LMFDB - Embedded newform 315.4.a.o.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,4,Mod(1,315)] 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("315.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,2,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.5856016518$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.37989$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.37989 q^{2} +3.42368 q^{4} +5.00000 q^{5} +7.00000 q^{7} +15.4675 q^{8} -16.8995 q^{10} +45.0132 q^{11} -35.4937 q^{13} -23.6593 q^{14} -79.6679 q^{16} +29.4413 q^{17} +3.18839 q^{19} +17.1184 q^{20} -152.140 q^{22} -23.6429 q^{23} +25.0000 q^{25} +119.965 q^{26} +23.9658 q^{28} +9.22109 q^{29} -80.2011 q^{31} +145.529 q^{32} -99.5084 q^{34} +35.0000 q^{35} -61.1957 q^{37} -10.7764 q^{38} +77.3373 q^{40} +282.085 q^{41} -58.8504 q^{43} +154.111 q^{44} +79.9105 q^{46} +371.773 q^{47} +49.0000 q^{49} -84.4973 q^{50} -121.519 q^{52} -256.184 q^{53} +225.066 q^{55} +108.272 q^{56} -31.1663 q^{58} +571.131 q^{59} +835.587 q^{61} +271.071 q^{62} +145.470 q^{64} -177.468 q^{65} +933.953 q^{67} +100.798 q^{68} -118.296 q^{70} +378.339 q^{71} -494.430 q^{73} +206.835 q^{74} +10.9160 q^{76} +315.093 q^{77} +1078.92 q^{79} -398.339 q^{80} -953.418 q^{82} -722.570 q^{83} +147.206 q^{85} +198.908 q^{86} +696.241 q^{88} +89.5596 q^{89} -248.456 q^{91} -80.9458 q^{92} -1256.55 q^{94} +15.9420 q^{95} -101.540 q^{97} -165.615 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 14 q^{4} + 15 q^{5} + 21 q^{7} + 66 q^{8} + 10 q^{10} - 20 q^{11} + 14 q^{14} + 114 q^{16} + 234 q^{17} - 82 q^{19} + 70 q^{20} - 236 q^{22} + 30 q^{23} + 75 q^{25} + 76 q^{26} + 98 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 14 * q^4 + 15 * q^5 + 21 * q^7 + 66 * q^8 + 10 * q^10 - 20 * q^11 + 14 * q^14 + 114 * q^16 + 234 * q^17 - 82 * q^19 + 70 * q^20 - 236 * q^22 + 30 * q^23 + 75 * q^25 + 76 * q^26 + 98 * q^28 - 32 * q^29 - 362 * q^31 + 430 * q^32 + 596 * q^34 + 105 * q^35 + 302 * q^37 + 160 * q^38 + 330 * q^40 + 626 * q^41 - 68 * q^43 + 300 * q^44 - 412 * q^46 + 936 * q^47 + 147 * q^49 + 50 * q^50 - 684 * q^52 + 298 * q^53 - 100 * q^55 + 462 * q^56 - 1184 * q^58 + 428 * q^59 + 1050 * q^61 - 608 * q^62 - 154 * q^64 - 52 * q^67 + 1964 * q^68 + 70 * q^70 + 416 * q^71 - 312 * q^73 + 1668 * q^74 + 1712 * q^76 - 140 * q^77 - 488 * q^79 + 570 * q^80 - 1476 * q^82 + 948 * q^83 + 1170 * q^85 - 1928 * q^86 + 124 * q^88 - 142 * q^89 - 3220 * q^92 + 328 * q^94 - 410 * q^95 - 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 $

$2$	−3.37989	−1.19497	−0.597486	−	0.801879i	$-0.703834\pi$
$2$	−3.37989	−1.19497	−0.597486	+	0.801879i	$0.703834\pi$
$3$	0	0
$3$	0	0
$4$	3.42368	0.427960
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	15.4675	0.683572
$9$	0	0
$10$	−16.8995	−0.534408
$11$	45.0132	1.23382	0.616909	−	0.787034i	$-0.288385\pi$
$11$	45.0132	1.23382	0.616909	+	0.787034i	$0.288385\pi$
$12$	0	0
$13$	−35.4937	−0.757244	−0.378622	−	0.925551i	$-0.623602\pi$
$13$	−35.4937	−0.757244	−0.378622	+	0.925551i	$0.623602\pi$
$14$	−23.6593	−0.451657
$15$	0	0
$16$	−79.6679	−1.24481
$17$	29.4413	0.420033	0.210016	−	0.977698i	$-0.432648\pi$
$17$	29.4413	0.420033	0.210016	+	0.977698i	$0.432648\pi$
$18$	0	0
$19$	3.18839	0.0384983	0.0192491	−	0.999815i	$-0.493872\pi$
$19$	3.18839	0.0384983	0.0192491	+	0.999815i	$0.493872\pi$
$20$	17.1184	0.191390
$21$	0	0
$22$	−152.140	−1.47438
$23$	−23.6429	−0.214343	−0.107171	−	0.994241i	$-0.534179\pi$
$23$	−23.6429	−0.214343	−0.107171	+	0.994241i	$0.534179\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	119.965	0.904886
$27$	0	0
$28$	23.9658	0.161754
$29$	9.22109	0.0590453	0.0295227	−	0.999564i	$-0.490601\pi$
$29$	9.22109	0.0590453	0.0295227	+	0.999564i	$0.490601\pi$
$30$	0	0
$31$	−80.2011	−0.464662	−0.232331	−	0.972637i	$-0.574635\pi$
$31$	−80.2011	−0.464662	−0.232331	+	0.972637i	$0.574635\pi$
$32$	145.529	0.803942
$33$	0	0
$34$	−99.5084	−0.501928
$35$	35.0000	0.169031
$36$	0	0
$37$	−61.1957	−0.271906	−0.135953	−	0.990715i	$-0.543410\pi$
$37$	−61.1957	−0.271906	−0.135953	+	0.990715i	$0.543410\pi$
$38$	−10.7764	−0.0460044
$39$	0	0
$40$	77.3373	0.305703
$41$	282.085	1.07449	0.537247	−	0.843425i	$-0.319464\pi$
$41$	282.085	1.07449	0.537247	+	0.843425i	$0.319464\pi$
$42$	0	0
$43$	−58.8504	−0.208712	−0.104356	−	0.994540i	$-0.533278\pi$
$43$	−58.8504	−0.208712	−0.104356	+	0.994540i	$0.533278\pi$
$44$	154.111	0.528025
$45$	0	0
$46$	79.9105	0.256134
$47$	371.773	1.15380	0.576901	−	0.816814i	$-0.304262\pi$
$47$	371.773	1.15380	0.576901	+	0.816814i	$0.304262\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−84.4973	−0.238995
$51$	0	0
$52$	−121.519	−0.324070
$53$	−256.184	−0.663953	−0.331977	−	0.943288i	$-0.607716\pi$
$53$	−256.184	−0.663953	−0.331977	+	0.943288i	$0.607716\pi$
$54$	0	0
$55$	225.066	0.551780
$56$	108.272	0.258366
$57$	0	0
$58$	−31.1663	−0.0705575
$59$	571.131	1.26025	0.630127	−	0.776492i	$-0.283003\pi$
$59$	571.131	1.26025	0.630127	+	0.776492i	$0.283003\pi$
$60$	0	0
$61$	835.587	1.75387	0.876934	−	0.480610i	$-0.159585\pi$
$61$	835.587	1.75387	0.876934	+	0.480610i	$0.159585\pi$
$62$	271.071	0.555259
$63$	0	0
$64$	145.470	0.284121
$65$	−177.468	−0.338650
$66$	0	0
$67$	933.953	1.70299	0.851497	−	0.524360i	$-0.175695\pi$
$67$	933.953	1.70299	0.851497	+	0.524360i	$0.175695\pi$
$68$	100.798	0.179757
$69$	0	0
$70$	−118.296	−0.201987
$71$	378.339	0.632403	0.316201	−	0.948692i	$-0.397592\pi$
$71$	378.339	0.632403	0.316201	+	0.948692i	$0.397592\pi$
$72$	0	0
$73$	−494.430	−0.792722	−0.396361	−	0.918095i	$-0.629727\pi$
$73$	−494.430	−0.792722	−0.396361	+	0.918095i	$0.629727\pi$
$74$	206.835	0.324920
$75$	0	0
$76$	10.9160	0.0164757
$77$	315.093	0.466340
$78$	0	0
$79$	1078.92	1.53656	0.768279	−	0.640115i	$-0.221113\pi$
$79$	1078.92	1.53656	0.768279	+	0.640115i	$0.221113\pi$
$80$	−398.339	−0.556696
$81$	0	0
$82$	−953.418	−1.28399
$83$	−722.570	−0.955571	−0.477785	−	0.878477i	$-0.658560\pi$
$83$	−722.570	−0.955571	−0.477785	+	0.878477i	$0.658560\pi$
$84$	0	0
$85$	147.206	0.187844
$86$	198.908	0.249405
$87$	0	0
$88$	696.241	0.843404
$89$	89.5596	0.106666	0.0533332	−	0.998577i	$-0.483015\pi$
$89$	89.5596	0.106666	0.0533332	+	0.998577i	$0.483015\pi$
$90$	0	0
$91$	−248.456	−0.286211
$92$	−80.9458	−0.0917302
$93$	0	0
$94$	−1256.55	−1.37876
$95$	15.9420	0.0172169
$96$	0	0
$97$	−101.540	−0.106287	−0.0531436	−	0.998587i	$-0.516924\pi$
$97$	−101.540	−0.106287	−0.0531436	+	0.998587i	$0.516924\pi$
$98$	−165.615	−0.170710
$99$	0	0
$100$	85.5920	0.0855920
$101$	1024.92	1.00974	0.504868	−	0.863197i	$-0.331541\pi$
$101$	1024.92	1.00974	0.504868	+	0.863197i	$0.331541\pi$
$102$	0	0
$103$	607.877	0.581514	0.290757	−	0.956797i	$-0.406093\pi$
$103$	607.877	0.581514	0.290757	+	0.956797i	$0.406093\pi$
$104$	−548.997	−0.517631
$105$	0	0
$106$	865.873	0.793406
$107$	169.320	0.152980	0.0764898	−	0.997070i	$-0.475629\pi$
$107$	169.320	0.152980	0.0764898	+	0.997070i	$0.475629\pi$
$108$	0	0
$109$	455.114	0.399926	0.199963	−	0.979803i	$-0.435918\pi$
$109$	455.114	0.399926	0.199963	+	0.979803i	$0.435918\pi$
$110$	−760.700	−0.659363
$111$	0	0
$112$	−557.675	−0.470494
$113$	−2039.49	−1.69787	−0.848934	−	0.528498i	$-0.822755\pi$
$113$	−2039.49	−1.69787	−0.848934	+	0.528498i	$0.822755\pi$
$114$	0	0
$115$	−118.215	−0.0958571
$116$	31.5701	0.0252690
$117$	0	0
$118$	−1930.36	−1.50597
$119$	206.089	0.158757
$120$	0	0
$121$	695.192	0.522308
$122$	−2824.20	−2.09583
$123$	0	0
$124$	−274.583	−0.198857
$125$	125.000	0.0894427
$126$	0	0
$127$	−2506.64	−1.75141	−0.875703	−	0.482849i	$-0.839602\pi$
$127$	−2506.64	−1.75141	−0.875703	+	0.482849i	$0.839602\pi$
$128$	−1655.91	−1.14346
$129$	0	0
$130$	599.824	0.404677
$131$	1485.93	0.991039	0.495519	−	0.868597i	$-0.334978\pi$
$131$	1485.93	0.991039	0.495519	+	0.868597i	$0.334978\pi$
$132$	0	0
$133$	22.3187	0.0145510
$134$	−3156.66	−2.03503
$135$	0	0
$136$	455.382	0.287123
$137$	1238.20	0.772165	0.386082	−	0.922464i	$-0.373828\pi$
$137$	1238.20	0.772165	0.386082	+	0.922464i	$0.373828\pi$
$138$	0	0
$139$	−2861.84	−1.74632	−0.873158	−	0.487437i	$-0.837932\pi$
$139$	−2861.84	−1.74632	−0.873158	+	0.487437i	$0.837932\pi$
$140$	119.829	0.0723385
$141$	0	0
$142$	−1278.75	−0.755704
$143$	−1597.68	−0.934301
$144$	0	0
$145$	46.1055	0.0264059
$146$	1671.12	0.947281
$147$	0	0
$148$	−209.515	−0.116365
$149$	1536.08	0.844569	0.422284	−	0.906463i	$-0.361228\pi$
$149$	1536.08	0.844569	0.422284	+	0.906463i	$0.361228\pi$
$150$	0	0
$151$	3520.06	1.89707	0.948537	−	0.316665i	$-0.102563\pi$
$151$	3520.06	1.89707	0.948537	+	0.316665i	$0.102563\pi$
$152$	49.3163	0.0263163
$153$	0	0
$154$	−1064.98	−0.557263
$155$	−401.005	−0.207803
$156$	0	0
$157$	−962.671	−0.489360	−0.244680	−	0.969604i	$-0.578683\pi$
$157$	−962.671	−0.489360	−0.244680	+	0.969604i	$0.578683\pi$
$158$	−3646.64	−1.83615
$159$	0	0
$160$	727.646	0.359534
$161$	−165.500	−0.0810140
$162$	0	0
$163$	−1558.10	−0.748709	−0.374354	−	0.927286i	$-0.622136\pi$
$163$	−1558.10	−0.748709	−0.374354	+	0.927286i	$0.622136\pi$
$164$	965.770	0.459841
$165$	0	0
$166$	2442.21	1.14188
$167$	1145.95	0.530996	0.265498	−	0.964111i	$-0.414464\pi$
$167$	1145.95	0.530996	0.265498	+	0.964111i	$0.414464\pi$
$168$	0	0
$169$	−937.200	−0.426582
$170$	−497.542	−0.224469
$171$	0	0
$172$	−201.485	−0.0893203
$173$	1277.72	0.561522	0.280761	−	0.959778i	$-0.409413\pi$
$173$	1277.72	0.561522	0.280761	+	0.959778i	$0.409413\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	−3586.11	−1.53587
$177$	0	0
$178$	−302.702	−0.127463
$179$	−1888.13	−0.788412	−0.394206	−	0.919022i	$-0.628980\pi$
$179$	−1888.13	−0.788412	−0.394206	+	0.919022i	$0.628980\pi$
$180$	0	0
$181$	−1193.79	−0.490242	−0.245121	−	0.969493i	$-0.578828\pi$
$181$	−1193.79	−0.490242	−0.245121	+	0.969493i	$0.578828\pi$
$182$	839.754	0.342015
$183$	0	0
$184$	−365.696	−0.146519
$185$	−305.979	−0.121600
$186$	0	0
$187$	1325.25	0.518244
$188$	1272.83	0.493781
$189$	0	0
$190$	−53.8821	−0.0205738
$191$	401.072	0.151940	0.0759700	−	0.997110i	$-0.475795\pi$
$191$	401.072	0.151940	0.0759700	+	0.997110i	$0.475795\pi$
$192$	0	0
$193$	3984.55	1.48608	0.743042	−	0.669245i	$-0.233382\pi$
$193$	3984.55	1.48608	0.743042	+	0.669245i	$0.233382\pi$
$194$	343.195	0.127010
$195$	0	0
$196$	167.760	0.0611372
$197$	3478.55	1.25805	0.629027	−	0.777383i	$-0.283453\pi$
$197$	3478.55	1.25805	0.629027	+	0.777383i	$0.283453\pi$
$198$	0	0
$199$	−2082.09	−0.741684	−0.370842	−	0.928696i	$-0.620931\pi$
$199$	−2082.09	−0.741684	−0.370842	+	0.928696i	$0.620931\pi$
$200$	386.687	0.136714
$201$	0	0
$202$	−3464.12	−1.20661
$203$	64.5477	0.0223170
$204$	0	0
$205$	1410.43	0.480529
$206$	−2054.56	−0.694893
$207$	0	0
$208$	2827.70	0.942625
$209$	143.520	0.0474999
$210$	0	0
$211$	−2423.61	−0.790751	−0.395375	−	0.918520i	$-0.629385\pi$
$211$	−2423.61	−0.790751	−0.395375	+	0.918520i	$0.629385\pi$
$212$	−877.091	−0.284146
$213$	0	0
$214$	−572.285	−0.182807
$215$	−294.252	−0.0933387
$216$	0	0
$217$	−561.407	−0.175626
$218$	−1538.24	−0.477901
$219$	0	0
$220$	770.555	0.236140
$221$	−1044.98	−0.318067
$222$	0	0
$223$	959.968	0.288270	0.144135	−	0.989558i	$-0.453960\pi$
$223$	959.968	0.288270	0.144135	+	0.989558i	$0.453960\pi$
$224$	1018.70	0.303862
$225$	0	0
$226$	6893.26	2.02891
$227$	−1018.79	−0.297883	−0.148942	−	0.988846i	$-0.547587\pi$
$227$	−1018.79	−0.297883	−0.148942	+	0.988846i	$0.547587\pi$
$228$	0	0
$229$	3281.03	0.946799	0.473399	−	0.880848i	$-0.343027\pi$
$229$	3281.03	0.946799	0.473399	+	0.880848i	$0.343027\pi$
$230$	399.553	0.114547
$231$	0	0
$232$	142.627	0.0403617
$233$	4682.68	1.31662	0.658311	−	0.752746i	$-0.271271\pi$
$233$	4682.68	1.31662	0.658311	+	0.752746i	$0.271271\pi$
$234$	0	0
$235$	1858.87	0.515996
$236$	1955.37	0.539339
$237$	0	0
$238$	−696.559	−0.189711
$239$	−3146.54	−0.851602	−0.425801	−	0.904817i	$-0.640008\pi$
$239$	−3146.54	−0.851602	−0.425801	+	0.904817i	$0.640008\pi$
$240$	0	0
$241$	7082.09	1.89294	0.946468	−	0.322798i	$-0.104624\pi$
$241$	7082.09	1.89294	0.946468	+	0.322798i	$0.104624\pi$
$242$	−2349.67	−0.624144
$243$	0	0
$244$	2860.78	0.750586
$245$	245.000	0.0638877
$246$	0	0
$247$	−113.168	−0.0291526
$248$	−1240.51	−0.317630
$249$	0	0
$250$	−422.487	−0.106882
$251$	7196.53	1.80972	0.904862	−	0.425705i	$-0.139974\pi$
$251$	7196.53	1.80972	0.904862	+	0.425705i	$0.139974\pi$
$252$	0	0
$253$	−1064.24	−0.264460
$254$	8472.19	2.09288
$255$	0	0
$256$	4433.03	1.08228
$257$	3886.10	0.943222	0.471611	−	0.881807i	$-0.343673\pi$
$257$	3886.10	0.943222	0.471611	+	0.881807i	$0.343673\pi$
$258$	0	0
$259$	−428.370	−0.102771
$260$	−607.595	−0.144929
$261$	0	0
$262$	−5022.28	−1.18426
$263$	2066.88	0.484597	0.242299	−	0.970202i	$-0.422099\pi$
$263$	2066.88	0.484597	0.242299	+	0.970202i	$0.422099\pi$
$264$	0	0
$265$	−1280.92	−0.296929
$266$	−75.4350	−0.0173880
$267$	0	0
$268$	3197.56	0.728813
$269$	−2230.84	−0.505638	−0.252819	−	0.967514i	$-0.581358\pi$
$269$	−2230.84	−0.505638	−0.252819	+	0.967514i	$0.581358\pi$
$270$	0	0
$271$	−836.594	−0.187526	−0.0937629	−	0.995595i	$-0.529890\pi$
$271$	−836.594	−0.187526	−0.0937629	+	0.995595i	$0.529890\pi$
$272$	−2345.52	−0.522861
$273$	0	0
$274$	−4184.98	−0.922716
$275$	1125.33	0.246764
$276$	0	0
$277$	−3674.18	−0.796968	−0.398484	−	0.917175i	$-0.630464\pi$
$277$	−3674.18	−0.796968	−0.398484	+	0.917175i	$0.630464\pi$
$278$	9672.71	2.08680
$279$	0	0
$280$	541.361	0.115545
$281$	−4363.28	−0.926305	−0.463152	−	0.886279i	$-0.653282\pi$
$281$	−4363.28	−0.926305	−0.463152	+	0.886279i	$0.653282\pi$
$282$	0	0
$283$	−7166.82	−1.50538	−0.752691	−	0.658374i	$-0.771245\pi$
$283$	−7166.82	−1.50538	−0.752691	+	0.658374i	$0.771245\pi$
$284$	1295.31	0.270643
$285$	0	0
$286$	5400.00	1.11646
$287$	1974.60	0.406121
$288$	0	0
$289$	−4046.21	−0.823572
$290$	−155.832	−0.0315543
$291$	0	0
$292$	−1692.77	−0.339253
$293$	5529.75	1.10257	0.551283	−	0.834319i	$-0.314139\pi$
$293$	5529.75	1.10257	0.551283	+	0.834319i	$0.314139\pi$
$294$	0	0
$295$	2855.66	0.563603
$296$	−946.543	−0.185867
$297$	0	0
$298$	−5191.79	−1.00924
$299$	839.173	0.162310
$300$	0	0
$301$	−411.953	−0.0788856
$302$	−11897.4	−2.26695
$303$	0	0
$304$	−254.012	−0.0479230
$305$	4177.94	0.784354
$306$	0	0
$307$	−8431.95	−1.56755	−0.783774	−	0.621047i	$-0.786708\pi$
$307$	−8431.95	−1.56755	−0.783774	+	0.621047i	$0.786708\pi$
$308$	1078.78	0.199575
$309$	0	0
$310$	1355.36	0.248319
$311$	−3579.03	−0.652566	−0.326283	−	0.945272i	$-0.605796\pi$
$311$	−3579.03	−0.652566	−0.326283	+	0.945272i	$0.605796\pi$
$312$	0	0
$313$	−7766.04	−1.40244	−0.701218	−	0.712947i	$-0.747360\pi$
$313$	−7766.04	−1.40244	−0.701218	+	0.712947i	$0.747360\pi$
$314$	3253.72	0.584772
$315$	0	0
$316$	3693.88	0.657586
$317$	10110.0	1.79128	0.895639	−	0.444781i	$-0.146719\pi$
$317$	10110.0	1.79128	0.895639	+	0.444781i	$0.146719\pi$
$318$	0	0
$319$	415.071	0.0728512
$320$	727.349	0.127063
$321$	0	0
$322$	559.374	0.0968095
$323$	93.8703	0.0161705
$324$	0	0
$325$	−887.342	−0.151449
$326$	5266.20	0.894686
$327$	0	0
$328$	4363.14	0.734495
$329$	2602.41	0.436096
$330$	0	0
$331$	−3124.35	−0.518821	−0.259411	−	0.965767i	$-0.583528\pi$
$331$	−3124.35	−0.518821	−0.259411	+	0.965767i	$0.583528\pi$
$332$	−2473.85	−0.408946
$333$	0	0
$334$	−3873.19	−0.634525
$335$	4669.77	0.761602
$336$	0	0
$337$	−4502.27	−0.727757	−0.363879	−	0.931446i	$-0.618548\pi$
$337$	−4502.27	−0.727757	−0.363879	+	0.931446i	$0.618548\pi$
$338$	3167.64	0.509753
$339$	0	0
$340$	503.988	0.0803899
$341$	−3610.11	−0.573309
$342$	0	0
$343$	343.000	0.0539949
$344$	−910.267	−0.142670
$345$	0	0
$346$	−4318.56	−0.671003
$347$	−11504.3	−1.77978	−0.889888	−	0.456179i	$-0.849218\pi$
$347$	−11504.3	−1.77978	−0.889888	+	0.456179i	$0.849218\pi$
$348$	0	0
$349$	917.499	0.140724	0.0703619	−	0.997522i	$-0.477585\pi$
$349$	917.499	0.140724	0.0703619	+	0.997522i	$0.477585\pi$
$350$	−591.481	−0.0903315
$351$	0	0
$352$	6550.74	0.991919
$353$	3548.38	0.535017	0.267509	−	0.963555i	$-0.413800\pi$
$353$	3548.38	0.535017	0.267509	+	0.963555i	$0.413800\pi$
$354$	0	0
$355$	1891.70	0.282819
$356$	306.624	0.0456489
$357$	0	0
$358$	6381.69	0.942130
$359$	−7875.13	−1.15775	−0.578876	−	0.815415i	$-0.696509\pi$
$359$	−7875.13	−1.15775	−0.578876	+	0.815415i	$0.696509\pi$
$360$	0	0
$361$	−6848.83	−0.998518
$362$	4034.89	0.585826
$363$	0	0
$364$	−850.633	−0.122487
$365$	−2472.15	−0.354516
$366$	0	0
$367$	6090.90	0.866327	0.433164	−	0.901315i	$-0.357397\pi$
$367$	6090.90	0.866327	0.433164	+	0.901315i	$0.357397\pi$
$368$	1883.58	0.266816
$369$	0	0
$370$	1034.18	0.145309
$371$	−1793.29	−0.250951
$372$	0	0
$373$	−11782.2	−1.63555	−0.817777	−	0.575535i	$-0.804794\pi$
$373$	−11782.2	−1.63555	−0.817777	+	0.575535i	$0.804794\pi$
$374$	−4479.19	−0.619288
$375$	0	0
$376$	5750.39	0.788706
$377$	−327.290	−0.0447117
$378$	0	0
$379$	6408.54	0.868561	0.434280	−	0.900778i	$-0.357003\pi$
$379$	6408.54	0.868561	0.434280	+	0.900778i	$0.357003\pi$
$380$	54.5802	0.00736817
$381$	0	0
$382$	−1355.58	−0.181564
$383$	−3769.38	−0.502889	−0.251444	−	0.967872i	$-0.580906\pi$
$383$	−3769.38	−0.502889	−0.251444	+	0.967872i	$0.580906\pi$
$384$	0	0
$385$	1575.46	0.208553
$386$	−13467.4	−1.77583
$387$	0	0
$388$	−347.642	−0.0454867
$389$	−3205.72	−0.417831	−0.208915	−	0.977934i	$-0.566993\pi$
$389$	−3205.72	−0.417831	−0.208915	+	0.977934i	$0.566993\pi$
$390$	0	0
$391$	−696.077	−0.0900311
$392$	757.906	0.0976531
$393$	0	0
$394$	−11757.1	−1.50334
$395$	5394.61	0.687170
$396$	0	0
$397$	−8989.91	−1.13650	−0.568250	−	0.822856i	$-0.692380\pi$
$397$	−8989.91	−1.13650	−0.568250	+	0.822856i	$0.692380\pi$
$398$	7037.23	0.886293
$399$	0	0
$400$	−1991.70	−0.248962
$401$	−15270.8	−1.90172	−0.950859	−	0.309625i	$-0.899797\pi$
$401$	−15270.8	−1.90172	−0.950859	+	0.309625i	$0.899797\pi$
$402$	0	0
$403$	2846.63	0.351863
$404$	3509.00	0.432127
$405$	0	0
$406$	−218.164	−0.0266682
$407$	−2754.62	−0.335483
$408$	0	0
$409$	13450.1	1.62608	0.813040	−	0.582208i	$-0.197811\pi$
$409$	13450.1	1.62608	0.813040	+	0.582208i	$0.197811\pi$
$410$	−4767.09	−0.574219
$411$	0	0
$412$	2081.18	0.248865
$413$	3997.92	0.476331
$414$	0	0
$415$	−3612.85	−0.427344
$416$	−5165.36	−0.608781
$417$	0	0
$418$	−485.082	−0.0567611
$419$	−7525.80	−0.877468	−0.438734	−	0.898617i	$-0.644573\pi$
$419$	−7525.80	−0.877468	−0.438734	+	0.898617i	$0.644573\pi$
$420$	0	0
$421$	6677.45	0.773014	0.386507	−	0.922286i	$-0.373681\pi$
$421$	6677.45	0.773014	0.386507	+	0.922286i	$0.373681\pi$
$422$	8191.55	0.944926
$423$	0	0
$424$	−3962.51	−0.453860
$425$	736.032	0.0840066
$426$	0	0
$427$	5849.11	0.662900
$428$	579.699	0.0654692
$429$	0	0
$430$	994.541	0.111537
$431$	−7676.15	−0.857882	−0.428941	−	0.903332i	$-0.641113\pi$
$431$	−7676.15	−0.857882	−0.428941	+	0.903332i	$0.641113\pi$
$432$	0	0
$433$	4853.67	0.538690	0.269345	−	0.963044i	$-0.413193\pi$
$433$	4853.67	0.538690	0.269345	+	0.963044i	$0.413193\pi$
$434$	1897.50	0.209868
$435$	0	0
$436$	1558.16	0.171153
$437$	−75.3828	−0.00825183
$438$	0	0
$439$	5301.90	0.576414	0.288207	−	0.957568i	$-0.406941\pi$
$439$	5301.90	0.576414	0.288207	+	0.957568i	$0.406941\pi$
$440$	3481.20	0.377182
$441$	0	0
$442$	3531.92	0.380082
$443$	−1994.30	−0.213887	−0.106944	−	0.994265i	$-0.534106\pi$
$443$	−1994.30	−0.213887	−0.106944	+	0.994265i	$0.534106\pi$
$444$	0	0
$445$	447.798	0.0477026
$446$	−3244.59	−0.344475
$447$	0	0
$448$	1018.29	0.107388
$449$	−432.941	−0.0455051	−0.0227525	−	0.999741i	$-0.507243\pi$
$449$	−432.941	−0.0455051	−0.0227525	+	0.999741i	$0.507243\pi$
$450$	0	0
$451$	12697.6	1.32573
$452$	−6982.57	−0.726620
$453$	0	0
$454$	3443.40	0.355962
$455$	−1242.28	−0.127998
$456$	0	0
$457$	−4192.20	−0.429109	−0.214555	−	0.976712i	$-0.568830\pi$
$457$	−4192.20	−0.429109	−0.214555	+	0.976712i	$0.568830\pi$
$458$	−11089.5	−1.13140
$459$	0	0
$460$	−404.729	−0.0410230
$461$	16787.0	1.69598	0.847992	−	0.530009i	$-0.177811\pi$
$461$	16787.0	1.69598	0.847992	+	0.530009i	$0.177811\pi$
$462$	0	0
$463$	5604.79	0.562585	0.281292	−	0.959622i	$-0.409237\pi$
$463$	5604.79	0.562585	0.281292	+	0.959622i	$0.409237\pi$
$464$	−734.625	−0.0735002
$465$	0	0
$466$	−15827.0	−1.57333
$467$	13011.1	1.28926	0.644630	−	0.764495i	$-0.277012\pi$
$467$	13011.1	1.28926	0.644630	+	0.764495i	$0.277012\pi$
$468$	0	0
$469$	6537.67	0.643671
$470$	−6282.77	−0.616601
$471$	0	0
$472$	8833.96	0.861474
$473$	−2649.05	−0.257512
$474$	0	0
$475$	79.7098	0.00769965
$476$	705.583	0.0679419
$477$	0	0
$478$	10635.0	1.01764
$479$	−16424.3	−1.56670	−0.783348	−	0.621583i	$-0.786490\pi$
$479$	−16424.3	−1.56670	−0.783348	+	0.621583i	$0.786490\pi$
$480$	0	0
$481$	2172.06	0.205899
$482$	−23936.7	−2.26201
$483$	0	0
$484$	2380.12	0.223527
$485$	−507.701	−0.0475330
$486$	0	0
$487$	3448.18	0.320846	0.160423	−	0.987048i	$-0.448714\pi$
$487$	3448.18	0.320846	0.160423	+	0.987048i	$0.448714\pi$
$488$	12924.4	1.19890
$489$	0	0
$490$	−828.074	−0.0763440
$491$	−19657.5	−1.80679	−0.903393	−	0.428814i	$-0.858932\pi$
$491$	−19657.5	−1.80679	−0.903393	+	0.428814i	$0.858932\pi$
$492$	0	0
$493$	271.481	0.0248010
$494$	382.495	0.0348365
$495$	0	0
$496$	6389.45	0.578417
$497$	2648.37	0.239026
$498$	0	0
$499$	4259.62	0.382138	0.191069	−	0.981577i	$-0.438805\pi$
$499$	4259.62	0.382138	0.191069	+	0.981577i	$0.438805\pi$
$500$	427.960	0.0382779
$501$	0	0
$502$	−24323.5	−2.16257
$503$	−14981.4	−1.32801	−0.664005	−	0.747728i	$-0.731145\pi$
$503$	−14981.4	−1.32801	−0.664005	+	0.747728i	$0.731145\pi$
$504$	0	0
$505$	5124.60	0.451567
$506$	3597.03	0.316023
$507$	0	0
$508$	−8581.95	−0.749532
$509$	3433.20	0.298966	0.149483	−	0.988764i	$-0.452239\pi$
$509$	3433.20	0.298966	0.149483	+	0.988764i	$0.452239\pi$
$510$	0	0
$511$	−3461.01	−0.299621
$512$	−1735.91	−0.149838
$513$	0	0
$514$	−13134.6	−1.12713
$515$	3039.39	0.260061
$516$	0	0
$517$	16734.7	1.42358
$518$	1447.85	0.122808
$519$	0	0
$520$	−2744.99	−0.231492
$521$	10205.7	0.858194	0.429097	−	0.903258i	$-0.358832\pi$
$521$	10205.7	0.858194	0.429097	+	0.903258i	$0.358832\pi$
$522$	0	0
$523$	6563.35	0.548748	0.274374	−	0.961623i	$-0.411529\pi$
$523$	6563.35	0.548748	0.274374	+	0.961623i	$0.411529\pi$
$524$	5087.34	0.424125
$525$	0	0
$526$	−6985.82	−0.579081
$527$	−2361.22	−0.195173
$528$	0	0
$529$	−11608.0	−0.954057
$530$	4329.37	0.354822
$531$	0	0
$532$	76.4122	0.00622724
$533$	−10012.2	−0.813655
$534$	0	0
$535$	846.602	0.0684146
$536$	14445.9	1.16412
$537$	0	0
$538$	7540.00	0.604224
$539$	2205.65	0.176260
$540$	0	0
$541$	−8321.50	−0.661311	−0.330655	−	0.943752i	$-0.607270\pi$
$541$	−8321.50	−0.661311	−0.330655	+	0.943752i	$0.607270\pi$
$542$	2827.60	0.224088
$543$	0	0
$544$	4284.56	0.337682
$545$	2275.57	0.178853
$546$	0	0
$547$	−6107.98	−0.477438	−0.238719	−	0.971089i	$-0.576727\pi$
$547$	−6107.98	−0.477438	−0.238719	+	0.971089i	$0.576727\pi$
$548$	4239.20	0.330456
$549$	0	0
$550$	−3803.50	−0.294876
$551$	29.4004	0.00227314
$552$	0	0
$553$	7552.45	0.580765
$554$	12418.3	0.952355
$555$	0	0
$556$	−9798.02	−0.747354
$557$	−2474.65	−0.188248	−0.0941240	−	0.995560i	$-0.530005\pi$
$557$	−2474.65	−0.188248	−0.0941240	+	0.995560i	$0.530005\pi$
$558$	0	0
$559$	2088.82	0.158046
$560$	−2788.38	−0.210411
$561$	0	0
$562$	14747.4	1.10691
$563$	−2322.75	−0.173876	−0.0869381	−	0.996214i	$-0.527708\pi$
$563$	−2322.75	−0.173876	−0.0869381	+	0.996214i	$0.527708\pi$
$564$	0	0
$565$	−10197.5	−0.759310
$566$	24223.1	1.79889
$567$	0	0
$568$	5851.95	0.432293
$569$	3303.12	0.243364	0.121682	−	0.992569i	$-0.461171\pi$
$569$	3303.12	0.243364	0.121682	+	0.992569i	$0.461171\pi$
$570$	0	0
$571$	−13356.7	−0.978914	−0.489457	−	0.872028i	$-0.662805\pi$
$571$	−13356.7	−0.978914	−0.489457	+	0.872028i	$0.662805\pi$
$572$	−5469.96	−0.399844
$573$	0	0
$574$	−6673.92	−0.485303
$575$	−591.073	−0.0428686
$576$	0	0
$577$	19598.7	1.41405	0.707023	−	0.707191i	$-0.250038\pi$
$577$	19598.7	1.41405	0.707023	+	0.707191i	$0.250038\pi$
$578$	13675.8	0.984147
$579$	0	0
$580$	157.850	0.0113007
$581$	−5057.99	−0.361172
$582$	0	0
$583$	−11531.7	−0.819198
$584$	−7647.59	−0.541882
$585$	0	0
$586$	−18690.0	−1.31754
$587$	−20293.7	−1.42693	−0.713467	−	0.700689i	$-0.752876\pi$
$587$	−20293.7	−1.42693	−0.713467	+	0.700689i	$0.752876\pi$
$588$	0	0
$589$	−255.712	−0.0178887
$590$	−9651.82	−0.673490
$591$	0	0
$592$	4875.33	0.338471
$593$	−26600.4	−1.84207	−0.921035	−	0.389479i	$-0.872655\pi$
$593$	−26600.4	−1.84207	−0.921035	+	0.389479i	$0.872655\pi$
$594$	0	0
$595$	1030.44	0.0709985
$596$	5259.06	0.361442
$597$	0	0
$598$	−2836.32	−0.193956
$599$	−10224.5	−0.697430	−0.348715	−	0.937229i	$-0.613382\pi$
$599$	−10224.5	−0.697430	−0.348715	+	0.937229i	$0.613382\pi$
$600$	0	0
$601$	−9417.06	−0.639152	−0.319576	−	0.947561i	$-0.603540\pi$
$601$	−9417.06	−0.639152	−0.319576	+	0.947561i	$0.603540\pi$
$602$	1392.36	0.0942662
$603$	0	0
$604$	12051.6	0.811873
$605$	3475.96	0.233583
$606$	0	0
$607$	9840.73	0.658028	0.329014	−	0.944325i	$-0.393284\pi$
$607$	9840.73	0.658028	0.329014	+	0.944325i	$0.393284\pi$
$608$	464.004	0.0309504
$609$	0	0
$610$	−14121.0	−0.937282
$611$	−13195.6	−0.873709
$612$	0	0
$613$	−20819.3	−1.37175	−0.685876	−	0.727718i	$-0.740581\pi$
$613$	−20819.3	−1.37175	−0.685876	+	0.727718i	$0.740581\pi$
$614$	28499.1	1.87318
$615$	0	0
$616$	4873.69	0.318777
$617$	21750.5	1.41920	0.709598	−	0.704607i	$-0.248877\pi$
$617$	21750.5	1.41920	0.709598	+	0.704607i	$0.248877\pi$
$618$	0	0
$619$	4123.18	0.267730	0.133865	−	0.991000i	$-0.457261\pi$
$619$	4123.18	0.267730	0.133865	+	0.991000i	$0.457261\pi$
$620$	−1372.91	−0.0889316
$621$	0	0
$622$	12096.7	0.779798
$623$	626.917	0.0403161
$624$	0	0
$625$	625.000	0.0400000
$626$	26248.4	1.67587
$627$	0	0
$628$	−3295.88	−0.209427
$629$	−1801.68	−0.114209
$630$	0	0
$631$	−15561.6	−0.981773	−0.490887	−	0.871224i	$-0.663327\pi$
$631$	−15561.6	−0.981773	−0.490887	+	0.871224i	$0.663327\pi$
$632$	16688.2	1.05035
$633$	0	0
$634$	−34170.8	−2.14053
$635$	−12533.2	−0.783253
$636$	0	0
$637$	−1739.19	−0.108178
$638$	−1402.90	−0.0870552
$639$	0	0
$640$	−8279.53	−0.511370
$641$	15463.9	0.952866	0.476433	−	0.879211i	$-0.341930\pi$
$641$	15463.9	0.952866	0.476433	+	0.879211i	$0.341930\pi$
$642$	0	0
$643$	−6231.42	−0.382182	−0.191091	−	0.981572i	$-0.561203\pi$
$643$	−6231.42	−0.382182	−0.191091	+	0.981572i	$0.561203\pi$
$644$	−566.620	−0.0346708
$645$	0	0
$646$	−317.272	−0.0193234
$647$	15001.4	0.911538	0.455769	−	0.890098i	$-0.349364\pi$
$647$	15001.4	0.911538	0.455769	+	0.890098i	$0.349364\pi$
$648$	0	0
$649$	25708.5	1.55492
$650$	2999.12	0.180977
$651$	0	0
$652$	−5334.42	−0.320417
$653$	6309.85	0.378137	0.189069	−	0.981964i	$-0.439453\pi$
$653$	6309.85	0.378137	0.189069	+	0.981964i	$0.439453\pi$
$654$	0	0
$655$	7429.64	0.443206
$656$	−22473.1	−1.33754
$657$	0	0
$658$	−8795.87	−0.521123
$659$	6287.74	0.371678	0.185839	−	0.982580i	$-0.440500\pi$
$659$	6287.74	0.371678	0.185839	+	0.982580i	$0.440500\pi$
$660$	0	0
$661$	131.962	0.00776511	0.00388256	−	0.999992i	$-0.498764\pi$
$661$	131.962	0.00776511	0.00388256	+	0.999992i	$0.498764\pi$
$662$	10560.0	0.619977
$663$	0	0
$664$	−11176.3	−0.653201
$665$	111.594	0.00650739
$666$	0	0
$667$	−218.013	−0.0126559
$668$	3923.37	0.227245
$669$	0	0
$670$	−15783.3	−0.910094
$671$	37612.5	2.16396
$672$	0	0
$673$	1586.31	0.0908583	0.0454291	−	0.998968i	$-0.485534\pi$
$673$	1586.31	0.0908583	0.0454291	+	0.998968i	$0.485534\pi$
$674$	15217.2	0.869650
$675$	0	0
$676$	−3208.67	−0.182560
$677$	17191.5	0.975958	0.487979	−	0.872855i	$-0.337734\pi$
$677$	17191.5	0.975958	0.487979	+	0.872855i	$0.337734\pi$
$678$	0	0
$679$	−710.782	−0.0401728
$680$	2276.91	0.128405
$681$	0	0
$682$	12201.8	0.685089
$683$	−28445.7	−1.59362	−0.796812	−	0.604227i	$-0.793482\pi$
$683$	−28445.7	−1.59362	−0.796812	+	0.604227i	$0.793482\pi$
$684$	0	0
$685$	6191.00	0.345323
$686$	−1159.30	−0.0645225
$687$	0	0
$688$	4688.49	0.259806
$689$	9092.90	0.502775
$690$	0	0
$691$	−5503.89	−0.303007	−0.151503	−	0.988457i	$-0.548411\pi$
$691$	−5503.89	−0.303007	−0.151503	+	0.988457i	$0.548411\pi$
$692$	4374.51	0.240309
$693$	0	0
$694$	38883.3	2.12678
$695$	−14309.2	−0.780976
$696$	0	0
$697$	8304.94	0.451323
$698$	−3101.05	−0.168161
$699$	0	0
$700$	599.144	0.0323508
$701$	9737.40	0.524646	0.262323	−	0.964980i	$-0.415511\pi$
$701$	9737.40	0.524646	0.262323	+	0.964980i	$0.415511\pi$
$702$	0	0
$703$	−195.116	−0.0104679
$704$	6548.07	0.350553
$705$	0	0
$706$	−11993.1	−0.639331
$707$	7174.43	0.381644
$708$	0	0
$709$	−11496.7	−0.608981	−0.304490	−	0.952515i	$-0.598486\pi$
$709$	−11496.7	−0.608981	−0.304490	+	0.952515i	$0.598486\pi$
$710$	−6393.73	−0.337961
$711$	0	0
$712$	1385.26	0.0729141
$713$	1896.19	0.0995971
$714$	0	0
$715$	−7988.42	−0.417832
$716$	−6464.37	−0.337409
$717$	0	0
$718$	26617.1	1.38348
$719$	−9513.43	−0.493451	−0.246725	−	0.969085i	$-0.579355\pi$
$719$	−9513.43	−0.493451	−0.246725	+	0.969085i	$0.579355\pi$
$720$	0	0
$721$	4255.14	0.219792
$722$	23148.3	1.19320
$723$	0	0
$724$	−4087.16	−0.209804
$725$	230.527	0.0118091
$726$	0	0
$727$	−3298.37	−0.168266	−0.0841332	−	0.996455i	$-0.526812\pi$
$727$	−3298.37	−0.168266	−0.0841332	+	0.996455i	$0.526812\pi$
$728$	−3842.98	−0.195646
$729$	0	0
$730$	8355.61	0.423637
$731$	−1732.63	−0.0876658
$732$	0	0
$733$	22811.5	1.14947	0.574735	−	0.818339i	$-0.305105\pi$
$733$	22811.5	1.14947	0.574735	+	0.818339i	$0.305105\pi$
$734$	−20586.6	−1.03524
$735$	0	0
$736$	−3440.73	−0.172319
$737$	42040.3	2.10118
$738$	0	0
$739$	24860.4	1.23749	0.618746	−	0.785591i	$-0.287641\pi$
$739$	24860.4	1.23749	0.618746	+	0.785591i	$0.287641\pi$
$740$	−1047.57	−0.0520400
$741$	0	0
$742$	6061.11	0.299879
$743$	−6616.66	−0.326705	−0.163352	−	0.986568i	$-0.552231\pi$
$743$	−6616.66	−0.326705	−0.163352	+	0.986568i	$0.552231\pi$
$744$	0	0
$745$	7680.41	0.377703
$746$	39822.7	1.95444
$747$	0	0
$748$	4537.22	0.221788
$749$	1185.24	0.0578209
$750$	0	0
$751$	−18976.4	−0.922051	−0.461025	−	0.887387i	$-0.652518\pi$
$751$	−18976.4	−0.922051	−0.461025	+	0.887387i	$0.652518\pi$
$752$	−29618.4	−1.43626
$753$	0	0
$754$	1106.21	0.0534293
$755$	17600.3	0.848398
$756$	0	0
$757$	−7069.89	−0.339444	−0.169722	−	0.985492i	$-0.554287\pi$
$757$	−7069.89	−0.339444	−0.169722	+	0.985492i	$0.554287\pi$
$758$	−21660.2	−1.03791
$759$	0	0
$760$	246.582	0.0117690
$761$	1345.22	0.0640793	0.0320397	−	0.999487i	$-0.489800\pi$
$761$	1345.22	0.0640793	0.0320397	+	0.999487i	$0.489800\pi$
$762$	0	0
$763$	3185.80	0.151158
$764$	1373.14	0.0650242
$765$	0	0
$766$	12740.1	0.600939
$767$	−20271.5	−0.954320
$768$	0	0
$769$	36811.4	1.72621	0.863103	−	0.505027i	$-0.168518\pi$
$769$	36811.4	1.72621	0.863103	+	0.505027i	$0.168518\pi$
$770$	−5324.90	−0.249216
$771$	0	0
$772$	13641.8	0.635985
$773$	11439.2	0.532265	0.266133	−	0.963936i	$-0.414254\pi$
$773$	11439.2	0.532265	0.266133	+	0.963936i	$0.414254\pi$
$774$	0	0
$775$	−2005.03	−0.0929325
$776$	−1570.57	−0.0726549
$777$	0	0
$778$	10835.0	0.499297
$779$	899.398	0.0413662
$780$	0	0
$781$	17030.3	0.780270
$782$	2352.67	0.107585
$783$	0	0
$784$	−3903.73	−0.177830
$785$	−4813.35	−0.218848
$786$	0	0
$787$	−1200.30	−0.0543662	−0.0271831	−	0.999630i	$-0.508654\pi$
$787$	−1200.30	−0.0543662	−0.0271831	+	0.999630i	$0.508654\pi$
$788$	11909.5	0.538397
$789$	0	0
$790$	−18233.2	−0.821150
$791$	−14276.4	−0.641734
$792$	0	0
$793$	−29658.1	−1.32811
$794$	30385.0	1.35809
$795$	0	0
$796$	−7128.40	−0.317411
$797$	−28945.9	−1.28647	−0.643236	−	0.765668i	$-0.722409\pi$
$797$	−28945.9	−1.28647	−0.643236	+	0.765668i	$0.722409\pi$
$798$	0	0
$799$	10945.5	0.484634
$800$	3638.23	0.160788
$801$	0	0
$802$	51613.8	2.27250
$803$	−22255.9	−0.978075
$804$	0	0
$805$	−827.502	−0.0362306
$806$	−9621.31	−0.420467
$807$	0	0
$808$	15852.9	0.690227
$809$	−43557.5	−1.89296	−0.946478	−	0.322770i	$-0.895386\pi$
$809$	−43557.5	−1.89296	−0.946478	+	0.322770i	$0.895386\pi$
$810$	0	0
$811$	−36631.5	−1.58608	−0.793038	−	0.609172i	$-0.791502\pi$
$811$	−36631.5	−1.58608	−0.793038	+	0.609172i	$0.791502\pi$
$812$	220.991	0.00955080
$813$	0	0
$814$	9310.32	0.400893
$815$	−7790.48	−0.334833
$816$	0	0
$817$	−187.638	−0.00803504
$818$	−45460.0	−1.94312
$819$	0	0
$820$	4828.85	0.205647
$821$	−17226.4	−0.732287	−0.366143	−	0.930558i	$-0.619322\pi$
$821$	−17226.4	−0.732287	−0.366143	+	0.930558i	$0.619322\pi$
$822$	0	0
$823$	−7294.54	−0.308957	−0.154479	−	0.987996i	$-0.549370\pi$
$823$	−7294.54	−0.308957	−0.154479	+	0.987996i	$0.549370\pi$
$824$	9402.32	0.397507
$825$	0	0
$826$	−13512.5	−0.569203
$827$	46248.5	1.94464	0.972321	−	0.233649i	$-0.0750666\pi$
$827$	46248.5	1.94464	0.972321	+	0.233649i	$0.0750666\pi$
$828$	0	0
$829$	−8189.74	−0.343114	−0.171557	−	0.985174i	$-0.554880\pi$
$829$	−8189.74	−0.343114	−0.171557	+	0.985174i	$0.554880\pi$
$830$	12211.1	0.510665
$831$	0	0
$832$	−5163.26	−0.215149
$833$	1442.62	0.0600047
$834$	0	0
$835$	5729.75	0.237468
$836$	491.366	0.0203281
$837$	0	0
$838$	25436.4	1.04855
$839$	−34877.9	−1.43518	−0.717592	−	0.696464i	$-0.754756\pi$
$839$	−34877.9	−1.43518	−0.717592	+	0.696464i	$0.754756\pi$
$840$	0	0
$841$	−24304.0	−0.996514
$842$	−22569.1	−0.923731
$843$	0	0
$844$	−8297.68	−0.338410
$845$	−4686.00	−0.190773
$846$	0	0
$847$	4866.34	0.197414
$848$	20409.6	0.826496
$849$	0	0
$850$	−2487.71	−0.100386
$851$	1446.85	0.0582811
$852$	0	0
$853$	−36422.3	−1.46199	−0.730995	−	0.682383i	$-0.760944\pi$
$853$	−36422.3	−1.46199	−0.730995	+	0.682383i	$0.760944\pi$
$854$	−19769.4	−0.792148
$855$	0	0
$856$	2618.96	0.104573
$857$	−27699.1	−1.10407	−0.552033	−	0.833823i	$-0.686148\pi$
$857$	−27699.1	−1.10407	−0.552033	+	0.833823i	$0.686148\pi$
$858$	0	0
$859$	45673.0	1.81413	0.907067	−	0.420986i	$-0.138316\pi$
$859$	45673.0	1.81413	0.907067	+	0.420986i	$0.138316\pi$
$860$	−1007.43	−0.0399453
$861$	0	0
$862$	25944.6	1.02515
$863$	7734.49	0.305081	0.152541	−	0.988297i	$-0.451254\pi$
$863$	7734.49	0.305081	0.152541	+	0.988297i	$0.451254\pi$
$864$	0	0
$865$	6388.60	0.251120
$866$	−16404.9	−0.643720
$867$	0	0
$868$	−1922.08	−0.0751609
$869$	48565.8	1.89583
$870$	0	0
$871$	−33149.4	−1.28958
$872$	7039.46	0.273379
$873$	0	0
$874$	254.786	0.00986071
$875$	875.000	0.0338062
$876$	0	0
$877$	16893.9	0.650475	0.325237	−	0.945632i	$-0.394556\pi$
$877$	16893.9	0.650475	0.325237	+	0.945632i	$0.394556\pi$
$878$	−17919.9	−0.688799
$879$	0	0
$880$	−17930.5	−0.686862
$881$	44345.0	1.69582	0.847912	−	0.530136i	$-0.177859\pi$
$881$	44345.0	1.69582	0.847912	+	0.530136i	$0.177859\pi$
$882$	0	0
$883$	−17457.4	−0.665332	−0.332666	−	0.943045i	$-0.607948\pi$
$883$	−17457.4	−0.665332	−0.332666	+	0.943045i	$0.607948\pi$
$884$	−3577.67	−0.136120
$885$	0	0
$886$	6740.52	0.255589
$887$	−28981.4	−1.09707	−0.548535	−	0.836128i	$-0.684814\pi$
$887$	−28981.4	−1.09707	−0.548535	+	0.836128i	$0.684814\pi$
$888$	0	0
$889$	−17546.5	−0.661970
$890$	−1513.51	−0.0570033
$891$	0	0
$892$	3286.63	0.123368
$893$	1185.36	0.0444194
$894$	0	0
$895$	−9440.66	−0.352588
$896$	−11591.3	−0.432187
$897$	0	0
$898$	1463.30	0.0543773
$899$	−739.541	−0.0274361
$900$	0	0
$901$	−7542.37	−0.278882
$902$	−42916.4	−1.58421
$903$	0	0
$904$	−31545.8	−1.16062
$905$	−5968.95	−0.219243
$906$	0	0
$907$	31108.2	1.13884	0.569422	−	0.822046i	$-0.307167\pi$
$907$	31108.2	1.13884	0.569422	+	0.822046i	$0.307167\pi$
$908$	−3488.01	−0.127482
$909$	0	0
$910$	4198.77	0.152954
$911$	−4115.73	−0.149682	−0.0748409	−	0.997195i	$-0.523845\pi$
$911$	−4115.73	−0.149682	−0.0748409	+	0.997195i	$0.523845\pi$
$912$	0	0
$913$	−32525.2	−1.17900
$914$	14169.2	0.512774
$915$	0	0
$916$	11233.2	0.405192
$917$	10401.5	0.374577
$918$	0	0
$919$	−13909.3	−0.499266	−0.249633	−	0.968341i	$-0.580310\pi$
$919$	−13909.3	−0.499266	−0.249633	+	0.968341i	$0.580310\pi$
$920$	−1828.48	−0.0655252
$921$	0	0
$922$	−56738.3	−2.02666
$923$	−13428.6	−0.478883
$924$	0	0
$925$	−1529.89	−0.0543812
$926$	−18943.6	−0.672273
$927$	0	0
$928$	1341.94	0.0474690
$929$	−22380.1	−0.790383	−0.395192	−	0.918599i	$-0.629322\pi$
$929$	−22380.1	−0.790383	−0.395192	+	0.918599i	$0.629322\pi$
$930$	0	0
$931$	156.231	0.00549975
$932$	16032.0	0.563461
$933$	0	0
$934$	−43976.3	−1.54063
$935$	6626.24	0.231766
$936$	0	0
$937$	−52911.1	−1.84475	−0.922374	−	0.386297i	$-0.873754\pi$
$937$	−52911.1	−1.84475	−0.922374	+	0.386297i	$0.873754\pi$
$938$	−22096.6	−0.769169
$939$	0	0
$940$	6364.16	0.220826
$941$	42984.9	1.48913	0.744564	−	0.667551i	$-0.232658\pi$
$941$	42984.9	1.48913	0.744564	+	0.667551i	$0.232658\pi$
$942$	0	0
$943$	−6669.31	−0.230310
$944$	−45500.8	−1.56878
$945$	0	0
$946$	8953.50	0.307720
$947$	−25210.3	−0.865075	−0.432537	−	0.901616i	$-0.642382\pi$
$947$	−25210.3	−0.865075	−0.432537	+	0.901616i	$0.642382\pi$
$948$	0	0
$949$	17549.1	0.600284
$950$	−269.411	−0.00920088
$951$	0	0
$952$	3187.67	0.108522
$953$	32792.0	1.11462	0.557312	−	0.830303i	$-0.311833\pi$
$953$	32792.0	1.11462	0.557312	+	0.830303i	$0.311833\pi$
$954$	0	0
$955$	2005.36	0.0679496
$956$	−10772.8	−0.364452
$957$	0	0
$958$	55512.5	1.87216
$959$	8667.40	0.291851
$960$	0	0
$961$	−23358.8	−0.784089
$962$	−7341.34	−0.246044
$963$	0	0
$964$	24246.8	0.810101
$965$	19922.8	0.664597
$966$	0	0
$967$	−22158.8	−0.736897	−0.368449	−	0.929648i	$-0.620111\pi$
$967$	−22158.8	−0.736897	−0.368449	+	0.929648i	$0.620111\pi$
$968$	10752.9	0.357035
$969$	0	0
$970$	1715.98	0.0568007
$971$	−45086.4	−1.49010	−0.745052	−	0.667007i	$-0.767575\pi$
$971$	−45086.4	−1.49010	−0.745052	+	0.667007i	$0.767575\pi$
$972$	0	0
$973$	−20032.9	−0.660045
$974$	−11654.5	−0.383402
$975$	0	0
$976$	−66569.4	−2.18323
$977$	−31577.0	−1.03402	−0.517010	−	0.855980i	$-0.672955\pi$
$977$	−31577.0	−1.03402	−0.517010	+	0.855980i	$0.672955\pi$
$978$	0	0
$979$	4031.37	0.131607
$980$	838.802	0.0273414
$981$	0	0
$982$	66440.3	2.15906
$983$	47938.2	1.55543	0.777716	−	0.628615i	$-0.216378\pi$
$983$	47938.2	1.55543	0.777716	+	0.628615i	$0.216378\pi$
$984$	0	0
$985$	17392.8	0.562619
$986$	−917.576	−0.0296365
$987$	0	0
$988$	−387.450	−0.0124761
$989$	1391.39	0.0447359
$990$	0	0
$991$	−10906.5	−0.349603	−0.174801	−	0.984604i	$-0.555928\pi$
$991$	−10906.5	−0.349603	−0.174801	+	0.984604i	$0.555928\pi$
$992$	−11671.6	−0.373562
$993$	0	0
$994$	−8951.22	−0.285629
$995$	−10410.4	−0.331691
$996$	0	0
$997$	−12617.3	−0.400796	−0.200398	−	0.979715i	$-0.564224\pi$
$997$	−12617.3	−0.400796	−0.200398	+	0.979715i	$0.564224\pi$
$998$	−14397.1	−0.456644
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.a.n.1.3	✓	3	3.2	odd	2
315.4.a.o.1.1	yes	3	1.1	even	1	trivial
1575.4.a.bb.1.3		3	5.4	even	2
1575.4.a.be.1.1		3	15.14	odd	2
2205.4.a.bk.1.3		3	21.20	even	2
2205.4.a.bl.1.1		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-3.37989 q^{2} +3.42368 q^{4} +5.00000 q^{5} +7.00000 q^{7} +15.4675 q^{8} -16.8995 q^{10} +45.0132 q^{11} -35.4937 q^{13} -23.6593 q^{14} -79.6679 q^{16} +29.4413 q^{17} +3.18839 q^{19} +17.1184 q^{20} -152.140 q^{22} -23.6429 q^{23} +25.0000 q^{25} +119.965 q^{26} +23.9658 q^{28} +9.22109 q^{29} -80.2011 q^{31} +145.529 q^{32} -99.5084 q^{34} +35.0000 q^{35} -61.1957 q^{37} -10.7764 q^{38} +77.3373 q^{40} +282.085 q^{41} -58.8504 q^{43} +154.111 q^{44} +79.9105 q^{46} +371.773 q^{47} +49.0000 q^{49} -84.4973 q^{50} -121.519 q^{52} -256.184 q^{53} +225.066 q^{55} +108.272 q^{56} -31.1663 q^{58} +571.131 q^{59} +835.587 q^{61} +271.071 q^{62} +145.470 q^{64} -177.468 q^{65} +933.953 q^{67} +100.798 q^{68} -118.296 q^{70} +378.339 q^{71} -494.430 q^{73} +206.835 q^{74} +10.9160 q^{76} +315.093 q^{77} +1078.92 q^{79} -398.339 q^{80} -953.418 q^{82} -722.570 q^{83} +147.206 q^{85} +198.908 q^{86} +696.241 q^{88} +89.5596 q^{89} -248.456 q^{91} -80.9458 q^{92} -1256.55 q^{94} +15.9420 q^{95} -101.540 q^{97} -165.615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 14 q^{4} + 15 q^{5} + 21 q^{7} + 66 q^{8} + 10 q^{10} - 20 q^{11} + 14 q^{14} + 114 q^{16} + 234 q^{17} - 82 q^{19} + 70 q^{20} - 236 q^{22} + 30 q^{23} + 75 q^{25} + 76 q^{26} + 98 q^{28}+ \cdots + 98 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 14 * q^4 + 15 * q^5 + 21 * q^7 + 66 * q^8 + 10 * q^10 - 20 * q^11 + 14 * q^14 + 114 * q^16 + 234 * q^17 - 82 * q^19 + 70 * q^20 - 236 * q^22 + 30 * q^23 + 75 * q^25 + 76 * q^26 + 98 * q^28 - 32 * q^29 - 362 * q^31 + 430 * q^32 + 596 * q^34 + 105 * q^35 + 302 * q^37 + 160 * q^38 + 330 * q^40 + 626 * q^41 - 68 * q^43 + 300 * q^44 - 412 * q^46 + 936 * q^47 + 147 * q^49 + 50 * q^50 - 684 * q^52 + 298 * q^53 - 100 * q^55 + 462 * q^56 - 1184 * q^58 + 428 * q^59 + 1050 * q^61 - 608 * q^62 - 154 * q^64 - 52 * q^67 + 1964 * q^68 + 70 * q^70 + 416 * q^71 - 312 * q^73 + 1668 * q^74 + 1712 * q^76 - 140 * q^77 - 488 * q^79 + 570 * q^80 - 1476 * q^82 + 948 * q^83 + 1170 * q^85 - 1928 * q^86 + 124 * q^88 - 142 * q^89 - 3220 * q^92 + 328 * q^94 - 410 * q^95 - 3052 * q^97 + 98 * q^98

Introduction

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