LMFDB - Embedded newform 1575.4.a.bh.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] 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("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [4,0,0,10,0,0,-28,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3030748.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 21x^{2} + 28 $ x^4 - 21*x^2 + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.19617$ 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)

$f(q)$	$=$	$q-1.19617 q^{2} -6.56918 q^{4} -7.00000 q^{7} +17.4272 q^{8} +4.52701 q^{11} +14.5692 q^{13} +8.37319 q^{14} +31.7075 q^{16} +15.5502 q^{17} -14.8616 q^{19} -5.41507 q^{22} -166.672 q^{23} -17.4272 q^{26} +45.9843 q^{28} -194.865 q^{29} +104.538 q^{31} -177.345 q^{32} -18.6007 q^{34} -47.5377 q^{37} +17.7770 q^{38} +378.633 q^{41} +194.969 q^{43} -29.7387 q^{44} +199.368 q^{46} +488.661 q^{47} +49.0000 q^{49} -95.7075 q^{52} +316.432 q^{53} -121.990 q^{56} +233.091 q^{58} +350.256 q^{59} +115.613 q^{61} -125.045 q^{62} -41.5253 q^{64} -434.890 q^{67} -102.152 q^{68} -410.119 q^{71} +464.182 q^{73} +56.8631 q^{74} +97.6288 q^{76} -31.6891 q^{77} -290.399 q^{79} -452.909 q^{82} +578.005 q^{83} -233.215 q^{86} +78.8931 q^{88} -937.022 q^{89} -101.984 q^{91} +1094.90 q^{92} -584.522 q^{94} -839.629 q^{97} -58.6123 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} - 28 q^{7} + 22 q^{13} + 18 q^{16} - 132 q^{19} + 196 q^{22} - 70 q^{28} - 126 q^{31} - 546 q^{34} + 354 q^{37} + 272 q^{43} - 182 q^{46} + 196 q^{49} - 274 q^{52} + 98 q^{58} - 1170 q^{61}+ \cdots - 1980 q^{97}+O(q^{100}) $ 4 * q + 10 * q^4 - 28 * q^7 + 22 * q^13 + 18 * q^16 - 132 * q^19 + 196 * q^22 - 70 * q^28 - 126 * q^31 - 546 * q^34 + 354 * q^37 + 272 * q^43 - 182 * q^46 + 196 * q^49 - 274 * q^52 + 98 * q^58 - 1170 * q^61 - 1726 * q^64 + 38 * q^67 + 188 * q^73 - 988 * q^76 - 690 * q^79 - 2646 * q^82 + 896 * q^88 - 154 * q^91 - 1540 * q^94 - 1980 * q^97

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 $

$2$	−1.19617	−0.422910	−0.211455	−	0.977388i	$-0.567820\pi$
$2$	−1.19617	−0.422910	−0.211455	+	0.977388i	$0.567820\pi$
$3$	0	0
$3$	0	0
$4$	−6.56918	−0.821147
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	17.4272	0.770181
$9$	0	0
$10$	0	0
$11$	4.52701	0.124086	0.0620430	−	0.998073i	$-0.480238\pi$
$11$	4.52701	0.124086	0.0620430	+	0.998073i	$0.480238\pi$
$12$	0	0
$13$	14.5692	0.310828	0.155414	−	0.987849i	$-0.450329\pi$
$13$	14.5692	0.310828	0.155414	+	0.987849i	$0.450329\pi$
$14$	8.37319	0.159845
$15$	0	0
$16$	31.7075	0.495430
$17$	15.5502	0.221852	0.110926	−	0.993829i	$-0.464618\pi$
$17$	15.5502	0.221852	0.110926	+	0.993829i	$0.464618\pi$
$18$	0	0
$19$	−14.8616	−0.179447	−0.0897235	−	0.995967i	$-0.528598\pi$
$19$	−14.8616	−0.179447	−0.0897235	+	0.995967i	$0.528598\pi$
$20$	0	0
$21$	0	0
$22$	−5.41507	−0.0524771
$23$	−166.672	−1.51102	−0.755511	−	0.655136i	$-0.772611\pi$
$23$	−166.672	−1.51102	−0.755511	+	0.655136i	$0.772611\pi$
$24$	0	0
$25$	0	0
$26$	−17.4272	−0.131452
$27$	0	0
$28$	45.9843	0.310365
$29$	−194.865	−1.24777	−0.623887	−	0.781514i	$-0.714447\pi$
$29$	−194.865	−1.24777	−0.623887	+	0.781514i	$0.714447\pi$
$30$	0	0
$31$	104.538	0.605662	0.302831	−	0.953044i	$-0.402068\pi$
$31$	104.538	0.605662	0.302831	+	0.953044i	$0.402068\pi$
$32$	−177.345	−0.979703
$33$	0	0
$34$	−18.6007	−0.0938232
$35$	0	0
$36$	0	0
$37$	−47.5377	−0.211220	−0.105610	−	0.994408i	$-0.533680\pi$
$37$	−47.5377	−0.211220	−0.105610	+	0.994408i	$0.533680\pi$
$38$	17.7770	0.0758899
$39$	0	0
$40$	0	0
$41$	378.633	1.44226	0.721128	−	0.692802i	$-0.243624\pi$
$41$	378.633	1.44226	0.721128	+	0.692802i	$0.243624\pi$
$42$	0	0
$43$	194.969	0.691452	0.345726	−	0.938336i	$-0.387633\pi$
$43$	194.969	0.691452	0.345726	+	0.938336i	$0.387633\pi$
$44$	−29.7387	−0.101893
$45$	0	0
$46$	199.368	0.639026
$47$	488.661	1.51657	0.758283	−	0.651926i	$-0.226039\pi$
$47$	488.661	1.51657	0.758283	+	0.651926i	$0.226039\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−95.7075	−0.255236
$53$	316.432	0.820099	0.410050	−	0.912063i	$-0.365511\pi$
$53$	316.432	0.820099	0.410050	+	0.912063i	$0.365511\pi$
$54$	0	0
$55$	0	0
$56$	−121.990	−0.291101
$57$	0	0
$58$	233.091	0.527696
$59$	350.256	0.772871	0.386436	−	0.922316i	$-0.373706\pi$
$59$	350.256	0.772871	0.386436	+	0.922316i	$0.373706\pi$
$60$	0	0
$61$	115.613	0.242668	0.121334	−	0.992612i	$-0.461283\pi$
$61$	115.613	0.242668	0.121334	+	0.992612i	$0.461283\pi$
$62$	−125.045	−0.256140
$63$	0	0
$64$	−41.5253	−0.0811041
$65$	0	0
$66$	0	0
$67$	−434.890	−0.792989	−0.396494	−	0.918037i	$-0.629773\pi$
$67$	−434.890	−0.792989	−0.396494	+	0.918037i	$0.629773\pi$
$68$	−102.152	−0.182173
$69$	0	0
$70$	0	0
$71$	−410.119	−0.685523	−0.342761	−	0.939422i	$-0.611362\pi$
$71$	−410.119	−0.685523	−0.342761	+	0.939422i	$0.611362\pi$
$72$	0	0
$73$	464.182	0.744225	0.372112	−	0.928188i	$-0.378634\pi$
$73$	464.182	0.744225	0.372112	+	0.928188i	$0.378634\pi$
$74$	56.8631	0.0893271
$75$	0	0
$76$	97.6288	0.147352
$77$	−31.6891	−0.0469001
$78$	0	0
$79$	−290.399	−0.413576	−0.206788	−	0.978386i	$-0.566301\pi$
$79$	−290.399	−0.413576	−0.206788	+	0.978386i	$0.566301\pi$
$80$	0	0
$81$	0	0
$82$	−452.909	−0.609944
$83$	578.005	0.764390	0.382195	−	0.924082i	$-0.375168\pi$
$83$	578.005	0.764390	0.382195	+	0.924082i	$0.375168\pi$
$84$	0	0
$85$	0	0
$86$	−233.215	−0.292422
$87$	0	0
$88$	78.8931	0.0955686
$89$	−937.022	−1.11600	−0.558001	−	0.829841i	$-0.688431\pi$
$89$	−937.022	−1.11600	−0.558001	+	0.829841i	$0.688431\pi$
$90$	0	0
$91$	−101.984	−0.117482
$92$	1094.90	1.24077
$93$	0	0
$94$	−584.522	−0.641371
$95$	0	0
$96$	0	0
$97$	−839.629	−0.878880	−0.439440	−	0.898272i	$-0.644823\pi$
$97$	−839.629	−0.878880	−0.439440	+	0.898272i	$0.644823\pi$
$98$	−58.6123	−0.0604157
$99$	0	0
$100$	0	0
$101$	1012.69	0.997690	0.498845	−	0.866691i	$-0.333758\pi$
$101$	1012.69	0.997690	0.498845	+	0.866691i	$0.333758\pi$
$102$	0	0
$103$	758.506	0.725610	0.362805	−	0.931865i	$-0.381819\pi$
$103$	758.506	0.725610	0.362805	+	0.931865i	$0.381819\pi$
$104$	253.900	0.239394
$105$	0	0
$106$	−378.506	−0.346828
$107$	264.372	0.238858	0.119429	−	0.992843i	$-0.461894\pi$
$107$	264.372	0.238858	0.119429	+	0.992843i	$0.461894\pi$
$108$	0	0
$109$	−775.538	−0.681496	−0.340748	−	0.940155i	$-0.610680\pi$
$109$	−775.538	−0.681496	−0.340748	+	0.940155i	$0.610680\pi$
$110$	0	0
$111$	0	0
$112$	−221.953	−0.187255
$113$	−437.649	−0.364341	−0.182171	−	0.983267i	$-0.558312\pi$
$113$	−437.649	−0.364341	−0.182171	+	0.983267i	$0.558312\pi$
$114$	0	0
$115$	0	0
$116$	1280.10	1.02461
$117$	0	0
$118$	−418.965	−0.326855
$119$	−108.851	−0.0838520
$120$	0	0
$121$	−1310.51	−0.984603
$122$	−138.293	−0.102627
$123$	0	0
$124$	−686.727	−0.497338
$125$	0	0
$126$	0	0
$127$	−991.412	−0.692705	−0.346353	−	0.938104i	$-0.612580\pi$
$127$	−991.412	−0.692705	−0.346353	+	0.938104i	$0.612580\pi$
$128$	1468.43	1.01400
$129$	0	0
$130$	0	0
$131$	−1529.62	−1.02018	−0.510089	−	0.860122i	$-0.670388\pi$
$131$	−1529.62	−1.02018	−0.510089	+	0.860122i	$0.670388\pi$
$132$	0	0
$133$	104.031	0.0678246
$134$	520.202	0.335363
$135$	0	0
$136$	270.997	0.170866
$137$	220.095	0.137255	0.0686277	−	0.997642i	$-0.478138\pi$
$137$	220.095	0.137255	0.0686277	+	0.997642i	$0.478138\pi$
$138$	0	0
$139$	−2041.16	−1.24553	−0.622767	−	0.782407i	$-0.713991\pi$
$139$	−2041.16	−1.24553	−0.622767	+	0.782407i	$0.713991\pi$
$140$	0	0
$141$	0	0
$142$	490.571	0.289914
$143$	65.9548	0.0385694
$144$	0	0
$145$	0	0
$146$	−555.241	−0.314740
$147$	0	0
$148$	312.283	0.173443
$149$	−591.993	−0.325490	−0.162745	−	0.986668i	$-0.552035\pi$
$149$	−591.993	−0.325490	−0.162745	+	0.986668i	$0.552035\pi$
$150$	0	0
$151$	−1962.54	−1.05768	−0.528839	−	0.848722i	$-0.677373\pi$
$151$	−1962.54	−1.05768	−0.528839	+	0.848722i	$0.677373\pi$
$152$	−258.997	−0.138207
$153$	0	0
$154$	37.9055	0.0198345
$155$	0	0
$156$	0	0
$157$	684.081	0.347743	0.173871	−	0.984768i	$-0.444372\pi$
$157$	684.081	0.347743	0.173871	+	0.984768i	$0.444372\pi$
$158$	347.367	0.174905
$159$	0	0
$160$	0	0
$161$	1166.70	0.571112
$162$	0	0
$163$	1199.68	0.576478	0.288239	−	0.957559i	$-0.406930\pi$
$163$	1199.68	0.576478	0.288239	+	0.957559i	$0.406930\pi$
$164$	−2487.31	−1.18430
$165$	0	0
$166$	−691.393	−0.323268
$167$	2211.33	1.02466	0.512328	−	0.858790i	$-0.328783\pi$
$167$	2211.33	1.02466	0.512328	+	0.858790i	$0.328783\pi$
$168$	0	0
$169$	−1984.74	−0.903386
$170$	0	0
$171$	0	0
$172$	−1280.78	−0.567784
$173$	−1973.01	−0.867082	−0.433541	−	0.901134i	$-0.642736\pi$
$173$	−1973.01	−0.867082	−0.433541	+	0.901134i	$0.642736\pi$
$174$	0	0
$175$	0	0
$176$	143.540	0.0614759
$177$	0	0
$178$	1120.84	0.471968
$179$	30.8784	0.0128936	0.00644681	−	0.999979i	$-0.497948\pi$
$179$	30.8784	0.0128936	0.00644681	+	0.999979i	$0.497948\pi$
$180$	0	0
$181$	−2459.16	−1.00988	−0.504938	−	0.863155i	$-0.668485\pi$
$181$	−2459.16	−1.00988	−0.504938	+	0.863155i	$0.668485\pi$
$182$	121.990	0.0496843
$183$	0	0
$184$	−2904.63	−1.16376
$185$	0	0
$186$	0	0
$187$	70.3959	0.0275287
$188$	−3210.10	−1.24532
$189$	0	0
$190$	0	0
$191$	4163.26	1.57719	0.788595	−	0.614913i	$-0.210809\pi$
$191$	4163.26	1.57719	0.788595	+	0.614913i	$0.210809\pi$
$192$	0	0
$193$	2057.39	0.767326	0.383663	−	0.923473i	$-0.374662\pi$
$193$	2057.39	0.767326	0.383663	+	0.923473i	$0.374662\pi$
$194$	1004.34	0.371687
$195$	0	0
$196$	−321.890	−0.117307
$197$	−2431.48	−0.879368	−0.439684	−	0.898153i	$-0.644910\pi$
$197$	−2431.48	−0.879368	−0.439684	+	0.898153i	$0.644910\pi$
$198$	0	0
$199$	−5291.18	−1.88483	−0.942417	−	0.334441i	$-0.891453\pi$
$199$	−5291.18	−1.88483	−0.942417	+	0.334441i	$0.891453\pi$
$200$	0	0
$201$	0	0
$202$	−1211.35	−0.421933
$203$	1364.05	0.471614
$204$	0	0
$205$	0	0
$206$	−907.302	−0.306868
$207$	0	0
$208$	461.953	0.153994
$209$	−67.2788	−0.0222669
$210$	0	0
$211$	−3586.37	−1.17012	−0.585061	−	0.810989i	$-0.698929\pi$
$211$	−3586.37	−1.17012	−0.585061	+	0.810989i	$0.698929\pi$
$212$	−2078.70	−0.673422
$213$	0	0
$214$	−316.234	−0.101015
$215$	0	0
$216$	0	0
$217$	−731.764	−0.228919
$218$	927.675	0.288211
$219$	0	0
$220$	0	0
$221$	226.554	0.0689577
$222$	0	0
$223$	1127.13	0.338467	0.169234	−	0.985576i	$-0.445871\pi$
$223$	1127.13	0.338467	0.169234	+	0.985576i	$0.445871\pi$
$224$	1241.42	0.370293
$225$	0	0
$226$	523.503	0.154084
$227$	674.839	0.197315	0.0986577	−	0.995121i	$-0.468545\pi$
$227$	674.839	0.197315	0.0986577	+	0.995121i	$0.468545\pi$
$228$	0	0
$229$	−5906.65	−1.70447	−0.852233	−	0.523163i	$-0.824752\pi$
$229$	−5906.65	−1.70447	−0.852233	+	0.523163i	$0.824752\pi$
$230$	0	0
$231$	0	0
$232$	−3395.95	−0.961012
$233$	−305.559	−0.0859136	−0.0429568	−	0.999077i	$-0.513678\pi$
$233$	−305.559	−0.0859136	−0.0429568	+	0.999077i	$0.513678\pi$
$234$	0	0
$235$	0	0
$236$	−2300.89	−0.634641
$237$	0	0
$238$	130.205	0.0354619
$239$	−3268.07	−0.884494	−0.442247	−	0.896893i	$-0.645819\pi$
$239$	−3268.07	−0.884494	−0.442247	+	0.896893i	$0.645819\pi$
$240$	0	0
$241$	477.426	0.127609	0.0638044	−	0.997962i	$-0.479677\pi$
$241$	477.426	0.127609	0.0638044	+	0.997962i	$0.479677\pi$
$242$	1567.59	0.416398
$243$	0	0
$244$	−759.483	−0.199266
$245$	0	0
$246$	0	0
$247$	−216.522	−0.0557772
$248$	1821.80	0.466469
$249$	0	0
$250$	0	0
$251$	−4439.63	−1.11644	−0.558221	−	0.829692i	$-0.688516\pi$
$251$	−4439.63	−1.11644	−0.558221	+	0.829692i	$0.688516\pi$
$252$	0	0
$253$	−754.525	−0.187496
$254$	1185.90	0.292952
$255$	0	0
$256$	−1424.29	−0.347728
$257$	2125.51	0.515897	0.257949	−	0.966159i	$-0.416953\pi$
$257$	2125.51	0.515897	0.257949	+	0.966159i	$0.416953\pi$
$258$	0	0
$259$	332.764	0.0798337
$260$	0	0
$261$	0	0
$262$	1829.68	0.431443
$263$	993.198	0.232864	0.116432	−	0.993199i	$-0.462854\pi$
$263$	993.198	0.232864	0.116432	+	0.993199i	$0.462854\pi$
$264$	0	0
$265$	0	0
$266$	−124.439	−0.0286837
$267$	0	0
$268$	2856.87	0.651161
$269$	6458.23	1.46381	0.731905	−	0.681406i	$-0.238631\pi$
$269$	6458.23	1.46381	0.731905	+	0.681406i	$0.238631\pi$
$270$	0	0
$271$	417.093	0.0934930	0.0467465	−	0.998907i	$-0.485115\pi$
$271$	417.093	0.0934930	0.0467465	+	0.998907i	$0.485115\pi$
$272$	493.059	0.109912
$273$	0	0
$274$	−263.271	−0.0580467
$275$	0	0
$276$	0	0
$277$	−4862.87	−1.05481	−0.527404	−	0.849615i	$-0.676834\pi$
$277$	−4862.87	−1.05481	−0.527404	+	0.849615i	$0.676834\pi$
$278$	2441.58	0.526748
$279$	0	0
$280$	0	0
$281$	−2026.16	−0.430144	−0.215072	−	0.976598i	$-0.568999\pi$
$281$	−2026.16	−0.430144	−0.215072	+	0.976598i	$0.568999\pi$
$282$	0	0
$283$	6445.96	1.35397	0.676983	−	0.735998i	$-0.263287\pi$
$283$	6445.96	1.35397	0.676983	+	0.735998i	$0.263287\pi$
$284$	2694.14	0.562915
$285$	0	0
$286$	−78.8931	−0.0163114
$287$	−2650.43	−0.545122
$288$	0	0
$289$	−4671.19	−0.950782
$290$	0	0
$291$	0	0
$292$	−3049.30	−0.611118
$293$	−541.850	−0.108038	−0.0540191	−	0.998540i	$-0.517203\pi$
$293$	−541.850	−0.108038	−0.0540191	+	0.998540i	$0.517203\pi$
$294$	0	0
$295$	0	0
$296$	−828.449	−0.162678
$297$	0	0
$298$	708.124	0.137653
$299$	−2428.27	−0.469668
$300$	0	0
$301$	−1364.78	−0.261344
$302$	2347.53	0.447303
$303$	0	0
$304$	−471.226	−0.0889035
$305$	0	0
$306$	0	0
$307$	−4494.53	−0.835558	−0.417779	−	0.908549i	$-0.637191\pi$
$307$	−4494.53	−0.835558	−0.417779	+	0.908549i	$0.637191\pi$
$308$	208.171	0.0385119
$309$	0	0
$310$	0	0
$311$	2869.51	0.523199	0.261599	−	0.965177i	$-0.415750\pi$
$311$	2869.51	0.523199	0.261599	+	0.965177i	$0.415750\pi$
$312$	0	0
$313$	−1653.23	−0.298550	−0.149275	−	0.988796i	$-0.547694\pi$
$313$	−1653.23	−0.298550	−0.149275	+	0.988796i	$0.547694\pi$
$314$	−818.277	−0.147064
$315$	0	0
$316$	1907.69	0.339607
$317$	−3953.00	−0.700387	−0.350194	−	0.936677i	$-0.613884\pi$
$317$	−3953.00	−0.700387	−0.350194	+	0.936677i	$0.613884\pi$
$318$	0	0
$319$	−882.154	−0.154831
$320$	0	0
$321$	0	0
$322$	−1395.57	−0.241529
$323$	−231.102	−0.0398106
$324$	0	0
$325$	0	0
$326$	−1435.02	−0.243798
$327$	0	0
$328$	6598.51	1.11080
$329$	−3420.63	−0.573208
$330$	0	0
$331$	2464.07	0.409177	0.204589	−	0.978848i	$-0.434414\pi$
$331$	2464.07	0.409177	0.204589	+	0.978848i	$0.434414\pi$
$332$	−3797.02	−0.627677
$333$	0	0
$334$	−2645.12	−0.433337
$335$	0	0
$336$	0	0
$337$	−10587.0	−1.71131	−0.855656	−	0.517545i	$-0.826846\pi$
$337$	−10587.0	−1.71131	−0.855656	+	0.517545i	$0.826846\pi$
$338$	2374.08	0.382051
$339$	0	0
$340$	0	0
$341$	473.243	0.0751541
$342$	0	0
$343$	−343.000	−0.0539949
$344$	3397.76	0.532543
$345$	0	0
$346$	2360.05	0.366697
$347$	−8188.81	−1.26685	−0.633427	−	0.773803i	$-0.718352\pi$
$347$	−8188.81	−1.26685	−0.633427	+	0.773803i	$0.718352\pi$
$348$	0	0
$349$	2023.68	0.310387	0.155194	−	0.987884i	$-0.450400\pi$
$349$	2023.68	0.310387	0.155194	+	0.987884i	$0.450400\pi$
$350$	0	0
$351$	0	0
$352$	−802.844	−0.121567
$353$	−6824.30	−1.02895	−0.514477	−	0.857504i	$-0.672014\pi$
$353$	−6824.30	−1.02895	−0.514477	+	0.857504i	$0.672014\pi$
$354$	0	0
$355$	0	0
$356$	6155.46	0.916401
$357$	0	0
$358$	−36.9358	−0.00545284
$359$	1034.05	0.152020	0.0760100	−	0.997107i	$-0.475782\pi$
$359$	1034.05	0.152020	0.0760100	+	0.997107i	$0.475782\pi$
$360$	0	0
$361$	−6638.13	−0.967799
$362$	2941.57	0.427087
$363$	0	0
$364$	669.953	0.0964700
$365$	0	0
$366$	0	0
$367$	9573.55	1.36168	0.680838	−	0.732434i	$-0.261616\pi$
$367$	9573.55	1.36168	0.680838	+	0.732434i	$0.261616\pi$
$368$	−5284.75	−0.748606
$369$	0	0
$370$	0	0
$371$	−2215.02	−0.309968
$372$	0	0
$373$	5540.42	0.769094	0.384547	−	0.923105i	$-0.374358\pi$
$373$	5540.42	0.769094	0.384547	+	0.923105i	$0.374358\pi$
$374$	−84.2055	−0.0116421
$375$	0	0
$376$	8516.00	1.16803
$377$	−2839.02	−0.387843
$378$	0	0
$379$	−1601.43	−0.217045	−0.108522	−	0.994094i	$-0.534612\pi$
$379$	−1601.43	−0.217045	−0.108522	+	0.994094i	$0.534612\pi$
$380$	0	0
$381$	0	0
$382$	−4979.97	−0.667009
$383$	−5034.28	−0.671644	−0.335822	−	0.941926i	$-0.609014\pi$
$383$	−5034.28	−0.671644	−0.335822	+	0.941926i	$0.609014\pi$
$384$	0	0
$385$	0	0
$386$	−2460.98	−0.324510
$387$	0	0
$388$	5515.67	0.721690
$389$	4065.87	0.529943	0.264971	−	0.964256i	$-0.414638\pi$
$389$	4065.87	0.529943	0.264971	+	0.964256i	$0.414638\pi$
$390$	0	0
$391$	−2591.78	−0.335223
$392$	853.933	0.110026
$393$	0	0
$394$	2908.46	0.371893
$395$	0	0
$396$	0	0
$397$	−285.834	−0.0361349	−0.0180675	−	0.999837i	$-0.505751\pi$
$397$	−285.834	−0.0361349	−0.0180675	+	0.999837i	$0.505751\pi$
$398$	6329.15	0.797115
$399$	0	0
$400$	0	0
$401$	−12459.4	−1.55160	−0.775800	−	0.630979i	$-0.782654\pi$
$401$	−12459.4	−1.55160	−0.775800	+	0.630979i	$0.782654\pi$
$402$	0	0
$403$	1523.03	0.188257
$404$	−6652.56	−0.819250
$405$	0	0
$406$	−1631.64	−0.199450
$407$	−215.204	−0.0262094
$408$	0	0
$409$	−8756.93	−1.05868	−0.529342	−	0.848408i	$-0.677561\pi$
$409$	−8756.93	−1.05868	−0.529342	+	0.848408i	$0.677561\pi$
$410$	0	0
$411$	0	0
$412$	−4982.76	−0.595833
$413$	−2451.79	−0.292118
$414$	0	0
$415$	0	0
$416$	−2583.77	−0.304519
$417$	0	0
$418$	80.4769	0.00941687
$419$	12419.7	1.44807	0.724033	−	0.689765i	$-0.242286\pi$
$419$	12419.7	1.44807	0.724033	+	0.689765i	$0.242286\pi$
$420$	0	0
$421$	−793.783	−0.0918922	−0.0459461	−	0.998944i	$-0.514630\pi$
$421$	−793.783	−0.0918922	−0.0459461	+	0.998944i	$0.514630\pi$
$422$	4289.90	0.494856
$423$	0	0
$424$	5514.52	0.631625
$425$	0	0
$426$	0	0
$427$	−809.291	−0.0917198
$428$	−1736.71	−0.196138
$429$	0	0
$430$	0	0
$431$	8485.37	0.948320	0.474160	−	0.880439i	$-0.342752\pi$
$431$	8485.37	0.948320	0.474160	+	0.880439i	$0.342752\pi$
$432$	0	0
$433$	−9098.11	−1.00976	−0.504881	−	0.863189i	$-0.668464\pi$
$433$	−9098.11	−1.00976	−0.504881	+	0.863189i	$0.668464\pi$
$434$	875.313	0.0968120
$435$	0	0
$436$	5094.65	0.559608
$437$	2477.02	0.271148
$438$	0	0
$439$	−4232.27	−0.460126	−0.230063	−	0.973176i	$-0.573893\pi$
$439$	−4232.27	−0.460126	−0.230063	+	0.973176i	$0.573893\pi$
$440$	0	0
$441$	0	0
$442$	−270.997	−0.0291629
$443$	−2348.20	−0.251843	−0.125922	−	0.992040i	$-0.540189\pi$
$443$	−2348.20	−0.251843	−0.125922	+	0.992040i	$0.540189\pi$
$444$	0	0
$445$	0	0
$446$	−1348.24	−0.143141
$447$	0	0
$448$	290.677	0.0306545
$449$	−10423.7	−1.09560	−0.547798	−	0.836610i	$-0.684534\pi$
$449$	−10423.7	−1.09560	−0.547798	+	0.836610i	$0.684534\pi$
$450$	0	0
$451$	1714.07	0.178964
$452$	2875.00	0.299178
$453$	0	0
$454$	−807.221	−0.0834467
$455$	0	0
$456$	0	0
$457$	10221.8	1.04630	0.523148	−	0.852242i	$-0.324758\pi$
$457$	10221.8	1.04630	0.523148	+	0.852242i	$0.324758\pi$
$458$	7065.36	0.720835
$459$	0	0
$460$	0	0
$461$	13341.4	1.34788	0.673941	−	0.738786i	$-0.264600\pi$
$461$	13341.4	1.34788	0.673941	+	0.738786i	$0.264600\pi$
$462$	0	0
$463$	−9155.72	−0.919012	−0.459506	−	0.888175i	$-0.651973\pi$
$463$	−9155.72	−0.919012	−0.459506	+	0.888175i	$0.651973\pi$
$464$	−6178.68	−0.618185
$465$	0	0
$466$	365.501	0.0363337
$467$	−13759.6	−1.36342	−0.681710	−	0.731623i	$-0.738763\pi$
$467$	−13759.6	−1.36342	−0.681710	+	0.731623i	$0.738763\pi$
$468$	0	0
$469$	3044.23	0.299722
$470$	0	0
$471$	0	0
$472$	6103.98	0.595251
$473$	882.624	0.0857994
$474$	0	0
$475$	0	0
$476$	715.064	0.0688549
$477$	0	0
$478$	3909.17	0.374061
$479$	−7423.91	−0.708157	−0.354079	−	0.935216i	$-0.615205\pi$
$479$	−7423.91	−0.708157	−0.354079	+	0.935216i	$0.615205\pi$
$480$	0	0
$481$	−692.585	−0.0656531
$482$	−571.083	−0.0539670
$483$	0	0
$484$	8608.95	0.808504
$485$	0	0
$486$	0	0
$487$	7663.90	0.713110	0.356555	−	0.934274i	$-0.383951\pi$
$487$	7663.90	0.713110	0.356555	+	0.934274i	$0.383951\pi$
$488$	2014.81	0.186898
$489$	0	0
$490$	0	0
$491$	−1257.19	−0.115552	−0.0577760	−	0.998330i	$-0.518401\pi$
$491$	−1257.19	−0.115552	−0.0577760	+	0.998330i	$0.518401\pi$
$492$	0	0
$493$	−3030.18	−0.276821
$494$	258.997	0.0235887
$495$	0	0
$496$	3314.63	0.300063
$497$	2870.83	0.259103
$498$	0	0
$499$	402.799	0.0361358	0.0180679	−	0.999837i	$-0.494248\pi$
$499$	402.799	0.0361358	0.0180679	+	0.999837i	$0.494248\pi$
$500$	0	0
$501$	0	0
$502$	5310.55	0.472154
$503$	−15462.0	−1.37061	−0.685304	−	0.728257i	$-0.740331\pi$
$503$	−15462.0	−1.37061	−0.685304	+	0.728257i	$0.740331\pi$
$504$	0	0
$505$	0	0
$506$	902.540	0.0792941
$507$	0	0
$508$	6512.76	0.568813
$509$	12974.1	1.12980	0.564899	−	0.825160i	$-0.308915\pi$
$509$	12974.1	1.12980	0.564899	+	0.825160i	$0.308915\pi$
$510$	0	0
$511$	−3249.28	−0.281291
$512$	−10043.8	−0.866946
$513$	0	0
$514$	−2542.47	−0.218178
$515$	0	0
$516$	0	0
$517$	2212.18	0.188184
$518$	−398.042	−0.0337625
$519$	0	0
$520$	0	0
$521$	−4290.85	−0.360817	−0.180409	−	0.983592i	$-0.557742\pi$
$521$	−4290.85	−0.360817	−0.180409	+	0.983592i	$0.557742\pi$
$522$	0	0
$523$	−23233.7	−1.94252	−0.971261	−	0.238018i	$-0.923502\pi$
$523$	−23233.7	−1.94252	−0.971261	+	0.238018i	$0.923502\pi$
$524$	10048.3	0.837717
$525$	0	0
$526$	−1188.03	−0.0984804
$527$	1625.58	0.134367
$528$	0	0
$529$	15612.5	1.28319
$530$	0	0
$531$	0	0
$532$	−683.402	−0.0556940
$533$	5516.37	0.448293
$534$	0	0
$535$	0	0
$536$	−7578.91	−0.610745
$537$	0	0
$538$	−7725.13	−0.619060
$539$	221.823	0.0177266
$540$	0	0
$541$	−10788.7	−0.857378	−0.428689	−	0.903452i	$-0.641024\pi$
$541$	−10788.7	−0.857378	−0.428689	+	0.903452i	$0.641024\pi$
$542$	−498.914	−0.0395391
$543$	0	0
$544$	−2757.75	−0.217349
$545$	0	0
$546$	0	0
$547$	4512.07	0.352691	0.176346	−	0.984328i	$-0.443572\pi$
$547$	4512.07	0.352691	0.176346	+	0.984328i	$0.443572\pi$
$548$	−1445.84	−0.112707
$549$	0	0
$550$	0	0
$551$	2896.01	0.223909
$552$	0	0
$553$	2032.80	0.156317
$554$	5816.82	0.446088
$555$	0	0
$556$	13408.8	1.02277
$557$	22436.2	1.70674	0.853370	−	0.521306i	$-0.174555\pi$
$557$	22436.2	1.70674	0.853370	+	0.521306i	$0.174555\pi$
$558$	0	0
$559$	2840.53	0.214922
$560$	0	0
$561$	0	0
$562$	2423.63	0.181912
$563$	4773.27	0.357317	0.178658	−	0.983911i	$-0.442824\pi$
$563$	4773.27	0.357317	0.178658	+	0.983911i	$0.442824\pi$
$564$	0	0
$565$	0	0
$566$	−7710.46	−0.572606
$567$	0	0
$568$	−7147.22	−0.527977
$569$	−21212.5	−1.56287	−0.781435	−	0.623987i	$-0.785512\pi$
$569$	−21212.5	−1.56287	−0.781435	+	0.623987i	$0.785512\pi$
$570$	0	0
$571$	−16931.8	−1.24093	−0.620467	−	0.784232i	$-0.713057\pi$
$571$	−16931.8	−1.24093	−0.620467	+	0.784232i	$0.713057\pi$
$572$	−433.269	−0.0316711
$573$	0	0
$574$	3170.36	0.230537
$575$	0	0
$576$	0	0
$577$	21721.5	1.56721	0.783604	−	0.621261i	$-0.213379\pi$
$577$	21721.5	1.56721	0.783604	+	0.621261i	$0.213379\pi$
$578$	5587.54	0.402095
$579$	0	0
$580$	0	0
$581$	−4046.04	−0.288912
$582$	0	0
$583$	1432.49	0.101763
$584$	8089.40	0.573188
$585$	0	0
$586$	648.144	0.0456904
$587$	12620.8	0.887421	0.443710	−	0.896170i	$-0.353662\pi$
$587$	12620.8	0.887421	0.443710	+	0.896170i	$0.353662\pi$
$588$	0	0
$589$	−1553.60	−0.108684
$590$	0	0
$591$	0	0
$592$	−1507.30	−0.104645
$593$	25590.5	1.77214	0.886069	−	0.463554i	$-0.153426\pi$
$593$	25590.5	1.77214	0.886069	+	0.463554i	$0.153426\pi$
$594$	0	0
$595$	0	0
$596$	3888.91	0.267275
$597$	0	0
$598$	2904.63	0.198627
$599$	−8933.79	−0.609390	−0.304695	−	0.952450i	$-0.598555\pi$
$599$	−8933.79	−0.609390	−0.304695	+	0.952450i	$0.598555\pi$
$600$	0	0
$601$	2249.31	0.152665	0.0763323	−	0.997082i	$-0.475679\pi$
$601$	2249.31	0.152665	0.0763323	+	0.997082i	$0.475679\pi$
$602$	1632.51	0.110525
$603$	0	0
$604$	12892.3	0.868510
$605$	0	0
$606$	0	0
$607$	5437.17	0.363572	0.181786	−	0.983338i	$-0.441812\pi$
$607$	5437.17	0.363572	0.181786	+	0.983338i	$0.441812\pi$
$608$	2635.64	0.175805
$609$	0	0
$610$	0	0
$611$	7119.40	0.471391
$612$	0	0
$613$	−7432.72	−0.489731	−0.244865	−	0.969557i	$-0.578744\pi$
$613$	−7432.72	−0.489731	−0.244865	+	0.969557i	$0.578744\pi$
$614$	5376.22	0.353365
$615$	0	0
$616$	−552.252	−0.0361215
$617$	−5304.71	−0.346126	−0.173063	−	0.984911i	$-0.555366\pi$
$617$	−5304.71	−0.346126	−0.173063	+	0.984911i	$0.555366\pi$
$618$	0	0
$619$	16716.9	1.08548	0.542738	−	0.839902i	$-0.317388\pi$
$619$	16716.9	1.08548	0.542738	+	0.839902i	$0.317388\pi$
$620$	0	0
$621$	0	0
$622$	−3432.42	−0.221266
$623$	6559.15	0.421809
$624$	0	0
$625$	0	0
$626$	1977.55	0.126260
$627$	0	0
$628$	−4493.85	−0.285548
$629$	−739.221	−0.0468595
$630$	0	0
$631$	−7013.44	−0.442473	−0.221237	−	0.975220i	$-0.571009\pi$
$631$	−7013.44	−0.442473	−0.221237	+	0.975220i	$0.571009\pi$
$632$	−5060.85	−0.318528
$633$	0	0
$634$	4728.46	0.296201
$635$	0	0
$636$	0	0
$637$	713.890	0.0444040
$638$	1055.21	0.0654796
$639$	0	0
$640$	0	0
$641$	−9981.90	−0.615072	−0.307536	−	0.951536i	$-0.599504\pi$
$641$	−9981.90	−0.615072	−0.307536	+	0.951536i	$0.599504\pi$
$642$	0	0
$643$	−24635.5	−1.51093	−0.755465	−	0.655188i	$-0.772589\pi$
$643$	−24635.5	−1.51093	−0.755465	+	0.655188i	$0.772589\pi$
$644$	−7664.28	−0.468967
$645$	0	0
$646$	276.437	0.0168363
$647$	18252.3	1.10908	0.554538	−	0.832158i	$-0.312895\pi$
$647$	18252.3	1.10908	0.554538	+	0.832158i	$0.312895\pi$
$648$	0	0
$649$	1585.61	0.0959024
$650$	0	0
$651$	0	0
$652$	−7880.89	−0.473373
$653$	11455.4	0.686501	0.343251	−	0.939244i	$-0.388472\pi$
$653$	11455.4	0.686501	0.343251	+	0.939244i	$0.388472\pi$
$654$	0	0
$655$	0	0
$656$	12005.5	0.714537
$657$	0	0
$658$	4091.65	0.242415
$659$	4647.50	0.274721	0.137360	−	0.990521i	$-0.456138\pi$
$659$	4647.50	0.274721	0.137360	+	0.990521i	$0.456138\pi$
$660$	0	0
$661$	−11966.6	−0.704158	−0.352079	−	0.935970i	$-0.614525\pi$
$661$	−11966.6	−0.704158	−0.352079	+	0.935970i	$0.614525\pi$
$662$	−2947.45	−0.173045
$663$	0	0
$664$	10073.0	0.588718
$665$	0	0
$666$	0	0
$667$	32478.5	1.88541
$668$	−14526.6	−0.841393
$669$	0	0
$670$	0	0
$671$	523.381	0.0301116
$672$	0	0
$673$	−17972.2	−1.02939	−0.514693	−	0.857374i	$-0.672094\pi$
$673$	−17972.2	−1.02939	−0.514693	+	0.857374i	$0.672094\pi$
$674$	12663.9	0.723731
$675$	0	0
$676$	13038.1	0.741813
$677$	−16225.6	−0.921125	−0.460562	−	0.887627i	$-0.652352\pi$
$677$	−16225.6	−0.921125	−0.460562	+	0.887627i	$0.652352\pi$
$678$	0	0
$679$	5877.40	0.332186
$680$	0	0
$681$	0	0
$682$	−566.079	−0.0317834
$683$	−32680.3	−1.83086	−0.915430	−	0.402478i	$-0.868149\pi$
$683$	−32680.3	−1.83086	−0.915430	+	0.402478i	$0.868149\pi$
$684$	0	0
$685$	0	0
$686$	410.286	0.0228350
$687$	0	0
$688$	6181.97	0.342566
$689$	4610.15	0.254910
$690$	0	0
$691$	30421.0	1.67477	0.837386	−	0.546611i	$-0.184083\pi$
$691$	30421.0	1.67477	0.837386	+	0.546611i	$0.184083\pi$
$692$	12961.1	0.712002
$693$	0	0
$694$	9795.20	0.535765
$695$	0	0
$696$	0	0
$697$	5887.82	0.319967
$698$	−2420.66	−0.131266
$699$	0	0
$700$	0	0
$701$	24865.9	1.33976	0.669881	−	0.742469i	$-0.266345\pi$
$701$	24865.9	1.33976	0.669881	+	0.742469i	$0.266345\pi$
$702$	0	0
$703$	706.488	0.0379028
$704$	−187.986	−0.0100639
$705$	0	0
$706$	8163.02	0.435155
$707$	−7088.85	−0.377091
$708$	0	0
$709$	22833.5	1.20949	0.604746	−	0.796418i	$-0.293275\pi$
$709$	22833.5	1.20949	0.604746	+	0.796418i	$0.293275\pi$
$710$	0	0
$711$	0	0
$712$	−16329.7	−0.859523
$713$	−17423.5	−0.915168
$714$	0	0
$715$	0	0
$716$	−202.846	−0.0105876
$717$	0	0
$718$	−1236.90	−0.0642907
$719$	−32998.1	−1.71157	−0.855786	−	0.517329i	$-0.826926\pi$
$719$	−32998.1	−1.71157	−0.855786	+	0.517329i	$0.826926\pi$
$720$	0	0
$721$	−5309.54	−0.274255
$722$	7940.33	0.409292
$723$	0	0
$724$	16154.6	0.829257
$725$	0	0
$726$	0	0
$727$	13769.6	0.702455	0.351228	−	0.936290i	$-0.385764\pi$
$727$	13769.6	0.702455	0.351228	+	0.936290i	$0.385764\pi$
$728$	−1777.30	−0.0904823
$729$	0	0
$730$	0	0
$731$	3031.80	0.153400
$732$	0	0
$733$	−33872.8	−1.70685	−0.853425	−	0.521216i	$-0.825479\pi$
$733$	−33872.8	−1.70685	−0.853425	+	0.521216i	$0.825479\pi$
$734$	−11451.6	−0.575866
$735$	0	0
$736$	29558.5	1.48035
$737$	−1968.75	−0.0983987
$738$	0	0
$739$	19929.6	0.992048	0.496024	−	0.868309i	$-0.334793\pi$
$739$	19929.6	0.992048	0.496024	+	0.868309i	$0.334793\pi$
$740$	0	0
$741$	0	0
$742$	2649.54	0.131089
$743$	22285.3	1.10036	0.550182	−	0.835045i	$-0.314558\pi$
$743$	22285.3	1.10036	0.550182	+	0.835045i	$0.314558\pi$
$744$	0	0
$745$	0	0
$746$	−6627.28	−0.325257
$747$	0	0
$748$	−462.443	−0.0226051
$749$	−1850.61	−0.0902799
$750$	0	0
$751$	−9443.37	−0.458846	−0.229423	−	0.973327i	$-0.573684\pi$
$751$	−9443.37	−0.458846	−0.229423	+	0.973327i	$0.573684\pi$
$752$	15494.3	0.751353
$753$	0	0
$754$	3395.95	0.164023
$755$	0	0
$756$	0	0
$757$	9430.33	0.452776	0.226388	−	0.974037i	$-0.427308\pi$
$757$	9430.33	0.452776	0.226388	+	0.974037i	$0.427308\pi$
$758$	1915.58	0.0917903
$759$	0	0
$760$	0	0
$761$	24810.4	1.18184	0.590918	−	0.806732i	$-0.298766\pi$
$761$	24810.4	1.18184	0.590918	+	0.806732i	$0.298766\pi$
$762$	0	0
$763$	5428.76	0.257581
$764$	−27349.2	−1.29510
$765$	0	0
$766$	6021.85	0.284045
$767$	5102.94	0.240230
$768$	0	0
$769$	4174.16	0.195740	0.0978699	−	0.995199i	$-0.468797\pi$
$769$	4174.16	0.195740	0.0978699	+	0.995199i	$0.468797\pi$
$770$	0	0
$771$	0	0
$772$	−13515.3	−0.630088
$773$	−25457.8	−1.18455	−0.592274	−	0.805737i	$-0.701770\pi$
$773$	−25457.8	−1.18455	−0.592274	+	0.805737i	$0.701770\pi$
$774$	0	0
$775$	0	0
$776$	−14632.4	−0.676897
$777$	0	0
$778$	−4863.47	−0.224118
$779$	−5627.10	−0.258809
$780$	0	0
$781$	−1856.61	−0.0850637
$782$	3100.21	0.141769
$783$	0	0
$784$	1553.67	0.0707757
$785$	0	0
$786$	0	0
$787$	5696.75	0.258027	0.129013	−	0.991643i	$-0.458819\pi$
$787$	5696.75	0.258027	0.129013	+	0.991643i	$0.458819\pi$
$788$	15972.8	0.722091
$789$	0	0
$790$	0	0
$791$	3063.55	0.137708
$792$	0	0
$793$	1684.39	0.0754279
$794$	341.905	0.0152818
$795$	0	0
$796$	34758.7	1.54773
$797$	13044.2	0.579736	0.289868	−	0.957067i	$-0.406389\pi$
$797$	13044.2	0.579736	0.289868	+	0.957067i	$0.406389\pi$
$798$	0	0
$799$	7598.79	0.336453
$800$	0	0
$801$	0	0
$802$	14903.5	0.656187
$803$	2101.36	0.0923478
$804$	0	0
$805$	0	0
$806$	−1821.80	−0.0796156
$807$	0	0
$808$	17648.4	0.768402
$809$	20548.8	0.893026	0.446513	−	0.894777i	$-0.352666\pi$
$809$	20548.8	0.893026	0.446513	+	0.894777i	$0.352666\pi$
$810$	0	0
$811$	−1266.15	−0.0548220	−0.0274110	−	0.999624i	$-0.508726\pi$
$811$	−1266.15	−0.0548220	−0.0274110	+	0.999624i	$0.508726\pi$
$812$	−8960.70	−0.387265
$813$	0	0
$814$	257.420	0.0110842
$815$	0	0
$816$	0	0
$817$	−2897.55	−0.124079
$818$	10474.8	0.447728
$819$	0	0
$820$	0	0
$821$	−29978.9	−1.27439	−0.637193	−	0.770704i	$-0.719905\pi$
$821$	−29978.9	−1.27439	−0.637193	+	0.770704i	$0.719905\pi$
$822$	0	0
$823$	19172.5	0.812043	0.406021	−	0.913864i	$-0.366916\pi$
$823$	19172.5	0.812043	0.406021	+	0.913864i	$0.366916\pi$
$824$	13218.6	0.558851
$825$	0	0
$826$	2932.76	0.123539
$827$	−13488.9	−0.567175	−0.283588	−	0.958946i	$-0.591525\pi$
$827$	−13488.9	−0.567175	−0.283588	+	0.958946i	$0.591525\pi$
$828$	0	0
$829$	16585.9	0.694876	0.347438	−	0.937703i	$-0.387052\pi$
$829$	16585.9	0.694876	0.347438	+	0.937703i	$0.387052\pi$
$830$	0	0
$831$	0	0
$832$	−604.990	−0.0252094
$833$	761.960	0.0316931
$834$	0	0
$835$	0	0
$836$	441.967	0.0182844
$837$	0	0
$838$	−14856.0	−0.612401
$839$	37812.6	1.55594	0.777971	−	0.628301i	$-0.216249\pi$
$839$	37812.6	1.55594	0.777971	+	0.628301i	$0.216249\pi$
$840$	0	0
$841$	13583.2	0.556940
$842$	949.499	0.0388621
$843$	0	0
$844$	23559.5	0.960842
$845$	0	0
$846$	0	0
$847$	9173.54	0.372145
$848$	10033.3	0.406302
$849$	0	0
$850$	0	0
$851$	7923.19	0.319158
$852$	0	0
$853$	5118.65	0.205462	0.102731	−	0.994709i	$-0.467242\pi$
$853$	5118.65	0.205462	0.102731	+	0.994709i	$0.467242\pi$
$854$	968.049	0.0387892
$855$	0	0
$856$	4607.27	0.183964
$857$	−34615.7	−1.37975	−0.689877	−	0.723927i	$-0.742335\pi$
$857$	−34615.7	−1.37975	−0.689877	+	0.723927i	$0.742335\pi$
$858$	0	0
$859$	−24854.2	−0.987210	−0.493605	−	0.869686i	$-0.664321\pi$
$859$	−24854.2	−0.987210	−0.493605	+	0.869686i	$0.664321\pi$
$860$	0	0
$861$	0	0
$862$	−10149.9	−0.401054
$863$	9637.52	0.380145	0.190072	−	0.981770i	$-0.439128\pi$
$863$	9637.52	0.380145	0.190072	+	0.981770i	$0.439128\pi$
$864$	0	0
$865$	0	0
$866$	10882.9	0.427038
$867$	0	0
$868$	4807.09	0.187976
$869$	−1314.64	−0.0513189
$870$	0	0
$871$	−6335.99	−0.246483
$872$	−13515.5	−0.524875
$873$	0	0
$874$	−2962.93	−0.114671
$875$	0	0
$876$	0	0
$877$	13503.3	0.519924	0.259962	−	0.965619i	$-0.416290\pi$
$877$	13503.3	0.519924	0.259962	+	0.965619i	$0.416290\pi$
$878$	5062.52	0.194592
$879$	0	0
$880$	0	0
$881$	43808.4	1.67530	0.837652	−	0.546205i	$-0.183928\pi$
$881$	43808.4	1.67530	0.837652	+	0.546205i	$0.183928\pi$
$882$	0	0
$883$	−24875.4	−0.948044	−0.474022	−	0.880513i	$-0.657198\pi$
$883$	−24875.4	−0.948044	−0.474022	+	0.880513i	$0.657198\pi$
$884$	−1488.27	−0.0566244
$885$	0	0
$886$	2808.85	0.106507
$887$	39120.3	1.48087	0.740435	−	0.672128i	$-0.234619\pi$
$887$	39120.3	1.48087	0.740435	+	0.672128i	$0.234619\pi$
$888$	0	0
$889$	6939.88	0.261818
$890$	0	0
$891$	0	0
$892$	−7404.31	−0.277931
$893$	−7262.31	−0.272143
$894$	0	0
$895$	0	0
$896$	−10279.0	−0.383257
$897$	0	0
$898$	12468.5	0.463338
$899$	−20370.7	−0.755729
$900$	0	0
$901$	4920.58	0.181940
$902$	−2050.32	−0.0756855
$903$	0	0
$904$	−7627.01	−0.280609
$905$	0	0
$906$	0	0
$907$	19658.5	0.719681	0.359840	−	0.933014i	$-0.382831\pi$
$907$	19658.5	0.719681	0.359840	+	0.933014i	$0.382831\pi$
$908$	−4433.14	−0.162025
$909$	0	0
$910$	0	0
$911$	−39577.9	−1.43938	−0.719690	−	0.694295i	$-0.755716\pi$
$911$	−39577.9	−1.43938	−0.719690	+	0.694295i	$0.755716\pi$
$912$	0	0
$913$	2616.64	0.0948500
$914$	−12227.0	−0.442489
$915$	0	0
$916$	38801.9	1.39962
$917$	10707.3	0.385591
$918$	0	0
$919$	24855.9	0.892188	0.446094	−	0.894986i	$-0.352815\pi$
$919$	24855.9	0.892188	0.446094	+	0.894986i	$0.352815\pi$
$920$	0	0
$921$	0	0
$922$	−15958.6	−0.570032
$923$	−5975.09	−0.213080
$924$	0	0
$925$	0	0
$926$	10951.8	0.388659
$927$	0	0
$928$	34558.3	1.22245
$929$	−4466.02	−0.157724	−0.0788619	−	0.996886i	$-0.525129\pi$
$929$	−4466.02	−0.157724	−0.0788619	+	0.996886i	$0.525129\pi$
$930$	0	0
$931$	−728.220	−0.0256353
$932$	2007.27	0.0705477
$933$	0	0
$934$	16458.8	0.576604
$935$	0	0
$936$	0	0
$937$	3768.86	0.131402	0.0657009	−	0.997839i	$-0.479072\pi$
$937$	3768.86	0.131402	0.0657009	+	0.997839i	$0.479072\pi$
$938$	−3641.41	−0.126755
$939$	0	0
$940$	0	0
$941$	−15423.4	−0.534312	−0.267156	−	0.963653i	$-0.586084\pi$
$941$	−15423.4	−0.534312	−0.267156	+	0.963653i	$0.586084\pi$
$942$	0	0
$943$	−63107.4	−2.17928
$944$	11105.7	0.382904
$945$	0	0
$946$	−1055.77	−0.0362854
$947$	−11755.0	−0.403365	−0.201682	−	0.979451i	$-0.564641\pi$
$947$	−11755.0	−0.403365	−0.201682	+	0.979451i	$0.564641\pi$
$948$	0	0
$949$	6762.75	0.231326
$950$	0	0
$951$	0	0
$952$	−1896.98	−0.0645813
$953$	−7321.10	−0.248850	−0.124425	−	0.992229i	$-0.539709\pi$
$953$	−7321.10	−0.248850	−0.124425	+	0.992229i	$0.539709\pi$
$954$	0	0
$955$	0	0
$956$	21468.6	0.726300
$957$	0	0
$958$	8880.26	0.299487
$959$	−1540.67	−0.0518777
$960$	0	0
$961$	−18862.9	−0.633174
$962$	828.449	0.0277653
$963$	0	0
$964$	−3136.30	−0.104786
$965$	0	0
$966$	0	0
$967$	21155.8	0.703542	0.351771	−	0.936086i	$-0.385580\pi$
$967$	21155.8	0.703542	0.351771	+	0.936086i	$0.385580\pi$
$968$	−22838.5	−0.758322
$969$	0	0
$970$	0	0
$971$	−59824.6	−1.97720	−0.988600	−	0.150566i	$-0.951891\pi$
$971$	−59824.6	−1.97720	−0.988600	+	0.150566i	$0.951891\pi$
$972$	0	0
$973$	14288.1	0.470768
$974$	−9167.33	−0.301581
$975$	0	0
$976$	3665.80	0.120225
$977$	19245.9	0.630227	0.315113	−	0.949054i	$-0.397957\pi$
$977$	19245.9	0.630227	0.315113	+	0.949054i	$0.397957\pi$
$978$	0	0
$979$	−4241.91	−0.138480
$980$	0	0
$981$	0	0
$982$	1503.81	0.0488681
$983$	52661.2	1.70868	0.854339	−	0.519716i	$-0.173962\pi$
$983$	52661.2	1.70868	0.854339	+	0.519716i	$0.173962\pi$
$984$	0	0
$985$	0	0
$986$	3624.61	0.117070
$987$	0	0
$988$	1422.37	0.0458013
$989$	−32495.8	−1.04480
$990$	0	0
$991$	−39065.5	−1.25223	−0.626114	−	0.779732i	$-0.715356\pi$
$991$	−39065.5	−1.25223	−0.626114	+	0.779732i	$0.715356\pi$
$992$	−18539.3	−0.593369
$993$	0	0
$994$	−3434.00	−0.109577
$995$	0	0
$996$	0	0
$997$	−31074.5	−0.987099	−0.493550	−	0.869718i	$-0.664301\pi$
$997$	−31074.5	−0.987099	−0.493550	+	0.869718i	$0.664301\pi$
$998$	−481.816	−0.0152822
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bh.1.2	✓	4
3.2	odd	2	inner	1575.4.a.bh.1.3	yes	4
5.4	even	2		1575.4.a.bi.1.3	yes	4
15.14	odd	2		1575.4.a.bi.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.4.a.bh.1.2	✓	4	1.1	even	1	trivial
1575.4.a.bh.1.3	yes	4	3.2	odd	2	inner
1575.4.a.bi.1.2	yes	4	15.14	odd	2
1575.4.a.bi.1.3	yes	4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.19617 q^{2} -6.56918 q^{4} -7.00000 q^{7} +17.4272 q^{8} +4.52701 q^{11} +14.5692 q^{13} +8.37319 q^{14} +31.7075 q^{16} +15.5502 q^{17} -14.8616 q^{19} -5.41507 q^{22} -166.672 q^{23} -17.4272 q^{26} +45.9843 q^{28} -194.865 q^{29} +104.538 q^{31} -177.345 q^{32} -18.6007 q^{34} -47.5377 q^{37} +17.7770 q^{38} +378.633 q^{41} +194.969 q^{43} -29.7387 q^{44} +199.368 q^{46} +488.661 q^{47} +49.0000 q^{49} -95.7075 q^{52} +316.432 q^{53} -121.990 q^{56} +233.091 q^{58} +350.256 q^{59} +115.613 q^{61} -125.045 q^{62} -41.5253 q^{64} -434.890 q^{67} -102.152 q^{68} -410.119 q^{71} +464.182 q^{73} +56.8631 q^{74} +97.6288 q^{76} -31.6891 q^{77} -290.399 q^{79} -452.909 q^{82} +578.005 q^{83} -233.215 q^{86} +78.8931 q^{88} -937.022 q^{89} -101.984 q^{91} +1094.90 q^{92} -584.522 q^{94} -839.629 q^{97} -58.6123 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} - 28 q^{7} + 22 q^{13} + 18 q^{16} - 132 q^{19} + 196 q^{22} - 70 q^{28} - 126 q^{31} - 546 q^{34} + 354 q^{37} + 272 q^{43} - 182 q^{46} + 196 q^{49} - 274 q^{52} + 98 q^{58} - 1170 q^{61}+ \cdots - 1980 q^{97}+O(q^{100}) \) 4 * q + 10 * q^4 - 28 * q^7 + 22 * q^13 + 18 * q^16 - 132 * q^19 + 196 * q^22 - 70 * q^28 - 126 * q^31 - 546 * q^34 + 354 * q^37 + 272 * q^43 - 182 * q^46 + 196 * q^49 - 274 * q^52 + 98 * q^58 - 1170 * q^61 - 1726 * q^64 + 38 * q^67 + 188 * q^73 - 988 * q^76 - 690 * q^79 - 2646 * q^82 + 896 * q^88 - 154 * q^91 - 1540 * q^94 - 1980 * q^97

Introduction

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