LMFDB - Embedded newform 675.4.a.z.1.2

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			1.2
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	$-0.575553\pi$
$2$	−1.33012	−0.470268	−0.235134	+	0.971963i	$0.575553\pi$
$3$	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	$-0.593261\pi$
$7$	−10.6979	−0.577630	−0.288815	+	0.957385i	$0.593261\pi$
$8$	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	$-0.490694\pi$
$13$	2.74029	0.0584630	0.0292315	+	0.999573i	$0.490694\pi$
$14$	14.2294	0.271641
$15$	0	0
$16$	24.6689	0.385452
$17$	−29.5692	−0.421858	−0.210929	−	0.977501i	$-0.567649\pi$
$17$	−29.5692	−0.421858	−0.210929	+	0.977501i	$0.567649\pi$
$18$	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	$-0.322155\pi$
$23$	116.944	1.06020	0.530098	+	0.847936i	$0.322155\pi$
$24$	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	$-0.284227\pi$
$37$	282.289	1.25427	0.627136	+	0.778910i	$0.284227\pi$
$38$	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	$-0.312799\pi$
$43$	312.868	1.10958	0.554789	+	0.831991i	$0.312799\pi$
$44$	−70.1509	−0.240355
$45$	0	0
$46$	−155.549	−0.498576
$47$	−193.619	−0.600900	−0.300450	−	0.953798i	$-0.597137\pi$
$47$	−193.619	−0.600900	−0.300450	+	0.953798i	$0.597137\pi$
$48$	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	$-0.457242\pi$
$53$	103.349	0.267850	0.133925	+	0.990991i	$0.457242\pi$
$54$	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	$-0.679647\pi$
$67$	−586.687	−1.06978	−0.534889	+	0.844922i	$0.679647\pi$
$68$	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	$-0.422016\pi$
$73$	302.564	0.485102	0.242551	+	0.970139i	$0.422016\pi$
$74$	−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	$-0.267481\pi$
$83$	1009.07	1.33446	0.667228	+	0.744854i	$0.267481\pi$
$84$	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	$-0.495722\pi$
$97$	25.6808	0.0268814	0.0134407	+	0.999910i	$0.495722\pi$
$98$	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	$-0.701675\pi$
$103$	−1237.75	−1.18407	−0.592034	+	0.805913i	$0.701675\pi$
$104$	51.8699	0.0489064
$105$	0	0
$106$	−137.466	−0.125961
$107$	−465.820	−0.420865	−0.210432	−	0.977608i	$-0.567487\pi$
$107$	−465.820	−0.420865	−0.210432	+	0.977608i	$0.567487\pi$
$108$	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	$-0.456827\pi$
$113$	324.843	0.270431	0.135215	+	0.990816i	$0.456827\pi$
$114$	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	$-0.503333\pi$
$127$	−29.9720	−0.0209416	−0.0104708	+	0.999945i	$0.503333\pi$
$128$	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	$-0.763220\pi$
$137$	−2359.95	−1.47171	−0.735855	+	0.677139i	$0.763220\pi$
$138$	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	$-0.669671\pi$
$157$	−1999.27	−1.01630	−0.508151	+	0.861268i	$0.669671\pi$
$158$	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	$-0.593633\pi$
$163$	−1206.73	−0.579867	−0.289934	+	0.957047i	$0.593633\pi$
$164$	2651.48	1.26247
$165$	0	0
$166$	−1342.18	−0.627552
$167$	102.994	0.0477239	0.0238619	−	0.999715i	$-0.492404\pi$
$167$	102.994	0.0477239	0.0238619	+	0.999715i	$0.492404\pi$
$168$	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	$-0.478418\pi$
$173$	308.318	0.135497	0.0677485	+	0.997702i	$0.478418\pi$
$174$	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	$-0.851942\pi$
$193$	−4792.77	−1.78752	−0.893760	+	0.448545i	$0.851942\pi$
$194$	−34.1585	−0.0126414
$195$	0	0
$196$	1424.08	0.518981
$197$	2262.41	0.818223	0.409111	−	0.912484i	$-0.365839\pi$
$197$	2262.41	0.818223	0.409111	+	0.912484i	$0.365839\pi$
$198$	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	$-0.164800\pi$
$223$	5787.33	1.73789	0.868943	+	0.494912i	$0.164800\pi$
$224$	1970.99	0.587912
$225$	0	0
$226$	−432.080	−0.127175
$227$	4358.48	1.27437	0.637186	−	0.770710i	$-0.280098\pi$
$227$	4358.48	1.27437	0.637186	+	0.770710i	$0.280098\pi$
$228$	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	$-0.783506\pi$
$233$	−5530.41	−1.55497	−0.777487	+	0.628899i	$0.783506\pi$
$234$	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	$-0.364775\pi$
$257$	3396.20	0.824316	0.412158	+	0.911112i	$0.364775\pi$
$258$	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	$-0.684170\pi$
$263$	−4664.71	−1.09368	−0.546841	+	0.837236i	$0.684170\pi$
$264$	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	$-0.430654\pi$
$277$	1992.86	0.432273	0.216136	+	0.976363i	$0.430654\pi$
$278$	−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	$-0.674038\pi$
$283$	−4950.46	−1.03984	−0.519920	+	0.854215i	$0.674038\pi$
$284$	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	$-0.798518\pi$
$293$	−8087.47	−1.61254	−0.806272	+	0.591546i	$0.798518\pi$
$294$	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	$-0.491934\pi$
$307$	272.579	0.0506740	0.0253370	+	0.999679i	$0.491934\pi$
$308$	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	$-0.234064\pi$
$313$	8213.36	1.48322	0.741608	+	0.670834i	$0.234064\pi$
$314$	2659.27	0.477935
$315$	0	0
$316$	3273.37	0.582726
$317$	4859.27	0.860959	0.430480	−	0.902600i	$-0.358344\pi$
$317$	4859.27	0.860959	0.430480	+	0.902600i	$0.358344\pi$
$318$	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	$-0.625494\pi$
$337$	−4752.67	−0.768232	−0.384116	+	0.923285i	$0.625494\pi$
$338$	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	$-0.206427\pi$
$347$	10303.2	1.59397	0.796984	+	0.604000i	$0.206427\pi$
$348$	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	$-0.530649\pi$
$353$	−1275.21	−0.192273	−0.0961366	+	0.995368i	$0.530649\pi$
$354$	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	$-0.482909\pi$
$367$	754.620	0.107332	0.0536660	+	0.998559i	$0.482909\pi$
$368$	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	$-0.641051\pi$
$373$	−6177.49	−0.857529	−0.428765	+	0.903416i	$0.641051\pi$
$374$	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	$-0.601753\pi$
$383$	−4710.88	−0.628498	−0.314249	+	0.949341i	$0.601753\pi$
$384$	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	$-0.292059\pi$
$397$	9615.34	1.21557	0.607783	+	0.794103i	$0.292059\pi$
$398$	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	$-0.503180\pi$
$433$	−179.997	−0.0199771	−0.00998856	+	0.999950i	$0.503180\pi$
$434$	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	$-0.816971\pi$
$443$	−15649.3	−1.67838	−0.839190	+	0.543838i	$0.816971\pi$
$444$	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	$-0.571574\pi$
$457$	−4356.56	−0.445933	−0.222967	+	0.974826i	$0.571574\pi$
$458$	−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	$-0.887787\pi$
$463$	−18699.8	−1.87701	−0.938503	+	0.345270i	$0.887787\pi$
$464$	2673.72	0.267510
$465$	0	0
$466$	7356.10	0.731255
$467$	−5345.39	−0.529669	−0.264834	−	0.964294i	$-0.585317\pi$
$467$	−5345.39	−0.529669	−0.264834	+	0.964294i	$0.585317\pi$
$468$	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	$-0.454637\pi$
$487$	3052.84	0.284060	0.142030	+	0.989862i	$0.454637\pi$
$488$	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	$-0.531261\pi$
$503$	−2212.28	−0.196105	−0.0980523	+	0.995181i	$0.531261\pi$
$504$	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	$-0.473068\pi$
$523$	2021.57	0.169020	0.0845098	+	0.996423i	$0.473068\pi$
$524$	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	$-0.590323\pi$
$547$	−7163.31	−0.559929	−0.279964	+	0.960010i	$0.590323\pi$
$548$	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	$-0.691592\pi$
$557$	−14886.5	−1.13243	−0.566214	+	0.824258i	$0.691592\pi$
$558$	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	$-0.401528\pi$
$563$	8134.04	0.608897	0.304448	+	0.952529i	$0.401528\pi$
$564$	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	$-0.437848\pi$
$577$	5378.18	0.388036	0.194018	+	0.980998i	$0.437848\pi$
$578$	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	$-0.569415\pi$
$587$	−6153.77	−0.432697	−0.216349	+	0.976316i	$0.569415\pi$
$588$	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	$-0.335396\pi$
$593$	14278.1	0.988753	0.494376	+	0.869248i	$0.335396\pi$
$594$	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	$-0.221369\pi$
$607$	22963.7	1.53553	0.767765	+	0.640732i	$0.221369\pi$
$608$	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	$-0.362054\pi$
$613$	12746.7	0.839862	0.419931	+	0.907556i	$0.362054\pi$
$614$	−362.563	−0.0238304
$615$	0	0
$616$	−2279.85	−0.149120
$617$	−13959.6	−0.910844	−0.455422	−	0.890276i	$-0.650512\pi$
$617$	−13959.6	−0.910844	−0.455422	+	0.890276i	$0.650512\pi$
$618$	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	$-0.217428\pi$
$643$	25293.3	1.55128	0.775639	+	0.631177i	$0.217428\pi$
$644$	7795.02	0.476967
$645$	0	0
$646$	−1223.67	−0.0745276
$647$	−1442.91	−0.0876763	−0.0438381	−	0.999039i	$-0.513959\pi$
$647$	−1442.91	−0.0876763	−0.0438381	+	0.999039i	$0.513959\pi$
$648$	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	$-0.412665\pi$
$653$	9042.22	0.541883	0.270941	+	0.962596i	$0.412665\pi$
$654$	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	$-0.706632\pi$
$673$	−21108.5	−1.20902	−0.604512	+	0.796596i	$0.706632\pi$
$674$	6321.61	0.361275
$675$	0	0
$676$	13642.2	0.776186
$677$	−19793.4	−1.12367	−0.561834	−	0.827250i	$-0.689904\pi$
$677$	−19793.4	−1.12367	−0.561834	+	0.827250i	$0.689904\pi$
$678$	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	$-0.564883\pi$
$683$	−7226.49	−0.404852	−0.202426	+	0.979298i	$0.564883\pi$
$684$	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	$-0.748520\pi$
$727$	−27592.3	−1.40762	−0.703811	+	0.710387i	$0.748520\pi$
$728$	−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	$-0.242033\pi$
$733$	28759.0	1.44916	0.724582	+	0.689189i	$0.242033\pi$
$734$	−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	$-0.882299\pi$
$743$	−37767.7	−1.86482	−0.932411	+	0.361399i	$0.882299\pi$
$744$	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	$-0.602834\pi$
$757$	−13224.5	−0.634943	−0.317472	+	0.948268i	$0.602834\pi$
$758$	−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	$-0.353352\pi$
$773$	19109.6	0.889164	0.444582	+	0.895738i	$0.353352\pi$
$774$	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	$-0.372565\pi$
$787$	17209.4	0.779480	0.389740	+	0.920925i	$0.372565\pi$
$788$	−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	$-0.218866\pi$
$797$	34775.4	1.54556	0.772779	+	0.634676i	$0.218866\pi$
$798$	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	$-0.240011\pi$
$823$	34421.1	1.45789	0.728945	+	0.684572i	$0.240011\pi$
$824$	−23428.9	−0.990515
$825$	0	0
$826$	−7036.89	−0.296422
$827$	30778.8	1.29417	0.647087	−	0.762416i	$-0.275987\pi$
$827$	30778.8	1.29417	0.647087	+	0.762416i	$0.275987\pi$
$828$	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	$-0.729730\pi$
$853$	−32918.6	−1.32135	−0.660675	+	0.750672i	$0.729730\pi$
$854$	6044.21	0.242188
$855$	0	0
$856$	−8817.33	−0.352068
$857$	−8865.73	−0.353381	−0.176691	−	0.984266i	$-0.556539\pi$
$857$	−8865.73	−0.353381	−0.176691	+	0.984266i	$0.556539\pi$
$858$	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	$-0.796472\pi$
$863$	−40687.9	−1.60491	−0.802453	+	0.596715i	$0.796472\pi$
$864$	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	$-0.384474\pi$
$877$	18440.9	0.710041	0.355021	+	0.934858i	$0.384474\pi$
$878$	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	$-0.489518\pi$
$883$	1727.71	0.0658461	0.0329231	+	0.999458i	$0.489518\pi$
$884$	504.868	0.0192088
$885$	0	0
$886$	20815.5	0.789289
$887$	−7445.07	−0.281827	−0.140914	−	0.990022i	$-0.545004\pi$
$887$	−7445.07	−0.281827	−0.140914	+	0.990022i	$0.545004\pi$
$888$	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	$-0.594920\pi$
$907$	−16050.6	−0.587598	−0.293799	+	0.955867i	$0.594920\pi$
$908$	−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	$-0.299065\pi$
$937$	33853.9	1.18032	0.590159	+	0.807287i	$0.299065\pi$
$938$	−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	$-0.231529\pi$
$947$	43534.3	1.49385	0.746925	+	0.664908i	$0.231529\pi$
$948$	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	$-0.479259\pi$
$953$	3831.21	0.130226	0.0651128	+	0.997878i	$0.479259\pi$
$954$	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	$-0.654004\pi$
$967$	−27975.3	−0.930324	−0.465162	+	0.885225i	$0.654004\pi$
$968$	−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	$-0.719875\pi$
$977$	−38912.9	−1.27424	−0.637121	+	0.770764i	$0.719875\pi$
$978$	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	$-0.646233\pi$
$983$	−27331.9	−0.886829	−0.443415	+	0.896317i	$0.646233\pi$
$984$	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	$-0.902188\pi$
$997$	−60011.9	−1.90632	−0.953158	+	0.302473i	$0.902188\pi$
$998$	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.z.1.2	yes	4
3.2	odd	2		675.4.a.v.1.3	yes	4
5.2	odd	4		675.4.b.p.649.4		8
5.3	odd	4		675.4.b.p.649.5		8
5.4	even	2		675.4.a.u.1.3	✓	4
15.2	even	4		675.4.b.q.649.5		8
15.8	even	4		675.4.b.q.649.4		8
15.14	odd	2		675.4.a.y.1.2	yes	4

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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more