2087.74	−0.457803
$276$	0	0
$277$	−1270.70	−0.275628	−0.137814	−	0.990458i	$-0.544008\pi$
$277$	−1270.70	−0.275628	−0.137814	+	0.990458i	$0.544008\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7651.80	−1.62444	−0.812221	−	0.583350i	$-0.801742\pi$
$281$	−7651.80	−1.62444	−0.812221	+	0.583350i	$0.801742\pi$
$282$	0	0
$283$	−7.45013	−0.00156489	−0.000782446	−	1.00000i	$-0.500249\pi$
$283$	−7.45013	−0.00156489	−0.000782446		1.00000i	$0.500249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	252.384	0.0519086
$288$	0	0
$289$	−1015.34	−0.206663
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2814.02	0.561082	0.280541	−	0.959842i	$-0.409486\pi$
$293$	2814.02	0.561082	0.280541	+	0.959842i	$0.409486\pi$
$294$	0	0
$295$	−3230.63	−0.637609
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9867.76	−1.90859
$300$	0	0
$301$	6474.59	1.23983
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6645.83	−1.24767
$306$	0	0
$307$	−5096.55	−0.947477	−0.473739	−	0.880666i	$-0.657096\pi$
$307$	−5096.55	−0.947477	−0.473739	+	0.880666i	$0.657096\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7784.81	−1.41941	−0.709705	−	0.704499i	$-0.751172\pi$
$311$	−7784.81	−1.41941	−0.709705	+	0.704499i	$0.751172\pi$
$312$	0	0
$313$	−246.469	−0.0445088	−0.0222544	−	0.999752i	$-0.507084\pi$
$313$	−246.469	−0.0445088	−0.0222544	+	0.999752i	$0.507084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	59.3201	0.0105102	0.00525512	−	0.999986i	$-0.498327\pi$
$317$	59.3201	0.0105102	0.00525512	+	0.999986i	$0.498327\pi$
$318$	0	0
$319$	2537.27	0.445329
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2484.08	−0.427920
$324$	0	0
$325$	−4102.94	−0.700277
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8350.46	−1.39932
$330$	0	0
$331$	4409.10	0.732163	0.366082	−	0.930583i	$-0.380699\pi$
$331$	4409.10	0.732163	0.366082	+	0.930583i	$0.380699\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7721.67	1.25934
$336$	0	0
$337$	6032.30	0.975075	0.487537	−	0.873102i	$-0.337895\pi$
$337$	6032.30	0.975075	0.487537	+	0.873102i	$0.337895\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	357.032	0.0566991
$342$	0	0
$343$	6889.64	1.08456
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6240.78	0.965484	0.482742	−	0.875763i	$-0.339641\pi$
$347$	6240.78	0.965484	0.482742	+	0.875763i	$0.339641\pi$
$348$	0	0
$349$	5634.61	0.864223	0.432111	−	0.901820i	$-0.357769\pi$
$349$	5634.61	0.864223	0.432111	+	0.901820i	$0.357769\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3441.32	0.518875	0.259437	−	0.965760i	$-0.416463\pi$
$353$	3441.32	0.518875	0.259437	+	0.965760i	$0.416463\pi$
$354$	0	0
$355$	−8828.54	−1.31992
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7808.79	−1.14800	−0.574000	−	0.818855i	$-0.694609\pi$
$359$	−7808.79	−1.14800	−0.574000	+	0.818855i	$0.694609\pi$
$360$	0	0
$361$	−5275.83	−0.769183
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9304.51	−1.33430
$366$	0	0
$367$	13608.1	1.93553	0.967763	−	0.251862i	$-0.0810429\pi$
$367$	13608.1	1.93553	0.967763	+	0.251862i	$0.0810429\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4730.18	0.661938
$372$	0	0
$373$	−3748.17	−0.520303	−0.260151	−	0.965568i	$-0.583772\pi$
$373$	−3748.17	−0.520303	−0.260151	+	0.965568i	$0.583772\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4986.37	0.681197
$378$	0	0
$379$	−10210.3	−1.38382	−0.691909	−	0.721985i	$-0.743230\pi$
$379$	−10210.3	−1.38382	−0.691909	+	0.721985i	$0.743230\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6129.49	−0.817761	−0.408880	−	0.912588i	$-0.634081\pi$
$383$	−6129.49	−0.817761	−0.408880	+	0.912588i	$0.634081\pi$
$384$	0	0
$385$	−7493.86	−0.992007
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4865.97	0.634227	0.317114	−	0.948388i	$-0.397286\pi$
$389$	4865.97	0.634227	0.317114	+	0.948388i	$0.397286\pi$
$390$	0	0
$391$	8022.71	1.03766
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9754.78	−1.24257
$396$	0	0
$397$	−438.311	−0.0554110	−0.0277055	−	0.999616i	$-0.508820\pi$
$397$	−438.311	−0.0554110	−0.0277055	+	0.999616i	$0.508820\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11753.7	1.46371	0.731857	−	0.681458i	$-0.238654\pi$
$401$	11753.7	1.46371	0.731857	+	0.681458i	$0.238654\pi$
$402$	0	0
$403$	701.658	0.0867297
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12484.2	−1.52043
$408$	0	0
$409$	11246.5	1.35967	0.679833	−	0.733367i	$-0.262052\pi$
$409$	11246.5	1.35967	0.679833	+	0.733367i	$0.262052\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3472.48	−0.413728
$414$	0	0
$415$	14566.4	1.72298
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12316.4	1.43602	0.718012	−	0.696031i	$-0.245052\pi$
$419$	12316.4	1.43602	0.718012	+	0.696031i	$0.245052\pi$
$420$	0	0
$421$	−4702.60	−0.544396	−0.272198	−	0.962241i	$-0.587751\pi$
$421$	−4702.60	−0.544396	−0.272198	+	0.962241i	$0.587751\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3335.78	0.380727
$426$	0	0
$427$	−7143.35	−0.809581
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2204.89	0.246417	0.123208	−	0.992381i	$-0.460682\pi$
$431$	2204.89	0.246417	0.123208	+	0.992381i	$0.460682\pi$
$432$	0	0
$433$	9426.46	1.04620	0.523102	−	0.852270i	$-0.324775\pi$
$433$	9426.46	1.04620	0.523102	+	0.852270i	$0.324775\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5113.08	−0.559707
$438$	0	0
$439$	−7684.59	−0.835457	−0.417728	−	0.908572i	$-0.637174\pi$
$439$	−7684.59	−0.835457	−0.417728	+	0.908572i	$0.637174\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12394.3	1.32928	0.664640	−	0.747164i	$-0.268585\pi$
$443$	12394.3	1.32928	0.664640	+	0.747164i	$0.268585\pi$
$444$	0	0
$445$	−4242.28	−0.451917
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14568.7	1.53127	0.765635	−	0.643275i	$-0.222425\pi$
$449$	14568.7	1.53127	0.765635	+	0.643275i	$0.222425\pi$
$450$	0	0
$451$	686.840	0.0717118
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−14727.3	−1.51742
$456$	0	0
$457$	−646.047	−0.0661287	−0.0330643	−	0.999453i	$-0.510527\pi$
$457$	−646.047	−0.0661287	−0.0330643	+	0.999453i	$0.510527\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−19001.0	−1.91966	−0.959831	−	0.280579i	$-0.909473\pi$
$461$	−19001.0	−1.91966	−0.959831	+	0.280579i	$0.909473\pi$
$462$	0	0
$463$	13665.7	1.37171	0.685853	−	0.727740i	$-0.259429\pi$
$463$	13665.7	1.37171	0.685853	+	0.727740i	$0.259429\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3762.83	−0.372855	−0.186427	−	0.982469i	$-0.559691\pi$
$467$	−3762.83	−0.372855	−0.186427	+	0.982469i	$0.559691\pi$
$468$	0	0
$469$	8299.74	0.817156
$470$	0	0
$471$	0	0
$472$	0	0
$473$	17620.0	1.71283
$474$	0	0
$475$	−2125.98	−0.205361
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6763.87	0.645197	0.322598	−	0.946536i	$-0.395444\pi$
$479$	6763.87	0.645197	0.322598	+	0.946536i	$0.395444\pi$
$480$	0	0
$481$	−24534.5	−2.32573
$482$	0	0
$483$	0	0
$484$	0	0
$485$	19845.1	1.85798
$486$	0	0
$487$	−1447.72	−0.134708	−0.0673538	−	0.997729i	$-0.521456\pi$
$487$	−1447.72	−0.134708	−0.0673538	+	0.997729i	$0.521456\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7838.28	0.720441	0.360220	−	0.932867i	$-0.382701\pi$
$491$	7838.28	0.720441	0.360220	+	0.932867i	$0.382701\pi$
$492$	0	0
$493$	−4054.04	−0.370354
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9489.46	−0.856460
$498$	0	0
$499$	−4245.34	−0.380857	−0.190428	−	0.981701i	$-0.560988\pi$
$499$	−4245.34	−0.380857	−0.190428	+	0.981701i	$0.560988\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17914.9	1.58804	0.794021	−	0.607891i	$-0.207984\pi$
$503$	17914.9	1.58804	0.794021	+	0.607891i	$0.207984\pi$
$504$	0	0
$505$	−13401.4	−1.18090
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14520.3	1.26444	0.632220	−	0.774789i	$-0.282144\pi$
$509$	14520.3	1.26444	0.632220	+	0.774789i	$0.282144\pi$
$510$	0	0
$511$	−10001.1	−0.865795
$512$	0	0
$513$	0	0
$514$	0	0
$515$	22943.3	1.96312
$516$	0	0
$517$	−22725.0	−1.93316
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3372.49	−0.283592	−0.141796	−	0.989896i	$-0.545288\pi$
$521$	−3372.49	−0.283592	−0.141796	+	0.989896i	$0.545288\pi$
$522$	0	0
$523$	4339.40	0.362808	0.181404	−	0.983409i	$-0.441936\pi$
$523$	4339.40	0.362808	0.181404	+	0.983409i	$0.441936\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−570.464	−0.0471533
$528$	0	0
$529$	4346.46	0.357234
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1349.81	0.109694
$534$	0	0
$535$	−13441.6	−1.08623
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5347.32	0.427320
$540$	0	0
$541$	−3831.98	−0.304528	−0.152264	−	0.988340i	$-0.548656\pi$
$541$	−3831.98	−0.304528	−0.152264	+	0.988340i	$0.548656\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−23041.4	−1.81098
$546$	0	0
$547$	13402.7	1.04764	0.523818	−	0.851830i	$-0.324507\pi$
$547$	13402.7	1.04764	0.523818	+	0.851830i	$0.324507\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2583.74	0.199766
$552$	0	0
$553$	−10485.0	−0.806274
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14810.0	−1.12661	−0.563304	−	0.826249i	$-0.690470\pi$
$557$	−14810.0	−1.12661	−0.563304	+	0.826249i	$0.690470\pi$
$558$	0	0
$559$	34627.7	2.62003
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12233.2	0.915753	0.457877	−	0.889016i	$-0.348610\pi$
$563$	12233.2	0.915753	0.457877	+	0.889016i	$0.348610\pi$
$564$	0	0
$565$	−8375.60	−0.623654
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9711.68	−0.715527	−0.357763	−	0.933812i	$-0.616461\pi$
$569$	−9711.68	−0.715527	−0.357763	+	0.933812i	$0.616461\pi$
$570$	0	0
$571$	8529.61	0.625137	0.312568	−	0.949895i	$-0.398811\pi$
$571$	8529.61	0.625137	0.312568	+	0.949895i	$0.398811\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6866.17	0.497981
$576$	0	0
$577$	15314.0	1.10491	0.552453	−	0.833544i	$-0.313692\pi$
$577$	15314.0	1.10491	0.552453	+	0.833544i	$0.313692\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15656.9	1.11800
$582$	0	0
$583$	12872.8	0.914468
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19675.1	1.38343	0.691717	−	0.722168i	$-0.256854\pi$
$587$	19675.1	1.38343	0.691717	+	0.722168i	$0.256854\pi$
$588$	0	0
$589$	363.571	0.0254341
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15635.9	1.08278	0.541391	−	0.840771i	$-0.317898\pi$
$593$	15635.9	1.08278	0.541391	+	0.840771i	$0.317898\pi$
$594$	0	0
$595$	11973.6	0.824994
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−485.189	−0.0330956	−0.0165478	−	0.999863i	$-0.505268\pi$
$599$	−485.189	−0.0330956	−0.0165478	+	0.999863i	$0.505268\pi$
$600$	0	0
$601$	−14796.2	−1.00425	−0.502123	−	0.864796i	$-0.667447\pi$
$601$	−14796.2	−1.00425	−0.502123	+	0.864796i	$0.667447\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2614.59	−0.175699
$606$	0	0
$607$	3547.98	0.237245	0.118623	−	0.992939i	$-0.462152\pi$
$607$	3547.98	0.237245	0.118623	+	0.992939i	$0.462152\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−44660.3	−2.95706
$612$	0	0
$613$	20450.9	1.34748	0.673740	−	0.738968i	$-0.264687\pi$
$613$	20450.9	1.34748	0.673740	+	0.738968i	$0.264687\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15305.2	−0.998646	−0.499323	−	0.866416i	$-0.666418\pi$
$617$	−15305.2	−0.998646	−0.499323	+	0.866416i	$0.666418\pi$
$618$	0	0
$619$	−4151.19	−0.269548	−0.134774	−	0.990876i	$-0.543031\pi$
$619$	−4151.19	−0.269548	−0.134774	+	0.990876i	$0.543031\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4559.86	−0.293238
$624$	0	0
$625$	−19449.0	−1.24474
$626$	0	0
$627$	0	0
$628$	0	0
$629$	19947.1	1.26446
$630$	0	0
$631$	25954.4	1.63745	0.818724	−	0.574187i	$-0.194682\pi$
$631$	25954.4	1.63745	0.818724	+	0.574187i	$0.194682\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	26582.7	1.66126
$636$	0	0
$637$	10508.8	0.653650
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17261.3	1.06362	0.531811	−	0.846863i	$-0.321512\pi$
$641$	17261.3	1.06362	0.531811	+	0.846863i	$0.321512\pi$
$642$	0	0
$643$	2860.36	0.175430	0.0877152	−	0.996146i	$-0.472043\pi$
$643$	2860.36	0.175430	0.0877152	+	0.996146i	$0.472043\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28384.9	1.72477	0.862384	−	0.506255i	$-0.168971\pi$
$647$	28384.9	1.72477	0.862384	+	0.506255i	$0.168971\pi$
$648$	0	0
$649$	−9450.04	−0.571566
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11036.4	0.661393	0.330696	−	0.943737i	$-0.392716\pi$
$653$	11036.4	0.661393	0.330696	+	0.943737i	$0.392716\pi$
$654$	0	0
$655$	2089.36	0.124638
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21294.9	1.25877	0.629387	−	0.777092i	$-0.283306\pi$
$659$	21294.9	1.25877	0.629387	+	0.777092i	$0.283306\pi$
$660$	0	0
$661$	12324.7	0.725229	0.362615	−	0.931939i	$-0.381884\pi$
$661$	12324.7	0.725229	0.362615	+	0.931939i	$0.381884\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7631.11	−0.444995
$666$	0	0
$667$	−8344.58	−0.484413
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−19440.0	−1.11844
$672$	0	0
$673$	3050.94	0.174748	0.0873738	−	0.996176i	$-0.472153\pi$
$673$	3050.94	0.174748	0.0873738	+	0.996176i	$0.472153\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1685.47	0.0956835	0.0478418	−	0.998855i	$-0.484766\pi$
$677$	1685.47	0.0956835	0.0478418	+	0.998855i	$0.484766\pi$
$678$	0	0
$679$	21330.7	1.20559
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34715.5	−1.94488	−0.972440	−	0.233155i	$-0.925095\pi$
$683$	−34715.5	−1.94488	−0.972440	+	0.233155i	$0.925095\pi$
$684$	0	0
$685$	−28309.4	−1.57905
$686$	0	0
$687$	0	0
$688$	0	0
$689$	25298.2	1.39882
$690$	0	0
$691$	−8295.37	−0.456687	−0.228343	−	0.973581i	$-0.573331\pi$
$691$	−8295.37	−0.456687	−0.228343	+	0.973581i	$0.573331\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	35889.4	1.95880
$696$	0	0
$697$	−1097.43	−0.0596385
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23420.5	1.26188	0.630942	−	0.775830i	$-0.282669\pi$
$701$	23420.5	1.26188	0.630942	+	0.775830i	$0.282669\pi$
$702$	0	0
$703$	−12712.8	−0.682037
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14404.7	−0.766258
$708$	0	0
$709$	−5282.67	−0.279823	−0.139912	−	0.990164i	$-0.544682\pi$
$709$	−5282.67	−0.279823	−0.139912	+	0.990164i	$0.544682\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1174.21	−0.0616752
$714$	0	0
$715$	−40079.0	−2.09632
$716$	0	0
$717$	0	0
$718$	0	0
$719$	714.975	0.0370849	0.0185425	−	0.999828i	$-0.494097\pi$
$719$	714.975	0.0370849	0.0185425	+	0.999828i	$0.494097\pi$
$720$	0	0
$721$	24660.9	1.27382
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3469.61	−0.177735
$726$	0	0
$727$	25795.5	1.31596	0.657979	−	0.753036i	$-0.271412\pi$
$727$	25795.5	1.31596	0.657979	+	0.753036i	$0.271412\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−28153.1	−1.42446
$732$	0	0
$733$	3631.55	0.182994	0.0914969	−	0.995805i	$-0.470835\pi$
$733$	3631.55	0.182994	0.0914969	+	0.995805i	$0.470835\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22587.0	1.12890
$738$	0	0
$739$	−24758.3	−1.23241	−0.616203	−	0.787588i	$-0.711330\pi$
$739$	−24758.3	−1.23241	−0.616203	+	0.787588i	$0.711330\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8845.33	−0.436748	−0.218374	−	0.975865i	$-0.570075\pi$
$743$	−8845.33	−0.436748	−0.218374	+	0.975865i	$0.570075\pi$
$744$	0	0
$745$	−20725.5	−1.01922
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−14447.9	−0.704827
$750$	0	0
$751$	1660.40	0.0806775	0.0403387	−	0.999186i	$-0.487156\pi$
$751$	1660.40	0.0806775	0.0403387	+	0.999186i	$0.487156\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	241.899	0.0116604
$756$	0	0
$757$	19937.2	0.957239	0.478619	−	0.878022i	$-0.341137\pi$
$757$	19937.2	0.957239	0.478619	+	0.878022i	$0.341137\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8443.74	−0.402215	−0.201107	−	0.979569i	$-0.564454\pi$
$761$	−8443.74	−0.402215	−0.201107	+	0.979569i	$0.564454\pi$
$762$	0	0
$763$	−24766.3	−1.17510
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18571.7	−0.874295
$768$	0	0
$769$	17107.9	0.802243	0.401122	−	0.916025i	$-0.368620\pi$
$769$	17107.9	0.802243	0.401122	+	0.916025i	$0.368620\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9821.05	0.456971	0.228486	−	0.973547i	$-0.426623\pi$
$773$	9821.05	0.456971	0.228486	+	0.973547i	$0.426623\pi$
$774$	0	0
$775$	−488.226	−0.0226292
$776$	0	0
$777$	0	0
$778$	0	0
$779$	699.419	0.0321685
$780$	0	0
$781$	−25824.7	−1.18320
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−49978.4	−2.27236
$786$	0	0
$787$	−5245.57	−0.237591	−0.118796	−	0.992919i	$-0.537903\pi$
$787$	−5245.57	−0.237591	−0.118796	+	0.992919i	$0.537903\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9002.62	−0.404673
$792$	0	0
$793$	−38204.4	−1.71082
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9218.52	0.409707	0.204854	−	0.978793i	$-0.434328\pi$
$797$	9218.52	0.409707	0.204854	+	0.978793i	$0.434328\pi$
$798$	0	0
$799$	36309.8	1.60770
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−27217.0	−1.19610
$804$	0	0
$805$	24645.8	1.07907
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16397.4	−0.712611	−0.356305	−	0.934370i	$-0.615964\pi$
$809$	−16397.4	−0.712611	−0.356305	+	0.934370i	$0.615964\pi$
$810$	0	0
$811$	8468.88	0.366686	0.183343	−	0.983049i	$-0.441308\pi$
$811$	8468.88	0.366686	0.183343	+	0.983049i	$0.441308\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−34849.3	−1.49781
$816$	0	0
$817$	17942.7	0.768342
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5233.46	0.222471	0.111236	−	0.993794i	$-0.464519\pi$
$821$	5233.46	0.222471	0.111236	+	0.993794i	$0.464519\pi$
$822$	0	0
$823$	−3124.68	−0.132344	−0.0661722	−	0.997808i	$-0.521079\pi$
$823$	−3124.68	−0.132344	−0.0661722	+	0.997808i	$0.521079\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11037.6	−0.464103	−0.232052	−	0.972703i	$-0.574544\pi$
$827$	−11037.6	−0.464103	−0.232052	+	0.972703i	$0.574544\pi$
$828$	0	0
$829$	−23941.9	−1.00306	−0.501529	−	0.865141i	$-0.667229\pi$
$829$	−23941.9	−1.00306	−0.501529	+	0.865141i	$0.667229\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8543.91	−0.355377
$834$	0	0
$835$	−20876.8	−0.865237
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−21259.4	−0.874798	−0.437399	−	0.899268i	$-0.644100\pi$
$839$	−21259.4	−0.874798	−0.437399	+	0.899268i	$0.644100\pi$
$840$	0	0
$841$	−20172.3	−0.827107
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−49418.1	−2.01187
$846$	0	0
$847$	−2810.32	−0.114007
$848$	0	0
$849$	0	0
$850$	0	0
$851$	41057.9	1.65387
$852$	0	0
$853$	8828.74	0.354385	0.177192	−	0.984176i	$-0.443299\pi$
$853$	8828.74	0.354385	0.177192	+	0.984176i	$0.443299\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38014.8	1.51524	0.757621	−	0.652695i	$-0.226362\pi$
$857$	38014.8	1.51524	0.757621	+	0.652695i	$0.226362\pi$
$858$	0	0
$859$	4917.61	0.195328	0.0976640	−	0.995219i	$-0.468863\pi$
$859$	4917.61	0.195328	0.0976640	+	0.995219i	$0.468863\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20474.1	0.807585	0.403792	−	0.914851i	$-0.367692\pi$
$863$	20474.1	0.807585	0.403792	+	0.914851i	$0.367692\pi$
$864$	0	0
$865$	−20073.7	−0.789049
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−28534.1	−1.11387
$870$	0	0
$871$	44389.0	1.72682
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−13726.1	−0.530316
$876$	0	0
$877$	14280.7	0.549858	0.274929	−	0.961464i	$-0.411346\pi$
$877$	14280.7	0.549858	0.274929	+	0.961464i	$0.411346\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12505.2	−0.478217	−0.239109	−	0.970993i	$-0.576855\pi$
$881$	−12505.2	−0.478217	−0.239109	+	0.970993i	$0.576855\pi$
$882$	0	0
$883$	−34255.9	−1.30555	−0.652776	−	0.757551i	$-0.726396\pi$
$883$	−34255.9	−1.30555	−0.652776	+	0.757551i	$0.726396\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24372.4	0.922600	0.461300	−	0.887244i	$-0.347383\pi$
$887$	24372.4	0.922600	0.461300	+	0.887244i	$0.347383\pi$
$888$	0	0
$889$	28572.8	1.07795
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−23141.2	−0.867179
$894$	0	0
$895$	55121.0	2.05865
$896$	0	0
$897$	0	0
$898$	0	0
$899$	593.350	0.0220126
$900$	0	0
$901$	−20568.0	−0.760510
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21136.5	−0.776355
$906$	0	0
$907$	−42836.1	−1.56819	−0.784095	−	0.620641i	$-0.786873\pi$
$907$	−42836.1	−1.56819	−0.784095	+	0.620641i	$0.786873\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36145.3	1.31454	0.657271	−	0.753655i	$-0.271711\pi$
$911$	36145.3	1.31454	0.657271	+	0.753655i	$0.271711\pi$
$912$	0	0
$913$	42608.8	1.54452
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2245.77	0.0808744
$918$	0	0
$919$	−23283.2	−0.835738	−0.417869	−	0.908507i	$-0.637223\pi$
$919$	−23283.2	−0.835738	−0.417869	+	0.908507i	$0.637223\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−50751.9	−1.80988
$924$	0	0
$925$	17071.5	0.606820
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39190.5	1.38407	0.692033	−	0.721866i	$-0.256715\pi$
$929$	39190.5	1.38407	0.692033	+	0.721866i	$0.256715\pi$
$930$	0	0
$931$	5445.26	0.191688
$932$	0	0
$933$	0	0
$934$	0	0
$935$	32585.1	1.13973
$936$	0	0
$937$	−36892.8	−1.28627	−0.643135	−	0.765753i	$-0.722366\pi$
$937$	−36892.8	−1.28627	−0.643135	+	0.765753i	$0.722366\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26605.8	−0.921705	−0.460853	−	0.887477i	$-0.652456\pi$
$941$	−26605.8	−0.921705	−0.460853	+	0.887477i	$0.652456\pi$
$942$	0	0
$943$	−2258.88	−0.0780055
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12841.3	−0.440641	−0.220321	−	0.975427i	$-0.570710\pi$
$947$	−12841.3	−0.440641	−0.220321	+	0.975427i	$0.570710\pi$
$948$	0	0
$949$	−53488.2	−1.82961
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−33977.5	−1.15492	−0.577460	−	0.816419i	$-0.695956\pi$
$953$	−33977.5	−1.15492	−0.577460	+	0.816419i	$0.695956\pi$
$954$	0	0
$955$	34408.1	1.16589
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−30428.7	−1.02460
$960$	0	0
$961$	−29707.5	−0.997197
$962$	0	0
$963$	0	0
$964$	0	0
$965$	18638.1	0.621744
$966$	0	0
$967$	49342.9	1.64091	0.820455	−	0.571712i	$-0.193720\pi$
$967$	49342.9	1.64091	0.820455	+	0.571712i	$0.193720\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−43375.7	−1.43356	−0.716782	−	0.697297i	$-0.754386\pi$
$971$	−43375.7	−1.43356	−0.716782	+	0.697297i	$0.754386\pi$
$972$	0	0
$973$	38576.2	1.27101
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39694.6	1.29984	0.649919	−	0.760004i	$-0.274803\pi$
$977$	39694.6	1.29984	0.649919	+	0.760004i	$0.274803\pi$
$978$	0	0
$979$	−12409.2	−0.405108
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9058.89	−0.293931	−0.146965	−	0.989142i	$-0.546951\pi$
$983$	−9058.89	−0.293931	−0.146965	+	0.989142i	$0.546951\pi$
$984$	0	0
$985$	67425.2	2.18106
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−57948.6	−1.86315
$990$	0	0
$991$	−33006.3	−1.05800	−0.529001	−	0.848621i	$-0.677433\pi$
$991$	−33006.3	−1.05800	−0.529001	+	0.848621i	$0.677433\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	32612.7	1.03909
$996$	0	0
$997$	44324.4	1.40799	0.703996	−	0.710204i	$-0.251397\pi$
$997$	44324.4	1.40799	0.703996	+	0.710204i	$0.251397\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.a.d.1.1	✓	2
3.2	odd	2		648.4.a.e.1.2	yes	2
4.3	odd	2		1296.4.a.n.1.1		2
9.2	odd	6		648.4.i.p.433.1		4
9.4	even	3		648.4.i.q.217.2		4
9.5	odd	6		648.4.i.p.217.1		4
9.7	even	3		648.4.i.q.433.2		4
12.11	even	2		1296.4.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.d.1.1	✓	2	1.1	even	1	trivial
648.4.a.e.1.2	yes	2	3.2	odd	2
648.4.i.p.217.1		4	9.5	odd	6
648.4.i.p.433.1		4	9.2	odd	6
648.4.i.q.217.2		4	9.4	even	3
648.4.i.q.433.2		4	9.7	even	3
1296.4.a.n.1.1		2	4.3	odd	2
1296.4.a.p.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q-13.3578 q^{5} -14.3578 q^{7} +O(q^{10})\)	\(q-13.3578 q^{5} -14.3578 q^{7} -39.0735 q^{11} -76.7891 q^{13} +62.4313 q^{17} -39.7891 q^{19} +128.505 q^{23} +53.4313 q^{25} -64.9360 q^{29} -9.13747 q^{31} +191.789 q^{35} +319.505 q^{37} -17.5782 q^{41} -450.945 q^{43} +581.597 q^{47} -136.853 q^{49} -329.450 q^{53} +521.936 q^{55} +241.853 q^{59} +497.524 q^{61} +1025.73 q^{65} -578.064 q^{67} +660.927 q^{71} +696.559 q^{73} +561.009 q^{77} +730.268 q^{79} -1090.48 q^{83} -833.945 q^{85} +317.588 q^{89} +1102.52 q^{91} +531.495 q^{95} -1485.65 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 - 6 * q^7	\( 2 q - 4 q^{5} - 6 q^{7} - 10 q^{11} - 40 q^{13} + 34 q^{17} + 34 q^{19} + 98 q^{23} + 16 q^{25} + 120 q^{29} - 200 q^{31} + 270 q^{35} + 480 q^{37} + 192 q^{41} - 334 q^{43} + 300 q^{47} - 410 q^{49} + 68 q^{53} + 794 q^{55} + 620 q^{59} + 200 q^{61} + 1370 q^{65} - 1406 q^{67} + 1390 q^{71} + 1802 q^{73} + 804 q^{77} - 334 q^{79} - 500 q^{83} - 1100 q^{85} + 90 q^{89} + 1410 q^{91} + 1222 q^{95} - 200 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 6 * q^7 - 10 * q^11 - 40 * q^13 + 34 * q^17 + 34 * q^19 + 98 * q^23 + 16 * q^25 + 120 * q^29 - 200 * q^31 + 270 * q^35 + 480 * q^37 + 192 * q^41 - 334 * q^43 + 300 * q^47 - 410 * q^49 + 68 * q^53 + 794 * q^55 + 620 * q^59 + 200 * q^61 + 1370 * q^65 - 1406 * q^67 + 1390 * q^71 + 1802 * q^73 + 804 * q^77 - 334 * q^79 - 500 * q^83 - 1100 * q^85 + 90 * q^89 + 1410 * q^91 + 1222 * q^95 - 200 * q^97

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