LMFDB - Embedded newform 675.4.a.t.1.1

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,-6,0,-6,0,0,0] 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:	$0$
Dimension:	$4$
Coefficient field:	4.4.467024.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 26x^{2} + 101 $ x^4 - 26*x^2 + 101
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.60936$ 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)

$f(q)$	$=$	$q-3.56155 q^{2} +4.68466 q^{4} -6.06288 q^{7} +11.8078 q^{8} -61.3752 q^{11} -70.8427 q^{13} +21.5933 q^{14} -79.5312 q^{16} +51.5464 q^{17} +15.7926 q^{19} +218.591 q^{22} +114.885 q^{23} +252.310 q^{26} -28.4025 q^{28} -39.7819 q^{29} -240.170 q^{31} +188.793 q^{32} -183.585 q^{34} -129.560 q^{37} -56.2462 q^{38} -262.524 q^{41} +127.648 q^{43} -287.522 q^{44} -409.170 q^{46} +85.2140 q^{47} -306.241 q^{49} -331.874 q^{52} +328.184 q^{53} -71.5891 q^{56} +141.685 q^{58} +687.580 q^{59} -455.685 q^{61} +855.380 q^{62} -36.1449 q^{64} -955.421 q^{67} +241.477 q^{68} +160.201 q^{71} +702.364 q^{73} +461.434 q^{74} +73.9830 q^{76} +372.111 q^{77} +65.3608 q^{79} +934.993 q^{82} -996.623 q^{83} -454.624 q^{86} -724.704 q^{88} -1337.71 q^{89} +429.511 q^{91} +538.199 q^{92} -303.494 q^{94} +1404.40 q^{97} +1090.70 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} - 6 q^{4} + 6 q^{8} - 46 q^{16} + 66 q^{17} - 110 q^{19} + 6 q^{23} - 268 q^{31} + 582 q^{32} - 388 q^{34} - 192 q^{38} - 944 q^{46} + 1116 q^{47} + 482 q^{49} + 1824 q^{53} - 1798 q^{61}+ \cdots + 2796 q^{98}+O(q^{100}) $ 4 * q - 6 * q^2 - 6 * q^4 + 6 * q^8 - 46 * q^16 + 66 * q^17 - 110 * q^19 + 6 * q^23 - 268 * q^31 + 582 * q^32 - 388 * q^34 - 192 * q^38 - 944 * q^46 + 1116 * q^47 + 482 * q^49 + 1824 * q^53 - 1798 * q^61 + 1830 * q^62 - 862 * q^64 + 768 * q^68 + 1236 * q^76 + 4086 * q^77 + 682 * q^79 + 384 * q^83 - 360 * q^91 + 2796 * q^92 - 76 * q^94 + 2796 * 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 $

$2$	−3.56155	−1.25920	−0.629600	−	0.776920i	$-0.716781\pi$
$2$	−3.56155	−1.25920	−0.629600	+	0.776920i	$0.716781\pi$
$3$	0	0
$3$	0	0
$4$	4.68466	0.585582
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−6.06288	−0.327365	−0.163682	−	0.986513i	$-0.552337\pi$
$7$	−6.06288	−0.327365	−0.163682	+	0.986513i	$0.552337\pi$
$8$	11.8078	0.521834
$9$	0	0
$10$	0	0
$11$	−61.3752	−1.68230	−0.841151	−	0.540800i	$-0.818122\pi$
$11$	−61.3752	−1.68230	−0.841151	+	0.540800i	$0.818122\pi$
$12$	0	0
$13$	−70.8427	−1.51140	−0.755702	−	0.654916i	$-0.772704\pi$
$13$	−70.8427	−1.51140	−0.755702	+	0.654916i	$0.772704\pi$
$14$	21.5933	0.412218
$15$	0	0
$16$	−79.5312	−1.24268
$17$	51.5464	0.735402	0.367701	−	0.929944i	$-0.380145\pi$
$17$	51.5464	0.735402	0.367701	+	0.929944i	$0.380145\pi$
$18$	0	0
$19$	15.7926	0.190688	0.0953440	−	0.995444i	$-0.469605\pi$
$19$	15.7926	0.190688	0.0953440	+	0.995444i	$0.469605\pi$
$20$	0	0
$21$	0	0
$22$	218.591	2.11835
$23$	114.885	1.04153	0.520767	−	0.853699i	$-0.325646\pi$
$23$	114.885	1.04153	0.520767	+	0.853699i	$0.325646\pi$
$24$	0	0
$25$	0	0
$26$	252.310	1.90316
$27$	0	0
$28$	−28.4025	−0.191699
$29$	−39.7819	−0.254735	−0.127368	−	0.991856i	$-0.540653\pi$
$29$	−39.7819	−0.254735	−0.127368	+	0.991856i	$0.540653\pi$
$30$	0	0
$31$	−240.170	−1.39148	−0.695740	−	0.718294i	$-0.744923\pi$
$31$	−240.170	−1.39148	−0.695740	+	0.718294i	$0.744923\pi$
$32$	188.793	1.04294
$33$	0	0
$34$	−183.585	−0.926018
$35$	0	0
$36$	0	0
$37$	−129.560	−0.575662	−0.287831	−	0.957681i	$-0.592934\pi$
$37$	−129.560	−0.575662	−0.287831	+	0.957681i	$0.592934\pi$
$38$	−56.2462	−0.240114
$39$	0	0
$40$	0	0
$41$	−262.524	−0.999985	−0.499992	−	0.866030i	$-0.666664\pi$
$41$	−262.524	−0.999985	−0.499992	+	0.866030i	$0.666664\pi$
$42$	0	0
$43$	127.648	0.452700	0.226350	−	0.974046i	$-0.427321\pi$
$43$	127.648	0.452700	0.226350	+	0.974046i	$0.427321\pi$
$44$	−287.522	−0.985126
$45$	0	0
$46$	−409.170	−1.31150
$47$	85.2140	0.264463	0.132231	−	0.991219i	$-0.457786\pi$
$47$	85.2140	0.264463	0.132231	+	0.991219i	$0.457786\pi$
$48$	0	0
$49$	−306.241	−0.892832
$50$	0	0
$51$	0	0
$52$	−331.874	−0.885051
$53$	328.184	0.850557	0.425278	−	0.905063i	$-0.360176\pi$
$53$	328.184	0.850557	0.425278	+	0.905063i	$0.360176\pi$
$54$	0	0
$55$	0	0
$56$	−71.5891	−0.170830
$57$	0	0
$58$	141.685	0.320762
$59$	687.580	1.51721	0.758605	−	0.651551i	$-0.225881\pi$
$59$	687.580	1.51721	0.758605	+	0.651551i	$0.225881\pi$
$60$	0	0
$61$	−455.685	−0.956466	−0.478233	−	0.878233i	$-0.658723\pi$
$61$	−455.685	−0.956466	−0.478233	+	0.878233i	$0.658723\pi$
$62$	855.380	1.75215
$63$	0	0
$64$	−36.1449	−0.0705955
$65$	0	0
$66$	0	0
$67$	−955.421	−1.74214	−0.871069	−	0.491161i	$-0.836573\pi$
$67$	−955.421	−1.74214	−0.871069	+	0.491161i	$0.836573\pi$
$68$	241.477	0.430639
$69$	0	0
$70$	0	0
$71$	160.201	0.267780	0.133890	−	0.990996i	$-0.457253\pi$
$71$	160.201	0.267780	0.133890	+	0.990996i	$0.457253\pi$
$72$	0	0
$73$	702.364	1.12610	0.563052	−	0.826422i	$-0.309627\pi$
$73$	702.364	1.12610	0.563052	+	0.826422i	$0.309627\pi$
$74$	461.434	0.724873
$75$	0	0
$76$	73.9830	0.111664
$77$	372.111	0.550727
$78$	0	0
$79$	65.3608	0.0930844	0.0465422	−	0.998916i	$-0.485180\pi$
$79$	65.3608	0.0930844	0.0465422	+	0.998916i	$0.485180\pi$
$80$	0	0
$81$	0	0
$82$	934.993	1.25918
$83$	−996.623	−1.31800	−0.658998	−	0.752145i	$-0.729019\pi$
$83$	−996.623	−1.31800	−0.658998	+	0.752145i	$0.729019\pi$
$84$	0	0
$85$	0	0
$86$	−454.624	−0.570040
$87$	0	0
$88$	−724.704	−0.877883
$89$	−1337.71	−1.59322	−0.796612	−	0.604491i	$-0.793377\pi$
$89$	−1337.71	−1.59322	−0.796612	+	0.604491i	$0.793377\pi$
$90$	0	0
$91$	429.511	0.494780
$92$	538.199	0.609903
$93$	0	0
$94$	−303.494	−0.333011
$95$	0	0
$96$	0	0
$97$	1404.40	1.47006	0.735028	−	0.678037i	$-0.237169\pi$
$97$	1404.40	1.47006	0.735028	+	0.678037i	$0.237169\pi$
$98$	1090.70	1.12425
$99$	0	0
$100$	0	0
$101$	1110.40	1.09395	0.546974	−	0.837150i	$-0.315780\pi$
$101$	1110.40	1.09395	0.546974	+	0.837150i	$0.315780\pi$
$102$	0	0
$103$	509.937	0.487821	0.243911	−	0.969798i	$-0.421570\pi$
$103$	509.937	0.487821	0.243911	+	0.969798i	$0.421570\pi$
$104$	−836.494	−0.788702
$105$	0	0
$106$	−1168.84	−1.07102
$107$	1748.35	1.57962	0.789809	−	0.613353i	$-0.210180\pi$
$107$	1748.35	1.57962	0.789809	+	0.613353i	$0.210180\pi$
$108$	0	0
$109$	1357.35	1.19276	0.596381	−	0.802702i	$-0.296605\pi$
$109$	1357.35	1.19276	0.596381	+	0.802702i	$0.296605\pi$
$110$	0	0
$111$	0	0
$112$	482.189	0.406809
$113$	1153.33	0.960139	0.480070	−	0.877230i	$-0.340611\pi$
$113$	1153.33	0.960139	0.480070	+	0.877230i	$0.340611\pi$
$114$	0	0
$115$	0	0
$116$	−186.365	−0.149168
$117$	0	0
$118$	−2448.85	−1.91047
$119$	−312.520	−0.240745
$120$	0	0
$121$	2435.92	1.83014
$122$	1622.94	1.20438
$123$	0	0
$124$	−1125.12	−0.814826
$125$	0	0
$126$	0	0
$127$	−1978.79	−1.38259	−0.691297	−	0.722571i	$-0.742960\pi$
$127$	−1978.79	−1.38259	−0.691297	+	0.722571i	$0.742960\pi$
$128$	−1381.61	−0.954048
$129$	0	0
$130$	0	0
$131$	160.293	0.106908	0.0534538	−	0.998570i	$-0.482977\pi$
$131$	160.293	0.106908	0.0534538	+	0.998570i	$0.482977\pi$
$132$	0	0
$133$	−95.7488	−0.0624246
$134$	3402.78	2.19370
$135$	0	0
$136$	608.648	0.383758
$137$	661.330	0.412418	0.206209	−	0.978508i	$-0.433887\pi$
$137$	661.330	0.412418	0.206209	+	0.978508i	$0.433887\pi$
$138$	0	0
$139$	2126.16	1.29740	0.648699	−	0.761045i	$-0.275313\pi$
$139$	2126.16	1.29740	0.648699	+	0.761045i	$0.275313\pi$
$140$	0	0
$141$	0	0
$142$	−570.566	−0.337189
$143$	4347.99	2.54264
$144$	0	0
$145$	0	0
$146$	−2501.51	−1.41799
$147$	0	0
$148$	−606.943	−0.337097
$149$	670.649	0.368736	0.184368	−	0.982857i	$-0.440976\pi$
$149$	670.649	0.368736	0.184368	+	0.982857i	$0.440976\pi$
$150$	0	0
$151$	1440.42	0.776291	0.388146	−	0.921598i	$-0.373116\pi$
$151$	1440.42	0.776291	0.388146	+	0.921598i	$0.373116\pi$
$152$	186.475	0.0995076
$153$	0	0
$154$	−1325.29	−0.693475
$155$	0	0
$156$	0	0
$157$	3037.28	1.54396	0.771979	−	0.635648i	$-0.219267\pi$
$157$	3037.28	1.54396	0.771979	+	0.635648i	$0.219267\pi$
$158$	−232.786	−0.117212
$159$	0	0
$160$	0	0
$161$	−696.537	−0.340961
$162$	0	0
$163$	1243.27	0.597426	0.298713	−	0.954343i	$-0.403443\pi$
$163$	1243.27	0.597426	0.298713	+	0.954343i	$0.403443\pi$
$164$	−1229.84	−0.585573
$165$	0	0
$166$	3549.53	1.65962
$167$	−165.425	−0.0766526	−0.0383263	−	0.999265i	$-0.512203\pi$
$167$	−165.425	−0.0766526	−0.0383263	+	0.999265i	$0.512203\pi$
$168$	0	0
$169$	2821.69	1.28434
$170$	0	0
$171$	0	0
$172$	597.986	0.265093
$173$	−1931.11	−0.848666	−0.424333	−	0.905506i	$-0.639491\pi$
$173$	−1931.11	−0.848666	−0.424333	+	0.905506i	$0.639491\pi$
$174$	0	0
$175$	0	0
$176$	4881.25	2.09056
$177$	0	0
$178$	4764.32	2.00619
$179$	623.874	0.260506	0.130253	−	0.991481i	$-0.458421\pi$
$179$	623.874	0.260506	0.130253	+	0.991481i	$0.458421\pi$
$180$	0	0
$181$	1139.12	0.467792	0.233896	−	0.972262i	$-0.424853\pi$
$181$	1139.12	0.467792	0.233896	+	0.972262i	$0.424853\pi$
$182$	−1529.73	−0.623027
$183$	0	0
$184$	1356.54	0.543508
$185$	0	0
$186$	0	0
$187$	−3163.67	−1.23717
$188$	399.199	0.154865
$189$	0	0
$190$	0	0
$191$	468.059	0.177317	0.0886586	−	0.996062i	$-0.471742\pi$
$191$	468.059	0.177317	0.0886586	+	0.996062i	$0.471742\pi$
$192$	0	0
$193$	−2479.17	−0.924634	−0.462317	−	0.886715i	$-0.652982\pi$
$193$	−2479.17	−0.924634	−0.462317	+	0.886715i	$0.652982\pi$
$194$	−5001.85	−1.85109
$195$	0	0
$196$	−1434.64	−0.522827
$197$	2401.87	0.868660	0.434330	−	0.900754i	$-0.356985\pi$
$197$	2401.87	0.868660	0.434330	+	0.900754i	$0.356985\pi$
$198$	0	0
$199$	−1039.02	−0.370121	−0.185061	−	0.982727i	$-0.559248\pi$
$199$	−1039.02	−0.370121	−0.185061	+	0.982727i	$0.559248\pi$
$200$	0	0
$201$	0	0
$202$	−3954.74	−1.37750
$203$	241.193	0.0833914
$204$	0	0
$205$	0	0
$206$	−1816.17	−0.614264
$207$	0	0
$208$	5634.21	1.87818
$209$	−969.275	−0.320795
$210$	0	0
$211$	3311.99	1.08060	0.540301	−	0.841472i	$-0.318310\pi$
$211$	3311.99	1.08060	0.540301	+	0.841472i	$0.318310\pi$
$212$	1537.43	0.498071
$213$	0	0
$214$	−6226.83	−1.98905
$215$	0	0
$216$	0	0
$217$	1456.13	0.455522
$218$	−4834.29	−1.50192
$219$	0	0
$220$	0	0
$221$	−3651.69	−1.11149
$222$	0	0
$223$	−3985.66	−1.19686	−0.598429	−	0.801176i	$-0.704208\pi$
$223$	−3985.66	−1.19686	−0.598429	+	0.801176i	$0.704208\pi$
$224$	−1144.63	−0.341423
$225$	0	0
$226$	−4107.63	−1.20901
$227$	−4430.88	−1.29554	−0.647770	−	0.761836i	$-0.724298\pi$
$227$	−4430.88	−1.29554	−0.647770	+	0.761836i	$0.724298\pi$
$228$	0	0
$229$	4653.72	1.34291	0.671456	−	0.741045i	$-0.265669\pi$
$229$	4653.72	1.34291	0.671456	+	0.741045i	$0.265669\pi$
$230$	0	0
$231$	0	0
$232$	−469.736	−0.132930
$233$	2296.46	0.645692	0.322846	−	0.946452i	$-0.395360\pi$
$233$	2296.46	0.645692	0.322846	+	0.946452i	$0.395360\pi$
$234$	0	0
$235$	0	0
$236$	3221.08	0.888451
$237$	0	0
$238$	1113.06	0.303146
$239$	3418.55	0.925220	0.462610	−	0.886562i	$-0.346913\pi$
$239$	3418.55	0.925220	0.462610	+	0.886562i	$0.346913\pi$
$240$	0	0
$241$	−5904.16	−1.57809	−0.789047	−	0.614333i	$-0.789425\pi$
$241$	−5904.16	−1.57809	−0.789047	+	0.614333i	$0.789425\pi$
$242$	−8675.65	−2.30451
$243$	0	0
$244$	−2134.73	−0.560090
$245$	0	0
$246$	0	0
$247$	−1118.79	−0.288206
$248$	−2835.88	−0.726122
$249$	0	0
$250$	0	0
$251$	−2391.39	−0.601368	−0.300684	−	0.953724i	$-0.597215\pi$
$251$	−2391.39	−0.601368	−0.300684	+	0.953724i	$0.597215\pi$
$252$	0	0
$253$	−7051.12	−1.75217
$254$	7047.57	1.74096
$255$	0	0
$256$	5209.83	1.27193
$257$	−3580.72	−0.869102	−0.434551	−	0.900647i	$-0.643093\pi$
$257$	−3580.72	−0.869102	−0.434551	+	0.900647i	$0.643093\pi$
$258$	0	0
$259$	785.505	0.188451
$260$	0	0
$261$	0	0
$262$	−570.893	−0.134618
$263$	−1703.58	−0.399419	−0.199709	−	0.979855i	$-0.564000\pi$
$263$	−1703.58	−0.399419	−0.199709	+	0.979855i	$0.564000\pi$
$264$	0	0
$265$	0	0
$266$	341.014	0.0786050
$267$	0	0
$268$	−4475.82	−1.02017
$269$	4087.14	0.926385	0.463192	−	0.886258i	$-0.346704\pi$
$269$	4087.14	0.926385	0.463192	+	0.886258i	$0.346704\pi$
$270$	0	0
$271$	−4572.37	−1.02492	−0.512458	−	0.858713i	$-0.671265\pi$
$271$	−4572.37	−1.02492	−0.512458	+	0.858713i	$0.671265\pi$
$272$	−4099.55	−0.913866
$273$	0	0
$274$	−2355.36	−0.519316
$275$	0	0
$276$	0	0
$277$	8334.59	1.80786	0.903929	−	0.427682i	$-0.140670\pi$
$277$	8334.59	1.80786	0.903929	+	0.427682i	$0.140670\pi$
$278$	−7572.43	−1.63368
$279$	0	0
$280$	0	0
$281$	2490.50	0.528721	0.264360	−	0.964424i	$-0.414839\pi$
$281$	2490.50	0.528721	0.264360	+	0.964424i	$0.414839\pi$
$282$	0	0
$283$	7132.72	1.49822	0.749110	−	0.662445i	$-0.230481\pi$
$283$	7132.72	1.49822	0.749110	+	0.662445i	$0.230481\pi$
$284$	750.489	0.156807
$285$	0	0
$286$	−15485.6	−3.20169
$287$	1591.65	0.327360
$288$	0	0
$289$	−2255.97	−0.459184
$290$	0	0
$291$	0	0
$292$	3290.34	0.659426
$293$	4915.24	0.980039	0.490020	−	0.871711i	$-0.336990\pi$
$293$	4915.24	0.980039	0.490020	+	0.871711i	$0.336990\pi$
$294$	0	0
$295$	0	0
$296$	−1529.81	−0.300400
$297$	0	0
$298$	−2388.55	−0.464313
$299$	−8138.80	−1.57418
$300$	0	0
$301$	−773.914	−0.148198
$302$	−5130.14	−0.977505
$303$	0	0
$304$	−1256.01	−0.236963
$305$	0	0
$306$	0	0
$307$	5798.84	1.07804	0.539018	−	0.842294i	$-0.318795\pi$
$307$	5798.84	1.07804	0.539018	+	0.842294i	$0.318795\pi$
$308$	1743.21	0.322496
$309$	0	0
$310$	0	0
$311$	60.3935	0.0110116	0.00550579	−	0.999985i	$-0.498247\pi$
$311$	60.3935	0.0110116	0.00550579	+	0.999985i	$0.498247\pi$
$312$	0	0
$313$	605.307	0.109310	0.0546549	−	0.998505i	$-0.482594\pi$
$313$	605.307	0.109310	0.0546549	+	0.998505i	$0.482594\pi$
$314$	−10817.4	−1.94415
$315$	0	0
$316$	306.193	0.0545086
$317$	3144.61	0.557157	0.278578	−	0.960414i	$-0.410137\pi$
$317$	3144.61	0.557157	0.278578	+	0.960414i	$0.410137\pi$
$318$	0	0
$319$	2441.63	0.428542
$320$	0	0
$321$	0	0
$322$	2480.75	0.429338
$323$	814.052	0.140232
$324$	0	0
$325$	0	0
$326$	−4427.97	−0.752278
$327$	0	0
$328$	−3099.82	−0.521826
$329$	−516.643	−0.0865758
$330$	0	0
$331$	−2122.94	−0.352529	−0.176265	−	0.984343i	$-0.556401\pi$
$331$	−2122.94	−0.352529	−0.176265	+	0.984343i	$0.556401\pi$
$332$	−4668.84	−0.771795
$333$	0	0
$334$	589.170	0.0965209
$335$	0	0
$336$	0	0
$337$	−1486.39	−0.240263	−0.120132	−	0.992758i	$-0.538332\pi$
$337$	−1486.39	−0.240263	−0.120132	+	0.992758i	$0.538332\pi$
$338$	−10049.6	−1.61724
$339$	0	0
$340$	0	0
$341$	14740.5	2.34089
$342$	0	0
$343$	3936.28	0.619647
$344$	1507.24	0.236235
$345$	0	0
$346$	6877.74	1.06864
$347$	−7649.36	−1.18340	−0.591699	−	0.806159i	$-0.701543\pi$
$347$	−7649.36	−1.18340	−0.591699	+	0.806159i	$0.701543\pi$
$348$	0	0
$349$	−1123.61	−0.172337	−0.0861683	−	0.996281i	$-0.527462\pi$
$349$	−1123.61	−0.172337	−0.0861683	+	0.996281i	$0.527462\pi$
$350$	0	0
$351$	0	0
$352$	−11587.2	−1.75454
$353$	12564.5	1.89446	0.947228	−	0.320560i	$-0.103871\pi$
$353$	12564.5	1.89446	0.947228	+	0.320560i	$0.103871\pi$
$354$	0	0
$355$	0	0
$356$	−6266.71	−0.932964
$357$	0	0
$358$	−2221.96	−0.328029
$359$	−2491.02	−0.366214	−0.183107	−	0.983093i	$-0.558616\pi$
$359$	−2491.02	−0.366214	−0.183107	+	0.983093i	$0.558616\pi$
$360$	0	0
$361$	−6609.59	−0.963638
$362$	−4057.04	−0.589043
$363$	0	0
$364$	2012.11	0.289735
$365$	0	0
$366$	0	0
$367$	−12962.0	−1.84363	−0.921816	−	0.387629i	$-0.873294\pi$
$367$	−12962.0	−1.84363	−0.921816	+	0.387629i	$0.873294\pi$
$368$	−9136.98	−1.29429
$369$	0	0
$370$	0	0
$371$	−1989.74	−0.278442
$372$	0	0
$373$	−5045.73	−0.700424	−0.350212	−	0.936670i	$-0.613890\pi$
$373$	−5045.73	−0.700424	−0.350212	+	0.936670i	$0.613890\pi$
$374$	11267.6	1.55784
$375$	0	0
$376$	1006.19	0.138006
$377$	2818.26	0.385008
$378$	0	0
$379$	357.793	0.0484923	0.0242461	−	0.999706i	$-0.492281\pi$
$379$	357.793	0.0484923	0.0242461	+	0.999706i	$0.492281\pi$
$380$	0	0
$381$	0	0
$382$	−1667.02	−0.223278
$383$	886.550	0.118278	0.0591391	−	0.998250i	$-0.481164\pi$
$383$	886.550	0.118278	0.0591391	+	0.998250i	$0.481164\pi$
$384$	0	0
$385$	0	0
$386$	8829.69	1.16430
$387$	0	0
$388$	6579.14	0.860838
$389$	8284.92	1.07985	0.539926	−	0.841713i	$-0.318452\pi$
$389$	8284.92	1.07985	0.539926	+	0.841713i	$0.318452\pi$
$390$	0	0
$391$	5921.93	0.765946
$392$	−3616.03	−0.465911
$393$	0	0
$394$	−8554.38	−1.09382
$395$	0	0
$396$	0	0
$397$	−4474.56	−0.565672	−0.282836	−	0.959168i	$-0.591275\pi$
$397$	−4474.56	−0.565672	−0.282836	+	0.959168i	$0.591275\pi$
$398$	3700.52	0.466056
$399$	0	0
$400$	0	0
$401$	−10558.3	−1.31486	−0.657428	−	0.753517i	$-0.728356\pi$
$401$	−10558.3	−1.31486	−0.657428	+	0.753517i	$0.728356\pi$
$402$	0	0
$403$	17014.3	2.10309
$404$	5201.83	0.640596
$405$	0	0
$406$	−859.023	−0.105006
$407$	7951.75	0.968437
$408$	0	0
$409$	−5232.44	−0.632586	−0.316293	−	0.948662i	$-0.602438\pi$
$409$	−5232.44	−0.632586	−0.316293	+	0.948662i	$0.602438\pi$
$410$	0	0
$411$	0	0
$412$	2388.88	0.285659
$413$	−4168.72	−0.496681
$414$	0	0
$415$	0	0
$416$	−13374.6	−1.57631
$417$	0	0
$418$	3452.12	0.403945
$419$	−3903.81	−0.455164	−0.227582	−	0.973759i	$-0.573082\pi$
$419$	−3903.81	−0.455164	−0.227582	+	0.973759i	$0.573082\pi$
$420$	0	0
$421$	6274.83	0.726405	0.363203	−	0.931710i	$-0.381683\pi$
$421$	6274.83	0.726405	0.363203	+	0.931710i	$0.381683\pi$
$422$	−11795.8	−1.36069
$423$	0	0
$424$	3875.12	0.443850
$425$	0	0
$426$	0	0
$427$	2762.76	0.313114
$428$	8190.41	0.924996
$429$	0	0
$430$	0	0
$431$	−8040.40	−0.898590	−0.449295	−	0.893383i	$-0.648325\pi$
$431$	−8040.40	−0.898590	−0.449295	+	0.893383i	$0.648325\pi$
$432$	0	0
$433$	−3121.18	−0.346407	−0.173204	−	0.984886i	$-0.555412\pi$
$433$	−3121.18	−0.346407	−0.173204	+	0.984886i	$0.555412\pi$
$434$	−5186.07	−0.573593
$435$	0	0
$436$	6358.74	0.698460
$437$	1814.34	0.198608
$438$	0	0
$439$	−2373.05	−0.257994	−0.128997	−	0.991645i	$-0.541176\pi$
$439$	−2373.05	−0.257994	−0.128997	+	0.991645i	$0.541176\pi$
$440$	0	0
$441$	0	0
$442$	13005.7	1.39959
$443$	13152.7	1.41062	0.705311	−	0.708898i	$-0.250807\pi$
$443$	13152.7	1.41062	0.705311	+	0.708898i	$0.250807\pi$
$444$	0	0
$445$	0	0
$446$	14195.1	1.50708
$447$	0	0
$448$	219.142	0.0231105
$449$	−9265.76	−0.973893	−0.486947	−	0.873432i	$-0.661889\pi$
$449$	−9265.76	−0.973893	−0.486947	+	0.873432i	$0.661889\pi$
$450$	0	0
$451$	16112.5	1.68228
$452$	5402.94	0.562241
$453$	0	0
$454$	15780.8	1.63134
$455$	0	0
$456$	0	0
$457$	−6314.89	−0.646385	−0.323193	−	0.946333i	$-0.604756\pi$
$457$	−6314.89	−0.646385	−0.323193	+	0.946333i	$0.604756\pi$
$458$	−16574.5	−1.69099
$459$	0	0
$460$	0	0
$461$	4870.70	0.492084	0.246042	−	0.969259i	$-0.420870\pi$
$461$	4870.70	0.492084	0.246042	+	0.969259i	$0.420870\pi$
$462$	0	0
$463$	−4892.25	−0.491063	−0.245531	−	0.969389i	$-0.578962\pi$
$463$	−4892.25	−0.491063	−0.245531	+	0.969389i	$0.578962\pi$
$464$	3163.91	0.316553
$465$	0	0
$466$	−8178.97	−0.813055
$467$	6854.09	0.679164	0.339582	−	0.940577i	$-0.389714\pi$
$467$	6854.09	0.679164	0.339582	+	0.940577i	$0.389714\pi$
$468$	0	0
$469$	5792.61	0.570315
$470$	0	0
$471$	0	0
$472$	8118.79	0.791732
$473$	−7834.41	−0.761579
$474$	0	0
$475$	0	0
$476$	−1464.05	−0.140976
$477$	0	0
$478$	−12175.3	−1.16504
$479$	−8572.26	−0.817696	−0.408848	−	0.912602i	$-0.634069\pi$
$479$	−8572.26	−0.817696	−0.408848	+	0.912602i	$0.634069\pi$
$480$	0	0
$481$	9178.36	0.870057
$482$	21028.0	1.98713
$483$	0	0
$484$	11411.4	1.07170
$485$	0	0
$486$	0	0
$487$	5413.38	0.503704	0.251852	−	0.967766i	$-0.418960\pi$
$487$	5413.38	0.503704	0.251852	+	0.967766i	$0.418960\pi$
$488$	−5380.62	−0.499117
$489$	0	0
$490$	0	0
$491$	10314.3	0.948025	0.474012	−	0.880518i	$-0.342805\pi$
$491$	10314.3	0.948025	0.474012	+	0.880518i	$0.342805\pi$
$492$	0	0
$493$	−2050.62	−0.187333
$494$	3984.64	0.362909
$495$	0	0
$496$	19101.1	1.72916
$497$	−971.282	−0.0876619
$498$	0	0
$499$	−13527.2	−1.21355	−0.606774	−	0.794874i	$-0.707537\pi$
$499$	−13527.2	−1.21355	−0.606774	+	0.794874i	$0.707537\pi$
$500$	0	0
$501$	0	0
$502$	8517.08	0.757243
$503$	−245.421	−0.0217550	−0.0108775	−	0.999941i	$-0.503462\pi$
$503$	−245.421	−0.0217550	−0.0108775	+	0.999941i	$0.503462\pi$
$504$	0	0
$505$	0	0
$506$	25112.9	2.20634
$507$	0	0
$508$	−9269.96	−0.809622
$509$	−4680.90	−0.407617	−0.203809	−	0.979011i	$-0.565332\pi$
$509$	−4680.90	−0.407617	−0.203809	+	0.979011i	$0.565332\pi$
$510$	0	0
$511$	−4258.35	−0.368647
$512$	−7502.22	−0.647567
$513$	0	0
$514$	12752.9	1.09437
$515$	0	0
$516$	0	0
$517$	−5230.03	−0.444906
$518$	−2797.62	−0.237298
$519$	0	0
$520$	0	0
$521$	−10212.1	−0.858735	−0.429368	−	0.903130i	$-0.641264\pi$
$521$	−10212.1	−0.858735	−0.429368	+	0.903130i	$0.641264\pi$
$522$	0	0
$523$	8159.59	0.682207	0.341103	−	0.940026i	$-0.389199\pi$
$523$	8159.59	0.682207	0.341103	+	0.940026i	$0.389199\pi$
$524$	750.919	0.0626032
$525$	0	0
$526$	6067.38	0.502947
$527$	−12379.9	−1.02330
$528$	0	0
$529$	1031.66	0.0847913
$530$	0	0
$531$	0	0
$532$	−448.550	−0.0365547
$533$	18597.9	1.51138
$534$	0	0
$535$	0	0
$536$	−11281.4	−0.909108
$537$	0	0
$538$	−14556.6	−1.16650
$539$	18795.6	1.50201
$540$	0	0
$541$	−7889.11	−0.626949	−0.313474	−	0.949597i	$-0.601493\pi$
$541$	−7889.11	−0.626949	−0.313474	+	0.949597i	$0.601493\pi$
$542$	16284.8	1.29057
$543$	0	0
$544$	9731.58	0.766982
$545$	0	0
$546$	0	0
$547$	1394.62	0.109012	0.0545060	−	0.998513i	$-0.482642\pi$
$547$	1394.62	0.109012	0.0545060	+	0.998513i	$0.482642\pi$
$548$	3098.11	0.241505
$549$	0	0
$550$	0	0
$551$	−628.261	−0.0485750
$552$	0	0
$553$	−396.275	−0.0304726
$554$	−29684.1	−2.27645
$555$	0	0
$556$	9960.33	0.759734
$557$	19617.8	1.49234	0.746169	−	0.665756i	$-0.231891\pi$
$557$	19617.8	1.49234	0.746169	+	0.665756i	$0.231891\pi$
$558$	0	0
$559$	−9042.92	−0.684212
$560$	0	0
$561$	0	0
$562$	−8870.03	−0.665765
$563$	−1339.24	−0.100253	−0.0501263	−	0.998743i	$-0.515962\pi$
$563$	−1339.24	−0.100253	−0.0501263	+	0.998743i	$0.515962\pi$
$564$	0	0
$565$	0	0
$566$	−25403.6	−1.88656
$567$	0	0
$568$	1891.62	0.139737
$569$	26030.2	1.91783	0.958914	−	0.283696i	$-0.0915606\pi$
$569$	26030.2	1.91783	0.958914	+	0.283696i	$0.0915606\pi$
$570$	0	0
$571$	−10020.8	−0.734426	−0.367213	−	0.930137i	$-0.619688\pi$
$571$	−10020.8	−0.734426	−0.367213	+	0.930137i	$0.619688\pi$
$572$	20368.8	1.48892
$573$	0	0
$574$	−5668.76	−0.412211
$575$	0	0
$576$	0	0
$577$	−11192.7	−0.807550	−0.403775	−	0.914858i	$-0.632302\pi$
$577$	−11192.7	−0.807550	−0.403775	+	0.914858i	$0.632302\pi$
$578$	8034.75	0.578204
$579$	0	0
$580$	0	0
$581$	6042.41	0.431465
$582$	0	0
$583$	−20142.3	−1.43089
$584$	8293.35	0.587639
$585$	0	0
$586$	−17505.9	−1.23406
$587$	11087.1	0.779578	0.389789	−	0.920904i	$-0.372548\pi$
$587$	11087.1	0.779578	0.389789	+	0.920904i	$0.372548\pi$
$588$	0	0
$589$	−3792.92	−0.265339
$590$	0	0
$591$	0	0
$592$	10304.0	0.715361
$593$	14525.9	1.00592	0.502958	−	0.864311i	$-0.332245\pi$
$593$	14525.9	1.00592	0.502958	+	0.864311i	$0.332245\pi$
$594$	0	0
$595$	0	0
$596$	3141.76	0.215925
$597$	0	0
$598$	28986.8	1.98220
$599$	16502.5	1.12567	0.562833	−	0.826570i	$-0.309711\pi$
$599$	16502.5	1.12567	0.562833	+	0.826570i	$0.309711\pi$
$600$	0	0
$601$	15945.4	1.08224	0.541119	−	0.840946i	$-0.318001\pi$
$601$	15945.4	1.08224	0.541119	+	0.840946i	$0.318001\pi$
$602$	2756.34	0.186611
$603$	0	0
$604$	6747.89	0.454582
$605$	0	0
$606$	0	0
$607$	−935.699	−0.0625681	−0.0312841	−	0.999511i	$-0.509960\pi$
$607$	−935.699	−0.0625681	−0.0312841	+	0.999511i	$0.509960\pi$
$608$	2981.53	0.198877
$609$	0	0
$610$	0	0
$611$	−6036.80	−0.399710
$612$	0	0
$613$	375.796	0.0247606	0.0123803	−	0.999923i	$-0.496059\pi$
$613$	375.796	0.0247606	0.0123803	+	0.999923i	$0.496059\pi$
$614$	−20652.9	−1.35746
$615$	0	0
$616$	4393.80	0.287388
$617$	14932.2	0.974306	0.487153	−	0.873317i	$-0.338035\pi$
$617$	14932.2	0.974306	0.487153	+	0.873317i	$0.338035\pi$
$618$	0	0
$619$	11320.8	0.735090	0.367545	−	0.930006i	$-0.380198\pi$
$619$	11320.8	0.735090	0.367545	+	0.930006i	$0.380198\pi$
$620$	0	0
$621$	0	0
$622$	−215.095	−0.0138658
$623$	8110.38	0.521566
$624$	0	0
$625$	0	0
$626$	−2155.83	−0.137643
$627$	0	0
$628$	14228.6	0.904115
$629$	−6678.34	−0.423343
$630$	0	0
$631$	−15521.7	−0.979253	−0.489627	−	0.871932i	$-0.662867\pi$
$631$	−15521.7	−0.979253	−0.489627	+	0.871932i	$0.662867\pi$
$632$	771.765	0.0485746
$633$	0	0
$634$	−11199.7	−0.701571
$635$	0	0
$636$	0	0
$637$	21695.0	1.34943
$638$	−8695.98	−0.539619
$639$	0	0
$640$	0	0
$641$	−20110.1	−1.23916	−0.619579	−	0.784934i	$-0.712697\pi$
$641$	−20110.1	−1.23916	−0.619579	+	0.784934i	$0.712697\pi$
$642$	0	0
$643$	16073.4	0.985808	0.492904	−	0.870084i	$-0.335935\pi$
$643$	16073.4	0.985808	0.492904	+	0.870084i	$0.335935\pi$
$644$	−3263.04	−0.199661
$645$	0	0
$646$	−2899.29	−0.176581
$647$	−3841.57	−0.233427	−0.116714	−	0.993166i	$-0.537236\pi$
$647$	−3841.57	−0.233427	−0.116714	+	0.993166i	$0.537236\pi$
$648$	0	0
$649$	−42200.4	−2.55240
$650$	0	0
$651$	0	0
$652$	5824.30	0.349842
$653$	14694.4	0.880609	0.440304	−	0.897849i	$-0.354871\pi$
$653$	14694.4	0.880609	0.440304	+	0.897849i	$0.354871\pi$
$654$	0	0
$655$	0	0
$656$	20878.9	1.24266
$657$	0	0
$658$	1840.05	0.109016
$659$	−8601.76	−0.508463	−0.254231	−	0.967143i	$-0.581823\pi$
$659$	−8601.76	−0.508463	−0.254231	+	0.967143i	$0.581823\pi$
$660$	0	0
$661$	−21709.0	−1.27743	−0.638717	−	0.769442i	$-0.720534\pi$
$661$	−21709.0	−1.27743	−0.638717	+	0.769442i	$0.720534\pi$
$662$	7560.96	0.443905
$663$	0	0
$664$	−11767.9	−0.687775
$665$	0	0
$666$	0	0
$667$	−4570.36	−0.265315
$668$	−774.960	−0.0448864
$669$	0	0
$670$	0	0
$671$	27967.7	1.60907
$672$	0	0
$673$	13192.3	0.755611	0.377806	−	0.925885i	$-0.376679\pi$
$673$	13192.3	0.755611	0.377806	+	0.925885i	$0.376679\pi$
$674$	5293.85	0.302539
$675$	0	0
$676$	13218.7	0.752086
$677$	4885.74	0.277362	0.138681	−	0.990337i	$-0.455714\pi$
$677$	4885.74	0.277362	0.138681	+	0.990337i	$0.455714\pi$
$678$	0	0
$679$	−8514.72	−0.481245
$680$	0	0
$681$	0	0
$682$	−52499.1	−2.94765
$683$	17180.4	0.962502	0.481251	−	0.876583i	$-0.340182\pi$
$683$	17180.4	0.962502	0.481251	+	0.876583i	$0.340182\pi$
$684$	0	0
$685$	0	0
$686$	−14019.3	−0.780259
$687$	0	0
$688$	−10152.0	−0.562560
$689$	−23249.4	−1.28553
$690$	0	0
$691$	−21428.3	−1.17970	−0.589850	−	0.807513i	$-0.700813\pi$
$691$	−21428.3	−1.17970	−0.589850	+	0.807513i	$0.700813\pi$
$692$	−9046.57	−0.496964
$693$	0	0
$694$	27243.6	1.49013
$695$	0	0
$696$	0	0
$697$	−13532.2	−0.735391
$698$	4001.80	0.217006
$699$	0	0
$700$	0	0
$701$	9809.73	0.528543	0.264271	−	0.964448i	$-0.414869\pi$
$701$	9809.73	0.528543	0.264271	+	0.964448i	$0.414869\pi$
$702$	0	0
$703$	−2046.09	−0.109772
$704$	2218.40	0.118763
$705$	0	0
$706$	−44749.3	−2.38550
$707$	−6732.21	−0.358120
$708$	0	0
$709$	13541.9	0.717315	0.358658	−	0.933469i	$-0.383235\pi$
$709$	13541.9	0.717315	0.358658	+	0.933469i	$0.383235\pi$
$710$	0	0
$711$	0	0
$712$	−15795.4	−0.831399
$713$	−27592.1	−1.44927
$714$	0	0
$715$	0	0
$716$	2922.64	0.152548
$717$	0	0
$718$	8871.89	0.461137
$719$	−344.757	−0.0178822	−0.00894109	−	0.999960i	$-0.502846\pi$
$719$	−344.757	−0.0178822	−0.00894109	+	0.999960i	$0.502846\pi$
$720$	0	0
$721$	−3091.69	−0.159696
$722$	23540.4	1.21341
$723$	0	0
$724$	5336.40	0.273930
$725$	0	0
$726$	0	0
$727$	10425.2	0.531841	0.265921	−	0.963995i	$-0.414324\pi$
$727$	10425.2	0.531841	0.265921	+	0.963995i	$0.414324\pi$
$728$	5071.57	0.258193
$729$	0	0
$730$	0	0
$731$	6579.79	0.332917
$732$	0	0
$733$	−12660.4	−0.637956	−0.318978	−	0.947762i	$-0.603340\pi$
$733$	−12660.4	−0.637956	−0.318978	+	0.947762i	$0.603340\pi$
$734$	46165.0	2.32150
$735$	0	0
$736$	21689.5	1.08626
$737$	58639.2	2.93080
$738$	0	0
$739$	−35663.9	−1.77526	−0.887630	−	0.460557i	$-0.847650\pi$
$739$	−35663.9	−1.77526	−0.887630	+	0.460557i	$0.847650\pi$
$740$	0	0
$741$	0	0
$742$	7086.56	0.350614
$743$	−9880.98	−0.487884	−0.243942	−	0.969790i	$-0.578441\pi$
$743$	−9880.98	−0.487884	−0.243942	+	0.969790i	$0.578441\pi$
$744$	0	0
$745$	0	0
$746$	17970.6	0.881973
$747$	0	0
$748$	−14820.7	−0.724464
$749$	−10600.0	−0.517112
$750$	0	0
$751$	−7951.18	−0.386342	−0.193171	−	0.981165i	$-0.561877\pi$
$751$	−7951.18	−0.386342	−0.193171	+	0.981165i	$0.561877\pi$
$752$	−6777.18	−0.328641
$753$	0	0
$754$	−10037.4	−0.484801
$755$	0	0
$756$	0	0
$757$	26539.3	1.27422	0.637112	−	0.770771i	$-0.280129\pi$
$757$	26539.3	1.27422	0.637112	+	0.770771i	$0.280129\pi$
$758$	−1274.30	−0.0610614
$759$	0	0
$760$	0	0
$761$	−6411.70	−0.305419	−0.152710	−	0.988271i	$-0.548800\pi$
$761$	−6411.70	−0.305419	−0.152710	+	0.988271i	$0.548800\pi$
$762$	0	0
$763$	−8229.49	−0.390468
$764$	2192.70	0.103834
$765$	0	0
$766$	−3157.49	−0.148936
$767$	−48710.1	−2.29311
$768$	0	0
$769$	21232.0	0.995636	0.497818	−	0.867282i	$-0.334135\pi$
$769$	21232.0	0.995636	0.497818	+	0.867282i	$0.334135\pi$
$770$	0	0
$771$	0	0
$772$	−11614.1	−0.541450
$773$	12948.2	0.602478	0.301239	−	0.953549i	$-0.402600\pi$
$773$	12948.2	0.602478	0.301239	+	0.953549i	$0.402600\pi$
$774$	0	0
$775$	0	0
$776$	16582.8	0.767126
$777$	0	0
$778$	−29507.2	−1.35975
$779$	−4145.94	−0.190685
$780$	0	0
$781$	−9832.39	−0.450487
$782$	−21091.3	−0.964478
$783$	0	0
$784$	24355.8	1.10950
$785$	0	0
$786$	0	0
$787$	16388.9	0.742315	0.371158	−	0.928570i	$-0.378961\pi$
$787$	16388.9	0.742315	0.371158	+	0.928570i	$0.378961\pi$
$788$	11251.9	0.508672
$789$	0	0
$790$	0	0
$791$	−6992.48	−0.314316
$792$	0	0
$793$	32281.9	1.44561
$794$	15936.4	0.712294
$795$	0	0
$796$	−4867.45	−0.216737
$797$	−22291.8	−0.990734	−0.495367	−	0.868684i	$-0.664966\pi$
$797$	−22291.8	−0.990734	−0.495367	+	0.868684i	$0.664966\pi$
$798$	0	0
$799$	4392.48	0.194486
$800$	0	0
$801$	0	0
$802$	37604.0	1.65567
$803$	−43107.8	−1.89445
$804$	0	0
$805$	0	0
$806$	−60597.4	−2.64821
$807$	0	0
$808$	13111.3	0.570859
$809$	−36536.8	−1.58784	−0.793921	−	0.608020i	$-0.791964\pi$
$809$	−36536.8	−1.58784	−0.793921	+	0.608020i	$0.791964\pi$
$810$	0	0
$811$	−22132.8	−0.958309	−0.479155	−	0.877730i	$-0.659057\pi$
$811$	−22132.8	−0.958309	−0.479155	+	0.877730i	$0.659057\pi$
$812$	1129.91	0.0488325
$813$	0	0
$814$	−28320.6	−1.21945
$815$	0	0
$816$	0	0
$817$	2015.89	0.0863245
$818$	18635.6	0.796552
$819$	0	0
$820$	0	0
$821$	−28628.6	−1.21698	−0.608492	−	0.793560i	$-0.708225\pi$
$821$	−28628.6	−1.21698	−0.608492	+	0.793560i	$0.708225\pi$
$822$	0	0
$823$	−42695.0	−1.80833	−0.904164	−	0.427185i	$-0.859505\pi$
$823$	−42695.0	−1.80833	−0.904164	+	0.427185i	$0.859505\pi$
$824$	6021.21	0.254562
$825$	0	0
$826$	14847.1	0.625421
$827$	−25905.2	−1.08925	−0.544627	−	0.838678i	$-0.683329\pi$
$827$	−25905.2	−1.08925	−0.544627	+	0.838678i	$0.683329\pi$
$828$	0	0
$829$	−9952.93	−0.416984	−0.208492	−	0.978024i	$-0.566856\pi$
$829$	−9952.93	−0.416984	−0.208492	+	0.978024i	$0.566856\pi$
$830$	0	0
$831$	0	0
$832$	2560.60	0.106698
$833$	−15785.6	−0.656591
$834$	0	0
$835$	0	0
$836$	−4540.72	−0.187852
$837$	0	0
$838$	13903.6	0.573142
$839$	16474.0	0.677886	0.338943	−	0.940807i	$-0.389931\pi$
$839$	16474.0	0.677886	0.338943	+	0.940807i	$0.389931\pi$
$840$	0	0
$841$	−22806.4	−0.935110
$842$	−22348.1	−0.914689
$843$	0	0
$844$	15515.5	0.632781
$845$	0	0
$846$	0	0
$847$	−14768.7	−0.599124
$848$	−26100.9	−1.05697
$849$	0	0
$850$	0	0
$851$	−14884.5	−0.599571
$852$	0	0
$853$	−22391.0	−0.898772	−0.449386	−	0.893338i	$-0.648357\pi$
$853$	−22391.0	−0.898772	−0.449386	+	0.893338i	$0.648357\pi$
$854$	−9839.73	−0.394272
$855$	0	0
$856$	20644.1	0.824299
$857$	−15349.4	−0.611814	−0.305907	−	0.952061i	$-0.598960\pi$
$857$	−15349.4	−0.611814	−0.305907	+	0.952061i	$0.598960\pi$
$858$	0	0
$859$	−5218.87	−0.207294	−0.103647	−	0.994614i	$-0.533051\pi$
$859$	−5218.87	−0.207294	−0.103647	+	0.994614i	$0.533051\pi$
$860$	0	0
$861$	0	0
$862$	28636.3	1.13150
$863$	−12101.4	−0.477330	−0.238665	−	0.971102i	$-0.576710\pi$
$863$	−12101.4	−0.477330	−0.238665	+	0.971102i	$0.576710\pi$
$864$	0	0
$865$	0	0
$866$	11116.2	0.436196
$867$	0	0
$868$	6821.45	0.266746
$869$	−4011.53	−0.156596
$870$	0	0
$871$	67684.6	2.63307
$872$	16027.3	0.622424
$873$	0	0
$874$	−6461.87	−0.250087
$875$	0	0
$876$	0	0
$877$	24268.8	0.934435	0.467218	−	0.884142i	$-0.345256\pi$
$877$	24268.8	0.934435	0.467218	+	0.884142i	$0.345256\pi$
$878$	8451.75	0.324866
$879$	0	0
$880$	0	0
$881$	24286.6	0.928758	0.464379	−	0.885637i	$-0.346278\pi$
$881$	24286.6	0.928758	0.464379	+	0.885637i	$0.346278\pi$
$882$	0	0
$883$	−48922.2	−1.86451	−0.932255	−	0.361801i	$-0.882162\pi$
$883$	−48922.2	−1.86451	−0.932255	+	0.361801i	$0.882162\pi$
$884$	−17106.9	−0.650868
$885$	0	0
$886$	−46844.2	−1.77626
$887$	1678.06	0.0635215	0.0317608	−	0.999495i	$-0.489889\pi$
$887$	1678.06	0.0635215	0.0317608	+	0.999495i	$0.489889\pi$
$888$	0	0
$889$	11997.2	0.452613
$890$	0	0
$891$	0	0
$892$	−18671.4	−0.700859
$893$	1345.75	0.0504299
$894$	0	0
$895$	0	0
$896$	8376.54	0.312322
$897$	0	0
$898$	33000.5	1.22633
$899$	9554.45	0.354459
$900$	0	0
$901$	16916.7	0.625501
$902$	−57385.4	−2.11832
$903$	0	0
$904$	13618.2	0.501034
$905$	0	0
$906$	0	0
$907$	14710.6	0.538541	0.269271	−	0.963065i	$-0.413217\pi$
$907$	14710.6	0.538541	0.269271	+	0.963065i	$0.413217\pi$
$908$	−20757.2	−0.758646
$909$	0	0
$910$	0	0
$911$	30257.7	1.10042	0.550209	−	0.835027i	$-0.314548\pi$
$911$	30257.7	1.10042	0.550209	+	0.835027i	$0.314548\pi$
$912$	0	0
$913$	61168.0	2.21727
$914$	22490.8	0.813928
$915$	0	0
$916$	21801.1	0.786385
$917$	−971.839	−0.0349978
$918$	0	0
$919$	51369.1	1.84386	0.921931	−	0.387355i	$-0.126611\pi$
$919$	51369.1	1.84386	0.921931	+	0.387355i	$0.126611\pi$
$920$	0	0
$921$	0	0
$922$	−17347.2	−0.619632
$923$	−11349.1	−0.404724
$924$	0	0
$925$	0	0
$926$	17424.0	0.618346
$927$	0	0
$928$	−7510.54	−0.265674
$929$	40968.5	1.44686	0.723430	−	0.690398i	$-0.242564\pi$
$929$	40968.5	1.44686	0.723430	+	0.690398i	$0.242564\pi$
$930$	0	0
$931$	−4836.35	−0.170252
$932$	10758.1	0.378106
$933$	0	0
$934$	−24411.2	−0.855202
$935$	0	0
$936$	0	0
$937$	−26835.8	−0.935632	−0.467816	−	0.883826i	$-0.654959\pi$
$937$	−26835.8	−0.935632	−0.467816	+	0.883826i	$0.654959\pi$
$938$	−20630.7	−0.718140
$939$	0	0
$940$	0	0
$941$	41221.5	1.42803	0.714017	−	0.700128i	$-0.246874\pi$
$941$	41221.5	1.42803	0.714017	+	0.700128i	$0.246874\pi$
$942$	0	0
$943$	−30160.2	−1.04152
$944$	−54684.1	−1.88540
$945$	0	0
$946$	27902.7	0.958979
$947$	14488.5	0.497164	0.248582	−	0.968611i	$-0.420035\pi$
$947$	14488.5	0.497164	0.248582	+	0.968611i	$0.420035\pi$
$948$	0	0
$949$	−49757.4	−1.70200
$950$	0	0
$951$	0	0
$952$	−3690.16	−0.125629
$953$	32207.2	1.09475	0.547373	−	0.836889i	$-0.315628\pi$
$953$	32207.2	1.09475	0.547373	+	0.836889i	$0.315628\pi$
$954$	0	0
$955$	0	0
$956$	16014.7	0.541792
$957$	0	0
$958$	30530.5	1.02964
$959$	−4009.57	−0.135011
$960$	0	0
$961$	27890.8	0.936217
$962$	−32689.2	−1.09557
$963$	0	0
$964$	−27659.0	−0.924104
$965$	0	0
$966$	0	0
$967$	−20552.1	−0.683464	−0.341732	−	0.939797i	$-0.611014\pi$
$967$	−20552.1	−0.683464	−0.341732	+	0.939797i	$0.611014\pi$
$968$	28762.7	0.955030
$969$	0	0
$970$	0	0
$971$	−23686.9	−0.782852	−0.391426	−	0.920210i	$-0.628018\pi$
$971$	−23686.9	−0.782852	−0.391426	+	0.920210i	$0.628018\pi$
$972$	0	0
$973$	−12890.7	−0.424723
$974$	−19280.0	−0.634263
$975$	0	0
$976$	36241.2	1.18858
$977$	−12597.1	−0.412506	−0.206253	−	0.978499i	$-0.566127\pi$
$977$	−12597.1	−0.412506	−0.206253	+	0.978499i	$0.566127\pi$
$978$	0	0
$979$	82102.2	2.68028
$980$	0	0
$981$	0	0
$982$	−36735.1	−1.19375
$983$	−51076.9	−1.65728	−0.828638	−	0.559785i	$-0.810884\pi$
$983$	−51076.9	−1.65728	−0.828638	+	0.559785i	$0.810884\pi$
$984$	0	0
$985$	0	0
$986$	7303.38	0.235889
$987$	0	0
$988$	−5241.16	−0.168769
$989$	14664.9	0.471502
$990$	0	0
$991$	27888.6	0.893957	0.446979	−	0.894545i	$-0.352500\pi$
$991$	27888.6	0.893957	0.446979	+	0.894545i	$0.352500\pi$
$992$	−45342.4	−1.45123
$993$	0	0
$994$	3459.27	0.110384
$995$	0	0
$996$	0	0
$997$	−54930.0	−1.74488	−0.872442	−	0.488717i	$-0.837465\pi$
$997$	−54930.0	−1.74488	−0.872442	+	0.488717i	$0.837465\pi$
$998$	48177.8	1.52810
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.t.1.1		4
3.2	odd	2		675.4.a.ba.1.3		4
5.2	odd	4		135.4.b.b.109.1	✓	8
5.3	odd	4		135.4.b.b.109.7	yes	8
5.4	even	2		675.4.a.ba.1.4		4
15.2	even	4		135.4.b.b.109.8	yes	8
15.8	even	4		135.4.b.b.109.2	yes	8
15.14	odd	2	inner	675.4.a.t.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.b.109.1	✓	8	5.2	odd	4
135.4.b.b.109.2	yes	8	15.8	even	4
135.4.b.b.109.7	yes	8	5.3	odd	4
135.4.b.b.109.8	yes	8	15.2	even	4
675.4.a.t.1.1		4	1.1	even	1	trivial
675.4.a.t.1.2		4	15.14	odd	2	inner
675.4.a.ba.1.3		4	3.2	odd	2
675.4.a.ba.1.4		4	5.4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-3.56155 q^{2} +4.68466 q^{4} -6.06288 q^{7} +11.8078 q^{8} -61.3752 q^{11} -70.8427 q^{13} +21.5933 q^{14} -79.5312 q^{16} +51.5464 q^{17} +15.7926 q^{19} +218.591 q^{22} +114.885 q^{23} +252.310 q^{26} -28.4025 q^{28} -39.7819 q^{29} -240.170 q^{31} +188.793 q^{32} -183.585 q^{34} -129.560 q^{37} -56.2462 q^{38} -262.524 q^{41} +127.648 q^{43} -287.522 q^{44} -409.170 q^{46} +85.2140 q^{47} -306.241 q^{49} -331.874 q^{52} +328.184 q^{53} -71.5891 q^{56} +141.685 q^{58} +687.580 q^{59} -455.685 q^{61} +855.380 q^{62} -36.1449 q^{64} -955.421 q^{67} +241.477 q^{68} +160.201 q^{71} +702.364 q^{73} +461.434 q^{74} +73.9830 q^{76} +372.111 q^{77} +65.3608 q^{79} +934.993 q^{82} -996.623 q^{83} -454.624 q^{86} -724.704 q^{88} -1337.71 q^{89} +429.511 q^{91} +538.199 q^{92} -303.494 q^{94} +1404.40 q^{97} +1090.70 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} - 6 q^{4} + 6 q^{8} - 46 q^{16} + 66 q^{17} - 110 q^{19} + 6 q^{23} - 268 q^{31} + 582 q^{32} - 388 q^{34} - 192 q^{38} - 944 q^{46} + 1116 q^{47} + 482 q^{49} + 1824 q^{53} - 1798 q^{61}+ \cdots + 2796 q^{98}+O(q^{100}) \) 4 * q - 6 * q^2 - 6 * q^4 + 6 * q^8 - 46 * q^16 + 66 * q^17 - 110 * q^19 + 6 * q^23 - 268 * q^31 + 582 * q^32 - 388 * q^34 - 192 * q^38 - 944 * q^46 + 1116 * q^47 + 482 * q^49 + 1824 * q^53 - 1798 * q^61 + 1830 * q^62 - 862 * q^64 + 768 * q^68 + 1236 * q^76 + 4086 * q^77 + 682 * q^79 + 384 * q^83 - 360 * q^91 + 2796 * q^92 - 76 * q^94 + 2796 * q^98

Introduction

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