LMFDB - Embedded newform 675.4.a.z.1.4

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			1.4
Root			$-2.82516$ 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)

$f(q)$	$=$	$q+4.52654 q^{2} +12.4896 q^{4} -30.8119 q^{7} +20.3223 q^{8} +6.79573 q^{11} -31.2493 q^{13} -139.471 q^{14} -7.92702 q^{16} +112.418 q^{17} -60.9161 q^{19} +30.7612 q^{22} -31.1821 q^{23} -141.451 q^{26} -384.827 q^{28} -189.228 q^{29} -343.933 q^{31} -198.460 q^{32} +508.865 q^{34} -206.503 q^{37} -275.739 q^{38} -435.018 q^{41} -60.9569 q^{43} +84.8758 q^{44} -141.147 q^{46} +251.239 q^{47} +606.370 q^{49} -390.291 q^{52} +248.620 q^{53} -626.167 q^{56} -856.547 q^{58} +571.583 q^{59} -329.038 q^{61} -1556.83 q^{62} -834.922 q^{64} +677.273 q^{67} +1404.05 q^{68} +453.668 q^{71} -1024.66 q^{73} -934.745 q^{74} -760.817 q^{76} -209.389 q^{77} +238.142 q^{79} -1969.13 q^{82} +826.853 q^{83} -275.924 q^{86} +138.105 q^{88} +1140.55 q^{89} +962.849 q^{91} -389.451 q^{92} +1137.24 q^{94} +1308.77 q^{97} +2744.76 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.52654	1.60037	0.800187	−	0.599750i	$-0.204733\pi$
$2$	4.52654	1.60037	0.800187	+	0.599750i	$0.204733\pi$
$3$	0	0
$3$	0	0
$4$	12.4896	1.56120
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−30.8119	−1.66368	−0.831842	−	0.555013i	$-0.812713\pi$
$7$	−30.8119	−1.66368	−0.831842	+	0.555013i	$0.812713\pi$
$8$	20.3223	0.898126
$9$	0	0
$10$	0	0
$11$	6.79573	0.186272	0.0931359	−	0.995653i	$-0.470311\pi$
$11$	6.79573	0.186272	0.0931359	+	0.995653i	$0.470311\pi$
$12$	0	0
$13$	−31.2493	−0.666692	−0.333346	−	0.942805i	$-0.608178\pi$
$13$	−31.2493	−0.666692	−0.333346	+	0.942805i	$0.608178\pi$
$14$	−139.471	−2.66252
$15$	0	0
$16$	−7.92702	−0.123860
$17$	112.418	1.60385	0.801923	−	0.597427i	$-0.203810\pi$
$17$	112.418	1.60385	0.801923	+	0.597427i	$0.203810\pi$
$18$	0	0
$19$	−60.9161	−0.735533	−0.367766	−	0.929918i	$-0.619877\pi$
$19$	−60.9161	−0.735533	−0.367766	+	0.929918i	$0.619877\pi$
$20$	0	0
$21$	0	0
$22$	30.7612	0.298105
$23$	−31.1821	−0.282692	−0.141346	−	0.989960i	$-0.545143\pi$
$23$	−31.1821	−0.282692	−0.141346	+	0.989960i	$0.545143\pi$
$24$	0	0
$25$	0	0
$26$	−141.451	−1.06696
$27$	0	0
$28$	−384.827	−2.59734
$29$	−189.228	−1.21168	−0.605839	−	0.795587i	$-0.707163\pi$
$29$	−189.228	−1.21168	−0.605839	+	0.795587i	$0.707163\pi$
$30$	0	0
$31$	−343.933	−1.99265	−0.996327	−	0.0856357i	$-0.972708\pi$
$31$	−343.933	−1.99265	−0.996327	+	0.0856357i	$0.972708\pi$
$32$	−198.460	−1.09635
$33$	0	0
$34$	508.865	2.56675
$35$	0	0
$36$	0	0
$37$	−206.503	−0.917538	−0.458769	−	0.888556i	$-0.651709\pi$
$37$	−206.503	−0.917538	−0.458769	+	0.888556i	$0.651709\pi$
$38$	−275.739	−1.17713
$39$	0	0
$40$	0	0
$41$	−435.018	−1.65704	−0.828518	−	0.559963i	$-0.810816\pi$
$41$	−435.018	−1.65704	−0.828518	+	0.559963i	$0.810816\pi$
$42$	0	0
$43$	−60.9569	−0.216183	−0.108091	−	0.994141i	$-0.534474\pi$
$43$	−60.9569	−0.216183	−0.108091	+	0.994141i	$0.534474\pi$
$44$	84.8758	0.290807
$45$	0	0
$46$	−141.147	−0.452412
$47$	251.239	0.779721	0.389861	−	0.920874i	$-0.372523\pi$
$47$	251.239	0.779721	0.389861	+	0.920874i	$0.372523\pi$
$48$	0	0
$49$	606.370	1.76784
$50$	0	0
$51$	0	0
$52$	−390.291	−1.04084
$53$	248.620	0.644350	0.322175	−	0.946680i	$-0.395586\pi$
$53$	248.620	0.644350	0.322175	+	0.946680i	$0.395586\pi$
$54$	0	0
$55$	0	0
$56$	−626.167	−1.49420
$57$	0	0
$58$	−856.547	−1.93914
$59$	571.583	1.26125	0.630626	−	0.776087i	$-0.282798\pi$
$59$	571.583	1.26125	0.630626	+	0.776087i	$0.282798\pi$
$60$	0	0
$61$	−329.038	−0.690639	−0.345319	−	0.938485i	$-0.612229\pi$
$61$	−329.038	−0.690639	−0.345319	+	0.938485i	$0.612229\pi$
$62$	−1556.83	−3.18899
$63$	0	0
$64$	−834.922	−1.63071
$65$	0	0
$66$	0	0
$67$	677.273	1.23496	0.617478	−	0.786588i	$-0.288155\pi$
$67$	677.273	1.23496	0.617478	+	0.786588i	$0.288155\pi$
$68$	1404.05	2.50392
$69$	0	0
$70$	0	0
$71$	453.668	0.758317	0.379159	−	0.925332i	$-0.376213\pi$
$71$	453.668	0.758317	0.379159	+	0.925332i	$0.376213\pi$
$72$	0	0
$73$	−1024.66	−1.64283	−0.821417	−	0.570328i	$-0.806816\pi$
$73$	−1024.66	−1.64283	−0.821417	+	0.570328i	$0.806816\pi$
$74$	−934.745	−1.46840
$75$	0	0
$76$	−760.817	−1.14831
$77$	−209.389	−0.309897
$78$	0	0
$79$	238.142	0.339153	0.169576	−	0.985517i	$-0.445760\pi$
$79$	238.142	0.339153	0.169576	+	0.985517i	$0.445760\pi$
$80$	0	0
$81$	0	0
$82$	−1969.13	−2.65188
$83$	826.853	1.09348	0.546740	−	0.837302i	$-0.315869\pi$
$83$	826.853	1.09348	0.546740	+	0.837302i	$0.315869\pi$
$84$	0	0
$85$	0	0
$86$	−275.924	−0.345973
$87$	0	0
$88$	138.105	0.167296
$89$	1140.55	1.35840	0.679201	−	0.733953i	$-0.262327\pi$
$89$	1140.55	1.35840	0.679201	+	0.733953i	$0.262327\pi$
$90$	0	0
$91$	962.849	1.10916
$92$	−389.451	−0.441337
$93$	0	0
$94$	1137.24	1.24785
$95$	0	0
$96$	0	0
$97$	1308.77	1.36995	0.684977	−	0.728565i	$-0.259812\pi$
$97$	1308.77	1.36995	0.684977	+	0.728565i	$0.259812\pi$
$98$	2744.76	2.82921
$99$	0	0
$100$	0	0
$101$	−1448.64	−1.42718	−0.713589	−	0.700565i	$-0.752931\pi$
$101$	−1448.64	−1.42718	−0.713589	+	0.700565i	$0.752931\pi$
$102$	0	0
$103$	1355.83	1.29703	0.648513	−	0.761204i	$-0.275391\pi$
$103$	1355.83	1.29703	0.648513	+	0.761204i	$0.275391\pi$
$104$	−635.057	−0.598773
$105$	0	0
$106$	1125.39	1.03120
$107$	−390.807	−0.353091	−0.176545	−	0.984292i	$-0.556492\pi$
$107$	−390.807	−0.353091	−0.176545	+	0.984292i	$0.556492\pi$
$108$	0	0
$109$	770.512	0.677080	0.338540	−	0.940952i	$-0.390067\pi$
$109$	770.512	0.677080	0.338540	+	0.940952i	$0.390067\pi$
$110$	0	0
$111$	0	0
$112$	244.246	0.206063
$113$	−1625.10	−1.35289	−0.676446	−	0.736493i	$-0.736481\pi$
$113$	−1625.10	−1.35289	−0.676446	+	0.736493i	$0.736481\pi$
$114$	0	0
$115$	0	0
$116$	−2363.37	−1.89167
$117$	0	0
$118$	2587.30	2.01847
$119$	−3463.81	−2.66829
$120$	0	0
$121$	−1284.82	−0.965303
$122$	−1489.40	−1.10528
$123$	0	0
$124$	−4295.58	−3.11093
$125$	0	0
$126$	0	0
$127$	1615.40	1.12869	0.564346	−	0.825538i	$-0.309128\pi$
$127$	1615.40	1.12869	0.564346	+	0.825538i	$0.309128\pi$
$128$	−2191.63	−1.51339
$129$	0	0
$130$	0	0
$131$	−51.2371	−0.0341726	−0.0170863	−	0.999854i	$-0.505439\pi$
$131$	−51.2371	−0.0341726	−0.0170863	+	0.999854i	$0.505439\pi$
$132$	0	0
$133$	1876.94	1.22369
$134$	3065.70	1.97639
$135$	0	0
$136$	2284.59	1.44046
$137$	1601.46	0.998703	0.499351	−	0.866400i	$-0.333572\pi$
$137$	1601.46	0.998703	0.499351	+	0.866400i	$0.333572\pi$
$138$	0	0
$139$	−436.476	−0.266341	−0.133170	−	0.991093i	$-0.542516\pi$
$139$	−436.476	−0.266341	−0.133170	+	0.991093i	$0.542516\pi$
$140$	0	0
$141$	0	0
$142$	2053.55	1.21359
$143$	−212.362	−0.124186
$144$	0	0
$145$	0	0
$146$	−4638.15	−2.62915
$147$	0	0
$148$	−2579.14	−1.43246
$149$	−1447.05	−0.795615	−0.397807	−	0.917469i	$-0.630229\pi$
$149$	−1447.05	−0.795615	−0.397807	+	0.917469i	$0.630229\pi$
$150$	0	0
$151$	−748.431	−0.403354	−0.201677	−	0.979452i	$-0.564639\pi$
$151$	−748.431	−0.403354	−0.201677	+	0.979452i	$0.564639\pi$
$152$	−1237.95	−0.660601
$153$	0	0
$154$	−947.809	−0.495952
$155$	0	0
$156$	0	0
$157$	−987.755	−0.502111	−0.251056	−	0.967973i	$-0.580778\pi$
$157$	−987.755	−0.502111	−0.251056	+	0.967973i	$0.580778\pi$
$158$	1077.96	0.542771
$159$	0	0
$160$	0	0
$161$	960.777	0.470309
$162$	0	0
$163$	2947.21	1.41621	0.708107	−	0.706105i	$-0.249549\pi$
$163$	2947.21	1.41621	0.708107	+	0.706105i	$0.249549\pi$
$164$	−5433.20	−2.58696
$165$	0	0
$166$	3742.78	1.74998
$167$	−445.456	−0.206410	−0.103205	−	0.994660i	$-0.532910\pi$
$167$	−445.456	−0.206410	−0.103205	+	0.994660i	$0.532910\pi$
$168$	0	0
$169$	−1220.48	−0.555522
$170$	0	0
$171$	0	0
$172$	−761.327	−0.337504
$173$	−4548.52	−1.99895	−0.999473	−	0.0324524i	$-0.989668\pi$
$173$	−4548.52	−1.99895	−0.999473	+	0.0324524i	$0.989668\pi$
$174$	0	0
$175$	0	0
$176$	−53.8699	−0.0230716
$177$	0	0
$178$	5162.73	2.17395
$179$	−1866.03	−0.779181	−0.389591	−	0.920988i	$-0.627384\pi$
$179$	−1866.03	−0.779181	−0.389591	+	0.920988i	$0.627384\pi$
$180$	0	0
$181$	−273.461	−0.112300	−0.0561498	−	0.998422i	$-0.517882\pi$
$181$	−273.461	−0.112300	−0.0561498	+	0.998422i	$0.517882\pi$
$182$	4358.38	1.77508
$183$	0	0
$184$	−633.690	−0.253893
$185$	0	0
$186$	0	0
$187$	763.963	0.298751
$188$	3137.86	1.21730
$189$	0	0
$190$	0	0
$191$	1935.50	0.733235	0.366618	−	0.930372i	$-0.380516\pi$
$191$	1935.50	0.733235	0.366618	+	0.930372i	$0.380516\pi$
$192$	0	0
$193$	1428.51	0.532779	0.266389	−	0.963866i	$-0.414169\pi$
$193$	1428.51	0.532779	0.266389	+	0.963866i	$0.414169\pi$
$194$	5924.21	2.19244
$195$	0	0
$196$	7573.31	2.75995
$197$	−2779.85	−1.00536	−0.502680	−	0.864473i	$-0.667653\pi$
$197$	−2779.85	−1.00536	−0.502680	+	0.864473i	$0.667653\pi$
$198$	0	0
$199$	2155.10	0.767695	0.383847	−	0.923397i	$-0.374599\pi$
$199$	2155.10	0.767695	0.383847	+	0.923397i	$0.374599\pi$
$200$	0	0
$201$	0	0
$202$	−6557.32	−2.28402
$203$	5830.45	2.01585
$204$	0	0
$205$	0	0
$206$	6137.21	2.07573
$207$	0	0
$208$	247.714	0.0825763
$209$	−413.970	−0.137009
$210$	0	0
$211$	−4577.31	−1.49344	−0.746718	−	0.665140i	$-0.768372\pi$
$211$	−4577.31	−1.49344	−0.746718	+	0.665140i	$0.768372\pi$
$212$	3105.16	1.00596
$213$	0	0
$214$	−1769.00	−0.565078
$215$	0	0
$216$	0	0
$217$	10597.2	3.31514
$218$	3487.76	1.08358
$219$	0	0
$220$	0	0
$221$	−3512.99	−1.06927
$222$	0	0
$223$	−892.931	−0.268140	−0.134070	−	0.990972i	$-0.542805\pi$
$223$	−892.931	−0.268140	−0.134070	+	0.990972i	$0.542805\pi$
$224$	6114.93	1.82398
$225$	0	0
$226$	−7356.09	−2.16513
$227$	1036.61	0.303093	0.151546	−	0.988450i	$-0.451575\pi$
$227$	1036.61	0.303093	0.151546	+	0.988450i	$0.451575\pi$
$228$	0	0
$229$	1450.92	0.418688	0.209344	−	0.977842i	$-0.432867\pi$
$229$	1450.92	0.418688	0.209344	+	0.977842i	$0.432867\pi$
$230$	0	0
$231$	0	0
$232$	−3845.53	−1.08824
$233$	−974.736	−0.274065	−0.137032	−	0.990567i	$-0.543756\pi$
$233$	−974.736	−0.274065	−0.137032	+	0.990567i	$0.543756\pi$
$234$	0	0
$235$	0	0
$236$	7138.84	1.96906
$237$	0	0
$238$	−15679.1	−4.27027
$239$	−3761.95	−1.01816	−0.509080	−	0.860719i	$-0.670014\pi$
$239$	−3761.95	−1.01816	−0.509080	+	0.860719i	$0.670014\pi$
$240$	0	0
$241$	−356.827	−0.0953744	−0.0476872	−	0.998862i	$-0.515185\pi$
$241$	−356.827	−0.0953744	−0.0476872	+	0.998862i	$0.515185\pi$
$242$	−5815.78	−1.54485
$243$	0	0
$244$	−4109.54	−1.07822
$245$	0	0
$246$	0	0
$247$	1903.59	0.490374
$248$	−6989.51	−1.78965
$249$	0	0
$250$	0	0
$251$	3161.78	0.795098	0.397549	−	0.917581i	$-0.369861\pi$
$251$	3161.78	0.795098	0.397549	+	0.917581i	$0.369861\pi$
$252$	0	0
$253$	−211.905	−0.0526575
$254$	7312.19	1.80633
$255$	0	0
$256$	−3241.12	−0.791289
$257$	−1683.47	−0.408607	−0.204304	−	0.978908i	$-0.565493\pi$
$257$	−1683.47	−0.408607	−0.204304	+	0.978908i	$0.565493\pi$
$258$	0	0
$259$	6362.74	1.52649
$260$	0	0
$261$	0	0
$262$	−231.927	−0.0546889
$263$	−5030.48	−1.17944	−0.589720	−	0.807608i	$-0.700762\pi$
$263$	−5030.48	−1.17944	−0.589720	+	0.807608i	$0.700762\pi$
$264$	0	0
$265$	0	0
$266$	8496.04	1.95837
$267$	0	0
$268$	8458.85	1.92801
$269$	6050.81	1.37147	0.685733	−	0.727853i	$-0.259482\pi$
$269$	6050.81	1.37147	0.685733	+	0.727853i	$0.259482\pi$
$270$	0	0
$271$	−2836.60	−0.635834	−0.317917	−	0.948119i	$-0.602983\pi$
$271$	−2836.60	−0.635834	−0.317917	+	0.948119i	$0.602983\pi$
$272$	−891.140	−0.198652
$273$	0	0
$274$	7249.09	1.59830
$275$	0	0
$276$	0	0
$277$	−1917.59	−0.415946	−0.207973	−	0.978135i	$-0.566687\pi$
$277$	−1917.59	−0.415946	−0.207973	+	0.978135i	$0.566687\pi$
$278$	−1975.73	−0.426245
$279$	0	0
$280$	0	0
$281$	−4694.31	−0.996580	−0.498290	−	0.867011i	$-0.666038\pi$
$281$	−4694.31	−0.996580	−0.498290	+	0.867011i	$0.666038\pi$
$282$	0	0
$283$	−7108.90	−1.49322	−0.746608	−	0.665265i	$-0.768319\pi$
$283$	−7108.90	−1.49322	−0.746608	+	0.665265i	$0.768319\pi$
$284$	5666.13	1.18388
$285$	0	0
$286$	−961.265	−0.198744
$287$	13403.7	2.75678
$288$	0	0
$289$	7724.82	1.57232
$290$	0	0
$291$	0	0
$292$	−12797.5	−2.56479
$293$	−2401.20	−0.478770	−0.239385	−	0.970925i	$-0.576946\pi$
$293$	−2401.20	−0.478770	−0.239385	+	0.970925i	$0.576946\pi$
$294$	0	0
$295$	0	0
$296$	−4196.61	−0.824064
$297$	0	0
$298$	−6550.11	−1.27328
$299$	974.417	0.188468
$300$	0	0
$301$	1878.20	0.359659
$302$	−3387.80	−0.645517
$303$	0	0
$304$	482.883	0.0911029
$305$	0	0
$306$	0	0
$307$	−1623.62	−0.301840	−0.150920	−	0.988546i	$-0.548224\pi$
$307$	−1623.62	−0.301840	−0.150920	+	0.988546i	$0.548224\pi$
$308$	−2615.18	−0.483811
$309$	0	0
$310$	0	0
$311$	2062.80	0.376111	0.188056	−	0.982158i	$-0.439781\pi$
$311$	2062.80	0.376111	0.188056	+	0.982158i	$0.439781\pi$
$312$	0	0
$313$	207.093	0.0373980	0.0186990	−	0.999825i	$-0.494048\pi$
$313$	207.093	0.0373980	0.0186990	+	0.999825i	$0.494048\pi$
$314$	−4471.12	−0.803566
$315$	0	0
$316$	2974.29	0.529484
$317$	699.087	0.123863	0.0619316	−	0.998080i	$-0.480274\pi$
$317$	699.087	0.123863	0.0619316	+	0.998080i	$0.480274\pi$
$318$	0	0
$319$	−1285.94	−0.225702
$320$	0	0
$321$	0	0
$322$	4349.00	0.752671
$323$	−6848.07	−1.17968
$324$	0	0
$325$	0	0
$326$	13340.6	2.26647
$327$	0	0
$328$	−8840.56	−1.48823
$329$	−7741.12	−1.29721
$330$	0	0
$331$	2662.58	0.442140	0.221070	−	0.975258i	$-0.429045\pi$
$331$	2662.58	0.442140	0.221070	+	0.975258i	$0.429045\pi$
$332$	10327.0	1.70714
$333$	0	0
$334$	−2016.38	−0.330333
$335$	0	0
$336$	0	0
$337$	−2552.84	−0.412647	−0.206323	−	0.978484i	$-0.566150\pi$
$337$	−2552.84	−0.412647	−0.206323	+	0.978484i	$0.566150\pi$
$338$	−5524.56	−0.889043
$339$	0	0
$340$	0	0
$341$	−2337.28	−0.371175
$342$	0	0
$343$	−8114.93	−1.27745
$344$	−1238.78	−0.194159
$345$	0	0
$346$	−20589.1	−3.19906
$347$	8835.70	1.36693	0.683466	−	0.729983i	$-0.260472\pi$
$347$	8835.70	1.36693	0.683466	+	0.729983i	$0.260472\pi$
$348$	0	0
$349$	−8837.73	−1.35551	−0.677755	−	0.735288i	$-0.737047\pi$
$349$	−8837.73	−1.35551	−0.677755	+	0.735288i	$0.737047\pi$
$350$	0	0
$351$	0	0
$352$	−1348.68	−0.204219
$353$	3128.88	0.471766	0.235883	−	0.971781i	$-0.424202\pi$
$353$	3128.88	0.471766	0.235883	+	0.971781i	$0.424202\pi$
$354$	0	0
$355$	0	0
$356$	14245.0	2.12073
$357$	0	0
$358$	−8446.65	−1.24698
$359$	−10823.3	−1.59118	−0.795588	−	0.605838i	$-0.792838\pi$
$359$	−10823.3	−1.59118	−0.795588	+	0.605838i	$0.792838\pi$
$360$	0	0
$361$	−3148.22	−0.458992
$362$	−1237.83	−0.179721
$363$	0	0
$364$	12025.6	1.73163
$365$	0	0
$366$	0	0
$367$	426.157	0.0606137	0.0303068	−	0.999541i	$-0.490352\pi$
$367$	426.157	0.0606137	0.0303068	+	0.999541i	$0.490352\pi$
$368$	247.181	0.0350141
$369$	0	0
$370$	0	0
$371$	−7660.44	−1.07200
$372$	0	0
$373$	2607.13	0.361909	0.180955	−	0.983491i	$-0.442081\pi$
$373$	2607.13	0.361909	0.180955	+	0.983491i	$0.442081\pi$
$374$	3458.11	0.478114
$375$	0	0
$376$	5105.74	0.700288
$377$	5913.23	0.807816
$378$	0	0
$379$	−3453.71	−0.468087	−0.234044	−	0.972226i	$-0.575196\pi$
$379$	−3453.71	−0.468087	−0.234044	+	0.972226i	$0.575196\pi$
$380$	0	0
$381$	0	0
$382$	8761.13	1.17345
$383$	1424.88	0.190100	0.0950498	−	0.995473i	$-0.469699\pi$
$383$	1424.88	0.190100	0.0950498	+	0.995473i	$0.469699\pi$
$384$	0	0
$385$	0	0
$386$	6466.20	0.852645
$387$	0	0
$388$	16346.0	2.13877
$389$	−6040.34	−0.787294	−0.393647	−	0.919262i	$-0.628787\pi$
$389$	−6040.34	−0.787294	−0.393647	+	0.919262i	$0.628787\pi$
$390$	0	0
$391$	−3505.43	−0.453394
$392$	12322.8	1.58775
$393$	0	0
$394$	−12583.1	−1.60895
$395$	0	0
$396$	0	0
$397$	−10652.4	−1.34667	−0.673336	−	0.739337i	$-0.735139\pi$
$397$	−10652.4	−1.34667	−0.673336	+	0.739337i	$0.735139\pi$
$398$	9755.17	1.22860
$399$	0	0
$400$	0	0
$401$	−13110.4	−1.63267	−0.816334	−	0.577580i	$-0.803997\pi$
$401$	−13110.4	−1.63267	−0.816334	+	0.577580i	$0.803997\pi$
$402$	0	0
$403$	10747.7	1.32849
$404$	−18092.9	−2.22811
$405$	0	0
$406$	26391.8	3.22611
$407$	−1403.34	−0.170911
$408$	0	0
$409$	9195.93	1.11176	0.555880	−	0.831263i	$-0.312381\pi$
$409$	9195.93	1.11176	0.555880	+	0.831263i	$0.312381\pi$
$410$	0	0
$411$	0	0
$412$	16933.7	2.02491
$413$	−17611.5	−2.09832
$414$	0	0
$415$	0	0
$416$	6201.74	0.730926
$417$	0	0
$418$	−1873.85	−0.219266
$419$	5.87977	0.000685550 0	0.000342775	−	1.00000i	$-0.499891\pi$
$419$	5.87977	0.000685550 0	0.000342775		1.00000i	$0.499891\pi$
$420$	0	0
$421$	−4757.16	−0.550712	−0.275356	−	0.961342i	$-0.588796\pi$
$421$	−4757.16	−0.550712	−0.275356	+	0.961342i	$0.588796\pi$
$422$	−20719.4	−2.39006
$423$	0	0
$424$	5052.52	0.578708
$425$	0	0
$426$	0	0
$427$	10138.3	1.14900
$428$	−4881.01	−0.551245
$429$	0	0
$430$	0	0
$431$	12028.0	1.34424	0.672121	−	0.740441i	$-0.265383\pi$
$431$	12028.0	1.34424	0.672121	+	0.740441i	$0.265383\pi$
$432$	0	0
$433$	15071.2	1.67269	0.836346	−	0.548203i	$-0.184688\pi$
$433$	15071.2	1.67269	0.836346	+	0.548203i	$0.184688\pi$
$434$	47968.8	5.30547
$435$	0	0
$436$	9623.38	1.05706
$437$	1899.49	0.207929
$438$	0	0
$439$	−1394.31	−0.151588	−0.0757938	−	0.997124i	$-0.524149\pi$
$439$	−1394.31	−0.151588	−0.0757938	+	0.997124i	$0.524149\pi$
$440$	0	0
$441$	0	0
$442$	−15901.7	−1.71123
$443$	16363.0	1.75492	0.877462	−	0.479646i	$-0.159235\pi$
$443$	16363.0	1.75492	0.877462	+	0.479646i	$0.159235\pi$
$444$	0	0
$445$	0	0
$446$	−4041.89	−0.429124
$447$	0	0
$448$	25725.5	2.71298
$449$	−8133.05	−0.854838	−0.427419	−	0.904054i	$-0.640577\pi$
$449$	−8133.05	−0.854838	−0.427419	+	0.904054i	$0.640577\pi$
$450$	0	0
$451$	−2956.27	−0.308659
$452$	−20296.8	−2.11213
$453$	0	0
$454$	4692.25	0.485062
$455$	0	0
$456$	0	0
$457$	1780.59	0.182259	0.0911297	−	0.995839i	$-0.470952\pi$
$457$	1780.59	0.182259	0.0911297	+	0.995839i	$0.470952\pi$
$458$	6567.65	0.670057
$459$	0	0
$460$	0	0
$461$	6751.82	0.682133	0.341067	−	0.940039i	$-0.389212\pi$
$461$	6751.82	0.682133	0.341067	+	0.940039i	$0.389212\pi$
$462$	0	0
$463$	4496.61	0.451350	0.225675	−	0.974203i	$-0.427541\pi$
$463$	4496.61	0.451350	0.225675	+	0.974203i	$0.427541\pi$
$464$	1500.01	0.150078
$465$	0	0
$466$	−4412.18	−0.438606
$467$	12171.4	1.20605	0.603027	−	0.797721i	$-0.293961\pi$
$467$	12171.4	1.20605	0.603027	+	0.797721i	$0.293961\pi$
$468$	0	0
$469$	−20868.0	−2.05458
$470$	0	0
$471$	0	0
$472$	11615.9	1.13276
$473$	−414.247	−0.0402687
$474$	0	0
$475$	0	0
$476$	−43261.5	−4.16573
$477$	0	0
$478$	−17028.6	−1.62944
$479$	11601.8	1.10668	0.553338	−	0.832957i	$-0.313354\pi$
$479$	11601.8	1.10668	0.553338	+	0.832957i	$0.313354\pi$
$480$	0	0
$481$	6453.07	0.611715
$482$	−1615.19	−0.152635
$483$	0	0
$484$	−16046.8	−1.50703
$485$	0	0
$486$	0	0
$487$	−9713.05	−0.903779	−0.451889	−	0.892074i	$-0.649250\pi$
$487$	−9713.05	−0.903779	−0.451889	+	0.892074i	$0.649250\pi$
$488$	−6686.79	−0.620281
$489$	0	0
$490$	0	0
$491$	−15375.7	−1.41323	−0.706616	−	0.707597i	$-0.749779\pi$
$491$	−15375.7	−1.41323	−0.706616	+	0.707597i	$0.749779\pi$
$492$	0	0
$493$	−21272.6	−1.94335
$494$	8616.66	0.784781
$495$	0	0
$496$	2726.37	0.246809
$497$	−13978.4	−1.26160
$498$	0	0
$499$	−7115.58	−0.638351	−0.319176	−	0.947696i	$-0.603406\pi$
$499$	−7115.58	−0.638351	−0.319176	+	0.947696i	$0.603406\pi$
$500$	0	0
$501$	0	0
$502$	14311.9	1.27245
$503$	−13438.6	−1.19125	−0.595623	−	0.803264i	$-0.703095\pi$
$503$	−13438.6	−1.19125	−0.595623	+	0.803264i	$0.703095\pi$
$504$	0	0
$505$	0	0
$506$	−959.196	−0.0842717
$507$	0	0
$508$	20175.7	1.76211
$509$	−7175.20	−0.624823	−0.312412	−	0.949947i	$-0.601137\pi$
$509$	−7175.20	−0.624823	−0.312412	+	0.949947i	$0.601137\pi$
$510$	0	0
$511$	31571.5	2.73316
$512$	2861.96	0.247035
$513$	0	0
$514$	−7620.30	−0.653924
$515$	0	0
$516$	0	0
$517$	1707.35	0.145240
$518$	28801.2	2.44296
$519$	0	0
$520$	0	0
$521$	21486.2	1.80677	0.903385	−	0.428830i	$-0.141074\pi$
$521$	21486.2	1.80677	0.903385	+	0.428830i	$0.141074\pi$
$522$	0	0
$523$	11699.9	0.978205	0.489103	−	0.872226i	$-0.337324\pi$
$523$	11699.9	0.978205	0.489103	+	0.872226i	$0.337324\pi$
$524$	−639.930	−0.0533502
$525$	0	0
$526$	−22770.7	−1.88755
$527$	−38664.3	−3.19591
$528$	0	0
$529$	−11194.7	−0.920085
$530$	0	0
$531$	0	0
$532$	23442.2	1.91043
$533$	13594.0	1.10473
$534$	0	0
$535$	0	0
$536$	13763.7	1.10915
$537$	0	0
$538$	27389.2	2.19486
$539$	4120.73	0.329300
$540$	0	0
$541$	17032.0	1.35354	0.676769	−	0.736195i	$-0.263379\pi$
$541$	17032.0	1.35354	0.676769	+	0.736195i	$0.263379\pi$
$542$	−12840.0	−1.01757
$543$	0	0
$544$	−22310.5	−1.75837
$545$	0	0
$546$	0	0
$547$	−23325.5	−1.82327	−0.911633	−	0.411005i	$-0.865178\pi$
$547$	−23325.5	−1.82327	−0.911633	+	0.411005i	$0.865178\pi$
$548$	20001.6	1.55917
$549$	0	0
$550$	0	0
$551$	11527.0	0.891229
$552$	0	0
$553$	−7337.59	−0.564243
$554$	−8680.07	−0.665670
$555$	0	0
$556$	−5451.40	−0.415811
$557$	16735.1	1.27305	0.636525	−	0.771256i	$-0.280371\pi$
$557$	16735.1	1.27305	0.636525	+	0.771256i	$0.280371\pi$
$558$	0	0
$559$	1904.86	0.144127
$560$	0	0
$561$	0	0
$562$	−21249.0	−1.59490
$563$	−20154.6	−1.50873	−0.754366	−	0.656454i	$-0.772056\pi$
$563$	−20154.6	−1.50873	−0.754366	+	0.656454i	$0.772056\pi$
$564$	0	0
$565$	0	0
$566$	−32178.7	−2.38970
$567$	0	0
$568$	9219.57	0.681065
$569$	−13838.5	−1.01958	−0.509789	−	0.860299i	$-0.670277\pi$
$569$	−13838.5	−1.01958	−0.509789	+	0.860299i	$0.670277\pi$
$570$	0	0
$571$	16171.3	1.18520	0.592598	−	0.805499i	$-0.298102\pi$
$571$	16171.3	1.18520	0.592598	+	0.805499i	$0.298102\pi$
$572$	−2652.31	−0.193879
$573$	0	0
$574$	60672.5	4.41189
$575$	0	0
$576$	0	0
$577$	5270.22	0.380246	0.190123	−	0.981760i	$-0.439111\pi$
$577$	5270.22	0.380246	0.190123	+	0.981760i	$0.439111\pi$
$578$	34966.7	2.51630
$579$	0	0
$580$	0	0
$581$	−25476.9	−1.81921
$582$	0	0
$583$	1689.55	0.120024
$584$	−20823.3	−1.47547
$585$	0	0
$586$	−10869.1	−0.766211
$587$	16540.2	1.16301	0.581504	−	0.813544i	$-0.302465\pi$
$587$	16540.2	1.16301	0.581504	+	0.813544i	$0.302465\pi$
$588$	0	0
$589$	20951.1	1.46566
$590$	0	0
$591$	0	0
$592$	1636.95	0.113646
$593$	12053.1	0.834673	0.417336	−	0.908752i	$-0.362964\pi$
$593$	12053.1	0.834673	0.417336	+	0.908752i	$0.362964\pi$
$594$	0	0
$595$	0	0
$596$	−18073.0	−1.24211
$597$	0	0
$598$	4410.74	0.301620
$599$	−16683.3	−1.13800	−0.569000	−	0.822337i	$-0.692670\pi$
$599$	−16683.3	−1.13800	−0.569000	+	0.822337i	$0.692670\pi$
$600$	0	0
$601$	−4852.77	−0.329366	−0.164683	−	0.986347i	$-0.552660\pi$
$601$	−4852.77	−0.329366	−0.164683	+	0.986347i	$0.552660\pi$
$602$	8501.74	0.575590
$603$	0	0
$604$	−9347.59	−0.629715
$605$	0	0
$606$	0	0
$607$	868.698	0.0580879	0.0290440	−	0.999578i	$-0.490754\pi$
$607$	868.698	0.0580879	0.0290440	+	0.999578i	$0.490754\pi$
$608$	12089.4	0.806400
$609$	0	0
$610$	0	0
$611$	−7851.03	−0.519834
$612$	0	0
$613$	−22753.0	−1.49916	−0.749579	−	0.661915i	$-0.769744\pi$
$613$	−22753.0	−1.49916	−0.749579	+	0.661915i	$0.769744\pi$
$614$	−7349.38	−0.483057
$615$	0	0
$616$	−4255.26	−0.278327
$617$	−872.076	−0.0569019	−0.0284510	−	0.999595i	$-0.509057\pi$
$617$	−872.076	−0.0569019	−0.0284510	+	0.999595i	$0.509057\pi$
$618$	0	0
$619$	−10397.0	−0.675105	−0.337553	−	0.941307i	$-0.609599\pi$
$619$	−10397.0	−0.675105	−0.337553	+	0.941307i	$0.609599\pi$
$620$	0	0
$621$	0	0
$622$	9337.35	0.601919
$623$	−35142.4	−2.25995
$624$	0	0
$625$	0	0
$626$	937.414	0.0598508
$627$	0	0
$628$	−12336.7	−0.783895
$629$	−23214.7	−1.47159
$630$	0	0
$631$	−23474.9	−1.48102	−0.740508	−	0.672047i	$-0.765415\pi$
$631$	−23474.9	−1.48102	−0.740508	+	0.672047i	$0.765415\pi$
$632$	4839.59	0.304602
$633$	0	0
$634$	3164.45	0.198228
$635$	0	0
$636$	0	0
$637$	−18948.6	−1.17861
$638$	−5820.86	−0.361207
$639$	0	0
$640$	0	0
$641$	−7948.00	−0.489746	−0.244873	−	0.969555i	$-0.578746\pi$
$641$	−7948.00	−0.489746	−0.244873	+	0.969555i	$0.578746\pi$
$642$	0	0
$643$	−24194.7	−1.48390	−0.741949	−	0.670456i	$-0.766099\pi$
$643$	−24194.7	−1.48390	−0.741949	+	0.670456i	$0.766099\pi$
$644$	11999.7	0.734246
$645$	0	0
$646$	−30998.1	−1.88793
$647$	−11011.7	−0.669113	−0.334556	−	0.942376i	$-0.608586\pi$
$647$	−11011.7	−0.669113	−0.334556	+	0.942376i	$0.608586\pi$
$648$	0	0
$649$	3884.33	0.234936
$650$	0	0
$651$	0	0
$652$	36809.4	2.21099
$653$	23121.1	1.38560	0.692800	−	0.721130i	$-0.256377\pi$
$653$	23121.1	1.38560	0.692800	+	0.721130i	$0.256377\pi$
$654$	0	0
$655$	0	0
$656$	3448.40	0.205240
$657$	0	0
$658$	−35040.5	−2.07602
$659$	−13152.6	−0.777470	−0.388735	−	0.921350i	$-0.627088\pi$
$659$	−13152.6	−0.777470	−0.388735	+	0.921350i	$0.627088\pi$
$660$	0	0
$661$	13205.9	0.777077	0.388539	−	0.921432i	$-0.372980\pi$
$661$	13205.9	0.777077	0.388539	+	0.921432i	$0.372980\pi$
$662$	12052.3	0.707590
$663$	0	0
$664$	16803.5	0.982083
$665$	0	0
$666$	0	0
$667$	5900.50	0.342531
$668$	−5563.56	−0.322247
$669$	0	0
$670$	0	0
$671$	−2236.05	−0.128647
$672$	0	0
$673$	−15527.3	−0.889351	−0.444675	−	0.895692i	$-0.646681\pi$
$673$	−15527.3	−0.889351	−0.444675	+	0.895692i	$0.646681\pi$
$674$	−11555.5	−0.660389
$675$	0	0
$676$	−15243.3	−0.867279
$677$	−2917.34	−0.165616	−0.0828082	−	0.996566i	$-0.526389\pi$
$677$	−2917.34	−0.165616	−0.0828082	+	0.996566i	$0.526389\pi$
$678$	0	0
$679$	−40325.7	−2.27917
$680$	0	0
$681$	0	0
$682$	−10579.8	−0.594019
$683$	−30803.5	−1.72571	−0.862856	−	0.505449i	$-0.831327\pi$
$683$	−30803.5	−1.72571	−0.862856	+	0.505449i	$0.831327\pi$
$684$	0	0
$685$	0	0
$686$	−36732.6	−2.04440
$687$	0	0
$688$	483.207	0.0267763
$689$	−7769.20	−0.429583
$690$	0	0
$691$	6315.91	0.347711	0.173856	−	0.984771i	$-0.444377\pi$
$691$	6315.91	0.347711	0.173856	+	0.984771i	$0.444377\pi$
$692$	−56809.1	−3.12075
$693$	0	0
$694$	39995.2	2.18760
$695$	0	0
$696$	0	0
$697$	−48903.9	−2.65763
$698$	−40004.4	−2.16932
$699$	0	0
$700$	0	0
$701$	−3545.05	−0.191006	−0.0955028	−	0.995429i	$-0.530446\pi$
$701$	−3545.05	−0.191006	−0.0955028	+	0.995429i	$0.530446\pi$
$702$	0	0
$703$	12579.4	0.674879
$704$	−5673.91	−0.303755
$705$	0	0
$706$	14163.0	0.755002
$707$	44635.2	2.37437
$708$	0	0
$709$	−12214.3	−0.646991	−0.323495	−	0.946230i	$-0.604858\pi$
$709$	−12214.3	−0.646991	−0.323495	+	0.946230i	$0.604858\pi$
$710$	0	0
$711$	0	0
$712$	23178.5	1.22002
$713$	10724.5	0.563306
$714$	0	0
$715$	0	0
$716$	−23305.9	−1.21646
$717$	0	0
$718$	−48992.2	−2.54648
$719$	36175.4	1.87638	0.938189	−	0.346122i	$-0.112502\pi$
$719$	36175.4	1.87638	0.938189	+	0.346122i	$0.112502\pi$
$720$	0	0
$721$	−41775.5	−2.15784
$722$	−14250.6	−0.734559
$723$	0	0
$724$	−3415.42	−0.175322
$725$	0	0
$726$	0	0
$727$	−29209.3	−1.49012	−0.745058	−	0.667000i	$-0.767578\pi$
$727$	−29209.3	−1.49012	−0.745058	+	0.667000i	$0.767578\pi$
$728$	19567.3	0.996170
$729$	0	0
$730$	0	0
$731$	−6852.66	−0.346723
$732$	0	0
$733$	30189.6	1.52125	0.760625	−	0.649191i	$-0.224892\pi$
$733$	30189.6	1.52125	0.760625	+	0.649191i	$0.224892\pi$
$734$	1929.02	0.0970046
$735$	0	0
$736$	6188.40	0.309928
$737$	4602.56	0.230037
$738$	0	0
$739$	34356.0	1.71016	0.855079	−	0.518498i	$-0.173509\pi$
$739$	34356.0	1.71016	0.855079	+	0.518498i	$0.173509\pi$
$740$	0	0
$741$	0	0
$742$	−34675.3	−1.71559
$743$	12730.6	0.628589	0.314294	−	0.949326i	$-0.398232\pi$
$743$	12730.6	0.628589	0.314294	+	0.949326i	$0.398232\pi$
$744$	0	0
$745$	0	0
$746$	11801.3	0.579190
$747$	0	0
$748$	9541.58	0.466410
$749$	12041.5	0.587432
$750$	0	0
$751$	−5514.18	−0.267930	−0.133965	−	0.990986i	$-0.542771\pi$
$751$	−5514.18	−0.267930	−0.133965	+	0.990986i	$0.542771\pi$
$752$	−1991.57	−0.0965761
$753$	0	0
$754$	26766.5	1.29281
$755$	0	0
$756$	0	0
$757$	9205.95	0.442003	0.221001	−	0.975274i	$-0.429067\pi$
$757$	9205.95	0.442003	0.221001	+	0.975274i	$0.429067\pi$
$758$	−15633.4	−0.749115
$759$	0	0
$760$	0	0
$761$	5372.53	0.255918	0.127959	−	0.991779i	$-0.459157\pi$
$761$	5372.53	0.255918	0.127959	+	0.991779i	$0.459157\pi$
$762$	0	0
$763$	−23740.9	−1.12645
$764$	24173.6	1.14473
$765$	0	0
$766$	6449.79	0.304230
$767$	−17861.6	−0.840866
$768$	0	0
$769$	4467.86	0.209513	0.104756	−	0.994498i	$-0.466594\pi$
$769$	4467.86	0.209513	0.104756	+	0.994498i	$0.466594\pi$
$770$	0	0
$771$	0	0
$772$	17841.5	0.831773
$773$	23496.7	1.09329	0.546647	−	0.837363i	$-0.315904\pi$
$773$	23496.7	1.09329	0.546647	+	0.837363i	$0.315904\pi$
$774$	0	0
$775$	0	0
$776$	26597.2	1.23039
$777$	0	0
$778$	−27341.9	−1.25997
$779$	26499.6	1.21880
$780$	0	0
$781$	3083.01	0.141253
$782$	−15867.5	−0.725600
$783$	0	0
$784$	−4806.71	−0.218965
$785$	0	0
$786$	0	0
$787$	−5324.68	−0.241174	−0.120587	−	0.992703i	$-0.538478\pi$
$787$	−5324.68	−0.241174	−0.120587	+	0.992703i	$0.538478\pi$
$788$	−34719.1	−1.56957
$789$	0	0
$790$	0	0
$791$	50072.4	2.25078
$792$	0	0
$793$	10282.2	0.460443
$794$	−48218.6	−2.15518
$795$	0	0
$796$	26916.3	1.19852
$797$	−34413.0	−1.52945	−0.764724	−	0.644358i	$-0.777125\pi$
$797$	−34413.0	−1.52945	−0.764724	+	0.644358i	$0.777125\pi$
$798$	0	0
$799$	28243.7	1.25055
$800$	0	0
$801$	0	0
$802$	−59344.6	−2.61288
$803$	−6963.29	−0.306014
$804$	0	0
$805$	0	0
$806$	48649.8	2.12607
$807$	0	0
$808$	−29439.6	−1.28179
$809$	17213.2	0.748062	0.374031	−	0.927416i	$-0.377975\pi$
$809$	17213.2	0.748062	0.374031	+	0.927416i	$0.377975\pi$
$810$	0	0
$811$	10256.4	0.444084	0.222042	−	0.975037i	$-0.428728\pi$
$811$	10256.4	0.444084	0.222042	+	0.975037i	$0.428728\pi$
$812$	72819.9	3.14714
$813$	0	0
$814$	−6352.27	−0.273522
$815$	0	0
$816$	0	0
$817$	3713.26	0.159009
$818$	41625.8	1.77923
$819$	0	0
$820$	0	0
$821$	−46354.4	−1.97050	−0.985250	−	0.171122i	$-0.945261\pi$
$821$	−46354.4	−1.97050	−0.985250	+	0.171122i	$0.945261\pi$
$822$	0	0
$823$	−24429.4	−1.03470	−0.517348	−	0.855775i	$-0.673081\pi$
$823$	−24429.4	−1.03470	−0.517348	+	0.855775i	$0.673081\pi$
$824$	27553.5	1.16489
$825$	0	0
$826$	−79719.4	−3.35810
$827$	9681.38	0.407079	0.203540	−	0.979067i	$-0.434755\pi$
$827$	9681.38	0.407079	0.203540	+	0.979067i	$0.434755\pi$
$828$	0	0
$829$	13781.5	0.577382	0.288691	−	0.957422i	$-0.406780\pi$
$829$	13781.5	0.577382	0.288691	+	0.957422i	$0.406780\pi$
$830$	0	0
$831$	0	0
$832$	26090.7	1.08718
$833$	68167.0	2.83535
$834$	0	0
$835$	0	0
$836$	−5170.31	−0.213898
$837$	0	0
$838$	26.6150	0.00109714
$839$	−5290.63	−0.217703	−0.108852	−	0.994058i	$-0.534717\pi$
$839$	−5290.63	−0.217703	−0.108852	+	0.994058i	$0.534717\pi$
$840$	0	0
$841$	11418.1	0.468165
$842$	−21533.5	−0.881345
$843$	0	0
$844$	−57168.7	−2.33155
$845$	0	0
$846$	0	0
$847$	39587.6	1.60596
$848$	−1970.82	−0.0798091
$849$	0	0
$850$	0	0
$851$	6439.19	0.259380
$852$	0	0
$853$	13788.1	0.553451	0.276726	−	0.960949i	$-0.410751\pi$
$853$	13788.1	0.553451	0.276726	+	0.960949i	$0.410751\pi$
$854$	45891.3	1.83884
$855$	0	0
$856$	−7942.09	−0.317120
$857$	34665.3	1.38173	0.690865	−	0.722983i	$-0.257230\pi$
$857$	34665.3	1.38173	0.690865	+	0.722983i	$0.257230\pi$
$858$	0	0
$859$	−25914.7	−1.02933	−0.514667	−	0.857390i	$-0.672085\pi$
$859$	−25914.7	−1.02933	−0.514667	+	0.857390i	$0.672085\pi$
$860$	0	0
$861$	0	0
$862$	54445.3	2.15129
$863$	256.730	0.0101265	0.00506327	−	0.999987i	$-0.498388\pi$
$863$	256.730	0.0101265	0.00506327	+	0.999987i	$0.498388\pi$
$864$	0	0
$865$	0	0
$866$	68220.4	2.67693
$867$	0	0
$868$	132355.	5.17560
$869$	1618.35	0.0631746
$870$	0	0
$871$	−21164.3	−0.823335
$872$	15658.6	0.608103
$873$	0	0
$874$	8598.12	0.332764
$875$	0	0
$876$	0	0
$877$	35072.4	1.35041	0.675207	−	0.737629i	$-0.264054\pi$
$877$	35072.4	1.35041	0.675207	+	0.737629i	$0.264054\pi$
$878$	−6311.42	−0.242597
$879$	0	0
$880$	0	0
$881$	−9294.28	−0.355428	−0.177714	−	0.984082i	$-0.556870\pi$
$881$	−9294.28	−0.355428	−0.177714	+	0.984082i	$0.556870\pi$
$882$	0	0
$883$	−9642.26	−0.367483	−0.183742	−	0.982975i	$-0.558821\pi$
$883$	−9642.26	−0.367483	−0.183742	+	0.982975i	$0.558821\pi$
$884$	−43875.7	−1.66934
$885$	0	0
$886$	74068.0	2.80854
$887$	−7224.57	−0.273481	−0.136740	−	0.990607i	$-0.543663\pi$
$887$	−7224.57	−0.273481	−0.136740	+	0.990607i	$0.543663\pi$
$888$	0	0
$889$	−49773.6	−1.87779
$890$	0	0
$891$	0	0
$892$	−11152.3	−0.418619
$893$	−15304.5	−0.573510
$894$	0	0
$895$	0	0
$896$	67528.1	2.51781
$897$	0	0
$898$	−36814.6	−1.36806
$899$	65081.7	2.41446
$900$	0	0
$901$	27949.4	1.03344
$902$	−13381.7	−0.493970
$903$	0	0
$904$	−33025.8	−1.21507
$905$	0	0
$906$	0	0
$907$	−24903.1	−0.911681	−0.455840	−	0.890061i	$-0.650661\pi$
$907$	−24903.1	−0.911681	−0.455840	+	0.890061i	$0.650661\pi$
$908$	12946.8	0.473188
$909$	0	0
$910$	0	0
$911$	18055.5	0.656647	0.328324	−	0.944565i	$-0.393516\pi$
$911$	18055.5	0.656647	0.328324	+	0.944565i	$0.393516\pi$
$912$	0	0
$913$	5619.07	0.203685
$914$	8059.92	0.291683
$915$	0	0
$916$	18121.4	0.653654
$917$	1578.71	0.0568524
$918$	0	0
$919$	374.785	0.0134527	0.00672634	−	0.999977i	$-0.497859\pi$
$919$	374.785	0.0134527	0.00672634	+	0.999977i	$0.497859\pi$
$920$	0	0
$921$	0	0
$922$	30562.4	1.09167
$923$	−14176.8	−0.505564
$924$	0	0
$925$	0	0
$926$	20354.1	0.722329
$927$	0	0
$928$	37554.1	1.32842
$929$	−7051.65	−0.249039	−0.124519	−	0.992217i	$-0.539739\pi$
$929$	−7051.65	−0.249039	−0.124519	+	0.992217i	$0.539739\pi$
$930$	0	0
$931$	−36937.7	−1.30031
$932$	−12174.0	−0.427869
$933$	0	0
$934$	55094.5	1.93014
$935$	0	0
$936$	0	0
$937$	21558.0	0.751623	0.375811	−	0.926696i	$-0.377364\pi$
$937$	21558.0	0.751623	0.375811	+	0.926696i	$0.377364\pi$
$938$	−94460.0	−3.28809
$939$	0	0
$940$	0	0
$941$	16659.3	0.577130	0.288565	−	0.957460i	$-0.406822\pi$
$941$	16659.3	0.577130	0.288565	+	0.957460i	$0.406822\pi$
$942$	0	0
$943$	13564.8	0.468430
$944$	−4530.95	−0.156218
$945$	0	0
$946$	−1875.11	−0.0644450
$947$	26547.4	0.910956	0.455478	−	0.890247i	$-0.349468\pi$
$947$	26547.4	0.910956	0.455478	+	0.890247i	$0.349468\pi$
$948$	0	0
$949$	32019.8	1.09526
$950$	0	0
$951$	0	0
$952$	−70392.5	−2.39646
$953$	−32158.1	−1.09308	−0.546539	−	0.837434i	$-0.684055\pi$
$953$	−32158.1	−1.09308	−0.546539	+	0.837434i	$0.684055\pi$
$954$	0	0
$955$	0	0
$956$	−46985.2	−1.58955
$957$	0	0
$958$	52515.8	1.77110
$959$	−49344.1	−1.66153
$960$	0	0
$961$	88499.1	2.97067
$962$	29210.1	0.978973
$963$	0	0
$964$	−4456.62	−0.148898
$965$	0	0
$966$	0	0
$967$	27131.2	0.902254	0.451127	−	0.892460i	$-0.351022\pi$
$967$	27131.2	0.902254	0.451127	+	0.892460i	$0.351022\pi$
$968$	−26110.4	−0.866964
$969$	0	0
$970$	0	0
$971$	31358.9	1.03641	0.518206	−	0.855256i	$-0.326600\pi$
$971$	31358.9	1.03641	0.518206	+	0.855256i	$0.326600\pi$
$972$	0	0
$973$	13448.6	0.443107
$974$	−43966.5	−1.44638
$975$	0	0
$976$	2608.29	0.0855423
$977$	−23459.2	−0.768193	−0.384097	−	0.923293i	$-0.625487\pi$
$977$	−23459.2	−0.768193	−0.384097	+	0.923293i	$0.625487\pi$
$978$	0	0
$979$	7750.85	0.253032
$980$	0	0
$981$	0	0
$982$	−69598.9	−2.26170
$983$	−37086.9	−1.20334	−0.601672	−	0.798743i	$-0.705499\pi$
$983$	−37086.9	−1.20334	−0.601672	+	0.798743i	$0.705499\pi$
$984$	0	0
$985$	0	0
$986$	−96291.3	−3.11008
$987$	0	0
$988$	23775.0	0.765570
$989$	1900.76	0.0611130
$990$	0	0
$991$	45468.4	1.45747	0.728735	−	0.684796i	$-0.240109\pi$
$991$	45468.4	1.45747	0.728735	+	0.684796i	$0.240109\pi$
$992$	68257.1	2.18464
$993$	0	0
$994$	−63273.6	−2.01903
$995$	0	0
$996$	0	0
$997$	7102.47	0.225614	0.112807	−	0.993617i	$-0.464016\pi$
$997$	7102.47	0.225614	0.112807	+	0.993617i	$0.464016\pi$
$998$	−32209.0	−1.02160
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.z.1.4	yes	4
3.2	odd	2		675.4.a.v.1.1	yes	4
5.2	odd	4		675.4.b.p.649.7		8
5.3	odd	4		675.4.b.p.649.2		8
5.4	even	2		675.4.a.u.1.1	✓	4
15.2	even	4		675.4.b.q.649.2		8
15.8	even	4		675.4.b.q.649.7		8
15.14	odd	2		675.4.a.y.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.u.1.1	✓	4	5.4	even	2
675.4.a.v.1.1	yes	4	3.2	odd	2
675.4.a.y.1.4	yes	4	15.14	odd	2
675.4.a.z.1.4	yes	4	1.1	even	1	trivial
675.4.b.p.649.2		8	5.3	odd	4
675.4.b.p.649.7		8	5.2	odd	4
675.4.b.q.649.2		8	15.2	even	4
675.4.b.q.649.7		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)

\(f(q)\)	\(=\)	\(q+4.52654 q^{2} +12.4896 q^{4} -30.8119 q^{7} +20.3223 q^{8} +6.79573 q^{11} -31.2493 q^{13} -139.471 q^{14} -7.92702 q^{16} +112.418 q^{17} -60.9161 q^{19} +30.7612 q^{22} -31.1821 q^{23} -141.451 q^{26} -384.827 q^{28} -189.228 q^{29} -343.933 q^{31} -198.460 q^{32} +508.865 q^{34} -206.503 q^{37} -275.739 q^{38} -435.018 q^{41} -60.9569 q^{43} +84.8758 q^{44} -141.147 q^{46} +251.239 q^{47} +606.370 q^{49} -390.291 q^{52} +248.620 q^{53} -626.167 q^{56} -856.547 q^{58} +571.583 q^{59} -329.038 q^{61} -1556.83 q^{62} -834.922 q^{64} +677.273 q^{67} +1404.05 q^{68} +453.668 q^{71} -1024.66 q^{73} -934.745 q^{74} -760.817 q^{76} -209.389 q^{77} +238.142 q^{79} -1969.13 q^{82} +826.853 q^{83} -275.924 q^{86} +138.105 q^{88} +1140.55 q^{89} +962.849 q^{91} -389.451 q^{92} +1137.24 q^{94} +1308.77 q^{97} +2744.76 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 19 q^{4} + 4 q^{7} - 15 q^{8} - 52 q^{11} - 2 q^{13} - 138 q^{14} - 5 q^{16} + 64 q^{17} - 46 q^{19} + 87 q^{22} - 90 q^{23} - 469 q^{26} - 110 q^{28} - 470 q^{29} - 262 q^{31} - 199 q^{32}+ \cdots + 4301 q^{98}+O(q^{100}) \) 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