LMFDB - Embedded newform 675.4.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-1,0,19,0,0,-4] 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:	$39.8262892539$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.183945.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 3x + 18 $ x^4 - x^3 - 12x^2 + 3x + 18
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24486$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33012 q^{2} -6.23078 q^{4} +10.6979 q^{7} -18.9286 q^{8} +11.2588 q^{11} -2.74029 q^{13} +14.2294 q^{14} +24.6689 q^{16} +29.5692 q^{17} -31.1126 q^{19} +14.9755 q^{22} -116.944 q^{23} -3.64491 q^{26} -66.6560 q^{28} +108.384 q^{29} +70.7730 q^{31} +184.242 q^{32} +39.3305 q^{34} -282.289 q^{37} -41.3834 q^{38} -425.545 q^{41} -312.868 q^{43} -70.1509 q^{44} -155.549 q^{46} +193.619 q^{47} -228.556 q^{49} +17.0741 q^{52} -103.349 q^{53} -202.496 q^{56} +144.164 q^{58} -494.531 q^{59} +424.769 q^{61} +94.1365 q^{62} +47.7120 q^{64} +586.687 q^{67} -184.239 q^{68} -1139.86 q^{71} -302.564 q^{73} -375.478 q^{74} +193.856 q^{76} +120.445 q^{77} -525.354 q^{79} -566.026 q^{82} -1009.07 q^{83} -416.151 q^{86} -213.113 q^{88} -1424.57 q^{89} -29.3152 q^{91} +728.652 q^{92} +257.537 q^{94} -25.6808 q^{97} -304.007 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 19 q^{4} - 4 q^{7} + 15 q^{8} - 52 q^{11} + 2 q^{13} - 138 q^{14} - 5 q^{16} - 64 q^{17} - 46 q^{19} - 87 q^{22} + 90 q^{23} - 469 q^{26} + 110 q^{28} - 470 q^{29} - 262 q^{31} + 199 q^{32}+ \cdots - 4301 q^{98}+O(q^{100}) $ 4 * q - q^2 + 19 * q^4 - 4 * q^7 + 15 * q^8 - 52 * q^11 + 2 * q^13 - 138 * q^14 - 5 * q^16 - 64 * q^17 - 46 * q^19 - 87 * q^22 + 90 * q^23 - 469 * q^26 + 110 * q^28 - 470 * q^29 - 262 * q^31 + 199 * q^32 - 42 * q^34 + 542 * q^37 - 532 * q^38 - 698 * q^41 - 142 * q^43 - 419 * q^44 + 537 * q^46 + 542 * q^47 + 780 * q^49 - 409 * q^52 - 910 * q^53 - 2034 * q^56 + 576 * q^58 - 100 * q^59 + 74 * q^61 + 2406 * q^62 - 965 * q^64 - 928 * q^67 - 2810 * q^68 - 1622 * q^71 + 536 * q^73 + 253 * q^74 - 2068 * q^76 + 1932 * q^77 - 508 * q^79 + 1782 * q^82 - 1524 * q^83 - 3940 * q^86 - 2247 * q^88 - 756 * q^89 + 1120 * q^91 + 3645 * q^92 + 2847 * q^94 - 892 * q^97 - 4301 * q^98

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 $

$2$	1.33012	0.470268	0.235134	−	0.971963i	$-0.424447\pi$
$2$	1.33012	0.470268	0.235134	+	0.971963i	$0.424447\pi$
$3$	0	0
$3$	0	0
$4$	−6.23078	−0.778848
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.6979	0.577630	0.288815	−	0.957385i	$-0.406739\pi$
$7$	10.6979	0.577630	0.288815	+	0.957385i	$0.406739\pi$
$8$	−18.9286	−0.836535
$9$	0	0
$10$	0	0
$11$	11.2588	0.308604	0.154302	−	0.988024i	$-0.450687\pi$
$11$	11.2588	0.308604	0.154302	+	0.988024i	$0.450687\pi$
$12$	0	0
$13$	−2.74029	−0.0584630	−0.0292315	−	0.999573i	$-0.509306\pi$
$13$	−2.74029	−0.0584630	−0.0292315	+	0.999573i	$0.509306\pi$
$14$	14.2294	0.271641
$15$	0	0
$16$	24.6689	0.385452
$17$	29.5692	0.421858	0.210929	−	0.977501i	$-0.432351\pi$
$17$	29.5692	0.421858	0.210929	+	0.977501i	$0.432351\pi$
$18$	0	0
$19$	−31.1126	−0.375669	−0.187835	−	0.982201i	$-0.560147\pi$
$19$	−31.1126	−0.375669	−0.187835	+	0.982201i	$0.560147\pi$
$20$	0	0
$21$	0	0
$22$	14.9755	0.145127
$23$	−116.944	−1.06020	−0.530098	−	0.847936i	$-0.677845\pi$
$23$	−116.944	−1.06020	−0.530098	+	0.847936i	$0.677845\pi$
$24$	0	0
$25$	0	0
$26$	−3.64491	−0.0274933
$27$	0	0
$28$	−66.6560	−0.449886
$29$	108.384	0.694016	0.347008	−	0.937862i	$-0.387198\pi$
$29$	108.384	0.694016	0.347008	+	0.937862i	$0.387198\pi$
$30$	0	0
$31$	70.7730	0.410039	0.205019	−	0.978758i	$-0.434274\pi$
$31$	70.7730	0.410039	0.205019	+	0.978758i	$0.434274\pi$
$32$	184.242	1.01780
$33$	0	0
$34$	39.3305	0.198386
$35$	0	0
$36$	0	0
$37$	−282.289	−1.25427	−0.627136	−	0.778910i	$-0.715773\pi$
$37$	−282.289	−1.25427	−0.627136	+	0.778910i	$0.715773\pi$
$38$	−41.3834	−0.176665
$39$	0	0
$40$	0	0
$41$	−425.545	−1.62095	−0.810475	−	0.585773i	$-0.800791\pi$
$41$	−425.545	−1.62095	−0.810475	+	0.585773i	$0.800791\pi$
$42$	0	0
$43$	−312.868	−1.10958	−0.554789	−	0.831991i	$-0.687201\pi$
$43$	−312.868	−1.10958	−0.554789	+	0.831991i	$0.687201\pi$
$44$	−70.1509	−0.240355
$45$	0	0
$46$	−155.549	−0.498576
$47$	193.619	0.600900	0.300450	−	0.953798i	$-0.402863\pi$
$47$	193.619	0.600900	0.300450	+	0.953798i	$0.402863\pi$
$48$	0	0
$49$	−228.556	−0.666344
$50$	0	0
$51$	0	0
$52$	17.0741	0.0455338
$53$	−103.349	−0.267850	−0.133925	−	0.990991i	$-0.542758\pi$
$53$	−103.349	−0.267850	−0.133925	+	0.990991i	$0.542758\pi$
$54$	0	0
$55$	0	0
$56$	−202.496	−0.483208
$57$	0	0
$58$	144.164	0.326374
$59$	−494.531	−1.09123	−0.545614	−	0.838036i	$-0.683704\pi$
$59$	−494.531	−1.09123	−0.545614	+	0.838036i	$0.683704\pi$
$60$	0	0
$61$	424.769	0.891575	0.445787	−	0.895139i	$-0.352924\pi$
$61$	424.769	0.891575	0.445787	+	0.895139i	$0.352924\pi$
$62$	94.1365	0.192828
$63$	0	0
$64$	47.7120	0.0931875
$65$	0	0
$66$	0	0
$67$	586.687	1.06978	0.534889	−	0.844922i	$-0.320353\pi$
$67$	586.687	1.06978	0.534889	+	0.844922i	$0.320353\pi$
$68$	−184.239	−0.328563
$69$	0	0
$70$	0	0
$71$	−1139.86	−1.90531	−0.952653	−	0.304060i	$-0.901658\pi$
$71$	−1139.86	−1.90531	−0.952653	+	0.304060i	$0.901658\pi$
$72$	0	0
$73$	−302.564	−0.485102	−0.242551	−	0.970139i	$-0.577984\pi$
$73$	−302.564	−0.485102	−0.242551	+	0.970139i	$0.577984\pi$
$74$	−375.478	−0.589844
$75$	0	0
$76$	193.856	0.292589
$77$	120.445	0.178259
$78$	0	0
$79$	−525.354	−0.748189	−0.374095	−	0.927391i	$-0.622046\pi$
$79$	−525.354	−0.748189	−0.374095	+	0.927391i	$0.622046\pi$
$80$	0	0
$81$	0	0
$82$	−566.026	−0.762281
$83$	−1009.07	−1.33446	−0.667228	−	0.744854i	$-0.732519\pi$
$83$	−1009.07	−1.33446	−0.667228	+	0.744854i	$0.732519\pi$
$84$	0	0
$85$	0	0
$86$	−416.151	−0.521799
$87$	0	0
$88$	−213.113	−0.258158
$89$	−1424.57	−1.69668	−0.848340	−	0.529451i	$-0.822398\pi$
$89$	−1424.57	−1.69668	−0.848340	+	0.529451i	$0.822398\pi$
$90$	0	0
$91$	−29.3152	−0.0337700
$92$	728.652	0.825731
$93$	0	0
$94$	257.537	0.282584
$95$	0	0
$96$	0	0
$97$	−25.6808	−0.0268814	−0.0134407	−	0.999910i	$-0.504278\pi$
$97$	−25.6808	−0.0268814	−0.0134407	+	0.999910i	$0.504278\pi$
$98$	−304.007	−0.313360
$99$	0	0
$100$	0	0
$101$	−1523.35	−1.50078	−0.750392	−	0.660993i	$-0.770135\pi$
$101$	−1523.35	−1.50078	−0.750392	+	0.660993i	$0.770135\pi$
$102$	0	0
$103$	1237.75	1.18407	0.592034	−	0.805913i	$-0.298325\pi$
$103$	1237.75	1.18407	0.592034	+	0.805913i	$0.298325\pi$
$104$	51.8699	0.0489064
$105$	0	0
$106$	−137.466	−0.125961
$107$	465.820	0.420865	0.210432	−	0.977608i	$-0.432513\pi$
$107$	465.820	0.420865	0.210432	+	0.977608i	$0.432513\pi$
$108$	0	0
$109$	748.032	0.657325	0.328663	−	0.944447i	$-0.393402\pi$
$109$	748.032	0.657325	0.328663	+	0.944447i	$0.393402\pi$
$110$	0	0
$111$	0	0
$112$	263.905	0.222649
$113$	−324.843	−0.270431	−0.135215	−	0.990816i	$-0.543173\pi$
$113$	−324.843	−0.270431	−0.135215	+	0.990816i	$0.543173\pi$
$114$	0	0
$115$	0	0
$116$	−675.319	−0.540533
$117$	0	0
$118$	−657.786	−0.513170
$119$	316.327	0.243678
$120$	0	0
$121$	−1204.24	−0.904764
$122$	564.993	0.419279
$123$	0	0
$124$	−440.971	−0.319358
$125$	0	0
$126$	0	0
$127$	29.9720	0.0209416	0.0104708	−	0.999945i	$-0.496667\pi$
$127$	29.9720	0.0209416	0.0104708	+	0.999945i	$0.496667\pi$
$128$	−1410.47	−0.973978
$129$	0	0
$130$	0	0
$131$	−906.495	−0.604587	−0.302293	−	0.953215i	$-0.597752\pi$
$131$	−906.495	−0.604587	−0.302293	+	0.953215i	$0.597752\pi$
$132$	0	0
$133$	−332.838	−0.216998
$134$	780.363	0.503083
$135$	0	0
$136$	−559.704	−0.352899
$137$	2359.95	1.47171	0.735855	−	0.677139i	$-0.236780\pi$
$137$	2359.95	1.47171	0.735855	+	0.677139i	$0.236780\pi$
$138$	0	0
$139$	1709.46	1.04313	0.521563	−	0.853213i	$-0.325349\pi$
$139$	1709.46	1.04313	0.521563	+	0.853213i	$0.325349\pi$
$140$	0	0
$141$	0	0
$142$	−1516.15	−0.896005
$143$	−30.8522	−0.0180419
$144$	0	0
$145$	0	0
$146$	−402.446	−0.228128
$147$	0	0
$148$	1758.88	0.976886
$149$	−119.170	−0.0655221	−0.0327611	−	0.999463i	$-0.510430\pi$
$149$	−119.170	−0.0655221	−0.0327611	+	0.999463i	$0.510430\pi$
$150$	0	0
$151$	768.157	0.413985	0.206992	−	0.978343i	$-0.433632\pi$
$151$	768.157	0.413985	0.206992	+	0.978343i	$0.433632\pi$
$152$	588.919	0.314261
$153$	0	0
$154$	160.206	0.0838294
$155$	0	0
$156$	0	0
$157$	1999.27	1.01630	0.508151	−	0.861268i	$-0.330329\pi$
$157$	1999.27	1.01630	0.508151	+	0.861268i	$0.330329\pi$
$158$	−698.783	−0.351850
$159$	0	0
$160$	0	0
$161$	−1251.05	−0.612401
$162$	0	0
$163$	1206.73	0.579867	0.289934	−	0.957047i	$-0.406367\pi$
$163$	1206.73	0.579867	0.289934	+	0.957047i	$0.406367\pi$
$164$	2651.48	1.26247
$165$	0	0
$166$	−1342.18	−0.627552
$167$	−102.994	−0.0477239	−0.0238619	−	0.999715i	$-0.507596\pi$
$167$	−102.994	−0.0477239	−0.0238619	+	0.999715i	$0.507596\pi$
$168$	0	0
$169$	−2189.49	−0.996582
$170$	0	0
$171$	0	0
$172$	1949.41	0.864193
$173$	−308.318	−0.135497	−0.0677485	−	0.997702i	$-0.521582\pi$
$173$	−308.318	−0.135497	−0.0677485	+	0.997702i	$0.521582\pi$
$174$	0	0
$175$	0	0
$176$	277.741	0.118952
$177$	0	0
$178$	−1894.85	−0.797895
$179$	−3145.61	−1.31349	−0.656743	−	0.754115i	$-0.728066\pi$
$179$	−3145.61	−1.31349	−0.656743	+	0.754115i	$0.728066\pi$
$180$	0	0
$181$	2321.35	0.953284	0.476642	−	0.879098i	$-0.341854\pi$
$181$	2321.35	0.953284	0.476642	+	0.879098i	$0.341854\pi$
$182$	−38.9927	−0.0158809
$183$	0	0
$184$	2213.59	0.886891
$185$	0	0
$186$	0	0
$187$	332.912	0.130187
$188$	−1206.40	−0.468009
$189$	0	0
$190$	0	0
$191$	2261.68	0.856804	0.428402	−	0.903588i	$-0.359077\pi$
$191$	2261.68	0.856804	0.428402	+	0.903588i	$0.359077\pi$
$192$	0	0
$193$	4792.77	1.78752	0.893760	−	0.448545i	$-0.148058\pi$
$193$	4792.77	1.78752	0.893760	+	0.448545i	$0.148058\pi$
$194$	−34.1585	−0.0126414
$195$	0	0
$196$	1424.08	0.518981
$197$	−2262.41	−0.818223	−0.409111	−	0.912484i	$-0.634161\pi$
$197$	−2262.41	−0.818223	−0.409111	+	0.912484i	$0.634161\pi$
$198$	0	0
$199$	−2188.42	−0.779562	−0.389781	−	0.920908i	$-0.627449\pi$
$199$	−2188.42	−0.779562	−0.389781	+	0.920908i	$0.627449\pi$
$200$	0	0
$201$	0	0
$202$	−2026.24	−0.705771
$203$	1159.48	0.400884
$204$	0	0
$205$	0	0
$206$	1646.35	0.556830
$207$	0	0
$208$	−67.6000	−0.0225347
$209$	−350.289	−0.115933
$210$	0	0
$211$	−4907.42	−1.60114	−0.800570	−	0.599240i	$-0.795470\pi$
$211$	−4907.42	−1.60114	−0.800570	+	0.599240i	$0.795470\pi$
$212$	643.943	0.208614
$213$	0	0
$214$	619.596	0.197919
$215$	0	0
$216$	0	0
$217$	757.119	0.236851
$218$	994.971	0.309119
$219$	0	0
$220$	0	0
$221$	−81.0281	−0.0246631
$222$	0	0
$223$	−5787.33	−1.73789	−0.868943	−	0.494912i	$-0.835200\pi$
$223$	−5787.33	−1.73789	−0.868943	+	0.494912i	$0.835200\pi$
$224$	1970.99	0.587912
$225$	0	0
$226$	−432.080	−0.127175
$227$	−4358.48	−1.27437	−0.637186	−	0.770710i	$-0.719902\pi$
$227$	−4358.48	−1.27437	−0.637186	+	0.770710i	$0.719902\pi$
$228$	0	0
$229$	3944.00	1.13811	0.569054	−	0.822300i	$-0.307310\pi$
$229$	3944.00	1.13811	0.569054	+	0.822300i	$0.307310\pi$
$230$	0	0
$231$	0	0
$232$	−2051.57	−0.580569
$233$	5530.41	1.55497	0.777487	−	0.628899i	$-0.216494\pi$
$233$	5530.41	1.55497	0.777487	+	0.628899i	$0.216494\pi$
$234$	0	0
$235$	0	0
$236$	3081.32	0.849901
$237$	0	0
$238$	420.752	0.114594
$239$	−3823.19	−1.03473	−0.517367	−	0.855763i	$-0.673088\pi$
$239$	−3823.19	−1.03473	−0.517367	+	0.855763i	$0.673088\pi$
$240$	0	0
$241$	3976.26	1.06280	0.531398	−	0.847122i	$-0.321667\pi$
$241$	3976.26	1.06280	0.531398	+	0.847122i	$0.321667\pi$
$242$	−1601.78	−0.425481
$243$	0	0
$244$	−2646.64	−0.694401
$245$	0	0
$246$	0	0
$247$	85.2574	0.0219628
$248$	−1339.64	−0.343012
$249$	0	0
$250$	0	0
$251$	1758.83	0.442297	0.221148	−	0.975240i	$-0.429019\pi$
$251$	1758.83	0.442297	0.221148	+	0.975240i	$0.429019\pi$
$252$	0	0
$253$	−1316.64	−0.327181
$254$	39.8663	0.00984817
$255$	0	0
$256$	−2257.79	−0.551218
$257$	−3396.20	−0.824316	−0.412158	−	0.911112i	$-0.635225\pi$
$257$	−3396.20	−0.824316	−0.412158	+	0.911112i	$0.635225\pi$
$258$	0	0
$259$	−3019.89	−0.724504
$260$	0	0
$261$	0	0
$262$	−1205.75	−0.284318
$263$	4664.71	1.09368	0.546841	−	0.837236i	$-0.315830\pi$
$263$	4664.71	1.09368	0.546841	+	0.837236i	$0.315830\pi$
$264$	0	0
$265$	0	0
$266$	−442.714	−0.102047
$267$	0	0
$268$	−3655.52	−0.833195
$269$	214.904	0.0487099	0.0243549	−	0.999703i	$-0.492247\pi$
$269$	214.904	0.0487099	0.0243549	+	0.999703i	$0.492247\pi$
$270$	0	0
$271$	3260.69	0.730896	0.365448	−	0.930832i	$-0.380916\pi$
$271$	3260.69	0.730896	0.365448	+	0.930832i	$0.380916\pi$
$272$	729.440	0.162606
$273$	0	0
$274$	3139.02	0.692098
$275$	0	0
$276$	0	0
$277$	−1992.86	−0.432273	−0.216136	−	0.976363i	$-0.569346\pi$
$277$	−1992.86	−0.432273	−0.216136	+	0.976363i	$0.569346\pi$
$278$	2273.78	0.490549
$279$	0	0
$280$	0	0
$281$	3029.98	0.643251	0.321626	−	0.946867i	$-0.395771\pi$
$281$	3029.98	0.643251	0.321626	+	0.946867i	$0.395771\pi$
$282$	0	0
$283$	4950.46	1.03984	0.519920	−	0.854215i	$-0.325962\pi$
$283$	4950.46	1.03984	0.519920	+	0.854215i	$0.325962\pi$
$284$	7102.23	1.48394
$285$	0	0
$286$	−41.0372	−0.00848454
$287$	−4552.42	−0.936309
$288$	0	0
$289$	−4038.66	−0.822036
$290$	0	0
$291$	0	0
$292$	1885.21	0.377821
$293$	8087.47	1.61254	0.806272	−	0.591546i	$-0.201482\pi$
$293$	8087.47	1.61254	0.806272	+	0.591546i	$0.201482\pi$
$294$	0	0
$295$	0	0
$296$	5343.35	1.04924
$297$	0	0
$298$	−158.510	−0.0308130
$299$	320.460	0.0619822
$300$	0	0
$301$	−3347.01	−0.640926
$302$	1021.74	0.194684
$303$	0	0
$304$	−767.514	−0.144802
$305$	0	0
$306$	0	0
$307$	−272.579	−0.0506740	−0.0253370	−	0.999679i	$-0.508066\pi$
$307$	−272.579	−0.0506740	−0.0253370	+	0.999679i	$0.508066\pi$
$308$	−750.464	−0.138836
$309$	0	0
$310$	0	0
$311$	9093.68	1.65806	0.829028	−	0.559208i	$-0.188895\pi$
$311$	9093.68	1.65806	0.829028	+	0.559208i	$0.188895\pi$
$312$	0	0
$313$	−8213.36	−1.48322	−0.741608	−	0.670834i	$-0.765936\pi$
$313$	−8213.36	−1.48322	−0.741608	+	0.670834i	$0.765936\pi$
$314$	2659.27	0.477935
$315$	0	0
$316$	3273.37	0.582726
$317$	−4859.27	−0.860959	−0.430480	−	0.902600i	$-0.641656\pi$
$317$	−4859.27	−0.860959	−0.430480	+	0.902600i	$0.641656\pi$
$318$	0	0
$319$	1220.27	0.214176
$320$	0	0
$321$	0	0
$322$	−1664.04	−0.287992
$323$	−919.973	−0.158479
$324$	0	0
$325$	0	0
$326$	1605.09	0.272693
$327$	0	0
$328$	8054.99	1.35598
$329$	2071.31	0.347098
$330$	0	0
$331$	10784.9	1.79091	0.895454	−	0.445155i	$-0.146851\pi$
$331$	10784.9	1.79091	0.895454	+	0.445155i	$0.146851\pi$
$332$	6287.30	1.03934
$333$	0	0
$334$	−136.994	−0.0224430
$335$	0	0
$336$	0	0
$337$	4752.67	0.768232	0.384116	−	0.923285i	$-0.374506\pi$
$337$	4752.67	0.768232	0.384116	+	0.923285i	$0.374506\pi$
$338$	−2912.28	−0.468661
$339$	0	0
$340$	0	0
$341$	796.816	0.126540
$342$	0	0
$343$	−6114.42	−0.962530
$344$	5922.16	0.928202
$345$	0	0
$346$	−410.099	−0.0637199
$347$	−10303.2	−1.59397	−0.796984	−	0.604000i	$-0.793573\pi$
$347$	−10303.2	−1.59397	−0.796984	+	0.604000i	$0.793573\pi$
$348$	0	0
$349$	−9058.09	−1.38931	−0.694654	−	0.719344i	$-0.744442\pi$
$349$	−9058.09	−1.38931	−0.694654	+	0.719344i	$0.744442\pi$
$350$	0	0
$351$	0	0
$352$	2074.33	0.314097
$353$	1275.21	0.192273	0.0961366	−	0.995368i	$-0.469351\pi$
$353$	1275.21	0.192273	0.0961366	+	0.995368i	$0.469351\pi$
$354$	0	0
$355$	0	0
$356$	8876.21	1.32146
$357$	0	0
$358$	−4184.03	−0.617690
$359$	2556.89	0.375899	0.187949	−	0.982179i	$-0.439816\pi$
$359$	2556.89	0.375899	0.187949	+	0.982179i	$0.439816\pi$
$360$	0	0
$361$	−5891.01	−0.858873
$362$	3087.67	0.448299
$363$	0	0
$364$	182.657	0.0263017
$365$	0	0
$366$	0	0
$367$	−754.620	−0.107332	−0.0536660	−	0.998559i	$-0.517091\pi$
$367$	−754.620	−0.107332	−0.0536660	+	0.998559i	$0.517091\pi$
$368$	−2884.88	−0.408655
$369$	0	0
$370$	0	0
$371$	−1105.61	−0.154718
$372$	0	0
$373$	6177.49	0.857529	0.428765	−	0.903416i	$-0.358949\pi$
$373$	6177.49	0.857529	0.428765	+	0.903416i	$0.358949\pi$
$374$	442.813	0.0612227
$375$	0	0
$376$	−3664.95	−0.502674
$377$	−297.004	−0.0405743
$378$	0	0
$379$	8146.93	1.10417	0.552084	−	0.833789i	$-0.313833\pi$
$379$	8146.93	1.10417	0.552084	+	0.833789i	$0.313833\pi$
$380$	0	0
$381$	0	0
$382$	3008.30	0.402927
$383$	4710.88	0.628498	0.314249	−	0.949341i	$-0.398247\pi$
$383$	4710.88	0.628498	0.314249	+	0.949341i	$0.398247\pi$
$384$	0	0
$385$	0	0
$386$	6374.96	0.840614
$387$	0	0
$388$	160.012	0.0209365
$389$	−9508.93	−1.23939	−0.619694	−	0.784844i	$-0.712743\pi$
$389$	−9508.93	−1.23939	−0.619694	+	0.784844i	$0.712743\pi$
$390$	0	0
$391$	−3457.94	−0.447252
$392$	4326.25	0.557420
$393$	0	0
$394$	−3009.27	−0.384784
$395$	0	0
$396$	0	0
$397$	−9615.34	−1.21557	−0.607783	−	0.794103i	$-0.707941\pi$
$397$	−9615.34	−1.21557	−0.607783	+	0.794103i	$0.707941\pi$
$398$	−2910.86	−0.366603
$399$	0	0
$400$	0	0
$401$	−481.837	−0.0600045	−0.0300022	−	0.999550i	$-0.509551\pi$
$401$	−481.837	−0.0600045	−0.0300022	+	0.999550i	$0.509551\pi$
$402$	0	0
$403$	−193.938	−0.0239721
$404$	9491.68	1.16888
$405$	0	0
$406$	1542.25	0.188523
$407$	−3178.22	−0.387073
$408$	0	0
$409$	12331.6	1.49085	0.745423	−	0.666592i	$-0.232248\pi$
$409$	12331.6	1.49085	0.745423	+	0.666592i	$0.232248\pi$
$410$	0	0
$411$	0	0
$412$	−7712.15	−0.922209
$413$	−5290.42	−0.630326
$414$	0	0
$415$	0	0
$416$	−504.875	−0.0595037
$417$	0	0
$418$	−465.926	−0.0545196
$419$	553.776	0.0645674	0.0322837	−	0.999479i	$-0.489722\pi$
$419$	553.776	0.0645674	0.0322837	+	0.999479i	$0.489722\pi$
$420$	0	0
$421$	−522.671	−0.0605070	−0.0302535	−	0.999542i	$-0.509631\pi$
$421$	−522.671	−0.0605070	−0.0302535	+	0.999542i	$0.509631\pi$
$422$	−6527.45	−0.752965
$423$	0	0
$424$	1956.25	0.224066
$425$	0	0
$426$	0	0
$427$	4544.11	0.515000
$428$	−2902.42	−0.327789
$429$	0	0
$430$	0	0
$431$	−1294.01	−0.144618	−0.0723089	−	0.997382i	$-0.523037\pi$
$431$	−1294.01	−0.144618	−0.0723089	+	0.997382i	$0.523037\pi$
$432$	0	0
$433$	179.997	0.0199771	0.00998856	−	0.999950i	$-0.496820\pi$
$433$	179.997	0.0199771	0.00998856	+	0.999950i	$0.496820\pi$
$434$	1007.06	0.111383
$435$	0	0
$436$	−4660.82	−0.511956
$437$	3638.43	0.398283
$438$	0	0
$439$	−6068.09	−0.659714	−0.329857	−	0.944031i	$-0.607000\pi$
$439$	−6068.09	−0.659714	−0.329857	+	0.944031i	$0.607000\pi$
$440$	0	0
$441$	0	0
$442$	−107.777	−0.0115983
$443$	15649.3	1.67838	0.839190	−	0.543838i	$-0.183029\pi$
$443$	15649.3	1.67838	0.839190	+	0.543838i	$0.183029\pi$
$444$	0	0
$445$	0	0
$446$	−7697.84	−0.817272
$447$	0	0
$448$	510.416	0.0538279
$449$	−7274.81	−0.764631	−0.382316	−	0.924032i	$-0.624873\pi$
$449$	−7274.81	−0.764631	−0.382316	+	0.924032i	$0.624873\pi$
$450$	0	0
$451$	−4791.11	−0.500232
$452$	2024.03	0.210624
$453$	0	0
$454$	−5797.29	−0.599296
$455$	0	0
$456$	0	0
$457$	4356.56	0.445933	0.222967	−	0.974826i	$-0.428426\pi$
$457$	4356.56	0.445933	0.222967	+	0.974826i	$0.428426\pi$
$458$	5245.98	0.535216
$459$	0	0
$460$	0	0
$461$	−9318.22	−0.941416	−0.470708	−	0.882289i	$-0.656001\pi$
$461$	−9318.22	−0.941416	−0.470708	+	0.882289i	$0.656001\pi$
$462$	0	0
$463$	18699.8	1.87701	0.938503	−	0.345270i	$-0.112213\pi$
$463$	18699.8	1.87701	0.938503	+	0.345270i	$0.112213\pi$
$464$	2673.72	0.267510
$465$	0	0
$466$	7356.10	0.731255
$467$	5345.39	0.529669	0.264834	−	0.964294i	$-0.414683\pi$
$467$	5345.39	0.529669	0.264834	+	0.964294i	$0.414683\pi$
$468$	0	0
$469$	6276.29	0.617936
$470$	0	0
$471$	0	0
$472$	9360.81	0.912852
$473$	−3522.50	−0.342420
$474$	0	0
$475$	0	0
$476$	−1970.96	−0.189788
$477$	0	0
$478$	−5085.30	−0.486603
$479$	−16930.5	−1.61498	−0.807489	−	0.589883i	$-0.799174\pi$
$479$	−16930.5	−1.61498	−0.807489	+	0.589883i	$0.799174\pi$
$480$	0	0
$481$	773.553	0.0733285
$482$	5288.90	0.499799
$483$	0	0
$484$	7503.36	0.704673
$485$	0	0
$486$	0	0
$487$	−3052.84	−0.284060	−0.142030	−	0.989862i	$-0.545363\pi$
$487$	−3052.84	−0.284060	−0.142030	+	0.989862i	$0.545363\pi$
$488$	−8040.29	−0.745834
$489$	0	0
$490$	0	0
$491$	−7591.68	−0.697775	−0.348888	−	0.937165i	$-0.613440\pi$
$491$	−7591.68	−0.697775	−0.348888	+	0.937165i	$0.613440\pi$
$492$	0	0
$493$	3204.84	0.292776
$494$	113.403	0.0103284
$495$	0	0
$496$	1745.89	0.158050
$497$	−12194.1	−1.10056
$498$	0	0
$499$	−3688.97	−0.330944	−0.165472	−	0.986215i	$-0.552915\pi$
$499$	−3688.97	−0.330944	−0.165472	+	0.986215i	$0.552915\pi$
$500$	0	0
$501$	0	0
$502$	2339.46	0.207998
$503$	2212.28	0.196105	0.0980523	−	0.995181i	$-0.468739\pi$
$503$	2212.28	0.196105	0.0980523	+	0.995181i	$0.468739\pi$
$504$	0	0
$505$	0	0
$506$	−1751.29	−0.153863
$507$	0	0
$508$	−186.749	−0.0163103
$509$	−13842.8	−1.20544	−0.602722	−	0.797952i	$-0.705917\pi$
$509$	−13842.8	−1.20544	−0.602722	+	0.797952i	$0.705917\pi$
$510$	0	0
$511$	−3236.79	−0.280209
$512$	8280.64	0.714758
$513$	0	0
$514$	−4517.35	−0.387649
$515$	0	0
$516$	0	0
$517$	2179.91	0.185440
$518$	−4016.81	−0.340711
$519$	0	0
$520$	0	0
$521$	−21683.4	−1.82335	−0.911677	−	0.410908i	$-0.865212\pi$
$521$	−21683.4	−1.82335	−0.911677	+	0.410908i	$0.865212\pi$
$522$	0	0
$523$	−2021.57	−0.169020	−0.0845098	−	0.996423i	$-0.526932\pi$
$523$	−2021.57	−0.169020	−0.0845098	+	0.996423i	$0.526932\pi$
$524$	5648.17	0.470881
$525$	0	0
$526$	6204.62	0.514324
$527$	2092.70	0.172978
$528$	0	0
$529$	1508.89	0.124015
$530$	0	0
$531$	0	0
$532$	2073.84	0.169008
$533$	1166.12	0.0947657
$534$	0	0
$535$	0	0
$536$	−11105.2	−0.894908
$537$	0	0
$538$	285.848	0.0229067
$539$	−2573.26	−0.205636
$540$	0	0
$541$	−9538.79	−0.758049	−0.379025	−	0.925387i	$-0.623740\pi$
$541$	−9538.79	−0.758049	−0.379025	+	0.925387i	$0.623740\pi$
$542$	4337.11	0.343717
$543$	0	0
$544$	5447.88	0.429367
$545$	0	0
$546$	0	0
$547$	7163.31	0.559929	0.279964	−	0.960010i	$-0.409677\pi$
$547$	7163.31	0.559929	0.279964	+	0.960010i	$0.409677\pi$
$548$	−14704.3	−1.14624
$549$	0	0
$550$	0	0
$551$	−3372.12	−0.260720
$552$	0	0
$553$	−5620.16	−0.432176
$554$	−2650.75	−0.203284
$555$	0	0
$556$	−10651.3	−0.812436
$557$	14886.5	1.13243	0.566214	−	0.824258i	$-0.308408\pi$
$557$	14886.5	1.13243	0.566214	+	0.824258i	$0.308408\pi$
$558$	0	0
$559$	857.348	0.0648693
$560$	0	0
$561$	0	0
$562$	4030.24	0.302501
$563$	−8134.04	−0.608897	−0.304448	−	0.952529i	$-0.598472\pi$
$563$	−8134.04	−0.608897	−0.304448	+	0.952529i	$0.598472\pi$
$564$	0	0
$565$	0	0
$566$	6584.71	0.489003
$567$	0	0
$568$	21576.0	1.59386
$569$	−10031.8	−0.739115	−0.369557	−	0.929208i	$-0.620491\pi$
$569$	−10031.8	−0.739115	−0.369557	+	0.929208i	$0.620491\pi$
$570$	0	0
$571$	−2507.05	−0.183742	−0.0918712	−	0.995771i	$-0.529285\pi$
$571$	−2507.05	−0.183742	−0.0918712	+	0.995771i	$0.529285\pi$
$572$	192.234	0.0140519
$573$	0	0
$574$	−6055.26	−0.440316
$575$	0	0
$576$	0	0
$577$	−5378.18	−0.388036	−0.194018	−	0.980998i	$-0.562152\pi$
$577$	−5378.18	−0.388036	−0.194018	+	0.980998i	$0.562152\pi$
$578$	−5371.90	−0.386577
$579$	0	0
$580$	0	0
$581$	−10794.9	−0.770821
$582$	0	0
$583$	−1163.58	−0.0826594
$584$	5727.13	0.405805
$585$	0	0
$586$	10757.3	0.758328
$587$	6153.77	0.432697	0.216349	−	0.976316i	$-0.430585\pi$
$587$	6153.77	0.432697	0.216349	+	0.976316i	$0.430585\pi$
$588$	0	0
$589$	−2201.93	−0.154039
$590$	0	0
$591$	0	0
$592$	−6963.77	−0.483461
$593$	−14278.1	−0.988753	−0.494376	−	0.869248i	$-0.664604\pi$
$593$	−14278.1	−0.988753	−0.494376	+	0.869248i	$0.664604\pi$
$594$	0	0
$595$	0	0
$596$	742.523	0.0510318
$597$	0	0
$598$	426.250	0.0291483
$599$	−2071.22	−0.141282	−0.0706410	−	0.997502i	$-0.522504\pi$
$599$	−2071.22	−0.141282	−0.0706410	+	0.997502i	$0.522504\pi$
$600$	0	0
$601$	12307.3	0.835319	0.417660	−	0.908604i	$-0.362850\pi$
$601$	12307.3	0.835319	0.417660	+	0.908604i	$0.362850\pi$
$602$	−4451.93	−0.301407
$603$	0	0
$604$	−4786.22	−0.322431
$605$	0	0
$606$	0	0
$607$	−22963.7	−1.53553	−0.767765	−	0.640732i	$-0.778631\pi$
$607$	−22963.7	−1.53553	−0.767765	+	0.640732i	$0.778631\pi$
$608$	−5732.23	−0.382356
$609$	0	0
$610$	0	0
$611$	−530.573	−0.0351304
$612$	0	0
$613$	−12746.7	−0.839862	−0.419931	−	0.907556i	$-0.637946\pi$
$613$	−12746.7	−0.839862	−0.419931	+	0.907556i	$0.637946\pi$
$614$	−362.563	−0.0238304
$615$	0	0
$616$	−2279.85	−0.149120
$617$	13959.6	0.910844	0.455422	−	0.890276i	$-0.349488\pi$
$617$	13959.6	0.910844	0.455422	+	0.890276i	$0.349488\pi$
$618$	0	0
$619$	15542.0	1.00918	0.504591	−	0.863358i	$-0.331643\pi$
$619$	15542.0	1.00918	0.504591	+	0.863358i	$0.331643\pi$
$620$	0	0
$621$	0	0
$622$	12095.7	0.779730
$623$	−15239.9	−0.980053
$624$	0	0
$625$	0	0
$626$	−10924.7	−0.697509
$627$	0	0
$628$	−12457.0	−0.791545
$629$	−8347.06	−0.529124
$630$	0	0
$631$	−12297.5	−0.775844	−0.387922	−	0.921692i	$-0.626807\pi$
$631$	−12297.5	−0.775844	−0.387922	+	0.921692i	$0.626807\pi$
$632$	9944.24	0.625887
$633$	0	0
$634$	−6463.41	−0.404882
$635$	0	0
$636$	0	0
$637$	626.309	0.0389565
$638$	1623.11	0.100720
$639$	0	0
$640$	0	0
$641$	−16523.4	−1.01815	−0.509075	−	0.860722i	$-0.670013\pi$
$641$	−16523.4	−1.01815	−0.509075	+	0.860722i	$0.670013\pi$
$642$	0	0
$643$	−25293.3	−1.55128	−0.775639	−	0.631177i	$-0.782572\pi$
$643$	−25293.3	−1.55128	−0.775639	+	0.631177i	$0.782572\pi$
$644$	7795.02	0.476967
$645$	0	0
$646$	−1223.67	−0.0745276
$647$	1442.91	0.0876763	0.0438381	−	0.999039i	$-0.486041\pi$
$647$	1442.91	0.0876763	0.0438381	+	0.999039i	$0.486041\pi$
$648$	0	0
$649$	−5567.81	−0.336757
$650$	0	0
$651$	0	0
$652$	−7518.87	−0.451628
$653$	−9042.22	−0.541883	−0.270941	−	0.962596i	$-0.587335\pi$
$653$	−9042.22	−0.541883	−0.270941	+	0.962596i	$0.587335\pi$
$654$	0	0
$655$	0	0
$656$	−10497.7	−0.624799
$657$	0	0
$658$	2755.09	0.163229
$659$	10084.1	0.596085	0.298042	−	0.954553i	$-0.403666\pi$
$659$	10084.1	0.596085	0.298042	+	0.954553i	$0.403666\pi$
$660$	0	0
$661$	4181.32	0.246043	0.123022	−	0.992404i	$-0.460742\pi$
$661$	4181.32	0.246043	0.123022	+	0.992404i	$0.460742\pi$
$662$	14345.2	0.842207
$663$	0	0
$664$	19100.3	1.11632
$665$	0	0
$666$	0	0
$667$	−12674.9	−0.735793
$668$	641.731	0.0371696
$669$	0	0
$670$	0	0
$671$	4782.37	0.275143
$672$	0	0
$673$	21108.5	1.20902	0.604512	−	0.796596i	$-0.293368\pi$
$673$	21108.5	1.20902	0.604512	+	0.796596i	$0.293368\pi$
$674$	6321.61	0.361275
$675$	0	0
$676$	13642.2	0.776186
$677$	19793.4	1.12367	0.561834	−	0.827250i	$-0.310096\pi$
$677$	19793.4	1.12367	0.561834	+	0.827250i	$0.310096\pi$
$678$	0	0
$679$	−274.730	−0.0155275
$680$	0	0
$681$	0	0
$682$	1059.86	0.0595075
$683$	7226.49	0.404852	0.202426	−	0.979298i	$-0.435117\pi$
$683$	7226.49	0.404852	0.202426	+	0.979298i	$0.435117\pi$
$684$	0	0
$685$	0	0
$686$	−8132.91	−0.452647
$687$	0	0
$688$	−7718.11	−0.427689
$689$	283.205	0.0156593
$690$	0	0
$691$	−1152.43	−0.0634451	−0.0317225	−	0.999497i	$-0.510099\pi$
$691$	−1152.43	−0.0634451	−0.0317225	+	0.999497i	$0.510099\pi$
$692$	1921.06	0.105531
$693$	0	0
$694$	−13704.5	−0.749593
$695$	0	0
$696$	0	0
$697$	−12583.0	−0.683810
$698$	−12048.3	−0.653347
$699$	0	0
$700$	0	0
$701$	−13364.5	−0.720073	−0.360036	−	0.932938i	$-0.617236\pi$
$701$	−13364.5	−0.720073	−0.360036	+	0.932938i	$0.617236\pi$
$702$	0	0
$703$	8782.74	0.471191
$704$	537.178	0.0287580
$705$	0	0
$706$	1696.18	0.0904200
$707$	−16296.6	−0.866898
$708$	0	0
$709$	−15588.0	−0.825696	−0.412848	−	0.910800i	$-0.635466\pi$
$709$	−15588.0	−0.825696	−0.412848	+	0.910800i	$0.635466\pi$
$710$	0	0
$711$	0	0
$712$	26965.2	1.41933
$713$	−8276.47	−0.434721
$714$	0	0
$715$	0	0
$716$	19599.6	1.02301
$717$	0	0
$718$	3400.97	0.176773
$719$	−1086.42	−0.0563513	−0.0281757	−	0.999603i	$-0.508970\pi$
$719$	−1086.42	−0.0563513	−0.0281757	+	0.999603i	$0.508970\pi$
$720$	0	0
$721$	13241.3	0.683953
$722$	−7835.74	−0.403900
$723$	0	0
$724$	−14463.8	−0.742463
$725$	0	0
$726$	0	0
$727$	27592.3	1.40762	0.703811	−	0.710387i	$-0.251480\pi$
$727$	27592.3	1.40762	0.703811	+	0.710387i	$0.251480\pi$
$728$	554.897	0.0282498
$729$	0	0
$730$	0	0
$731$	−9251.24	−0.468084
$732$	0	0
$733$	−28759.0	−1.44916	−0.724582	−	0.689189i	$-0.757967\pi$
$733$	−28759.0	−1.44916	−0.724582	+	0.689189i	$0.757967\pi$
$734$	−1003.73	−0.0504748
$735$	0	0
$736$	−21546.0	−1.07907
$737$	6605.36	0.330138
$738$	0	0
$739$	113.264	0.00563800	0.00281900	−	0.999996i	$-0.499103\pi$
$739$	113.264	0.00563800	0.00281900	+	0.999996i	$0.499103\pi$
$740$	0	0
$741$	0	0
$742$	−1470.59	−0.0727589
$743$	37767.7	1.86482	0.932411	−	0.361399i	$-0.117701\pi$
$743$	37767.7	1.86482	0.932411	+	0.361399i	$0.117701\pi$
$744$	0	0
$745$	0	0
$746$	8216.80	0.403269
$747$	0	0
$748$	−2074.30	−0.101396
$749$	4983.27	0.243104
$750$	0	0
$751$	26372.9	1.28144	0.640720	−	0.767774i	$-0.278636\pi$
$751$	26372.9	1.28144	0.640720	+	0.767774i	$0.278636\pi$
$752$	4776.38	0.231618
$753$	0	0
$754$	−395.051	−0.0190808
$755$	0	0
$756$	0	0
$757$	13224.5	0.634943	0.317472	−	0.948268i	$-0.397166\pi$
$757$	13224.5	0.634943	0.317472	+	0.948268i	$0.397166\pi$
$758$	10836.4	0.519255
$759$	0	0
$760$	0	0
$761$	−1709.91	−0.0814511	−0.0407256	−	0.999170i	$-0.512967\pi$
$761$	−1709.91	−0.0814511	−0.0407256	+	0.999170i	$0.512967\pi$
$762$	0	0
$763$	8002.33	0.379691
$764$	−14092.0	−0.667320
$765$	0	0
$766$	6266.03	0.295563
$767$	1355.16	0.0637965
$768$	0	0
$769$	−4705.78	−0.220669	−0.110335	−	0.993894i	$-0.535192\pi$
$769$	−4705.78	−0.220669	−0.110335	+	0.993894i	$0.535192\pi$
$770$	0	0
$771$	0	0
$772$	−29862.7	−1.39221
$773$	−19109.6	−0.889164	−0.444582	−	0.895738i	$-0.646648\pi$
$773$	−19109.6	−0.889164	−0.444582	+	0.895738i	$0.646648\pi$
$774$	0	0
$775$	0	0
$776$	486.103	0.0224872
$777$	0	0
$778$	−12648.0	−0.582844
$779$	13239.8	0.608941
$780$	0	0
$781$	−12833.4	−0.587985
$782$	−4599.47	−0.210328
$783$	0	0
$784$	−5638.23	−0.256844
$785$	0	0
$786$	0	0
$787$	−17209.4	−0.779480	−0.389740	−	0.920925i	$-0.627435\pi$
$787$	−17209.4	−0.779480	−0.389740	+	0.920925i	$0.627435\pi$
$788$	14096.6	0.637271
$789$	0	0
$790$	0	0
$791$	−3475.12	−0.156209
$792$	0	0
$793$	−1163.99	−0.0521242
$794$	−12789.5	−0.571642
$795$	0	0
$796$	13635.6	0.607160
$797$	−34775.4	−1.54556	−0.772779	−	0.634676i	$-0.781134\pi$
$797$	−34775.4	−1.54556	−0.772779	+	0.634676i	$0.781134\pi$
$798$	0	0
$799$	5725.17	0.253494
$800$	0	0
$801$	0	0
$802$	−640.901	−0.0282182
$803$	−3406.50	−0.149704
$804$	0	0
$805$	0	0
$806$	−257.961	−0.0112733
$807$	0	0
$808$	28835.0	1.25546
$809$	−22233.7	−0.966250	−0.483125	−	0.875551i	$-0.660498\pi$
$809$	−22233.7	−0.966250	−0.483125	+	0.875551i	$0.660498\pi$
$810$	0	0
$811$	−3576.54	−0.154857	−0.0774287	−	0.996998i	$-0.524671\pi$
$811$	−3576.54	−0.154857	−0.0774287	+	0.996998i	$0.524671\pi$
$812$	−7224.47	−0.312228
$813$	0	0
$814$	−4227.42	−0.182028
$815$	0	0
$816$	0	0
$817$	9734.12	0.416834
$818$	16402.4	0.701097
$819$	0	0
$820$	0	0
$821$	29910.6	1.27148	0.635741	−	0.771902i	$-0.280694\pi$
$821$	29910.6	1.27148	0.635741	+	0.771902i	$0.280694\pi$
$822$	0	0
$823$	−34421.1	−1.45789	−0.728945	−	0.684572i	$-0.759989\pi$
$823$	−34421.1	−1.45789	−0.728945	+	0.684572i	$0.759989\pi$
$824$	−23428.9	−0.990515
$825$	0	0
$826$	−7036.89	−0.296422
$827$	−30778.8	−1.29417	−0.647087	−	0.762416i	$-0.724013\pi$
$827$	−30778.8	−1.29417	−0.647087	+	0.762416i	$0.724013\pi$
$828$	0	0
$829$	21156.8	0.886375	0.443188	−	0.896429i	$-0.353848\pi$
$829$	21156.8	0.886375	0.443188	+	0.896429i	$0.353848\pi$
$830$	0	0
$831$	0	0
$832$	−130.745	−0.00544802
$833$	−6758.21	−0.281102
$834$	0	0
$835$	0	0
$836$	2182.57	0.0902941
$837$	0	0
$838$	736.588	0.0303640
$839$	20004.9	0.823178	0.411589	−	0.911370i	$-0.364974\pi$
$839$	20004.9	0.823178	0.411589	+	0.911370i	$0.364974\pi$
$840$	0	0
$841$	−12641.8	−0.518342
$842$	−695.215	−0.0284545
$843$	0	0
$844$	30577.0	1.24704
$845$	0	0
$846$	0	0
$847$	−12882.8	−0.522618
$848$	−2549.50	−0.103243
$849$	0	0
$850$	0	0
$851$	33012.0	1.32977
$852$	0	0
$853$	32918.6	1.32135	0.660675	−	0.750672i	$-0.270270\pi$
$853$	32918.6	1.32135	0.660675	+	0.750672i	$0.270270\pi$
$854$	6044.21	0.242188
$855$	0	0
$856$	−8817.33	−0.352068
$857$	8865.73	0.353381	0.176691	−	0.984266i	$-0.443461\pi$
$857$	8865.73	0.353381	0.176691	+	0.984266i	$0.443461\pi$
$858$	0	0
$859$	−32088.1	−1.27454	−0.637271	−	0.770640i	$-0.719937\pi$
$859$	−32088.1	−1.27454	−0.637271	+	0.770640i	$0.719937\pi$
$860$	0	0
$861$	0	0
$862$	−1721.19	−0.0680091
$863$	40687.9	1.60491	0.802453	−	0.596715i	$-0.203528\pi$
$863$	40687.9	1.60491	0.802453	+	0.596715i	$0.203528\pi$
$864$	0	0
$865$	0	0
$866$	239.417	0.00939460
$867$	0	0
$868$	−4717.44	−0.184471
$869$	−5914.83	−0.230894
$870$	0	0
$871$	−1607.69	−0.0625425
$872$	−14159.2	−0.549876
$873$	0	0
$874$	4839.54	0.187300
$875$	0	0
$876$	0	0
$877$	−18440.9	−0.710041	−0.355021	−	0.934858i	$-0.615526\pi$
$877$	−18440.9	−0.710041	−0.355021	+	0.934858i	$0.615526\pi$
$878$	−8071.29	−0.310242
$879$	0	0
$880$	0	0
$881$	13603.8	0.520230	0.260115	−	0.965578i	$-0.416240\pi$
$881$	13603.8	0.520230	0.260115	+	0.965578i	$0.416240\pi$
$882$	0	0
$883$	−1727.71	−0.0658461	−0.0329231	−	0.999458i	$-0.510482\pi$
$883$	−1727.71	−0.0658461	−0.0329231	+	0.999458i	$0.510482\pi$
$884$	504.868	0.0192088
$885$	0	0
$886$	20815.5	0.789289
$887$	7445.07	0.281827	0.140914	−	0.990022i	$-0.454996\pi$
$887$	7445.07	0.281827	0.140914	+	0.990022i	$0.454996\pi$
$888$	0	0
$889$	320.636	0.0120965
$890$	0	0
$891$	0	0
$892$	36059.6	1.35355
$893$	−6024.00	−0.225739
$894$	0	0
$895$	0	0
$896$	−15089.0	−0.562599
$897$	0	0
$898$	−9676.36	−0.359582
$899$	7670.68	0.284574
$900$	0	0
$901$	−3055.94	−0.112994
$902$	−6372.74	−0.235243
$903$	0	0
$904$	6148.84	0.226225
$905$	0	0
$906$	0	0
$907$	16050.6	0.587598	0.293799	−	0.955867i	$-0.405080\pi$
$907$	16050.6	0.587598	0.293799	+	0.955867i	$0.405080\pi$
$908$	27156.7	0.992541
$909$	0	0
$910$	0	0
$911$	52398.4	1.90564	0.952818	−	0.303541i	$-0.0981689\pi$
$911$	52398.4	1.90564	0.952818	+	0.303541i	$0.0981689\pi$
$912$	0	0
$913$	−11360.9	−0.411818
$914$	5794.75	0.209708
$915$	0	0
$916$	−24574.2	−0.886413
$917$	−9697.55	−0.349227
$918$	0	0
$919$	34880.1	1.25200	0.626001	−	0.779822i	$-0.284691\pi$
$919$	34880.1	1.25200	0.626001	+	0.779822i	$0.284691\pi$
$920$	0	0
$921$	0	0
$922$	−12394.3	−0.442718
$923$	3123.55	0.111390
$924$	0	0
$925$	0	0
$926$	24873.0	0.882696
$927$	0	0
$928$	19968.9	0.706370
$929$	42172.6	1.48938	0.744692	−	0.667408i	$-0.232596\pi$
$929$	42172.6	1.48938	0.744692	+	0.667408i	$0.232596\pi$
$930$	0	0
$931$	7110.96	0.250325
$932$	−34458.8	−1.21109
$933$	0	0
$934$	7110.01	0.249086
$935$	0	0
$936$	0	0
$937$	−33853.9	−1.18032	−0.590159	−	0.807287i	$-0.700935\pi$
$937$	−33853.9	−1.18032	−0.590159	+	0.807287i	$0.700935\pi$
$938$	8348.21	0.290596
$939$	0	0
$940$	0	0
$941$	−14665.4	−0.508053	−0.254027	−	0.967197i	$-0.581755\pi$
$941$	−14665.4	−0.508053	−0.254027	+	0.967197i	$0.581755\pi$
$942$	0	0
$943$	49764.9	1.71853
$944$	−12199.6	−0.420616
$945$	0	0
$946$	−4685.35	−0.161029
$947$	−43534.3	−1.49385	−0.746925	−	0.664908i	$-0.768471\pi$
$947$	−43534.3	−1.49385	−0.746925	+	0.664908i	$0.768471\pi$
$948$	0	0
$949$	829.113	0.0283605
$950$	0	0
$951$	0	0
$952$	−5987.63	−0.203845
$953$	−3831.21	−0.130226	−0.0651128	−	0.997878i	$-0.520741\pi$
$953$	−3831.21	−0.130226	−0.0651128	+	0.997878i	$0.520741\pi$
$954$	0	0
$955$	0	0
$956$	23821.5	0.805901
$957$	0	0
$958$	−22519.6	−0.759473
$959$	25246.4	0.850104
$960$	0	0
$961$	−24782.2	−0.831868
$962$	1028.92	0.0344840
$963$	0	0
$964$	−24775.2	−0.827756
$965$	0	0
$966$	0	0
$967$	27975.3	0.930324	0.465162	−	0.885225i	$-0.345996\pi$
$967$	27975.3	0.930324	0.465162	+	0.885225i	$0.345996\pi$
$968$	22794.6	0.756867
$969$	0	0
$970$	0	0
$971$	39865.6	1.31756	0.658779	−	0.752337i	$-0.271073\pi$
$971$	39865.6	1.31756	0.658779	+	0.752337i	$0.271073\pi$
$972$	0	0
$973$	18287.5	0.602540
$974$	−4060.64	−0.133584
$975$	0	0
$976$	10478.6	0.343659
$977$	38912.9	1.27424	0.637121	−	0.770764i	$-0.280125\pi$
$977$	38912.9	1.27424	0.637121	+	0.770764i	$0.280125\pi$
$978$	0	0
$979$	−16038.9	−0.523602
$980$	0	0
$981$	0	0
$982$	−10097.8	−0.328141
$983$	27331.9	0.886829	0.443415	−	0.896317i	$-0.353767\pi$
$983$	27331.9	0.886829	0.443415	+	0.896317i	$0.353767\pi$
$984$	0	0
$985$	0	0
$986$	4262.81	0.137683
$987$	0	0
$988$	−531.221	−0.0171056
$989$	36588.0	1.17637
$990$	0	0
$991$	−47040.3	−1.50786	−0.753928	−	0.656957i	$-0.771843\pi$
$991$	−47040.3	−1.50786	−0.753928	+	0.656957i	$0.771843\pi$
$992$	13039.3	0.417338
$993$	0	0
$994$	−16219.6	−0.517559
$995$	0	0
$996$	0	0
$997$	60011.9	1.90632	0.953158	−	0.302473i	$-0.0978123\pi$
$997$	60011.9	1.90632	0.953158	+	0.302473i	$0.0978123\pi$
$998$	−4906.76	−0.155632
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.u.1.3	✓	4
3.2	odd	2		675.4.a.y.1.2	yes	4
5.2	odd	4		675.4.b.p.649.5		8
5.3	odd	4		675.4.b.p.649.4		8
5.4	even	2		675.4.a.z.1.2	yes	4
15.2	even	4		675.4.b.q.649.4		8
15.8	even	4		675.4.b.q.649.5		8
15.14	odd	2		675.4.a.v.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.u.1.3	✓	4	1.1	even	1	trivial
675.4.a.v.1.3	yes	4	15.14	odd	2
675.4.a.y.1.2	yes	4	3.2	odd	2
675.4.a.z.1.2	yes	4	5.4	even	2
675.4.b.p.649.4		8	5.3	odd	4
675.4.b.p.649.5		8	5.2	odd	4
675.4.b.q.649.4		8	15.2	even	4
675.4.b.q.649.5		8	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q+1.33012 q^{2} -6.23078 q^{4} +10.6979 q^{7} -18.9286 q^{8} +11.2588 q^{11} -2.74029 q^{13} +14.2294 q^{14} +24.6689 q^{16} +29.5692 q^{17} -31.1126 q^{19} +14.9755 q^{22} -116.944 q^{23} -3.64491 q^{26} -66.6560 q^{28} +108.384 q^{29} +70.7730 q^{31} +184.242 q^{32} +39.3305 q^{34} -282.289 q^{37} -41.3834 q^{38} -425.545 q^{41} -312.868 q^{43} -70.1509 q^{44} -155.549 q^{46} +193.619 q^{47} -228.556 q^{49} +17.0741 q^{52} -103.349 q^{53} -202.496 q^{56} +144.164 q^{58} -494.531 q^{59} +424.769 q^{61} +94.1365 q^{62} +47.7120 q^{64} +586.687 q^{67} -184.239 q^{68} -1139.86 q^{71} -302.564 q^{73} -375.478 q^{74} +193.856 q^{76} +120.445 q^{77} -525.354 q^{79} -566.026 q^{82} -1009.07 q^{83} -416.151 q^{86} -213.113 q^{88} -1424.57 q^{89} -29.3152 q^{91} +728.652 q^{92} +257.537 q^{94} -25.6808 q^{97} -304.007 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 19 q^{4} - 4 q^{7} + 15 q^{8} - 52 q^{11} + 2 q^{13} - 138 q^{14} - 5 q^{16} - 64 q^{17} - 46 q^{19} - 87 q^{22} + 90 q^{23} - 469 q^{26} + 110 q^{28} - 470 q^{29} - 262 q^{31} + 199 q^{32}+ \cdots - 4301 q^{98}+O(q^{100}) \) 4 * q - q^2 + 19 * q^4 - 4 * q^7 + 15 * q^8 - 52 * q^11 + 2 * q^13 - 138 * q^14 - 5 * q^16 - 64 * q^17 - 46 * q^19 - 87 * q^22 + 90 * q^23 - 469 * q^26 + 110 * q^28 - 470 * q^29 - 262 * q^31 + 199 * q^32 - 42 * q^34 + 542 * q^37 - 532 * q^38 - 698 * q^41 - 142 * q^43 - 419 * q^44 + 537 * q^46 + 542 * q^47 + 780 * q^49 - 409 * q^52 - 910 * q^53 - 2034 * q^56 + 576 * q^58 - 100 * q^59 + 74 * q^61 + 2406 * q^62 - 965 * q^64 - 928 * q^67 - 2810 * q^68 - 1622 * q^71 + 536 * q^73 + 253 * q^74 - 2068 * q^76 + 1932 * q^77 - 508 * q^79 + 1782 * q^82 - 1524 * q^83 - 3940 * q^86 - 2247 * q^88 - 756 * q^89 + 1120 * q^91 + 3645 * q^92 + 2847 * q^94 - 892 * q^97 - 4301 * q^98

Introduction

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