LMFDB - Embedded newform 1575.4.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 67x^{8} + 1523x^{6} - 13569x^{4} + 36944x^{2} - 256 $ x^10 - 67x^8 + 1523x^6 - 13569x^4 + 36944x^2 - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.73445$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.73445 q^{2} +5.94609 q^{4} -7.00000 q^{7} +7.67022 q^{8} +O(q^{10})$	$q-3.73445 q^{2} +5.94609 q^{4} -7.00000 q^{7} +7.67022 q^{8} +36.3609 q^{11} +75.8669 q^{13} +26.1411 q^{14} -76.2127 q^{16} -31.6691 q^{17} +25.1516 q^{19} -135.788 q^{22} -212.336 q^{23} -283.321 q^{26} -41.6226 q^{28} +235.331 q^{29} -270.177 q^{31} +223.251 q^{32} +118.267 q^{34} -362.170 q^{37} -93.9273 q^{38} -132.833 q^{41} +5.13906 q^{43} +216.205 q^{44} +792.959 q^{46} -216.748 q^{47} +49.0000 q^{49} +451.112 q^{52} +455.439 q^{53} -53.6915 q^{56} -878.831 q^{58} +689.881 q^{59} +130.923 q^{61} +1008.96 q^{62} -224.016 q^{64} -633.994 q^{67} -188.307 q^{68} +1060.84 q^{71} -1008.24 q^{73} +1352.51 q^{74} +149.554 q^{76} -254.526 q^{77} -381.510 q^{79} +496.056 q^{82} -48.5463 q^{83} -19.1915 q^{86} +278.896 q^{88} -53.5256 q^{89} -531.068 q^{91} -1262.57 q^{92} +809.432 q^{94} -968.013 q^{97} -182.988 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 54 q^{4} - 70 q^{7}+O(q^{10}) $ 10 * q + 54 * q^4 - 70 * q^7	$ 10 q + 54 q^{4} - 70 q^{7} - 104 q^{13} + 310 q^{16} + 36 q^{19} - 644 q^{22} - 378 q^{28} - 24 q^{31} + 116 q^{34} - 732 q^{37} - 212 q^{43} - 184 q^{46} + 490 q^{49} - 2692 q^{52} - 3196 q^{58} + 1024 q^{61} + 1590 q^{64} - 2388 q^{67} - 3432 q^{73} - 4184 q^{76} - 776 q^{79} - 1860 q^{82} - 10164 q^{88} + 728 q^{91} - 4028 q^{94} - 4024 q^{97}+O(q^{100}) $ 10 * q + 54 * q^4 - 70 * q^7 - 104 * q^13 + 310 * q^16 + 36 * q^19 - 644 * q^22 - 378 * q^28 - 24 * q^31 + 116 * q^34 - 732 * q^37 - 212 * q^43 - 184 * q^46 + 490 * q^49 - 2692 * q^52 - 3196 * q^58 + 1024 * q^61 + 1590 * q^64 - 2388 * q^67 - 3432 * q^73 - 4184 * q^76 - 776 * q^79 - 1860 * q^82 - 10164 * q^88 + 728 * q^91 - 4028 * q^94 - 4024 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.73445	−1.32033	−0.660163	−	0.751122i	$-0.729513\pi$
$2$	−3.73445	−1.32033	−0.660163	+	0.751122i	$0.729513\pi$
$3$	0	0
$3$	0	0
$4$	5.94609	0.743261
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	7.67022	0.338979
$9$	0	0
$10$	0	0
$11$	36.3609	0.996657	0.498328	−	0.866988i	$-0.333947\pi$
$11$	36.3609	0.996657	0.498328	+	0.866988i	$0.333947\pi$
$12$	0	0
$13$	75.8669	1.61859	0.809296	−	0.587401i	$-0.199849\pi$
$13$	75.8669	1.61859	0.809296	+	0.587401i	$0.199849\pi$
$14$	26.1411	0.499036
$15$	0	0
$16$	−76.2127	−1.19082
$17$	−31.6691	−0.451817	−0.225909	−	0.974149i	$-0.572535\pi$
$17$	−31.6691	−0.451817	−0.225909	+	0.974149i	$0.572535\pi$
$18$	0	0
$19$	25.1516	0.303693	0.151847	−	0.988404i	$-0.451478\pi$
$19$	25.1516	0.303693	0.151847	+	0.988404i	$0.451478\pi$
$20$	0	0
$21$	0	0
$22$	−135.788	−1.31591
$23$	−212.336	−1.92501	−0.962504	−	0.271268i	$-0.912557\pi$
$23$	−212.336	−1.92501	−0.962504	+	0.271268i	$0.912557\pi$
$24$	0	0
$25$	0	0
$26$	−283.321	−2.13707
$27$	0	0
$28$	−41.6226	−0.280926
$29$	235.331	1.50689	0.753446	−	0.657510i	$-0.228390\pi$
$29$	235.331	1.50689	0.753446	+	0.657510i	$0.228390\pi$
$30$	0	0
$31$	−270.177	−1.56533	−0.782665	−	0.622443i	$-0.786140\pi$
$31$	−270.177	−1.56533	−0.782665	+	0.622443i	$0.786140\pi$
$32$	223.251	1.23330
$33$	0	0
$34$	118.267	0.596546
$35$	0	0
$36$	0	0
$37$	−362.170	−1.60920	−0.804601	−	0.593816i	$-0.797621\pi$
$37$	−362.170	−1.60920	−0.804601	+	0.593816i	$0.797621\pi$
$38$	−93.9273	−0.400974
$39$	0	0
$40$	0	0
$41$	−132.833	−0.505975	−0.252987	−	0.967470i	$-0.581413\pi$
$41$	−132.833	−0.505975	−0.252987	+	0.967470i	$0.581413\pi$
$42$	0	0
$43$	5.13906	0.0182256	0.00911278	−	0.999958i	$-0.497099\pi$
$43$	5.13906	0.0182256	0.00911278	+	0.999958i	$0.497099\pi$
$44$	216.205	0.740776
$45$	0	0
$46$	792.959	2.54164
$47$	−216.748	−0.672679	−0.336339	−	0.941741i	$-0.609189\pi$
$47$	−216.748	−0.672679	−0.336339	+	0.941741i	$0.609189\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	451.112	1.20304
$53$	455.439	1.18036	0.590182	−	0.807270i	$-0.299056\pi$
$53$	455.439	1.18036	0.590182	+	0.807270i	$0.299056\pi$
$54$	0	0
$55$	0	0
$56$	−53.6915	−0.128122
$57$	0	0
$58$	−878.831	−1.98959
$59$	689.881	1.52228	0.761142	−	0.648585i	$-0.224639\pi$
$59$	689.881	1.52228	0.761142	+	0.648585i	$0.224639\pi$
$60$	0	0
$61$	130.923	0.274804	0.137402	−	0.990515i	$-0.456125\pi$
$61$	130.923	0.274804	0.137402	+	0.990515i	$0.456125\pi$
$62$	1008.96	2.06675
$63$	0	0
$64$	−224.016	−0.437531
$65$	0	0
$66$	0	0
$67$	−633.994	−1.15604	−0.578020	−	0.816022i	$-0.696175\pi$
$67$	−633.994	−1.15604	−0.578020	+	0.816022i	$0.696175\pi$
$68$	−188.307	−0.335818
$69$	0	0
$70$	0	0
$71$	1060.84	1.77322	0.886609	−	0.462520i	$-0.153055\pi$
$71$	1060.84	1.77322	0.886609	+	0.462520i	$0.153055\pi$
$72$	0	0
$73$	−1008.24	−1.61652	−0.808260	−	0.588826i	$-0.799591\pi$
$73$	−1008.24	−1.61652	−0.808260	+	0.588826i	$0.799591\pi$
$74$	1352.51	2.12467
$75$	0	0
$76$	149.554	0.225723
$77$	−254.526	−0.376701
$78$	0	0
$79$	−381.510	−0.543332	−0.271666	−	0.962392i	$-0.587575\pi$
$79$	−381.510	−0.543332	−0.271666	+	0.962392i	$0.587575\pi$
$80$	0	0
$81$	0	0
$82$	496.056	0.668052
$83$	−48.5463	−0.0642005	−0.0321003	−	0.999485i	$-0.510220\pi$
$83$	−48.5463	−0.0642005	−0.0321003	+	0.999485i	$0.510220\pi$
$84$	0	0
$85$	0	0
$86$	−19.1915	−0.0240637
$87$	0	0
$88$	278.896	0.337846
$89$	−53.5256	−0.0637494	−0.0318747	−	0.999492i	$-0.510148\pi$
$89$	−53.5256	−0.0637494	−0.0318747	+	0.999492i	$0.510148\pi$
$90$	0	0
$91$	−531.068	−0.611770
$92$	−1262.57	−1.43078
$93$	0	0
$94$	809.432	0.888155
$95$	0	0
$96$	0	0
$97$	−968.013	−1.01327	−0.506633	−	0.862162i	$-0.669110\pi$
$97$	−968.013	−1.01327	−0.506633	+	0.862162i	$0.669110\pi$
$98$	−182.988	−0.188618
$99$	0	0
$100$	0	0
$101$	−730.373	−0.719552	−0.359776	−	0.933039i	$-0.617147\pi$
$101$	−730.373	−0.719552	−0.359776	+	0.933039i	$0.617147\pi$
$102$	0	0
$103$	1231.60	1.17819	0.589094	−	0.808064i	$-0.299485\pi$
$103$	1231.60	1.17819	0.589094	+	0.808064i	$0.299485\pi$
$104$	581.916	0.548669
$105$	0	0
$106$	−1700.81	−1.55847
$107$	425.821	0.384726	0.192363	−	0.981324i	$-0.438385\pi$
$107$	425.821	0.384726	0.192363	+	0.981324i	$0.438385\pi$
$108$	0	0
$109$	−1682.52	−1.47850	−0.739249	−	0.673433i	$-0.764819\pi$
$109$	−1682.52	−1.47850	−0.739249	+	0.673433i	$0.764819\pi$
$110$	0	0
$111$	0	0
$112$	533.489	0.450089
$113$	−178.662	−0.148736	−0.0743678	−	0.997231i	$-0.523694\pi$
$113$	−178.662	−0.148736	−0.0743678	+	0.997231i	$0.523694\pi$
$114$	0	0
$115$	0	0
$116$	1399.30	1.12001
$117$	0	0
$118$	−2576.32	−2.00991
$119$	221.684	0.170771
$120$	0	0
$121$	−8.88538	−0.00667572
$122$	−488.926	−0.362831
$123$	0	0
$124$	−1606.50	−1.16345
$125$	0	0
$126$	0	0
$127$	−306.821	−0.214378	−0.107189	−	0.994239i	$-0.534185\pi$
$127$	−306.821	−0.214378	−0.107189	+	0.994239i	$0.534185\pi$
$128$	−949.431	−0.655614
$129$	0	0
$130$	0	0
$131$	1311.32	0.874584	0.437292	−	0.899320i	$-0.355938\pi$
$131$	1311.32	0.874584	0.437292	+	0.899320i	$0.355938\pi$
$132$	0	0
$133$	−176.061	−0.114785
$134$	2367.62	1.52635
$135$	0	0
$136$	−242.909	−0.153156
$137$	−545.169	−0.339978	−0.169989	−	0.985446i	$-0.554373\pi$
$137$	−545.169	−0.339978	−0.169989	+	0.985446i	$0.554373\pi$
$138$	0	0
$139$	3204.71	1.95554	0.977768	−	0.209688i	$-0.0672449\pi$
$139$	3204.71	1.95554	0.977768	+	0.209688i	$0.0672449\pi$
$140$	0	0
$141$	0	0
$142$	−3961.65	−2.34123
$143$	2758.59	1.61318
$144$	0	0
$145$	0	0
$146$	3765.23	2.13433
$147$	0	0
$148$	−2153.50	−1.19606
$149$	−780.819	−0.429310	−0.214655	−	0.976690i	$-0.568863\pi$
$149$	−780.819	−0.429310	−0.214655	+	0.976690i	$0.568863\pi$
$150$	0	0
$151$	2217.28	1.19496	0.597482	−	0.801882i	$-0.296168\pi$
$151$	2217.28	1.19496	0.597482	+	0.801882i	$0.296168\pi$
$152$	192.918	0.102946
$153$	0	0
$154$	950.515	0.497368
$155$	0	0
$156$	0	0
$157$	2202.33	1.11952	0.559762	−	0.828653i	$-0.310892\pi$
$157$	2202.33	1.11952	0.559762	+	0.828653i	$0.310892\pi$
$158$	1424.73	0.717376
$159$	0	0
$160$	0	0
$161$	1486.35	0.727585
$162$	0	0
$163$	−45.0182	−0.0216325	−0.0108163	−	0.999942i	$-0.503443\pi$
$163$	−45.0182	−0.0216325	−0.0108163	+	0.999942i	$0.503443\pi$
$164$	−789.835	−0.376072
$165$	0	0
$166$	181.293	0.0847656
$167$	−1157.73	−0.536455	−0.268228	−	0.963356i	$-0.586438\pi$
$167$	−1157.73	−0.536455	−0.268228	+	0.963356i	$0.586438\pi$
$168$	0	0
$169$	3558.79	1.61984
$170$	0	0
$171$	0	0
$172$	30.5573	0.0135464
$173$	−1231.12	−0.541041	−0.270521	−	0.962714i	$-0.587196\pi$
$173$	−1231.12	−0.541041	−0.270521	+	0.962714i	$0.587196\pi$
$174$	0	0
$175$	0	0
$176$	−2771.16	−1.18684
$177$	0	0
$178$	199.888	0.0841701
$179$	−2127.36	−0.888303	−0.444152	−	0.895952i	$-0.646495\pi$
$179$	−2127.36	−0.888303	−0.444152	+	0.895952i	$0.646495\pi$
$180$	0	0
$181$	−226.308	−0.0929358	−0.0464679	−	0.998920i	$-0.514797\pi$
$181$	−226.308	−0.0929358	−0.0464679	+	0.998920i	$0.514797\pi$
$182$	1983.25	0.807736
$183$	0	0
$184$	−1628.67	−0.652537
$185$	0	0
$186$	0	0
$187$	−1151.52	−0.450306
$188$	−1288.80	−0.499976
$189$	0	0
$190$	0	0
$191$	1210.78	0.458687	0.229344	−	0.973346i	$-0.426342\pi$
$191$	1210.78	0.458687	0.229344	+	0.973346i	$0.426342\pi$
$192$	0	0
$193$	−604.554	−0.225475	−0.112738	−	0.993625i	$-0.535962\pi$
$193$	−604.554	−0.225475	−0.112738	+	0.993625i	$0.535962\pi$
$194$	3614.99	1.33784
$195$	0	0
$196$	291.358	0.106180
$197$	−2735.83	−0.989442	−0.494721	−	0.869052i	$-0.664730\pi$
$197$	−2735.83	−0.989442	−0.494721	+	0.869052i	$0.664730\pi$
$198$	0	0
$199$	−2461.10	−0.876696	−0.438348	−	0.898805i	$-0.644436\pi$
$199$	−2461.10	−0.876696	−0.438348	+	0.898805i	$0.644436\pi$
$200$	0	0
$201$	0	0
$202$	2727.54	0.950044
$203$	−1647.32	−0.569552
$204$	0	0
$205$	0	0
$206$	−4599.35	−1.55559
$207$	0	0
$208$	−5782.03	−1.92746
$209$	914.535	0.302678
$210$	0	0
$211$	2199.14	0.717514	0.358757	−	0.933431i	$-0.383201\pi$
$211$	2199.14	0.717514	0.358757	+	0.933431i	$0.383201\pi$
$212$	2708.08	0.877319
$213$	0	0
$214$	−1590.21	−0.507964
$215$	0	0
$216$	0	0
$217$	1891.24	0.591639
$218$	6283.28	1.95210
$219$	0	0
$220$	0	0
$221$	−2402.64	−0.731308
$222$	0	0
$223$	−3121.61	−0.937392	−0.468696	−	0.883360i	$-0.655276\pi$
$223$	−3121.61	−0.937392	−0.468696	+	0.883360i	$0.655276\pi$
$224$	−1562.75	−0.466143
$225$	0	0
$226$	667.204	0.196380
$227$	−1251.78	−0.366008	−0.183004	−	0.983112i	$-0.558582\pi$
$227$	−1251.78	−0.366008	−0.183004	+	0.983112i	$0.558582\pi$
$228$	0	0
$229$	−3360.96	−0.969861	−0.484931	−	0.874553i	$-0.661155\pi$
$229$	−3360.96	−0.969861	−0.484931	+	0.874553i	$0.661155\pi$
$230$	0	0
$231$	0	0
$232$	1805.04	0.510805
$233$	618.821	0.173993	0.0869963	−	0.996209i	$-0.472273\pi$
$233$	618.821	0.173993	0.0869963	+	0.996209i	$0.472273\pi$
$234$	0	0
$235$	0	0
$236$	4102.09	1.13146
$237$	0	0
$238$	−827.867	−0.225473
$239$	1829.59	0.495172	0.247586	−	0.968866i	$-0.420363\pi$
$239$	1829.59	0.495172	0.247586	+	0.968866i	$0.420363\pi$
$240$	0	0
$241$	−5114.85	−1.36712	−0.683560	−	0.729894i	$-0.739569\pi$
$241$	−5114.85	−1.36712	−0.683560	+	0.729894i	$0.739569\pi$
$242$	33.1820	0.00881412
$243$	0	0
$244$	778.482	0.204251
$245$	0	0
$246$	0	0
$247$	1908.17	0.491556
$248$	−2072.32	−0.530614
$249$	0	0
$250$	0	0
$251$	−2379.15	−0.598290	−0.299145	−	0.954208i	$-0.596701\pi$
$251$	−2379.15	−0.598290	−0.299145	+	0.954208i	$0.596701\pi$
$252$	0	0
$253$	−7720.74	−1.91857
$254$	1145.81	0.283048
$255$	0	0
$256$	5337.72	1.30316
$257$	5432.85	1.31865	0.659323	−	0.751860i	$-0.270843\pi$
$257$	5432.85	1.31865	0.659323	+	0.751860i	$0.270843\pi$
$258$	0	0
$259$	2535.19	0.608221
$260$	0	0
$261$	0	0
$262$	−4897.05	−1.15474
$263$	−1786.64	−0.418893	−0.209447	−	0.977820i	$-0.567166\pi$
$263$	−1786.64	−0.418893	−0.209447	+	0.977820i	$0.567166\pi$
$264$	0	0
$265$	0	0
$266$	657.491	0.151554
$267$	0	0
$268$	−3769.79	−0.859240
$269$	260.564	0.0590591	0.0295295	−	0.999564i	$-0.490599\pi$
$269$	260.564	0.0590591	0.0295295	+	0.999564i	$0.490599\pi$
$270$	0	0
$271$	238.920	0.0535547	0.0267774	−	0.999641i	$-0.491475\pi$
$271$	238.920	0.0535547	0.0267774	+	0.999641i	$0.491475\pi$
$272$	2413.59	0.538035
$273$	0	0
$274$	2035.90	0.448881
$275$	0	0
$276$	0	0
$277$	−3525.33	−0.764681	−0.382341	−	0.924021i	$-0.624882\pi$
$277$	−3525.33	−0.764681	−0.382341	+	0.924021i	$0.624882\pi$
$278$	−11967.8	−2.58195
$279$	0	0
$280$	0	0
$281$	−5771.60	−1.22528	−0.612642	−	0.790361i	$-0.709893\pi$
$281$	−5771.60	−1.22528	−0.612642	+	0.790361i	$0.709893\pi$
$282$	0	0
$283$	−4557.06	−0.957205	−0.478602	−	0.878032i	$-0.658856\pi$
$283$	−4557.06	−0.957205	−0.478602	+	0.878032i	$0.658856\pi$
$284$	6307.85	1.31796
$285$	0	0
$286$	−10301.8	−2.12992
$287$	929.828	0.191241
$288$	0	0
$289$	−3910.07	−0.795861
$290$	0	0
$291$	0	0
$292$	−5995.11	−1.20150
$293$	5124.22	1.02171	0.510854	−	0.859668i	$-0.329329\pi$
$293$	5124.22	1.02171	0.510854	+	0.859668i	$0.329329\pi$
$294$	0	0
$295$	0	0
$296$	−2777.93	−0.545485
$297$	0	0
$298$	2915.93	0.566829
$299$	−16109.3	−3.11580
$300$	0	0
$301$	−35.9734	−0.00688862
$302$	−8280.31	−1.57774
$303$	0	0
$304$	−1916.87	−0.361645
$305$	0	0
$306$	0	0
$307$	−5319.55	−0.988933	−0.494467	−	0.869197i	$-0.664637\pi$
$307$	−5319.55	−0.988933	−0.494467	+	0.869197i	$0.664637\pi$
$308$	−1513.44	−0.279987
$309$	0	0
$310$	0	0
$311$	26.9763	0.00491861	0.00245930	−	0.999997i	$-0.499217\pi$
$311$	26.9763	0.00491861	0.00245930	+	0.999997i	$0.499217\pi$
$312$	0	0
$313$	−755.232	−0.136384	−0.0681920	−	0.997672i	$-0.521723\pi$
$313$	−755.232	−0.136384	−0.0681920	+	0.997672i	$0.521723\pi$
$314$	−8224.50	−1.47814
$315$	0	0
$316$	−2268.49	−0.403838
$317$	−799.246	−0.141609	−0.0708046	−	0.997490i	$-0.522557\pi$
$317$	−799.246	−0.141609	−0.0708046	+	0.997490i	$0.522557\pi$
$318$	0	0
$319$	8556.85	1.50185
$320$	0	0
$321$	0	0
$322$	−5550.71	−0.960649
$323$	−796.529	−0.137214
$324$	0	0
$325$	0	0
$326$	168.118	0.0285620
$327$	0	0
$328$	−1018.86	−0.171515
$329$	1517.23	0.254249
$330$	0	0
$331$	7526.10	1.24976	0.624882	−	0.780719i	$-0.285147\pi$
$331$	7526.10	1.24976	0.624882	+	0.780719i	$0.285147\pi$
$332$	−288.660	−0.0477178
$333$	0	0
$334$	4323.49	0.708296
$335$	0	0
$336$	0	0
$337$	−7984.88	−1.29069	−0.645347	−	0.763889i	$-0.723287\pi$
$337$	−7984.88	−1.29069	−0.645347	+	0.763889i	$0.723287\pi$
$338$	−13290.1	−2.13872
$339$	0	0
$340$	0	0
$341$	−9823.88	−1.56010
$342$	0	0
$343$	−343.000	−0.0539949
$344$	39.4177	0.00617808
$345$	0	0
$346$	4597.54	0.714351
$347$	977.847	0.151278	0.0756392	−	0.997135i	$-0.475900\pi$
$347$	977.847	0.151278	0.0756392	+	0.997135i	$0.475900\pi$
$348$	0	0
$349$	−7504.16	−1.15097	−0.575485	−	0.817812i	$-0.695187\pi$
$349$	−7504.16	−1.15097	−0.575485	+	0.817812i	$0.695187\pi$
$350$	0	0
$351$	0	0
$352$	8117.59	1.22917
$353$	9942.38	1.49909	0.749546	−	0.661952i	$-0.230272\pi$
$353$	9942.38	1.49909	0.749546	+	0.661952i	$0.230272\pi$
$354$	0	0
$355$	0	0
$356$	−318.268	−0.0473825
$357$	0	0
$358$	7944.51	1.17285
$359$	−4488.64	−0.659892	−0.329946	−	0.944000i	$-0.607031\pi$
$359$	−4488.64	−0.659892	−0.329946	+	0.944000i	$0.607031\pi$
$360$	0	0
$361$	−6226.40	−0.907770
$362$	845.137	0.122706
$363$	0	0
$364$	−3157.78	−0.454705
$365$	0	0
$366$	0	0
$367$	−5272.31	−0.749897	−0.374948	−	0.927046i	$-0.622340\pi$
$367$	−5272.31	−0.749897	−0.374948	+	0.927046i	$0.622340\pi$
$368$	16182.7	2.29235
$369$	0	0
$370$	0	0
$371$	−3188.07	−0.446136
$372$	0	0
$373$	3291.18	0.456865	0.228433	−	0.973560i	$-0.426640\pi$
$373$	3291.18	0.456865	0.228433	+	0.973560i	$0.426640\pi$
$374$	4300.28	0.594551
$375$	0	0
$376$	−1662.50	−0.228024
$377$	17853.8	2.43904
$378$	0	0
$379$	−26.7439	−0.00362465	−0.00181233	−	0.999998i	$-0.500577\pi$
$379$	−26.7439	−0.00362465	−0.00181233	+	0.999998i	$0.500577\pi$
$380$	0	0
$381$	0	0
$382$	−4521.61	−0.605617
$383$	−3090.76	−0.412352	−0.206176	−	0.978515i	$-0.566102\pi$
$383$	−3090.76	−0.412352	−0.206176	+	0.978515i	$0.566102\pi$
$384$	0	0
$385$	0	0
$386$	2257.67	0.297701
$387$	0	0
$388$	−5755.89	−0.753122
$389$	−10651.7	−1.38834	−0.694168	−	0.719813i	$-0.744227\pi$
$389$	−10651.7	−1.38834	−0.694168	+	0.719813i	$0.744227\pi$
$390$	0	0
$391$	6724.51	0.869752
$392$	375.841	0.0484256
$393$	0	0
$394$	10216.8	1.30639
$395$	0	0
$396$	0	0
$397$	747.137	0.0944528	0.0472264	−	0.998884i	$-0.484962\pi$
$397$	747.137	0.0944528	0.0472264	+	0.998884i	$0.484962\pi$
$398$	9190.83	1.15752
$399$	0	0
$400$	0	0
$401$	−1310.79	−0.163236	−0.0816179	−	0.996664i	$-0.526009\pi$
$401$	−1310.79	−0.163236	−0.0816179	+	0.996664i	$0.526009\pi$
$402$	0	0
$403$	−20497.5	−2.53363
$404$	−4342.86	−0.534815
$405$	0	0
$406$	6151.82	0.751994
$407$	−13168.8	−1.60382
$408$	0	0
$409$	−10588.7	−1.28015	−0.640073	−	0.768314i	$-0.721096\pi$
$409$	−10588.7	−1.28015	−0.640073	+	0.768314i	$0.721096\pi$
$410$	0	0
$411$	0	0
$412$	7323.22	0.875702
$413$	−4829.16	−0.575370
$414$	0	0
$415$	0	0
$416$	16937.3	1.99621
$417$	0	0
$418$	−3415.28	−0.399634
$419$	−11286.6	−1.31595	−0.657977	−	0.753038i	$-0.728588\pi$
$419$	−11286.6	−1.31595	−0.657977	+	0.753038i	$0.728588\pi$
$420$	0	0
$421$	6943.47	0.803810	0.401905	−	0.915681i	$-0.368348\pi$
$421$	6943.47	0.803810	0.401905	+	0.915681i	$0.368348\pi$
$422$	−8212.59	−0.947352
$423$	0	0
$424$	3493.31	0.400119
$425$	0	0
$426$	0	0
$427$	−916.464	−0.103866
$428$	2531.97	0.285952
$429$	0	0
$430$	0	0
$431$	−6303.13	−0.704434	−0.352217	−	0.935918i	$-0.614572\pi$
$431$	−6303.13	−0.704434	−0.352217	+	0.935918i	$0.614572\pi$
$432$	0	0
$433$	11367.1	1.26158	0.630792	−	0.775952i	$-0.282730\pi$
$433$	11367.1	1.26158	0.630792	+	0.775952i	$0.282730\pi$
$434$	−7062.73	−0.781157
$435$	0	0
$436$	−10004.4	−1.09891
$437$	−5340.60	−0.584612
$438$	0	0
$439$	−4567.07	−0.496525	−0.248262	−	0.968693i	$-0.579860\pi$
$439$	−4567.07	−0.496525	−0.248262	+	0.968693i	$0.579860\pi$
$440$	0	0
$441$	0	0
$442$	8972.53	0.965565
$443$	−14362.4	−1.54036	−0.770181	−	0.637825i	$-0.779834\pi$
$443$	−14362.4	−1.54036	−0.770181	+	0.637825i	$0.779834\pi$
$444$	0	0
$445$	0	0
$446$	11657.5	1.23766
$447$	0	0
$448$	1568.11	0.165371
$449$	−3702.87	−0.389197	−0.194598	−	0.980883i	$-0.562340\pi$
$449$	−3702.87	−0.389197	−0.194598	+	0.980883i	$0.562340\pi$
$450$	0	0
$451$	−4829.91	−0.504283
$452$	−1062.34	−0.110549
$453$	0	0
$454$	4674.71	0.483249
$455$	0	0
$456$	0	0
$457$	2093.71	0.214310	0.107155	−	0.994242i	$-0.465826\pi$
$457$	2093.71	0.214310	0.107155	+	0.994242i	$0.465826\pi$
$458$	12551.3	1.28053
$459$	0	0
$460$	0	0
$461$	−5892.10	−0.595276	−0.297638	−	0.954679i	$-0.596199\pi$
$461$	−5892.10	−0.595276	−0.297638	+	0.954679i	$0.596199\pi$
$462$	0	0
$463$	9673.70	0.971004	0.485502	−	0.874236i	$-0.338637\pi$
$463$	9673.70	0.971004	0.485502	+	0.874236i	$0.338637\pi$
$464$	−17935.2	−1.79444
$465$	0	0
$466$	−2310.95	−0.229727
$467$	−12286.7	−1.21747	−0.608735	−	0.793374i	$-0.708323\pi$
$467$	−12286.7	−1.21747	−0.608735	+	0.793374i	$0.708323\pi$
$468$	0	0
$469$	4437.96	0.436942
$470$	0	0
$471$	0	0
$472$	5291.53	0.516022
$473$	186.861	0.0181646
$474$	0	0
$475$	0	0
$476$	1318.15	0.126927
$477$	0	0
$478$	−6832.49	−0.653789
$479$	44.4456	0.00423961	0.00211980	−	0.999998i	$-0.499325\pi$
$479$	44.4456	0.00423961	0.00211980	+	0.999998i	$0.499325\pi$
$480$	0	0
$481$	−27476.8	−2.60464
$482$	19101.1	1.80505
$483$	0	0
$484$	−52.8333	−0.00496180
$485$	0	0
$486$	0	0
$487$	5871.85	0.546364	0.273182	−	0.961962i	$-0.411924\pi$
$487$	5871.85	0.546364	0.273182	+	0.961962i	$0.411924\pi$
$488$	1004.21	0.0931527
$489$	0	0
$490$	0	0
$491$	−12179.7	−1.11948	−0.559738	−	0.828670i	$-0.689098\pi$
$491$	−12179.7	−1.11948	−0.559738	+	0.828670i	$0.689098\pi$
$492$	0	0
$493$	−7452.73	−0.680840
$494$	−7125.98	−0.649014
$495$	0	0
$496$	20590.9	1.86403
$497$	−7425.88	−0.670213
$498$	0	0
$499$	−11654.7	−1.04556	−0.522779	−	0.852468i	$-0.675105\pi$
$499$	−11654.7	−1.04556	−0.522779	+	0.852468i	$0.675105\pi$
$500$	0	0
$501$	0	0
$502$	8884.82	0.789938
$503$	15940.4	1.41302	0.706508	−	0.707705i	$-0.250269\pi$
$503$	15940.4	1.41302	0.706508	+	0.707705i	$0.250269\pi$
$504$	0	0
$505$	0	0
$506$	28832.7	2.53314
$507$	0	0
$508$	−1824.39	−0.159339
$509$	14861.7	1.29417	0.647085	−	0.762418i	$-0.275988\pi$
$509$	14861.7	1.29417	0.647085	+	0.762418i	$0.275988\pi$
$510$	0	0
$511$	7057.71	0.610987
$512$	−12338.0	−1.06498
$513$	0	0
$514$	−20288.7	−1.74104
$515$	0	0
$516$	0	0
$517$	−7881.14	−0.670429
$518$	−9467.54	−0.803050
$519$	0	0
$520$	0	0
$521$	−16382.0	−1.37756	−0.688780	−	0.724970i	$-0.741853\pi$
$521$	−16382.0	−1.37756	−0.688780	+	0.724970i	$0.741853\pi$
$522$	0	0
$523$	10536.9	0.880966	0.440483	−	0.897761i	$-0.354807\pi$
$523$	10536.9	0.880966	0.440483	+	0.897761i	$0.354807\pi$
$524$	7797.22	0.650044
$525$	0	0
$526$	6672.11	0.553076
$527$	8556.27	0.707243
$528$	0	0
$529$	32919.7	2.70566
$530$	0	0
$531$	0	0
$532$	−1046.88	−0.0853155
$533$	−10077.6	−0.818967
$534$	0	0
$535$	0	0
$536$	−4862.87	−0.391873
$537$	0	0
$538$	−973.064	−0.0779772
$539$	1781.68	0.142380
$540$	0	0
$541$	−2373.66	−0.188635	−0.0943177	−	0.995542i	$-0.530067\pi$
$541$	−2373.66	−0.188635	−0.0943177	+	0.995542i	$0.530067\pi$
$542$	−892.233	−0.0707097
$543$	0	0
$544$	−7070.15	−0.557225
$545$	0	0
$546$	0	0
$547$	58.0430	0.00453700	0.00226850	−	0.999997i	$-0.499278\pi$
$547$	58.0430	0.00453700	0.00226850	+	0.999997i	$0.499278\pi$
$548$	−3241.62	−0.252692
$549$	0	0
$550$	0	0
$551$	5918.95	0.457633
$552$	0	0
$553$	2670.57	0.205360
$554$	13165.2	1.00963
$555$	0	0
$556$	19055.5	1.45347
$557$	10564.6	0.803653	0.401827	−	0.915716i	$-0.368375\pi$
$557$	10564.6	0.803653	0.401827	+	0.915716i	$0.368375\pi$
$558$	0	0
$559$	389.885	0.0294998
$560$	0	0
$561$	0	0
$562$	21553.7	1.61777
$563$	5200.12	0.389270	0.194635	−	0.980876i	$-0.437648\pi$
$563$	5200.12	0.389270	0.194635	+	0.980876i	$0.437648\pi$
$564$	0	0
$565$	0	0
$566$	17018.1	1.26382
$567$	0	0
$568$	8136.87	0.601084
$569$	17438.0	1.28478	0.642388	−	0.766380i	$-0.277944\pi$
$569$	17438.0	1.28478	0.642388	+	0.766380i	$0.277944\pi$
$570$	0	0
$571$	−9022.69	−0.661275	−0.330637	−	0.943758i	$-0.607264\pi$
$571$	−9022.69	−0.661275	−0.330637	+	0.943758i	$0.607264\pi$
$572$	16402.8	1.19901
$573$	0	0
$574$	−3472.39	−0.252500
$575$	0	0
$576$	0	0
$577$	−24717.4	−1.78336	−0.891681	−	0.452665i	$-0.850473\pi$
$577$	−24717.4	−1.78336	−0.891681	+	0.452665i	$0.850473\pi$
$578$	14601.9	1.05080
$579$	0	0
$580$	0	0
$581$	339.824	0.0242655
$582$	0	0
$583$	16560.2	1.17642
$584$	−7733.45	−0.547966
$585$	0	0
$586$	−19136.1	−1.34899
$587$	−12171.1	−0.855803	−0.427902	−	0.903825i	$-0.640747\pi$
$587$	−12171.1	−0.855803	−0.427902	+	0.903825i	$0.640747\pi$
$588$	0	0
$589$	−6795.39	−0.475380
$590$	0	0
$591$	0	0
$592$	27602.0	1.91628
$593$	−25236.5	−1.74762	−0.873811	−	0.486265i	$-0.838359\pi$
$593$	−25236.5	−1.74762	−0.873811	+	0.486265i	$0.838359\pi$
$594$	0	0
$595$	0	0
$596$	−4642.82	−0.319090
$597$	0	0
$598$	60159.3	4.11388
$599$	24921.8	1.69996	0.849982	−	0.526811i	$-0.176612\pi$
$599$	24921.8	1.69996	0.849982	+	0.526811i	$0.176612\pi$
$600$	0	0
$601$	15297.0	1.03823	0.519116	−	0.854704i	$-0.326261\pi$
$601$	15297.0	1.03823	0.519116	+	0.854704i	$0.326261\pi$
$602$	134.341	0.00909522
$603$	0	0
$604$	13184.1	0.888170
$605$	0	0
$606$	0	0
$607$	8715.89	0.582812	0.291406	−	0.956599i	$-0.405877\pi$
$607$	8715.89	0.582812	0.291406	+	0.956599i	$0.405877\pi$
$608$	5615.11	0.374544
$609$	0	0
$610$	0	0
$611$	−16444.0	−1.08879
$612$	0	0
$613$	24859.5	1.63796	0.818978	−	0.573825i	$-0.194541\pi$
$613$	24859.5	1.63796	0.818978	+	0.573825i	$0.194541\pi$
$614$	19865.6	1.30571
$615$	0	0
$616$	−1952.27	−0.127694
$617$	14557.6	0.949866	0.474933	−	0.880022i	$-0.342472\pi$
$617$	14557.6	0.949866	0.474933	+	0.880022i	$0.342472\pi$
$618$	0	0
$619$	4196.95	0.272519	0.136260	−	0.990673i	$-0.456492\pi$
$619$	4196.95	0.272519	0.136260	+	0.990673i	$0.456492\pi$
$620$	0	0
$621$	0	0
$622$	−100.742	−0.00649416
$623$	374.679	0.0240950
$624$	0	0
$625$	0	0
$626$	2820.37	0.180071
$627$	0	0
$628$	13095.3	0.832099
$629$	11469.6	0.727065
$630$	0	0
$631$	−20103.0	−1.26828	−0.634142	−	0.773216i	$-0.718647\pi$
$631$	−20103.0	−1.26828	−0.634142	+	0.773216i	$0.718647\pi$
$632$	−2926.26	−0.184178
$633$	0	0
$634$	2984.74	0.186970
$635$	0	0
$636$	0	0
$637$	3717.48	0.231227
$638$	−31955.1	−1.98294
$639$	0	0
$640$	0	0
$641$	3274.83	0.201791	0.100895	−	0.994897i	$-0.467829\pi$
$641$	3274.83	0.201791	0.100895	+	0.994897i	$0.467829\pi$
$642$	0	0
$643$	21839.6	1.33946	0.669729	−	0.742606i	$-0.266410\pi$
$643$	21839.6	1.33946	0.669729	+	0.742606i	$0.266410\pi$
$644$	8838.00	0.540785
$645$	0	0
$646$	2974.60	0.181167
$647$	2827.61	0.171816	0.0859079	−	0.996303i	$-0.472621\pi$
$647$	2827.61	0.171816	0.0859079	+	0.996303i	$0.472621\pi$
$648$	0	0
$649$	25084.7	1.51720
$650$	0	0
$651$	0	0
$652$	−267.682	−0.0160786
$653$	−29262.3	−1.75363	−0.876816	−	0.480826i	$-0.840337\pi$
$653$	−29262.3	−1.75363	−0.876816	+	0.480826i	$0.840337\pi$
$654$	0	0
$655$	0	0
$656$	10123.5	0.602527
$657$	0	0
$658$	−5666.03	−0.335691
$659$	−4458.49	−0.263548	−0.131774	−	0.991280i	$-0.542067\pi$
$659$	−4458.49	−0.263548	−0.131774	+	0.991280i	$0.542067\pi$
$660$	0	0
$661$	14714.8	0.865868	0.432934	−	0.901426i	$-0.357478\pi$
$661$	14714.8	0.865868	0.432934	+	0.901426i	$0.357478\pi$
$662$	−28105.8	−1.65010
$663$	0	0
$664$	−372.360	−0.0217626
$665$	0	0
$666$	0	0
$667$	−49969.3	−2.90078
$668$	−6883.98	−0.398727
$669$	0	0
$670$	0	0
$671$	4760.49	0.273885
$672$	0	0
$673$	−31419.5	−1.79960	−0.899802	−	0.436299i	$-0.856289\pi$
$673$	−31419.5	−1.79960	−0.899802	+	0.436299i	$0.856289\pi$
$674$	29819.1	1.70414
$675$	0	0
$676$	21160.9	1.20396
$677$	13711.0	0.778368	0.389184	−	0.921160i	$-0.372757\pi$
$677$	13711.0	0.778368	0.389184	+	0.921160i	$0.372757\pi$
$678$	0	0
$679$	6776.09	0.382979
$680$	0	0
$681$	0	0
$682$	36686.7	2.05984
$683$	24775.2	1.38799	0.693996	−	0.719979i	$-0.255849\pi$
$683$	24775.2	1.38799	0.693996	+	0.719979i	$0.255849\pi$
$684$	0	0
$685$	0	0
$686$	1280.92	0.0712909
$687$	0	0
$688$	−391.662	−0.0217034
$689$	34552.7	1.91053
$690$	0	0
$691$	−1661.87	−0.0914911	−0.0457456	−	0.998953i	$-0.514566\pi$
$691$	−1661.87	−0.0914911	−0.0457456	+	0.998953i	$0.514566\pi$
$692$	−7320.34	−0.402135
$693$	0	0
$694$	−3651.72	−0.199737
$695$	0	0
$696$	0	0
$697$	4206.69	0.228608
$698$	28023.9	1.51966
$699$	0	0
$700$	0	0
$701$	−19207.7	−1.03490	−0.517449	−	0.855714i	$-0.673118\pi$
$701$	−19207.7	−1.03490	−0.517449	+	0.855714i	$0.673118\pi$
$702$	0	0
$703$	−9109.17	−0.488704
$704$	−8145.41	−0.436068
$705$	0	0
$706$	−37129.3	−1.97929
$707$	5112.61	0.271965
$708$	0	0
$709$	−31511.5	−1.66916	−0.834582	−	0.550883i	$-0.814291\pi$
$709$	−31511.5	−1.66916	−0.834582	+	0.550883i	$0.814291\pi$
$710$	0	0
$711$	0	0
$712$	−410.553	−0.0216097
$713$	57368.4	3.01327
$714$	0	0
$715$	0	0
$716$	−12649.5	−0.660241
$717$	0	0
$718$	16762.6	0.871273
$719$	−14842.3	−0.769854	−0.384927	−	0.922947i	$-0.625773\pi$
$719$	−14842.3	−0.769854	−0.384927	+	0.922947i	$0.625773\pi$
$720$	0	0
$721$	−8621.22	−0.445313
$722$	23252.1	1.19855
$723$	0	0
$724$	−1345.65	−0.0690756
$725$	0	0
$726$	0	0
$727$	14698.6	0.749852	0.374926	−	0.927055i	$-0.377668\pi$
$727$	14698.6	0.749852	0.374926	+	0.927055i	$0.377668\pi$
$728$	−4073.41	−0.207377
$729$	0	0
$730$	0	0
$731$	−162.750	−0.00823462
$732$	0	0
$733$	−32319.1	−1.62856	−0.814280	−	0.580473i	$-0.802868\pi$
$733$	−32319.1	−1.62856	−0.814280	+	0.580473i	$0.802868\pi$
$734$	19689.1	0.990109
$735$	0	0
$736$	−47404.2	−2.37411
$737$	−23052.6	−1.15218
$738$	0	0
$739$	−9943.93	−0.494984	−0.247492	−	0.968890i	$-0.579606\pi$
$739$	−9943.93	−0.494984	−0.247492	+	0.968890i	$0.579606\pi$
$740$	0	0
$741$	0	0
$742$	11905.7	0.589045
$743$	−4740.07	−0.234046	−0.117023	−	0.993129i	$-0.537335\pi$
$743$	−4740.07	−0.234046	−0.117023	+	0.993129i	$0.537335\pi$
$744$	0	0
$745$	0	0
$746$	−12290.7	−0.603211
$747$	0	0
$748$	−6847.03	−0.334695
$749$	−2980.75	−0.145413
$750$	0	0
$751$	32678.9	1.58784	0.793921	−	0.608020i	$-0.208036\pi$
$751$	32678.9	1.58784	0.793921	+	0.608020i	$0.208036\pi$
$752$	16518.9	0.801042
$753$	0	0
$754$	−66674.2	−3.22033
$755$	0	0
$756$	0	0
$757$	−11280.9	−0.541627	−0.270814	−	0.962632i	$-0.587293\pi$
$757$	−11280.9	−0.541627	−0.270814	+	0.962632i	$0.587293\pi$
$758$	99.8737	0.00478572
$759$	0	0
$760$	0	0
$761$	−12303.2	−0.586060	−0.293030	−	0.956103i	$-0.594664\pi$
$761$	−12303.2	−0.586060	−0.293030	+	0.956103i	$0.594664\pi$
$762$	0	0
$763$	11777.6	0.558820
$764$	7199.43	0.340924
$765$	0	0
$766$	11542.3	0.544439
$767$	52339.1	2.46396
$768$	0	0
$769$	−13816.2	−0.647888	−0.323944	−	0.946076i	$-0.605009\pi$
$769$	−13816.2	−0.647888	−0.323944	+	0.946076i	$0.605009\pi$
$770$	0	0
$771$	0	0
$772$	−3594.73	−0.167587
$773$	−28337.7	−1.31855	−0.659274	−	0.751903i	$-0.729136\pi$
$773$	−28337.7	−1.31855	−0.659274	+	0.751903i	$0.729136\pi$
$774$	0	0
$775$	0	0
$776$	−7424.87	−0.343476
$777$	0	0
$778$	39778.2	1.83306
$779$	−3340.95	−0.153661
$780$	0	0
$781$	38573.1	1.76729
$782$	−25112.3	−1.14836
$783$	0	0
$784$	−3734.42	−0.170118
$785$	0	0
$786$	0	0
$787$	30356.0	1.37493	0.687467	−	0.726215i	$-0.258722\pi$
$787$	30356.0	1.37493	0.687467	+	0.726215i	$0.258722\pi$
$788$	−16267.5	−0.735414
$789$	0	0
$790$	0	0
$791$	1250.64	0.0562168
$792$	0	0
$793$	9932.76	0.444795
$794$	−2790.14	−0.124708
$795$	0	0
$796$	−14633.9	−0.651614
$797$	7859.19	0.349293	0.174647	−	0.984631i	$-0.444122\pi$
$797$	7859.19	0.349293	0.174647	+	0.984631i	$0.444122\pi$
$798$	0	0
$799$	6864.21	0.303928
$800$	0	0
$801$	0	0
$802$	4895.06	0.215525
$803$	−36660.6	−1.61112
$804$	0	0
$805$	0	0
$806$	76546.8	3.34522
$807$	0	0
$808$	−5602.12	−0.243913
$809$	23799.5	1.03429	0.517147	−	0.855896i	$-0.326994\pi$
$809$	23799.5	1.03429	0.517147	+	0.855896i	$0.326994\pi$
$810$	0	0
$811$	−11319.3	−0.490105	−0.245053	−	0.969510i	$-0.578805\pi$
$811$	−11319.3	−0.490105	−0.245053	+	0.969510i	$0.578805\pi$
$812$	−9795.10	−0.423326
$813$	0	0
$814$	49178.3	2.11757
$815$	0	0
$816$	0	0
$817$	129.256	0.00553498
$818$	39543.1	1.69021
$819$	0	0
$820$	0	0
$821$	2171.84	0.0923236	0.0461618	−	0.998934i	$-0.485301\pi$
$821$	2171.84	0.0923236	0.0461618	+	0.998934i	$0.485301\pi$
$822$	0	0
$823$	−11850.1	−0.501907	−0.250954	−	0.967999i	$-0.580744\pi$
$823$	−11850.1	−0.501907	−0.250954	+	0.967999i	$0.580744\pi$
$824$	9446.66	0.399381
$825$	0	0
$826$	18034.3	0.759676
$827$	−4612.78	−0.193956	−0.0969782	−	0.995287i	$-0.530918\pi$
$827$	−4612.78	−0.193956	−0.0969782	+	0.995287i	$0.530918\pi$
$828$	0	0
$829$	27833.8	1.16611	0.583057	−	0.812431i	$-0.301856\pi$
$829$	27833.8	1.16611	0.583057	+	0.812431i	$0.301856\pi$
$830$	0	0
$831$	0	0
$832$	−16995.4	−0.708184
$833$	−1551.79	−0.0645453
$834$	0	0
$835$	0	0
$836$	5437.91	0.224969
$837$	0	0
$838$	42149.1	1.73749
$839$	25136.5	1.03434	0.517168	−	0.855884i	$-0.326986\pi$
$839$	25136.5	1.03434	0.517168	+	0.855884i	$0.326986\pi$
$840$	0	0
$841$	30991.7	1.27072
$842$	−25930.0	−1.06129
$843$	0	0
$844$	13076.3	0.533300
$845$	0	0
$846$	0	0
$847$	62.1976	0.00252318
$848$	−34710.2	−1.40561
$849$	0	0
$850$	0	0
$851$	76901.9	3.09773
$852$	0	0
$853$	−17398.4	−0.698370	−0.349185	−	0.937054i	$-0.613542\pi$
$853$	−17398.4	−0.698370	−0.349185	+	0.937054i	$0.613542\pi$
$854$	3422.49	0.137137
$855$	0	0
$856$	3266.14	0.130414
$857$	−21963.3	−0.875440	−0.437720	−	0.899111i	$-0.644214\pi$
$857$	−21963.3	−0.875440	−0.437720	+	0.899111i	$0.644214\pi$
$858$	0	0
$859$	26796.4	1.06436	0.532178	−	0.846632i	$-0.321374\pi$
$859$	26796.4	1.06436	0.532178	+	0.846632i	$0.321374\pi$
$860$	0	0
$861$	0	0
$862$	23538.7	0.930082
$863$	−988.582	−0.0389939	−0.0194969	−	0.999810i	$-0.506206\pi$
$863$	−988.582	−0.0389939	−0.0194969	+	0.999810i	$0.506206\pi$
$864$	0	0
$865$	0	0
$866$	−42449.6	−1.66570
$867$	0	0
$868$	11245.5	0.439742
$869$	−13872.0	−0.541515
$870$	0	0
$871$	−48099.2	−1.87116
$872$	−12905.3	−0.501179
$873$	0	0
$874$	19944.2	0.771879
$875$	0	0
$876$	0	0
$877$	−11910.6	−0.458602	−0.229301	−	0.973356i	$-0.573644\pi$
$877$	−11910.6	−0.458602	−0.229301	+	0.973356i	$0.573644\pi$
$878$	17055.5	0.655575
$879$	0	0
$880$	0	0
$881$	−49025.0	−1.87479	−0.937397	−	0.348264i	$-0.886771\pi$
$881$	−49025.0	−1.87479	−0.937397	+	0.348264i	$0.886771\pi$
$882$	0	0
$883$	−3871.59	−0.147553	−0.0737766	−	0.997275i	$-0.523505\pi$
$883$	−3871.59	−0.147553	−0.0737766	+	0.997275i	$0.523505\pi$
$884$	−14286.3	−0.543553
$885$	0	0
$886$	53635.8	2.03378
$887$	−33233.8	−1.25804	−0.629019	−	0.777390i	$-0.716543\pi$
$887$	−33233.8	−1.25804	−0.629019	+	0.777390i	$0.716543\pi$
$888$	0	0
$889$	2147.75	0.0810271
$890$	0	0
$891$	0	0
$892$	−18561.4	−0.696727
$893$	−5451.55	−0.204288
$894$	0	0
$895$	0	0
$896$	6646.01	0.247799
$897$	0	0
$898$	13828.2	0.513867
$899$	−63581.0	−2.35878
$900$	0	0
$901$	−14423.3	−0.533309
$902$	18037.1	0.665818
$903$	0	0
$904$	−1370.38	−0.0504182
$905$	0	0
$906$	0	0
$907$	−43792.5	−1.60320	−0.801601	−	0.597859i	$-0.796018\pi$
$907$	−43792.5	−1.60320	−0.801601	+	0.597859i	$0.796018\pi$
$908$	−7443.21	−0.272039
$909$	0	0
$910$	0	0
$911$	40514.0	1.47342	0.736711	−	0.676208i	$-0.236378\pi$
$911$	40514.0	1.47342	0.736711	+	0.676208i	$0.236378\pi$
$912$	0	0
$913$	−1765.19	−0.0639859
$914$	−7818.86	−0.282960
$915$	0	0
$916$	−19984.5	−0.720860
$917$	−9179.24	−0.330562
$918$	0	0
$919$	−10001.2	−0.358987	−0.179493	−	0.983759i	$-0.557446\pi$
$919$	−10001.2	−0.358987	−0.179493	+	0.983759i	$0.557446\pi$
$920$	0	0
$921$	0	0
$922$	22003.7	0.785959
$923$	80482.6	2.87012
$924$	0	0
$925$	0	0
$926$	−36125.9	−1.28204
$927$	0	0
$928$	52537.8	1.85845
$929$	−30626.7	−1.08163	−0.540813	−	0.841143i	$-0.681883\pi$
$929$	−30626.7	−1.08163	−0.540813	+	0.841143i	$0.681883\pi$
$930$	0	0
$931$	1232.43	0.0433848
$932$	3679.56	0.129322
$933$	0	0
$934$	45883.8	1.60746
$935$	0	0
$936$	0	0
$937$	−42363.5	−1.47701	−0.738503	−	0.674250i	$-0.764467\pi$
$937$	−42363.5	−1.47701	−0.738503	+	0.674250i	$0.764467\pi$
$938$	−16573.3	−0.576906
$939$	0	0
$940$	0	0
$941$	22857.4	0.791848	0.395924	−	0.918283i	$-0.370424\pi$
$941$	22857.4	0.791848	0.395924	+	0.918283i	$0.370424\pi$
$942$	0	0
$943$	28205.2	0.974006
$944$	−52577.7	−1.81277
$945$	0	0
$946$	−697.822	−0.0239832
$947$	−31777.4	−1.09042	−0.545210	−	0.838300i	$-0.683550\pi$
$947$	−31777.4	−1.09042	−0.545210	+	0.838300i	$0.683550\pi$
$948$	0	0
$949$	−76492.3	−2.61649
$950$	0	0
$951$	0	0
$952$	1700.36	0.0578877
$953$	−37460.0	−1.27329	−0.636647	−	0.771156i	$-0.719679\pi$
$953$	−37460.0	−1.27329	−0.636647	+	0.771156i	$0.719679\pi$
$954$	0	0
$955$	0	0
$956$	10878.9	0.368042
$957$	0	0
$958$	−165.980	−0.00559767
$959$	3816.18	0.128499
$960$	0	0
$961$	43204.6	1.45026
$962$	102610.	3.43898
$963$	0	0
$964$	−30413.3	−1.01613
$965$	0	0
$966$	0	0
$967$	−36595.6	−1.21700	−0.608498	−	0.793555i	$-0.708228\pi$
$967$	−36595.6	−1.21700	−0.608498	+	0.793555i	$0.708228\pi$
$968$	−68.1528	−0.00226293
$969$	0	0
$970$	0	0
$971$	−13441.8	−0.444253	−0.222126	−	0.975018i	$-0.571300\pi$
$971$	−13441.8	−0.444253	−0.222126	+	0.975018i	$0.571300\pi$
$972$	0	0
$973$	−22432.9	−0.739123
$974$	−21928.1	−0.721378
$975$	0	0
$976$	−9978.03	−0.327243
$977$	17928.9	0.587101	0.293550	−	0.955944i	$-0.405163\pi$
$977$	17928.9	0.587101	0.293550	+	0.955944i	$0.405163\pi$
$978$	0	0
$979$	−1946.24	−0.0635363
$980$	0	0
$981$	0	0
$982$	45484.4	1.47807
$983$	44229.0	1.43508	0.717542	−	0.696515i	$-0.245267\pi$
$983$	44229.0	1.43508	0.717542	+	0.696515i	$0.245267\pi$
$984$	0	0
$985$	0	0
$986$	27831.8	0.898930
$987$	0	0
$988$	11346.2	0.365354
$989$	−1091.21	−0.0350844
$990$	0	0
$991$	8843.98	0.283490	0.141745	−	0.989903i	$-0.454729\pi$
$991$	8843.98	0.283490	0.141745	+	0.989903i	$0.454729\pi$
$992$	−60317.2	−1.93052
$993$	0	0
$994$	27731.5	0.884900
$995$	0	0
$996$	0	0
$997$	27943.9	0.887656	0.443828	−	0.896112i	$-0.353620\pi$
$997$	27943.9	0.887656	0.443828	+	0.896112i	$0.353620\pi$
$998$	43523.7	1.38048
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bt.1.3		10
3.2	odd	2	inner	1575.4.a.bt.1.8		10
5.2	odd	4		315.4.d.d.64.5	✓	20
5.3	odd	4		315.4.d.d.64.15	yes	20
5.4	even	2		1575.4.a.bu.1.8		10
15.2	even	4		315.4.d.d.64.16	yes	20
15.8	even	4		315.4.d.d.64.6	yes	20
15.14	odd	2		1575.4.a.bu.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.d.d.64.5	✓	20	5.2	odd	4
315.4.d.d.64.6	yes	20	15.8	even	4
315.4.d.d.64.15	yes	20	5.3	odd	4
315.4.d.d.64.16	yes	20	15.2	even	4
1575.4.a.bt.1.3		10	1.1	even	1	trivial
1575.4.a.bt.1.8		10	3.2	odd	2	inner
1575.4.a.bu.1.3		10	15.14	odd	2
1575.4.a.bu.1.8		10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-3.73445 q^{2} +5.94609 q^{4} -7.00000 q^{7} +7.67022 q^{8} +O(q^{10})\)	\(q-3.73445 q^{2} +5.94609 q^{4} -7.00000 q^{7} +7.67022 q^{8} +36.3609 q^{11} +75.8669 q^{13} +26.1411 q^{14} -76.2127 q^{16} -31.6691 q^{17} +25.1516 q^{19} -135.788 q^{22} -212.336 q^{23} -283.321 q^{26} -41.6226 q^{28} +235.331 q^{29} -270.177 q^{31} +223.251 q^{32} +118.267 q^{34} -362.170 q^{37} -93.9273 q^{38} -132.833 q^{41} +5.13906 q^{43} +216.205 q^{44} +792.959 q^{46} -216.748 q^{47} +49.0000 q^{49} +451.112 q^{52} +455.439 q^{53} -53.6915 q^{56} -878.831 q^{58} +689.881 q^{59} +130.923 q^{61} +1008.96 q^{62} -224.016 q^{64} -633.994 q^{67} -188.307 q^{68} +1060.84 q^{71} -1008.24 q^{73} +1352.51 q^{74} +149.554 q^{76} -254.526 q^{77} -381.510 q^{79} +496.056 q^{82} -48.5463 q^{83} -19.1915 q^{86} +278.896 q^{88} -53.5256 q^{89} -531.068 q^{91} -1262.57 q^{92} +809.432 q^{94} -968.013 q^{97} -182.988 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 54 q^{4} - 70 q^{7}+O(q^{10}) \) 10 * q + 54 * q^4 - 70 * q^7	\( 10 q + 54 q^{4} - 70 q^{7} - 104 q^{13} + 310 q^{16} + 36 q^{19} - 644 q^{22} - 378 q^{28} - 24 q^{31} + 116 q^{34} - 732 q^{37} - 212 q^{43} - 184 q^{46} + 490 q^{49} - 2692 q^{52} - 3196 q^{58} + 1024 q^{61} + 1590 q^{64} - 2388 q^{67} - 3432 q^{73} - 4184 q^{76} - 776 q^{79} - 1860 q^{82} - 10164 q^{88} + 728 q^{91} - 4028 q^{94} - 4024 q^{97}+O(q^{100}) \) 10 * q + 54 * q^4 - 70 * q^7 - 104 * q^13 + 310 * q^16 + 36 * q^19 - 644 * q^22 - 378 * q^28 - 24 * q^31 + 116 * q^34 - 732 * q^37 - 212 * q^43 - 184 * q^46 + 490 * q^49 - 2692 * q^52 - 3196 * q^58 + 1024 * q^61 + 1590 * q^64 - 2388 * q^67 - 3432 * q^73 - 4184 * q^76 - 776 * q^79 - 1860 * q^82 - 10164 * q^88 + 728 * q^91 - 4028 * q^94 - 4024 * q^97

Introduction

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