LMFDB - Embedded newform 1575.4.a.be.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.22952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 18x + 14 $ x^3 - x^2 - 18*x + 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.15010$ 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)

$f(q)$	$=$	$q+5.15010 q^{2} +18.5235 q^{4} -7.00000 q^{7} +54.1971 q^{8} +14.0073 q^{11} +10.5931 q^{13} -36.0507 q^{14} +130.932 q^{16} +131.801 q^{17} +38.6871 q^{19} +72.1391 q^{22} -102.481 q^{23} +54.5553 q^{26} -129.665 q^{28} +232.376 q^{29} -165.501 q^{31} +240.738 q^{32} +678.789 q^{34} -280.004 q^{37} +199.242 q^{38} +122.271 q^{41} +431.844 q^{43} +259.465 q^{44} -527.787 q^{46} +295.692 q^{47} +49.0000 q^{49} +196.221 q^{52} +243.639 q^{53} -379.380 q^{56} +1196.76 q^{58} -566.480 q^{59} -188.012 q^{61} -852.346 q^{62} +192.364 q^{64} +871.647 q^{67} +2441.42 q^{68} +176.800 q^{71} +220.524 q^{73} -1442.05 q^{74} +716.621 q^{76} -98.0513 q^{77} -190.494 q^{79} +629.710 q^{82} +518.241 q^{83} +2224.04 q^{86} +759.157 q^{88} +598.355 q^{89} -74.1515 q^{91} -1898.31 q^{92} +1522.85 q^{94} +1882.24 q^{97} +252.355 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 14 q^{4} - 21 q^{7} + 66 q^{8} + 20 q^{11} - 14 q^{14} + 114 q^{16} + 234 q^{17} - 82 q^{19} + 236 q^{22} + 30 q^{23} - 76 q^{26} - 98 q^{28} + 32 q^{29} - 362 q^{31} + 430 q^{32} + 596 q^{34}+ \cdots + 98 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 14 * q^4 - 21 * q^7 + 66 * q^8 + 20 * q^11 - 14 * q^14 + 114 * q^16 + 234 * q^17 - 82 * q^19 + 236 * q^22 + 30 * q^23 - 76 * q^26 - 98 * q^28 + 32 * q^29 - 362 * q^31 + 430 * q^32 + 596 * q^34 - 302 * q^37 + 160 * q^38 - 626 * q^41 + 68 * q^43 - 300 * q^44 - 412 * q^46 + 936 * q^47 + 147 * q^49 + 684 * q^52 + 298 * q^53 - 462 * q^56 + 1184 * q^58 - 428 * q^59 + 1050 * q^61 - 608 * q^62 - 154 * q^64 + 52 * q^67 + 1964 * q^68 - 416 * q^71 + 312 * q^73 - 1668 * q^74 + 1712 * q^76 - 140 * q^77 - 488 * q^79 + 1476 * q^82 + 948 * q^83 + 1928 * q^86 - 124 * q^88 + 142 * q^89 - 3220 * q^92 + 328 * q^94 + 3052 * q^97 + 98 * 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 $

$2$	5.15010	1.82083	0.910417	−	0.413691i	$-0.135761\pi$
$2$	5.15010	1.82083	0.910417	+	0.413691i	$0.135761\pi$
$3$	0	0
$3$	0	0
$4$	18.5235	2.31544
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	54.1971	2.39520
$9$	0	0
$10$	0	0
$11$	14.0073	0.383943	0.191971	−	0.981401i	$-0.438512\pi$
$11$	14.0073	0.383943	0.191971	+	0.981401i	$0.438512\pi$
$12$	0	0
$13$	10.5931	0.225999	0.113000	−	0.993595i	$-0.463954\pi$
$13$	10.5931	0.225999	0.113000	+	0.993595i	$0.463954\pi$
$14$	−36.0507	−0.688211
$15$	0	0
$16$	130.932	2.04582
$17$	131.801	1.88038	0.940191	−	0.340649i	$-0.110647\pi$
$17$	131.801	1.88038	0.940191	+	0.340649i	$0.110647\pi$
$18$	0	0
$19$	38.6871	0.467128	0.233564	−	0.972341i	$-0.424961\pi$
$19$	38.6871	0.467128	0.233564	+	0.972341i	$0.424961\pi$
$20$	0	0
$21$	0	0
$22$	72.1391	0.699096
$23$	−102.481	−0.929076	−0.464538	−	0.885553i	$-0.653780\pi$
$23$	−102.481	−0.929076	−0.464538	+	0.885553i	$0.653780\pi$
$24$	0	0
$25$	0	0
$26$	54.5553	0.411507
$27$	0	0
$28$	−129.665	−0.875154
$29$	232.376	1.48797	0.743986	−	0.668195i	$-0.232933\pi$
$29$	232.376	1.48797	0.743986	+	0.668195i	$0.232933\pi$
$30$	0	0
$31$	−165.501	−0.958866	−0.479433	−	0.877578i	$-0.659158\pi$
$31$	−165.501	−0.958866	−0.479433	+	0.877578i	$0.659158\pi$
$32$	240.738	1.32990
$33$	0	0
$34$	678.789	3.42386
$35$	0	0
$36$	0	0
$37$	−280.004	−1.24412	−0.622059	−	0.782970i	$-0.713704\pi$
$37$	−280.004	−1.24412	−0.622059	+	0.782970i	$0.713704\pi$
$38$	199.242	0.850563
$39$	0	0
$40$	0	0
$41$	122.271	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$41$	122.271	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$42$	0	0
$43$	431.844	1.53153	0.765764	−	0.643122i	$-0.222361\pi$
$43$	431.844	1.53153	0.765764	+	0.643122i	$0.222361\pi$
$44$	259.465	0.888996
$45$	0	0
$46$	−527.787	−1.69169
$47$	295.692	0.917685	0.458842	−	0.888518i	$-0.348264\pi$
$47$	295.692	0.917685	0.458842	+	0.888518i	$0.348264\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	196.221	0.523287
$53$	243.639	0.631441	0.315721	−	0.948852i	$-0.397754\pi$
$53$	243.639	0.631441	0.315721	+	0.948852i	$0.397754\pi$
$54$	0	0
$55$	0	0
$56$	−379.380	−0.905299
$57$	0	0
$58$	1196.76	2.70935
$59$	−566.480	−1.24999	−0.624995	−	0.780628i	$-0.714899\pi$
$59$	−566.480	−1.24999	−0.624995	+	0.780628i	$0.714899\pi$
$60$	0	0
$61$	−188.012	−0.394630	−0.197315	−	0.980340i	$-0.563222\pi$
$61$	−188.012	−0.394630	−0.197315	+	0.980340i	$0.563222\pi$
$62$	−852.346	−1.74594
$63$	0	0
$64$	192.364	0.375711
$65$	0	0
$66$	0	0
$67$	871.647	1.58938	0.794692	−	0.607013i	$-0.207633\pi$
$67$	871.647	1.58938	0.794692	+	0.607013i	$0.207633\pi$
$68$	2441.42	4.35391
$69$	0	0
$70$	0	0
$71$	176.800	0.295525	0.147762	−	0.989023i	$-0.452793\pi$
$71$	176.800	0.295525	0.147762	+	0.989023i	$0.452793\pi$
$72$	0	0
$73$	220.524	0.353566	0.176783	−	0.984250i	$-0.443431\pi$
$73$	220.524	0.353566	0.176783	+	0.984250i	$0.443431\pi$
$74$	−1442.05	−2.26533
$75$	0	0
$76$	716.621	1.08161
$77$	−98.0513	−0.145117
$78$	0	0
$79$	−190.494	−0.271294	−0.135647	−	0.990757i	$-0.543311\pi$
$79$	−190.494	−0.271294	−0.135647	+	0.990757i	$0.543311\pi$
$80$	0	0
$81$	0	0
$82$	629.710	0.848047
$83$	518.241	0.685353	0.342676	−	0.939453i	$-0.388667\pi$
$83$	518.241	0.685353	0.342676	+	0.939453i	$0.388667\pi$
$84$	0	0
$85$	0	0
$86$	2224.04	2.78866
$87$	0	0
$88$	759.157	0.919618
$89$	598.355	0.712646	0.356323	−	0.934363i	$-0.384030\pi$
$89$	598.355	0.712646	0.356323	+	0.934363i	$0.384030\pi$
$90$	0	0
$91$	−74.1515	−0.0854196
$92$	−1898.31	−2.15122
$93$	0	0
$94$	1522.85	1.67095
$95$	0	0
$96$	0	0
$97$	1882.24	1.97023	0.985116	−	0.171889i	$-0.0549871\pi$
$97$	1882.24	1.97023	0.985116	+	0.171889i	$0.0549871\pi$
$98$	252.355	0.260119
$99$	0	0
$100$	0	0
$101$	−190.791	−0.187964	−0.0939821	−	0.995574i	$-0.529960\pi$
$101$	−190.791	−0.187964	−0.0939821	+	0.995574i	$0.529960\pi$
$102$	0	0
$103$	1496.53	1.43163	0.715813	−	0.698292i	$-0.246056\pi$
$103$	1496.53	1.43163	0.715813	+	0.698292i	$0.246056\pi$
$104$	574.114	0.541312
$105$	0	0
$106$	1254.76	1.14975
$107$	−1965.85	−1.77613	−0.888066	−	0.459716i	$-0.847951\pi$
$107$	−1965.85	−1.77613	−0.888066	+	0.459716i	$0.847951\pi$
$108$	0	0
$109$	−870.482	−0.764927	−0.382464	−	0.923971i	$-0.624924\pi$
$109$	−870.482	−0.764927	−0.382464	+	0.923971i	$0.624924\pi$
$110$	0	0
$111$	0	0
$112$	−916.527	−0.773247
$113$	2051.20	1.70762	0.853809	−	0.520587i	$-0.174287\pi$
$113$	2051.20	1.70762	0.853809	+	0.520587i	$0.174287\pi$
$114$	0	0
$115$	0	0
$116$	4304.42	3.44531
$117$	0	0
$118$	−2917.43	−2.27603
$119$	−922.608	−0.710717
$120$	0	0
$121$	−1134.79	−0.852588
$122$	−968.279	−0.718556
$123$	0	0
$124$	−3065.66	−2.22020
$125$	0	0
$126$	0	0
$127$	−2501.15	−1.74757	−0.873785	−	0.486313i	$-0.838341\pi$
$127$	−2501.15	−1.74757	−0.873785	+	0.486313i	$0.838341\pi$
$128$	−935.209	−0.645793
$129$	0	0
$130$	0	0
$131$	−534.190	−0.356278	−0.178139	−	0.984005i	$-0.557008\pi$
$131$	−534.190	−0.356278	−0.178139	+	0.984005i	$0.557008\pi$
$132$	0	0
$133$	−270.810	−0.176558
$134$	4489.07	2.89400
$135$	0	0
$136$	7143.24	4.50388
$137$	503.159	0.313779	0.156890	−	0.987616i	$-0.449853\pi$
$137$	503.159	0.313779	0.156890	+	0.987616i	$0.449853\pi$
$138$	0	0
$139$	1424.87	0.869466	0.434733	−	0.900559i	$-0.356843\pi$
$139$	1424.87	0.869466	0.434733	+	0.900559i	$0.356843\pi$
$140$	0	0
$141$	0	0
$142$	910.536	0.538102
$143$	148.381	0.0867707
$144$	0	0
$145$	0	0
$146$	1135.72	0.643786
$147$	0	0
$148$	−5186.66	−2.88068
$149$	−1482.49	−0.815105	−0.407552	−	0.913182i	$-0.633618\pi$
$149$	−1482.49	−0.815105	−0.407552	+	0.913182i	$0.633618\pi$
$150$	0	0
$151$	−1267.34	−0.683008	−0.341504	−	0.939880i	$-0.610936\pi$
$151$	−1267.34	−0.683008	−0.341504	+	0.939880i	$0.610936\pi$
$152$	2096.73	1.11886
$153$	0	0
$154$	−504.974	−0.264233
$155$	0	0
$156$	0	0
$157$	−3234.00	−1.64396	−0.821979	−	0.569517i	$-0.807130\pi$
$157$	−3234.00	−1.64396	−0.821979	+	0.569517i	$0.807130\pi$
$158$	−981.063	−0.493982
$159$	0	0
$160$	0	0
$161$	717.367	0.351158
$162$	0	0
$163$	−1758.52	−0.845017	−0.422508	−	0.906359i	$-0.638850\pi$
$163$	−1758.52	−0.845017	−0.422508	+	0.906359i	$0.638850\pi$
$164$	2264.90	1.07841
$165$	0	0
$166$	2668.99	1.24791
$167$	1682.57	0.779646	0.389823	−	0.920890i	$-0.372536\pi$
$167$	1682.57	0.779646	0.389823	+	0.920890i	$0.372536\pi$
$168$	0	0
$169$	−2084.79	−0.948924
$170$	0	0
$171$	0	0
$172$	7999.28	3.54616
$173$	3211.53	1.41138	0.705688	−	0.708523i	$-0.250638\pi$
$173$	3211.53	1.41138	0.705688	+	0.708523i	$0.250638\pi$
$174$	0	0
$175$	0	0
$176$	1834.01	0.785477
$177$	0	0
$178$	3081.59	1.29761
$179$	−3311.97	−1.38295	−0.691476	−	0.722399i	$-0.743039\pi$
$179$	−3311.97	−1.38295	−0.691476	+	0.722399i	$0.743039\pi$
$180$	0	0
$181$	2062.42	0.846954	0.423477	−	0.905907i	$-0.360809\pi$
$181$	2062.42	0.846954	0.423477	+	0.905907i	$0.360809\pi$
$182$	−381.887	−0.155535
$183$	0	0
$184$	−5554.17	−2.22532
$185$	0	0
$186$	0	0
$187$	1846.18	0.721958
$188$	5477.26	2.12484
$189$	0	0
$190$	0	0
$191$	−2783.01	−1.05430	−0.527150	−	0.849772i	$-0.676739\pi$
$191$	−2783.01	−1.05430	−0.527150	+	0.849772i	$0.676739\pi$
$192$	0	0
$193$	−300.293	−0.111998	−0.0559988	−	0.998431i	$-0.517834\pi$
$193$	−300.293	−0.111998	−0.0559988	+	0.998431i	$0.517834\pi$
$194$	9693.72	3.58747
$195$	0	0
$196$	907.652	0.330777
$197$	−1485.64	−0.537295	−0.268648	−	0.963239i	$-0.586577\pi$
$197$	−1485.64	−0.537295	−0.268648	+	0.963239i	$0.586577\pi$
$198$	0	0
$199$	−3040.55	−1.08311	−0.541555	−	0.840665i	$-0.682164\pi$
$199$	−3040.55	−1.08311	−0.541555	+	0.840665i	$0.682164\pi$
$200$	0	0
$201$	0	0
$202$	−982.591	−0.342252
$203$	−1626.63	−0.562400
$204$	0	0
$205$	0	0
$206$	7707.28	2.60676
$207$	0	0
$208$	1386.98	0.462353
$209$	541.903	0.179350
$210$	0	0
$211$	−2886.74	−0.941855	−0.470928	−	0.882172i	$-0.656081\pi$
$211$	−2886.74	−0.941855	−0.470928	+	0.882172i	$0.656081\pi$
$212$	4513.05	1.46206
$213$	0	0
$214$	−10124.3	−3.23404
$215$	0	0
$216$	0	0
$217$	1158.51	0.362417
$218$	−4483.07	−1.39281
$219$	0	0
$220$	0	0
$221$	1396.18	0.424964
$222$	0	0
$223$	−4591.34	−1.37874	−0.689369	−	0.724410i	$-0.742112\pi$
$223$	−4591.34	−1.37874	−0.689369	+	0.724410i	$0.742112\pi$
$224$	−1685.16	−0.502655
$225$	0	0
$226$	10563.9	3.10929
$227$	−2547.05	−0.744729	−0.372364	−	0.928087i	$-0.621453\pi$
$227$	−2547.05	−0.744729	−0.372364	+	0.928087i	$0.621453\pi$
$228$	0	0
$229$	2007.40	0.579269	0.289635	−	0.957137i	$-0.406466\pi$
$229$	2007.40	0.579269	0.289635	+	0.957137i	$0.406466\pi$
$230$	0	0
$231$	0	0
$232$	12594.1	3.56399
$233$	2810.65	0.790266	0.395133	−	0.918624i	$-0.370699\pi$
$233$	2810.65	0.790266	0.395133	+	0.918624i	$0.370699\pi$
$234$	0	0
$235$	0	0
$236$	−10493.2	−2.89428
$237$	0	0
$238$	−4751.52	−1.29410
$239$	−1848.68	−0.500339	−0.250170	−	0.968202i	$-0.580486\pi$
$239$	−1848.68	−0.500339	−0.250170	+	0.968202i	$0.580486\pi$
$240$	0	0
$241$	5722.64	1.52958	0.764788	−	0.644283i	$-0.222844\pi$
$241$	5722.64	1.52958	0.764788	+	0.644283i	$0.222844\pi$
$242$	−5844.30	−1.55242
$243$	0	0
$244$	−3482.64	−0.913742
$245$	0	0
$246$	0	0
$247$	409.815	0.105571
$248$	−8969.68	−2.29667
$249$	0	0
$250$	0	0
$251$	−1723.38	−0.433381	−0.216691	−	0.976240i	$-0.569526\pi$
$251$	−1723.38	−0.433381	−0.216691	+	0.976240i	$0.569526\pi$
$252$	0	0
$253$	−1435.48	−0.356712
$254$	−12881.2	−3.18203
$255$	0	0
$256$	−6355.33	−1.55159
$257$	−5754.01	−1.39660	−0.698298	−	0.715807i	$-0.746059\pi$
$257$	−5754.01	−1.39660	−0.698298	+	0.715807i	$0.746059\pi$
$258$	0	0
$259$	1960.03	0.470232
$260$	0	0
$261$	0	0
$262$	−2751.13	−0.648723
$263$	−6680.33	−1.56626	−0.783130	−	0.621857i	$-0.786378\pi$
$263$	−6680.33	−1.56626	−0.783130	+	0.621857i	$0.786378\pi$
$264$	0	0
$265$	0	0
$266$	−1394.70	−0.321483
$267$	0	0
$268$	16146.0	3.68012
$269$	−7468.96	−1.69290	−0.846451	−	0.532466i	$-0.821265\pi$
$269$	−7468.96	−1.69290	−0.846451	+	0.532466i	$0.821265\pi$
$270$	0	0
$271$	−757.673	−0.169835	−0.0849176	−	0.996388i	$-0.527063\pi$
$271$	−757.673	−0.169835	−0.0849176	+	0.996388i	$0.527063\pi$
$272$	17257.0	3.84692
$273$	0	0
$274$	2591.32	0.571340
$275$	0	0
$276$	0	0
$277$	−1169.29	−0.253632	−0.126816	−	0.991926i	$-0.540476\pi$
$277$	−1169.29	−0.253632	−0.126816	+	0.991926i	$0.540476\pi$
$278$	7338.22	1.58315
$279$	0	0
$280$	0	0
$281$	−5735.20	−1.21756	−0.608778	−	0.793341i	$-0.708340\pi$
$281$	−5735.20	−1.21756	−0.608778	+	0.793341i	$0.708340\pi$
$282$	0	0
$283$	9010.69	1.89269	0.946343	−	0.323164i	$-0.104747\pi$
$283$	9010.69	1.89269	0.946343	+	0.323164i	$0.104747\pi$
$284$	3274.95	0.684270
$285$	0	0
$286$	764.174	0.157995
$287$	−855.900	−0.176035
$288$	0	0
$289$	12458.6	2.53583
$290$	0	0
$291$	0	0
$292$	4084.87	0.818662
$293$	5303.84	1.05752	0.528760	−	0.848771i	$-0.322657\pi$
$293$	5303.84	1.05752	0.528760	+	0.848771i	$0.322657\pi$
$294$	0	0
$295$	0	0
$296$	−15175.4	−2.97991
$297$	0	0
$298$	−7634.99	−1.48417
$299$	−1085.59	−0.209970
$300$	0	0
$301$	−3022.91	−0.578863
$302$	−6526.90	−1.24365
$303$	0	0
$304$	5065.40	0.955659
$305$	0	0
$306$	0	0
$307$	−8783.81	−1.63296	−0.816480	−	0.577374i	$-0.804078\pi$
$307$	−8783.81	−1.63296	−0.816480	+	0.577374i	$0.804078\pi$
$308$	−1816.25	−0.336009
$309$	0	0
$310$	0	0
$311$	2357.25	0.429799	0.214899	−	0.976636i	$-0.431058\pi$
$311$	2357.25	0.429799	0.214899	+	0.976636i	$0.431058\pi$
$312$	0	0
$313$	2633.84	0.475635	0.237817	−	0.971310i	$-0.423568\pi$
$313$	2633.84	0.475635	0.237817	+	0.971310i	$0.423568\pi$
$314$	−16655.4	−2.99338
$315$	0	0
$316$	−3528.62	−0.628166
$317$	−842.572	−0.149286	−0.0746428	−	0.997210i	$-0.523782\pi$
$317$	−842.572	−0.149286	−0.0746428	+	0.997210i	$0.523782\pi$
$318$	0	0
$319$	3254.97	0.571296
$320$	0	0
$321$	0	0
$322$	3694.51	0.639401
$323$	5099.01	0.878379
$324$	0	0
$325$	0	0
$326$	−9056.54	−1.53864
$327$	0	0
$328$	6626.76	1.11555
$329$	−2069.85	−0.346852
$330$	0	0
$331$	11058.8	1.83639	0.918197	−	0.396125i	$-0.129645\pi$
$331$	11058.8	1.83639	0.918197	+	0.396125i	$0.129645\pi$
$332$	9599.64	1.58689
$333$	0	0
$334$	8665.38	1.41961
$335$	0	0
$336$	0	0
$337$	1197.28	0.193531	0.0967657	−	0.995307i	$-0.469150\pi$
$337$	1197.28	0.193531	0.0967657	+	0.995307i	$0.469150\pi$
$338$	−10736.9	−1.72783
$339$	0	0
$340$	0	0
$341$	−2318.23	−0.368150
$342$	0	0
$343$	−343.000	−0.0539949
$344$	23404.7	3.66831
$345$	0	0
$346$	16539.7	2.56988
$347$	−518.087	−0.0801509	−0.0400755	−	0.999197i	$-0.512760\pi$
$347$	−518.087	−0.0801509	−0.0400755	+	0.999197i	$0.512760\pi$
$348$	0	0
$349$	−1776.08	−0.272412	−0.136206	−	0.990681i	$-0.543491\pi$
$349$	−1776.08	−0.272412	−0.136206	+	0.990681i	$0.543491\pi$
$350$	0	0
$351$	0	0
$352$	3372.09	0.510606
$353$	−7507.35	−1.13194	−0.565972	−	0.824425i	$-0.691499\pi$
$353$	−7507.35	−1.13194	−0.565972	+	0.824425i	$0.691499\pi$
$354$	0	0
$355$	0	0
$356$	11083.6	1.65009
$357$	0	0
$358$	−17057.0	−2.51813
$359$	4915.58	0.722658	0.361329	−	0.932438i	$-0.382323\pi$
$359$	4915.58	0.722658	0.361329	+	0.932438i	$0.382323\pi$
$360$	0	0
$361$	−5362.31	−0.781791
$362$	10621.7	1.54216
$363$	0	0
$364$	−1373.55	−0.197784
$365$	0	0
$366$	0	0
$367$	3628.98	0.516161	0.258080	−	0.966123i	$-0.416910\pi$
$367$	3628.98	0.516161	0.258080	+	0.966123i	$0.416910\pi$
$368$	−13418.1	−1.90072
$369$	0	0
$370$	0	0
$371$	−1705.47	−0.238662
$372$	0	0
$373$	2836.97	0.393814	0.196907	−	0.980422i	$-0.436910\pi$
$373$	2836.97	0.393814	0.196907	+	0.980422i	$0.436910\pi$
$374$	9508.02	1.31457
$375$	0	0
$376$	16025.7	2.19804
$377$	2461.58	0.336280
$378$	0	0
$379$	9703.96	1.31519	0.657597	−	0.753370i	$-0.271573\pi$
$379$	9703.96	1.31519	0.657597	+	0.753370i	$0.271573\pi$
$380$	0	0
$381$	0	0
$382$	−14332.8	−1.91971
$383$	−3855.90	−0.514431	−0.257216	−	0.966354i	$-0.582805\pi$
$383$	−3855.90	−0.514431	−0.257216	+	0.966354i	$0.582805\pi$
$384$	0	0
$385$	0	0
$386$	−1546.54	−0.203929
$387$	0	0
$388$	34865.7	4.56195
$389$	−7106.46	−0.926251	−0.463125	−	0.886293i	$-0.653272\pi$
$389$	−7106.46	−0.926251	−0.463125	+	0.886293i	$0.653272\pi$
$390$	0	0
$391$	−13507.1	−1.74702
$392$	2655.66	0.342171
$393$	0	0
$394$	−7651.17	−0.978326
$395$	0	0
$396$	0	0
$397$	6775.82	0.856596	0.428298	−	0.903637i	$-0.359113\pi$
$397$	6775.82	0.856596	0.428298	+	0.903637i	$0.359113\pi$
$398$	−15659.1	−1.97217
$399$	0	0
$400$	0	0
$401$	−6444.62	−0.802566	−0.401283	−	0.915954i	$-0.631436\pi$
$401$	−6444.62	−0.802566	−0.401283	+	0.915954i	$0.631436\pi$
$402$	0	0
$403$	−1753.16	−0.216703
$404$	−3534.11	−0.435220
$405$	0	0
$406$	−8377.32	−1.02404
$407$	−3922.11	−0.477670
$408$	0	0
$409$	4807.99	0.581270	0.290635	−	0.956834i	$-0.406133\pi$
$409$	4807.99	0.581270	0.290635	+	0.956834i	$0.406133\pi$
$410$	0	0
$411$	0	0
$412$	27721.0	3.31484
$413$	3965.36	0.472452
$414$	0	0
$415$	0	0
$416$	2550.15	0.300556
$417$	0	0
$418$	2790.85	0.326567
$419$	4521.51	0.527184	0.263592	−	0.964634i	$-0.415093\pi$
$419$	4521.51	0.527184	0.263592	+	0.964634i	$0.415093\pi$
$420$	0	0
$421$	−5821.82	−0.673962	−0.336981	−	0.941511i	$-0.609406\pi$
$421$	−5821.82	−0.673962	−0.336981	+	0.941511i	$0.609406\pi$
$422$	−14867.0	−1.71496
$423$	0	0
$424$	13204.5	1.51243
$425$	0	0
$426$	0	0
$427$	1316.08	0.149156
$428$	−36414.5	−4.11252
$429$	0	0
$430$	0	0
$431$	8382.59	0.936833	0.468417	−	0.883508i	$-0.344825\pi$
$431$	8382.59	0.936833	0.468417	+	0.883508i	$0.344825\pi$
$432$	0	0
$433$	−2628.05	−0.291677	−0.145838	−	0.989308i	$-0.546588\pi$
$433$	−2628.05	−0.291677	−0.145838	+	0.989308i	$0.546588\pi$
$434$	5966.42	0.659902
$435$	0	0
$436$	−16124.4	−1.77114
$437$	−3964.69	−0.433998
$438$	0	0
$439$	−4122.97	−0.448243	−0.224121	−	0.974561i	$-0.571951\pi$
$439$	−4122.97	−0.448243	−0.224121	+	0.974561i	$0.571951\pi$
$440$	0	0
$441$	0	0
$442$	7190.46	0.773790
$443$	−210.327	−0.0225574	−0.0112787	−	0.999936i	$-0.503590\pi$
$443$	−210.327	−0.0225574	−0.0112787	+	0.999936i	$0.503590\pi$
$444$	0	0
$445$	0	0
$446$	−23645.8	−2.51045
$447$	0	0
$448$	−1346.55	−0.142005
$449$	−4414.35	−0.463978	−0.231989	−	0.972718i	$-0.574523\pi$
$449$	−4414.35	−0.463978	−0.231989	+	0.972718i	$0.574523\pi$
$450$	0	0
$451$	1712.70	0.178820
$452$	37995.4	3.95388
$453$	0	0
$454$	−13117.5	−1.35603
$455$	0	0
$456$	0	0
$457$	−14355.4	−1.46941	−0.734704	−	0.678388i	$-0.762679\pi$
$457$	−14355.4	−1.46941	−0.734704	+	0.678388i	$0.762679\pi$
$458$	10338.3	1.05475
$459$	0	0
$460$	0	0
$461$	14366.9	1.45149	0.725744	−	0.687965i	$-0.241496\pi$
$461$	14366.9	1.45149	0.725744	+	0.687965i	$0.241496\pi$
$462$	0	0
$463$	−11634.1	−1.16778	−0.583892	−	0.811832i	$-0.698471\pi$
$463$	−11634.1	−1.16778	−0.583892	+	0.811832i	$0.698471\pi$
$464$	30425.6	3.04412
$465$	0	0
$466$	14475.1	1.43894
$467$	−12112.2	−1.20019	−0.600094	−	0.799929i	$-0.704870\pi$
$467$	−12112.2	−1.20019	−0.600094	+	0.799929i	$0.704870\pi$
$468$	0	0
$469$	−6101.53	−0.600730
$470$	0	0
$471$	0	0
$472$	−30701.6	−2.99397
$473$	6048.99	0.588018
$474$	0	0
$475$	0	0
$476$	−17089.9	−1.64562
$477$	0	0
$478$	−9520.88	−0.911035
$479$	−11380.8	−1.08560	−0.542801	−	0.839861i	$-0.682636\pi$
$479$	−11380.8	−1.08560	−0.542801	+	0.839861i	$0.682636\pi$
$480$	0	0
$481$	−2966.10	−0.281170
$482$	29472.2	2.78510
$483$	0	0
$484$	−21020.4	−1.97412
$485$	0	0
$486$	0	0
$487$	11615.5	1.08080	0.540401	−	0.841408i	$-0.318273\pi$
$487$	11615.5	1.08080	0.540401	+	0.841408i	$0.318273\pi$
$488$	−10189.7	−0.945217
$489$	0	0
$490$	0	0
$491$	6045.64	0.555674	0.277837	−	0.960628i	$-0.410383\pi$
$491$	6045.64	0.555674	0.277837	+	0.960628i	$0.410383\pi$
$492$	0	0
$493$	30627.5	2.79795
$494$	2110.59	0.192226
$495$	0	0
$496$	−21669.4	−1.96167
$497$	−1237.60	−0.111698
$498$	0	0
$499$	860.279	0.0771771	0.0385886	−	0.999255i	$-0.487714\pi$
$499$	860.279	0.0771771	0.0385886	+	0.999255i	$0.487714\pi$
$500$	0	0
$501$	0	0
$502$	−8875.57	−0.789115
$503$	3341.49	0.296202	0.148101	−	0.988972i	$-0.452684\pi$
$503$	3341.49	0.296202	0.148101	+	0.988972i	$0.452684\pi$
$504$	0	0
$505$	0	0
$506$	−7392.89	−0.649514
$507$	0	0
$508$	−46330.1	−4.04639
$509$	−19754.5	−1.72024	−0.860122	−	0.510088i	$-0.829613\pi$
$509$	−19754.5	−1.72024	−0.860122	+	0.510088i	$0.829613\pi$
$510$	0	0
$511$	−1543.67	−0.133636
$512$	−25248.9	−2.17940
$513$	0	0
$514$	−29633.7	−2.54297
$515$	0	0
$516$	0	0
$517$	4141.86	0.352338
$518$	10094.3	0.856215
$519$	0	0
$520$	0	0
$521$	−20754.2	−1.74522	−0.872609	−	0.488420i	$-0.837573\pi$
$521$	−20754.2	−1.74522	−0.872609	+	0.488420i	$0.837573\pi$
$522$	0	0
$523$	9495.26	0.793879	0.396940	−	0.917845i	$-0.370072\pi$
$523$	9495.26	0.793879	0.396940	+	0.917845i	$0.370072\pi$
$524$	−9895.07	−0.824939
$525$	0	0
$526$	−34404.3	−2.85190
$527$	−21813.2	−1.80303
$528$	0	0
$529$	−1664.65	−0.136817
$530$	0	0
$531$	0	0
$532$	−5016.35	−0.408809
$533$	1295.23	0.105258
$534$	0	0
$535$	0	0
$536$	47240.8	3.80689
$537$	0	0
$538$	−38465.9	−3.08250
$539$	686.359	0.0548489
$540$	0	0
$541$	15585.9	1.23862	0.619308	−	0.785148i	$-0.287413\pi$
$541$	15585.9	1.23862	0.619308	+	0.785148i	$0.287413\pi$
$542$	−3902.09	−0.309242
$543$	0	0
$544$	31729.5	2.50072
$545$	0	0
$546$	0	0
$547$	23104.6	1.80600	0.902998	−	0.429644i	$-0.141361\pi$
$547$	23104.6	1.80600	0.902998	+	0.429644i	$0.141361\pi$
$548$	9320.27	0.726537
$549$	0	0
$550$	0	0
$551$	8989.96	0.695073
$552$	0	0
$553$	1333.46	0.102540
$554$	−6021.98	−0.461822
$555$	0	0
$556$	26393.6	2.01320
$557$	−21731.2	−1.65311	−0.826553	−	0.562859i	$-0.809701\pi$
$557$	−21731.2	−1.65311	−0.826553	+	0.562859i	$0.809701\pi$
$558$	0	0
$559$	4574.56	0.346124
$560$	0	0
$561$	0	0
$562$	−29536.8	−2.21697
$563$	13595.1	1.01770	0.508850	−	0.860855i	$-0.330071\pi$
$563$	13595.1	1.01770	0.508850	+	0.860855i	$0.330071\pi$
$564$	0	0
$565$	0	0
$566$	46405.9	3.44627
$567$	0	0
$568$	9582.04	0.707840
$569$	10018.2	0.738107	0.369054	−	0.929408i	$-0.379682\pi$
$569$	10018.2	0.738107	0.369054	+	0.929408i	$0.379682\pi$
$570$	0	0
$571$	−6658.73	−0.488020	−0.244010	−	0.969773i	$-0.578463\pi$
$571$	−6658.73	−0.488020	−0.244010	+	0.969773i	$0.578463\pi$
$572$	2748.53	0.200912
$573$	0	0
$574$	−4407.97	−0.320531
$575$	0	0
$576$	0	0
$577$	12676.0	0.914570	0.457285	−	0.889320i	$-0.348822\pi$
$577$	12676.0	0.914570	0.457285	+	0.889320i	$0.348822\pi$
$578$	64162.8	4.61733
$579$	0	0
$580$	0	0
$581$	−3627.68	−0.259039
$582$	0	0
$583$	3412.73	0.242437
$584$	11951.8	0.846861
$585$	0	0
$586$	27315.3	1.92557
$587$	−18625.6	−1.30965	−0.654823	−	0.755782i	$-0.727257\pi$
$587$	−18625.6	−1.30965	−0.654823	+	0.755782i	$0.727257\pi$
$588$	0	0
$589$	−6402.76	−0.447913
$590$	0	0
$591$	0	0
$592$	−36661.6	−2.54524
$593$	−19417.3	−1.34465	−0.672323	−	0.740258i	$-0.734703\pi$
$593$	−19417.3	−1.34465	−0.672323	+	0.740258i	$0.734703\pi$
$594$	0	0
$595$	0	0
$596$	−27461.0	−1.88733
$597$	0	0
$598$	−5590.88	−0.382321
$599$	13376.9	0.912461	0.456231	−	0.889862i	$-0.349199\pi$
$599$	13376.9	0.912461	0.456231	+	0.889862i	$0.349199\pi$
$600$	0	0
$601$	14963.9	1.01562	0.507812	−	0.861468i	$-0.330454\pi$
$601$	14963.9	1.01562	0.507812	+	0.861468i	$0.330454\pi$
$602$	−15568.3	−1.05401
$603$	0	0
$604$	−23475.5	−1.58146
$605$	0	0
$606$	0	0
$607$	−20270.8	−1.35546	−0.677731	−	0.735310i	$-0.737037\pi$
$607$	−20270.8	−1.35546	−0.677731	+	0.735310i	$0.737037\pi$
$608$	9313.45	0.621234
$609$	0	0
$610$	0	0
$611$	3132.29	0.207396
$612$	0	0
$613$	−13005.1	−0.856888	−0.428444	−	0.903568i	$-0.640938\pi$
$613$	−13005.1	−0.856888	−0.428444	+	0.903568i	$0.640938\pi$
$614$	−45237.5	−2.97335
$615$	0	0
$616$	−5314.10	−0.347583
$617$	−11107.4	−0.724745	−0.362372	−	0.932033i	$-0.618033\pi$
$617$	−11107.4	−0.724745	−0.362372	+	0.932033i	$0.618033\pi$
$618$	0	0
$619$	348.077	0.0226016	0.0113008	−	0.999936i	$-0.496403\pi$
$619$	348.077	0.0226016	0.0113008	+	0.999936i	$0.496403\pi$
$620$	0	0
$621$	0	0
$622$	12140.1	0.782593
$623$	−4188.48	−0.269355
$624$	0	0
$625$	0	0
$626$	13564.6	0.866052
$627$	0	0
$628$	−59905.1	−3.80649
$629$	−36904.8	−2.33942
$630$	0	0
$631$	7548.88	0.476254	0.238127	−	0.971234i	$-0.423467\pi$
$631$	7548.88	0.476254	0.238127	+	0.971234i	$0.423467\pi$
$632$	−10324.2	−0.649804
$633$	0	0
$634$	−4339.33	−0.271825
$635$	0	0
$636$	0	0
$637$	519.060	0.0322856
$638$	16763.4	1.04023
$639$	0	0
$640$	0	0
$641$	−8720.39	−0.537340	−0.268670	−	0.963232i	$-0.586584\pi$
$641$	−8720.39	−0.537340	−0.268670	+	0.963232i	$0.586584\pi$
$642$	0	0
$643$	10517.6	0.645061	0.322530	−	0.946559i	$-0.395467\pi$
$643$	10517.6	0.645061	0.322530	+	0.946559i	$0.395467\pi$
$644$	13288.2	0.813085
$645$	0	0
$646$	26260.4	1.59938
$647$	−6194.54	−0.376402	−0.188201	−	0.982130i	$-0.560266\pi$
$647$	−6194.54	−0.376402	−0.188201	+	0.982130i	$0.560266\pi$
$648$	0	0
$649$	−7934.87	−0.479925
$650$	0	0
$651$	0	0
$652$	−32573.9	−1.95658
$653$	16726.9	1.00241	0.501205	−	0.865329i	$-0.332890\pi$
$653$	16726.9	1.00241	0.501205	+	0.865329i	$0.332890\pi$
$654$	0	0
$655$	0	0
$656$	16009.3	0.952832
$657$	0	0
$658$	−10659.9	−0.631561
$659$	9874.87	0.583718	0.291859	−	0.956461i	$-0.405726\pi$
$659$	9874.87	0.583718	0.291859	+	0.956461i	$0.405726\pi$
$660$	0	0
$661$	−26873.6	−1.58134	−0.790668	−	0.612245i	$-0.790267\pi$
$661$	−26873.6	−1.58134	−0.790668	+	0.612245i	$0.790267\pi$
$662$	56953.9	3.34377
$663$	0	0
$664$	28087.1	1.64156
$665$	0	0
$666$	0	0
$667$	−23814.1	−1.38244
$668$	31167.0	1.80522
$669$	0	0
$670$	0	0
$671$	−2633.54	−0.151515
$672$	0	0
$673$	−270.279	−0.0154807	−0.00774034	−	0.999970i	$-0.502464\pi$
$673$	−270.279	−0.0154807	−0.00774034	+	0.999970i	$0.502464\pi$
$674$	6166.12	0.352389
$675$	0	0
$676$	−38617.6	−2.19718
$677$	−5216.49	−0.296139	−0.148070	−	0.988977i	$-0.547306\pi$
$677$	−5216.49	−0.296139	−0.148070	+	0.988977i	$0.547306\pi$
$678$	0	0
$679$	−13175.7	−0.744678
$680$	0	0
$681$	0	0
$682$	−11939.1	−0.670339
$683$	597.154	0.0334546	0.0167273	−	0.999860i	$-0.494675\pi$
$683$	597.154	0.0334546	0.0167273	+	0.999860i	$0.494675\pi$
$684$	0	0
$685$	0	0
$686$	−1766.48	−0.0983158
$687$	0	0
$688$	56542.4	3.13323
$689$	2580.88	0.142705
$690$	0	0
$691$	−32491.9	−1.78878	−0.894392	−	0.447283i	$-0.852392\pi$
$691$	−32491.9	−1.78878	−0.894392	+	0.447283i	$0.852392\pi$
$692$	59488.8	3.26795
$693$	0	0
$694$	−2668.20	−0.145942
$695$	0	0
$696$	0	0
$697$	16115.5	0.875780
$698$	−9147.01	−0.496016
$699$	0	0
$700$	0	0
$701$	−25288.3	−1.36252	−0.681260	−	0.732041i	$-0.738568\pi$
$701$	−25288.3	−1.36252	−0.681260	+	0.732041i	$0.738568\pi$
$702$	0	0
$703$	−10832.5	−0.581162
$704$	2694.51	0.144251
$705$	0	0
$706$	−38663.6	−2.06108
$707$	1335.53	0.0710438
$708$	0	0
$709$	20110.2	1.06524	0.532618	−	0.846355i	$-0.321208\pi$
$709$	20110.2	1.06524	0.532618	+	0.846355i	$0.321208\pi$
$710$	0	0
$711$	0	0
$712$	32429.1	1.70693
$713$	16960.7	0.890860
$714$	0	0
$715$	0	0
$716$	−61349.4	−3.20214
$717$	0	0
$718$	25315.7	1.31584
$719$	5650.46	0.293083	0.146542	−	0.989205i	$-0.453186\pi$
$719$	5650.46	0.293083	0.146542	+	0.989205i	$0.453186\pi$
$720$	0	0
$721$	−10475.7	−0.541104
$722$	−27616.4	−1.42351
$723$	0	0
$724$	38203.3	1.96107
$725$	0	0
$726$	0	0
$727$	−6571.94	−0.335268	−0.167634	−	0.985849i	$-0.553613\pi$
$727$	−6571.94	−0.335268	−0.167634	+	0.985849i	$0.553613\pi$
$728$	−4018.80	−0.204597
$729$	0	0
$730$	0	0
$731$	56917.6	2.87985
$732$	0	0
$733$	31444.0	1.58446	0.792231	−	0.610222i	$-0.208920\pi$
$733$	31444.0	1.58446	0.792231	+	0.610222i	$0.208920\pi$
$734$	18689.6	0.939843
$735$	0	0
$736$	−24671.0	−1.23558
$737$	12209.4	0.610232
$738$	0	0
$739$	8157.80	0.406075	0.203038	−	0.979171i	$-0.434919\pi$
$739$	8157.80	0.406075	0.203038	+	0.979171i	$0.434919\pi$
$740$	0	0
$741$	0	0
$742$	−8783.35	−0.434565
$743$	−5705.69	−0.281725	−0.140862	−	0.990029i	$-0.544987\pi$
$743$	−5705.69	−0.281725	−0.140862	+	0.990029i	$0.544987\pi$
$744$	0	0
$745$	0	0
$746$	14610.7	0.717070
$747$	0	0
$748$	34197.8	1.67165
$749$	13761.0	0.671315
$750$	0	0
$751$	−7296.31	−0.354522	−0.177261	−	0.984164i	$-0.556724\pi$
$751$	−7296.31	−0.354522	−0.177261	+	0.984164i	$0.556724\pi$
$752$	38715.7	1.87742
$753$	0	0
$754$	12677.4	0.612311
$755$	0	0
$756$	0	0
$757$	−8240.48	−0.395648	−0.197824	−	0.980238i	$-0.563387\pi$
$757$	−8240.48	−0.395648	−0.197824	+	0.980238i	$0.563387\pi$
$758$	49976.3	2.39475
$759$	0	0
$760$	0	0
$761$	30887.8	1.47133	0.735665	−	0.677345i	$-0.236870\pi$
$761$	30887.8	1.47133	0.735665	+	0.677345i	$0.236870\pi$
$762$	0	0
$763$	6093.37	0.289115
$764$	−51551.1	−2.44117
$765$	0	0
$766$	−19858.2	−0.936694
$767$	−6000.76	−0.282497
$768$	0	0
$769$	30953.0	1.45149	0.725743	−	0.687966i	$-0.241496\pi$
$769$	30953.0	1.45149	0.725743	+	0.687966i	$0.241496\pi$
$770$	0	0
$771$	0	0
$772$	−5562.48	−0.259324
$773$	−6620.92	−0.308070	−0.154035	−	0.988065i	$-0.549227\pi$
$773$	−6620.92	−0.308070	−0.154035	+	0.988065i	$0.549227\pi$
$774$	0	0
$775$	0	0
$776$	102012.	4.71910
$777$	0	0
$778$	−36598.9	−1.68655
$779$	4730.33	0.217563
$780$	0	0
$781$	2476.49	0.113465
$782$	−69563.0	−3.18103
$783$	0	0
$784$	6415.69	0.292260
$785$	0	0
$786$	0	0
$787$	10068.6	0.456044	0.228022	−	0.973656i	$-0.426774\pi$
$787$	10068.6	0.456044	0.228022	+	0.973656i	$0.426774\pi$
$788$	−27519.2	−1.24407
$789$	0	0
$790$	0	0
$791$	−14358.4	−0.645419
$792$	0	0
$793$	−1991.62	−0.0891861
$794$	34896.1	1.55972
$795$	0	0
$796$	−56321.7	−2.50788
$797$	42016.5	1.86738	0.933690	−	0.358084i	$-0.116570\pi$
$797$	42016.5	1.86738	0.933690	+	0.358084i	$0.116570\pi$
$798$	0	0
$799$	38972.6	1.72560
$800$	0	0
$801$	0	0
$802$	−33190.4	−1.46134
$803$	3088.95	0.135749
$804$	0	0
$805$	0	0
$806$	−9028.96	−0.394580
$807$	0	0
$808$	−10340.3	−0.450211
$809$	32805.5	1.42568	0.712842	−	0.701324i	$-0.247407\pi$
$809$	32805.5	1.42568	0.712842	+	0.701324i	$0.247407\pi$
$810$	0	0
$811$	15395.7	0.666606	0.333303	−	0.942820i	$-0.391837\pi$
$811$	15395.7	0.666606	0.333303	+	0.942820i	$0.391837\pi$
$812$	−30131.0	−1.30220
$813$	0	0
$814$	−20199.2	−0.869758
$815$	0	0
$816$	0	0
$817$	16706.8	0.715419
$818$	24761.6	1.05840
$819$	0	0
$820$	0	0
$821$	−10626.7	−0.451734	−0.225867	−	0.974158i	$-0.572521\pi$
$821$	−10626.7	−0.451734	−0.225867	+	0.974158i	$0.572521\pi$
$822$	0	0
$823$	42578.1	1.80338	0.901689	−	0.432384i	$-0.142328\pi$
$823$	42578.1	1.80338	0.901689	+	0.432384i	$0.142328\pi$
$824$	81107.7	3.42903
$825$	0	0
$826$	20422.0	0.860257
$827$	733.624	0.0308472	0.0154236	−	0.999881i	$-0.495090\pi$
$827$	733.624	0.0308472	0.0154236	+	0.999881i	$0.495090\pi$
$828$	0	0
$829$	15778.0	0.661028	0.330514	−	0.943801i	$-0.392778\pi$
$829$	15778.0	0.661028	0.330514	+	0.943801i	$0.392778\pi$
$830$	0	0
$831$	0	0
$832$	2037.73	0.0849104
$833$	6458.26	0.268626
$834$	0	0
$835$	0	0
$836$	10037.9	0.415275
$837$	0	0
$838$	23286.2	0.959916
$839$	−16776.8	−0.690343	−0.345172	−	0.938540i	$-0.612179\pi$
$839$	−16776.8	−0.690343	−0.345172	+	0.938540i	$0.612179\pi$
$840$	0	0
$841$	29609.7	1.21406
$842$	−29983.0	−1.22717
$843$	0	0
$844$	−53472.6	−2.18081
$845$	0	0
$846$	0	0
$847$	7943.56	0.322248
$848$	31900.2	1.29181
$849$	0	0
$850$	0	0
$851$	28695.1	1.15588
$852$	0	0
$853$	−7646.57	−0.306933	−0.153466	−	0.988154i	$-0.549044\pi$
$853$	−7646.57	−0.306933	−0.153466	+	0.988154i	$0.549044\pi$
$854$	6777.95	0.271589
$855$	0	0
$856$	−106544.	−4.25419
$857$	26980.3	1.07541	0.537707	−	0.843132i	$-0.319291\pi$
$857$	26980.3	1.07541	0.537707	+	0.843132i	$0.319291\pi$
$858$	0	0
$859$	3034.02	0.120511	0.0602557	−	0.998183i	$-0.480808\pi$
$859$	3034.02	0.120511	0.0602557	+	0.998183i	$0.480808\pi$
$860$	0	0
$861$	0	0
$862$	43171.2	1.70582
$863$	−25954.5	−1.02376	−0.511879	−	0.859058i	$-0.671050\pi$
$863$	−25954.5	−1.02376	−0.511879	+	0.859058i	$0.671050\pi$
$864$	0	0
$865$	0	0
$866$	−13534.7	−0.531095
$867$	0	0
$868$	21459.6	0.839155
$869$	−2668.31	−0.104161
$870$	0	0
$871$	9233.42	0.359199
$872$	−47177.6	−1.83215
$873$	0	0
$874$	−20418.6	−0.790238
$875$	0	0
$876$	0	0
$877$	11235.7	0.432616	0.216308	−	0.976325i	$-0.430598\pi$
$877$	11235.7	0.432616	0.216308	+	0.976325i	$0.430598\pi$
$878$	−21233.7	−0.816176
$879$	0	0
$880$	0	0
$881$	4594.31	0.175694	0.0878468	−	0.996134i	$-0.472001\pi$
$881$	4594.31	0.175694	0.0878468	+	0.996134i	$0.472001\pi$
$882$	0	0
$883$	14162.1	0.539741	0.269871	−	0.962897i	$-0.413019\pi$
$883$	14162.1	0.539741	0.269871	+	0.962897i	$0.413019\pi$
$884$	25862.1	0.983979
$885$	0	0
$886$	−1083.20	−0.0410733
$887$	−31024.2	−1.17440	−0.587199	−	0.809442i	$-0.699770\pi$
$887$	−31024.2	−1.17440	−0.587199	+	0.809442i	$0.699770\pi$
$888$	0	0
$889$	17508.1	0.660519
$890$	0	0
$891$	0	0
$892$	−85047.7	−3.19238
$893$	11439.5	0.428676
$894$	0	0
$895$	0	0
$896$	6546.46	0.244087
$897$	0	0
$898$	−22734.4	−0.844828
$899$	−38458.5	−1.42677
$900$	0	0
$901$	32111.9	1.18735
$902$	8820.55	0.325601
$903$	0	0
$904$	111169.	4.09008
$905$	0	0
$906$	0	0
$907$	10594.5	0.387855	0.193928	−	0.981016i	$-0.437877\pi$
$907$	10594.5	0.387855	0.193928	+	0.981016i	$0.437877\pi$
$908$	−47180.2	−1.72437
$909$	0	0
$910$	0	0
$911$	−1469.37	−0.0534384	−0.0267192	−	0.999643i	$-0.508506\pi$
$911$	−1469.37	−0.0534384	−0.0267192	+	0.999643i	$0.508506\pi$
$912$	0	0
$913$	7259.17	0.263136
$914$	−73931.9	−2.67555
$915$	0	0
$916$	37184.1	1.34126
$917$	3739.33	0.134660
$918$	0	0
$919$	41834.6	1.50163	0.750813	−	0.660515i	$-0.229662\pi$
$919$	41834.6	1.50163	0.750813	+	0.660515i	$0.229662\pi$
$920$	0	0
$921$	0	0
$922$	73991.2	2.64292
$923$	1872.85	0.0667884
$924$	0	0
$925$	0	0
$926$	−59916.9	−2.12634
$927$	0	0
$928$	55941.7	1.97885
$929$	−8570.23	−0.302670	−0.151335	−	0.988483i	$-0.548357\pi$
$929$	−8570.23	−0.302670	−0.151335	+	0.988483i	$0.548357\pi$
$930$	0	0
$931$	1895.67	0.0667326
$932$	52063.1	1.82981
$933$	0	0
$934$	−62379.3	−2.18534
$935$	0	0
$936$	0	0
$937$	−41260.3	−1.43854	−0.719271	−	0.694730i	$-0.755524\pi$
$937$	−41260.3	−1.43854	−0.719271	+	0.694730i	$0.755524\pi$
$938$	−31423.5	−1.09383
$939$	0	0
$940$	0	0
$941$	−47969.6	−1.66181	−0.830905	−	0.556414i	$-0.812177\pi$
$941$	−47969.6	−1.66181	−0.830905	+	0.556414i	$0.812177\pi$
$942$	0	0
$943$	−12530.5	−0.432714
$944$	−74170.6	−2.55725
$945$	0	0
$946$	31152.9	1.07068
$947$	−15666.3	−0.537577	−0.268789	−	0.963199i	$-0.586623\pi$
$947$	−15666.3	−0.537577	−0.268789	+	0.963199i	$0.586623\pi$
$948$	0	0
$949$	2336.02	0.0799057
$950$	0	0
$951$	0	0
$952$	−50002.7	−1.70231
$953$	27495.7	0.934600	0.467300	−	0.884099i	$-0.345227\pi$
$953$	27495.7	0.934600	0.467300	+	0.884099i	$0.345227\pi$
$954$	0	0
$955$	0	0
$956$	−34244.0	−1.15851
$957$	0	0
$958$	−58612.4	−1.97670
$959$	−3522.11	−0.118597
$960$	0	0
$961$	−2400.42	−0.0805755
$962$	−15275.7	−0.511963
$963$	0	0
$964$	106003.	3.54164
$965$	0	0
$966$	0	0
$967$	−51246.2	−1.70421	−0.852103	−	0.523374i	$-0.824673\pi$
$967$	−51246.2	−1.70421	−0.852103	+	0.523374i	$0.824673\pi$
$968$	−61502.6	−2.04212
$969$	0	0
$970$	0	0
$971$	−40446.0	−1.33674	−0.668370	−	0.743829i	$-0.733008\pi$
$971$	−40446.0	−1.33674	−0.668370	+	0.743829i	$0.733008\pi$
$972$	0	0
$973$	−9974.08	−0.328627
$974$	59821.2	1.96796
$975$	0	0
$976$	−24616.8	−0.807342
$977$	21457.8	0.702656	0.351328	−	0.936253i	$-0.385730\pi$
$977$	21457.8	0.702656	0.351328	+	0.936253i	$0.385730\pi$
$978$	0	0
$979$	8381.35	0.273615
$980$	0	0
$981$	0	0
$982$	31135.6	1.01179
$983$	−8746.68	−0.283801	−0.141900	−	0.989881i	$-0.545321\pi$
$983$	−8746.68	−0.283801	−0.141900	+	0.989881i	$0.545321\pi$
$984$	0	0
$985$	0	0
$986$	157734.	5.09461
$987$	0	0
$988$	7591.22	0.244442
$989$	−44255.8	−1.42291
$990$	0	0
$991$	−29046.2	−0.931061	−0.465531	−	0.885032i	$-0.654137\pi$
$991$	−29046.2	−0.931061	−0.465531	+	0.885032i	$0.654137\pi$
$992$	−39842.3	−1.27520
$993$	0	0
$994$	−6373.75	−0.203383
$995$	0	0
$996$	0	0
$997$	4656.30	0.147910	0.0739552	−	0.997262i	$-0.476438\pi$
$997$	4656.30	0.147910	0.0739552	+	0.997262i	$0.476438\pi$
$998$	4430.52	0.140527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.be.1.3		3
3.2	odd	2		1575.4.a.bb.1.1		3
5.4	even	2		315.4.a.n.1.1	✓	3
15.14	odd	2		315.4.a.o.1.3	yes	3
35.34	odd	2		2205.4.a.bk.1.1		3
105.104	even	2		2205.4.a.bl.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.n.1.1	✓	3	5.4	even	2
315.4.a.o.1.3	yes	3	15.14	odd	2
1575.4.a.bb.1.1		3	3.2	odd	2
1575.4.a.be.1.3		3	1.1	even	1	trivial
2205.4.a.bk.1.1		3	35.34	odd	2
2205.4.a.bl.1.3		3	105.104	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)

\(f(q)\)	\(=\)	\(q+5.15010 q^{2} +18.5235 q^{4} -7.00000 q^{7} +54.1971 q^{8} +14.0073 q^{11} +10.5931 q^{13} -36.0507 q^{14} +130.932 q^{16} +131.801 q^{17} +38.6871 q^{19} +72.1391 q^{22} -102.481 q^{23} +54.5553 q^{26} -129.665 q^{28} +232.376 q^{29} -165.501 q^{31} +240.738 q^{32} +678.789 q^{34} -280.004 q^{37} +199.242 q^{38} +122.271 q^{41} +431.844 q^{43} +259.465 q^{44} -527.787 q^{46} +295.692 q^{47} +49.0000 q^{49} +196.221 q^{52} +243.639 q^{53} -379.380 q^{56} +1196.76 q^{58} -566.480 q^{59} -188.012 q^{61} -852.346 q^{62} +192.364 q^{64} +871.647 q^{67} +2441.42 q^{68} +176.800 q^{71} +220.524 q^{73} -1442.05 q^{74} +716.621 q^{76} -98.0513 q^{77} -190.494 q^{79} +629.710 q^{82} +518.241 q^{83} +2224.04 q^{86} +759.157 q^{88} +598.355 q^{89} -74.1515 q^{91} -1898.31 q^{92} +1522.85 q^{94} +1882.24 q^{97} +252.355 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 14 q^{4} - 21 q^{7} + 66 q^{8} + 20 q^{11} - 14 q^{14} + 114 q^{16} + 234 q^{17} - 82 q^{19} + 236 q^{22} + 30 q^{23} - 76 q^{26} - 98 q^{28} + 32 q^{29} - 362 q^{31} + 430 q^{32} + 596 q^{34}+ \cdots + 98 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 14 * q^4 - 21 * q^7 + 66 * q^8 + 20 * q^11 - 14 * q^14 + 114 * q^16 + 234 * q^17 - 82 * q^19 + 236 * q^22 + 30 * q^23 - 76 * q^26 - 98 * q^28 + 32 * q^29 - 362 * q^31 + 430 * q^32 + 596 * q^34 - 302 * q^37 + 160 * q^38 - 626 * q^41 + 68 * q^43 - 300 * q^44 - 412 * q^46 + 936 * q^47 + 147 * q^49 + 684 * q^52 + 298 * q^53 - 462 * q^56 + 1184 * q^58 - 428 * q^59 + 1050 * q^61 - 608 * q^62 - 154 * q^64 + 52 * q^67 + 1964 * q^68 - 416 * q^71 + 312 * q^73 - 1668 * q^74 + 1712 * q^76 - 140 * q^77 - 488 * q^79 + 1476 * q^82 + 948 * q^83 + 1928 * q^86 - 124 * q^88 + 142 * q^89 - 3220 * q^92 + 328 * q^94 + 3052 * q^97 + 98 * q^98

Introduction

L-functions

Modular forms

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