LMFDB - Embedded newform 1575.4.a.bf.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,-6,0,16,0,0,-28,-93,0,0,-57] 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:	$0$
Dimension:	$4$
Coefficient field:	4.4.26729725.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 18x^{2} + 19x + 44 $ x^4 - 2x^3 - 18x^2 + 19*x + 44
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.21734$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.21734 q^{2} +2.35129 q^{4} -7.00000 q^{7} +18.1738 q^{8} +2.09810 q^{11} +80.8215 q^{13} +22.5214 q^{14} -77.2818 q^{16} -101.965 q^{17} +143.319 q^{19} -6.75032 q^{22} +116.258 q^{23} -260.030 q^{26} -16.4590 q^{28} -181.194 q^{29} +303.614 q^{31} +103.251 q^{32} +328.058 q^{34} +158.336 q^{37} -461.106 q^{38} +379.372 q^{41} -238.980 q^{43} +4.93326 q^{44} -374.041 q^{46} -125.956 q^{47} +49.0000 q^{49} +190.035 q^{52} +43.4805 q^{53} -127.217 q^{56} +582.964 q^{58} -31.0944 q^{59} -812.675 q^{61} -976.830 q^{62} +286.059 q^{64} -426.225 q^{67} -239.750 q^{68} +1034.51 q^{71} -471.741 q^{73} -509.422 q^{74} +336.984 q^{76} -14.6867 q^{77} +1201.92 q^{79} -1220.57 q^{82} -1325.72 q^{83} +768.880 q^{86} +38.1306 q^{88} +886.226 q^{89} -565.751 q^{91} +273.356 q^{92} +405.245 q^{94} -134.908 q^{97} -157.650 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} + 16 q^{4} - 28 q^{7} - 93 q^{8} - 57 q^{11} + 43 q^{13} + 42 q^{14} + 216 q^{16} - 99 q^{17} - 12 q^{19} - 41 q^{22} - 156 q^{23} + 81 q^{26} - 112 q^{28} - 378 q^{29} - 93 q^{31} - 690 q^{32}+ \cdots - 294 q^{98}+O(q^{100}) $ 4 * q - 6 * q^2 + 16 * q^4 - 28 * q^7 - 93 * q^8 - 57 * q^11 + 43 * q^13 + 42 * q^14 + 216 * q^16 - 99 * q^17 - 12 * q^19 - 41 * q^22 - 156 * q^23 + 81 * q^26 - 112 * q^28 - 378 * q^29 - 93 * q^31 - 690 * q^32 + 783 * q^34 + 81 * q^37 - 216 * q^38 + 465 * q^41 - 64 * q^43 - 681 * q^44 + 310 * q^46 - 744 * q^47 + 196 * q^49 - 727 * q^52 - 729 * q^53 + 651 * q^56 + 1172 * q^58 - 231 * q^59 - 1353 * q^61 + 165 * q^62 + 3107 * q^64 + 1487 * q^67 - 2577 * q^68 + 1725 * q^71 + 512 * q^73 + 1953 * q^74 - 3046 * q^76 + 399 * q^77 + 1629 * q^79 - 693 * q^82 - 321 * q^83 + 4542 * q^86 + 3482 * q^88 + 978 * q^89 - 301 * q^91 - 852 * q^92 + 2480 * q^94 + 2616 * q^97 - 294 * 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 $

$2$	−3.21734	−1.13750	−0.568751	−	0.822510i	$-0.692573\pi$
$2$	−3.21734	−1.13750	−0.568751	+	0.822510i	$0.692573\pi$
$3$	0	0
$3$	0	0
$4$	2.35129	0.293911
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	18.1738	0.803177
$9$	0	0
$10$	0	0
$11$	2.09810	0.0575093	0.0287546	−	0.999586i	$-0.490846\pi$
$11$	2.09810	0.0575093	0.0287546	+	0.999586i	$0.490846\pi$
$12$	0	0
$13$	80.8215	1.72430	0.862148	−	0.506656i	$-0.169119\pi$
$13$	80.8215	1.72430	0.862148	+	0.506656i	$0.169119\pi$
$14$	22.5214	0.429935
$15$	0	0
$16$	−77.2818	−1.20753
$17$	−101.965	−1.45472	−0.727360	−	0.686256i	$-0.759253\pi$
$17$	−101.965	−1.45472	−0.727360	+	0.686256i	$0.759253\pi$
$18$	0	0
$19$	143.319	1.73050	0.865252	−	0.501337i	$-0.167158\pi$
$19$	143.319	1.73050	0.865252	+	0.501337i	$0.167158\pi$
$20$	0	0
$21$	0	0
$22$	−6.75032	−0.0654170
$23$	116.258	1.05398	0.526988	−	0.849873i	$-0.323321\pi$
$23$	116.258	1.05398	0.526988	+	0.849873i	$0.323321\pi$
$24$	0	0
$25$	0	0
$26$	−260.030	−1.96139
$27$	0	0
$28$	−16.4590	−0.111088
$29$	−181.194	−1.16024	−0.580119	−	0.814532i	$-0.696994\pi$
$29$	−181.194	−1.16024	−0.580119	+	0.814532i	$0.696994\pi$
$30$	0	0
$31$	303.614	1.75905	0.879527	−	0.475850i	$-0.157859\pi$
$31$	303.614	1.75905	0.879527	+	0.475850i	$0.157859\pi$
$32$	103.251	0.570388
$33$	0	0
$34$	328.058	1.65475
$35$	0	0
$36$	0	0
$37$	158.336	0.703522	0.351761	−	0.936090i	$-0.385583\pi$
$37$	158.336	0.703522	0.351761	+	0.936090i	$0.385583\pi$
$38$	−461.106	−1.96845
$39$	0	0
$40$	0	0
$41$	379.372	1.44507	0.722536	−	0.691333i	$-0.242976\pi$
$41$	379.372	1.44507	0.722536	+	0.691333i	$0.242976\pi$
$42$	0	0
$43$	−238.980	−0.847537	−0.423769	−	0.905770i	$-0.639293\pi$
$43$	−238.980	−0.847537	−0.423769	+	0.905770i	$0.639293\pi$
$44$	4.93326	0.0169026
$45$	0	0
$46$	−374.041	−1.19890
$47$	−125.956	−0.390907	−0.195453	−	0.980713i	$-0.562618\pi$
$47$	−125.956	−0.390907	−0.195453	+	0.980713i	$0.562618\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	190.035	0.506790
$53$	43.4805	0.112689	0.0563443	−	0.998411i	$-0.482056\pi$
$53$	43.4805	0.112689	0.0563443	+	0.998411i	$0.482056\pi$
$54$	0	0
$55$	0	0
$56$	−127.217	−0.303572
$57$	0	0
$58$	582.964	1.31977
$59$	−31.0944	−0.0686127	−0.0343063	−	0.999411i	$-0.510922\pi$
$59$	−31.0944	−0.0686127	−0.0343063	+	0.999411i	$0.510922\pi$
$60$	0	0
$61$	−812.675	−1.70578	−0.852889	−	0.522093i	$-0.825151\pi$
$61$	−812.675	−1.70578	−0.852889	+	0.522093i	$0.825151\pi$
$62$	−976.830	−2.00093
$63$	0	0
$64$	286.059	0.558710
$65$	0	0
$66$	0	0
$67$	−426.225	−0.777189	−0.388595	−	0.921409i	$-0.627039\pi$
$67$	−426.225	−0.777189	−0.388595	+	0.921409i	$0.627039\pi$
$68$	−239.750	−0.427559
$69$	0	0
$70$	0	0
$71$	1034.51	1.72921	0.864603	−	0.502455i	$-0.167570\pi$
$71$	1034.51	1.72921	0.864603	+	0.502455i	$0.167570\pi$
$72$	0	0
$73$	−471.741	−0.756344	−0.378172	−	0.925735i	$-0.623447\pi$
$73$	−471.741	−0.756344	−0.378172	+	0.925735i	$0.623447\pi$
$74$	−509.422	−0.800257
$75$	0	0
$76$	336.984	0.508615
$77$	−14.6867	−0.0217365
$78$	0	0
$79$	1201.92	1.71172	0.855861	−	0.517205i	$-0.173028\pi$
$79$	1201.92	1.71172	0.855861	+	0.517205i	$0.173028\pi$
$80$	0	0
$81$	0	0
$82$	−1220.57	−1.64377
$83$	−1325.72	−1.75321	−0.876606	−	0.481209i	$-0.840198\pi$
$83$	−1325.72	−1.75321	−0.876606	+	0.481209i	$0.840198\pi$
$84$	0	0
$85$	0	0
$86$	768.880	0.964076
$87$	0	0
$88$	38.1306	0.0461902
$89$	886.226	1.05550	0.527751	−	0.849399i	$-0.323035\pi$
$89$	886.226	1.05550	0.527751	+	0.849399i	$0.323035\pi$
$90$	0	0
$91$	−565.751	−0.651723
$92$	273.356	0.309776
$93$	0	0
$94$	405.245	0.444657
$95$	0	0
$96$	0	0
$97$	−134.908	−0.141215	−0.0706076	−	0.997504i	$-0.522494\pi$
$97$	−134.908	−0.141215	−0.0706076	+	0.997504i	$0.522494\pi$
$98$	−157.650	−0.162500
$99$	0	0
$100$	0	0
$101$	−585.617	−0.576941	−0.288471	−	0.957489i	$-0.593147\pi$
$101$	−585.617	−0.576941	−0.288471	+	0.957489i	$0.593147\pi$
$102$	0	0
$103$	850.822	0.813923	0.406961	−	0.913445i	$-0.366588\pi$
$103$	850.822	0.813923	0.406961	+	0.913445i	$0.366588\pi$
$104$	1468.84	1.38492
$105$	0	0
$106$	−139.892	−0.128184
$107$	−566.799	−0.512099	−0.256049	−	0.966664i	$-0.582421\pi$
$107$	−566.799	−0.512099	−0.256049	+	0.966664i	$0.582421\pi$
$108$	0	0
$109$	−1111.49	−0.976714	−0.488357	−	0.872644i	$-0.662404\pi$
$109$	−1111.49	−0.976714	−0.488357	+	0.872644i	$0.662404\pi$
$110$	0	0
$111$	0	0
$112$	540.972	0.456403
$113$	−192.740	−0.160455	−0.0802275	−	0.996777i	$-0.525565\pi$
$113$	−192.740	−0.160455	−0.0802275	+	0.996777i	$0.525565\pi$
$114$	0	0
$115$	0	0
$116$	−426.040	−0.341007
$117$	0	0
$118$	100.041	0.0780471
$119$	713.758	0.549833
$120$	0	0
$121$	−1326.60	−0.996693
$122$	2614.65	1.94033
$123$	0	0
$124$	713.885	0.517006
$125$	0	0
$126$	0	0
$127$	937.870	0.655296	0.327648	−	0.944800i	$-0.393744\pi$
$127$	937.870	0.655296	0.327648	+	0.944800i	$0.393744\pi$
$128$	−1746.36	−1.20592
$129$	0	0
$130$	0	0
$131$	−746.316	−0.497755	−0.248878	−	0.968535i	$-0.580062\pi$
$131$	−746.316	−0.497755	−0.248878	+	0.968535i	$0.580062\pi$
$132$	0	0
$133$	−1003.23	−0.654069
$134$	1371.31	0.884054
$135$	0	0
$136$	−1853.10	−1.16840
$137$	−492.514	−0.307141	−0.153570	−	0.988138i	$-0.549077\pi$
$137$	−492.514	−0.307141	−0.153570	+	0.988138i	$0.549077\pi$
$138$	0	0
$139$	−152.482	−0.0930456	−0.0465228	−	0.998917i	$-0.514814\pi$
$139$	−152.482	−0.0930456	−0.0465228	+	0.998917i	$0.514814\pi$
$140$	0	0
$141$	0	0
$142$	−3328.37	−1.96698
$143$	169.572	0.0991631
$144$	0	0
$145$	0	0
$146$	1517.75	0.860343
$147$	0	0
$148$	372.294	0.206773
$149$	1433.18	0.787989	0.393994	−	0.919113i	$-0.371093\pi$
$149$	1433.18	0.787989	0.393994	+	0.919113i	$0.371093\pi$
$150$	0	0
$151$	916.243	0.493794	0.246897	−	0.969042i	$-0.420589\pi$
$151$	916.243	0.493794	0.246897	+	0.969042i	$0.420589\pi$
$152$	2604.65	1.38990
$153$	0	0
$154$	47.2522	0.0247253
$155$	0	0
$156$	0	0
$157$	169.765	0.0862978	0.0431489	−	0.999069i	$-0.486261\pi$
$157$	169.765	0.0862978	0.0431489	+	0.999069i	$0.486261\pi$
$158$	−3866.98	−1.94709
$159$	0	0
$160$	0	0
$161$	−813.805	−0.398365
$162$	0	0
$163$	1051.63	0.505338	0.252669	−	0.967553i	$-0.418692\pi$
$163$	1051.63	0.505338	0.252669	+	0.967553i	$0.418692\pi$
$164$	892.014	0.424723
$165$	0	0
$166$	4265.29	1.99428
$167$	−2891.32	−1.33974	−0.669870	−	0.742478i	$-0.733650\pi$
$167$	−2891.32	−1.33974	−0.669870	+	0.742478i	$0.733650\pi$
$168$	0	0
$169$	4335.12	1.97320
$170$	0	0
$171$	0	0
$172$	−561.912	−0.249101
$173$	868.785	0.381806	0.190903	−	0.981609i	$-0.438858\pi$
$173$	868.785	0.381806	0.190903	+	0.981609i	$0.438858\pi$
$174$	0	0
$175$	0	0
$176$	−162.145	−0.0694441
$177$	0	0
$178$	−2851.29	−1.20064
$179$	2073.48	0.865804	0.432902	−	0.901441i	$-0.357490\pi$
$179$	2073.48	0.865804	0.432902	+	0.901441i	$0.357490\pi$
$180$	0	0
$181$	−2152.24	−0.883839	−0.441919	−	0.897055i	$-0.645702\pi$
$181$	−2152.24	−0.883839	−0.441919	+	0.897055i	$0.645702\pi$
$182$	1820.21	0.741336
$183$	0	0
$184$	2112.85	0.846529
$185$	0	0
$186$	0	0
$187$	−213.934	−0.0836599
$188$	−296.160	−0.114892
$189$	0	0
$190$	0	0
$191$	4149.19	1.57186	0.785929	−	0.618316i	$-0.212185\pi$
$191$	4149.19	1.57186	0.785929	+	0.618316i	$0.212185\pi$
$192$	0	0
$193$	−1860.51	−0.693899	−0.346949	−	0.937884i	$-0.612782\pi$
$193$	−1860.51	−0.693899	−0.346949	+	0.937884i	$0.612782\pi$
$194$	434.047	0.160633
$195$	0	0
$196$	115.213	0.0419874
$197$	97.0507	0.0350994	0.0175497	−	0.999846i	$-0.494413\pi$
$197$	97.0507	0.0350994	0.0175497	+	0.999846i	$0.494413\pi$
$198$	0	0
$199$	−936.970	−0.333769	−0.166885	−	0.985976i	$-0.553371\pi$
$199$	−936.970	−0.333769	−0.166885	+	0.985976i	$0.553371\pi$
$200$	0	0
$201$	0	0
$202$	1884.13	0.656272
$203$	1268.36	0.438529
$204$	0	0
$205$	0	0
$206$	−2737.39	−0.925839
$207$	0	0
$208$	−6246.03	−2.08214
$209$	300.698	0.0995201
$210$	0	0
$211$	−1342.19	−0.437915	−0.218958	−	0.975734i	$-0.570266\pi$
$211$	−1342.19	−0.437915	−0.218958	+	0.975734i	$0.570266\pi$
$212$	102.235	0.0331205
$213$	0	0
$214$	1823.59	0.582513
$215$	0	0
$216$	0	0
$217$	−2125.30	−0.664860
$218$	3576.06	1.11101
$219$	0	0
$220$	0	0
$221$	−8241.00	−2.50837
$222$	0	0
$223$	810.794	0.243474	0.121737	−	0.992562i	$-0.461153\pi$
$223$	810.794	0.243474	0.121737	+	0.992562i	$0.461153\pi$
$224$	−722.759	−0.215586
$225$	0	0
$226$	620.109	0.182518
$227$	−2584.59	−0.755707	−0.377853	−	0.925865i	$-0.623338\pi$
$227$	−2584.59	−0.755707	−0.377853	+	0.925865i	$0.623338\pi$
$228$	0	0
$229$	−1637.36	−0.472489	−0.236244	−	0.971694i	$-0.575917\pi$
$229$	−1637.36	−0.472489	−0.236244	+	0.971694i	$0.575917\pi$
$230$	0	0
$231$	0	0
$232$	−3292.99	−0.931877
$233$	−3336.49	−0.938115	−0.469057	−	0.883168i	$-0.655406\pi$
$233$	−3336.49	−0.938115	−0.469057	+	0.883168i	$0.655406\pi$
$234$	0	0
$235$	0	0
$236$	−73.1121	−0.0201661
$237$	0	0
$238$	−2296.40	−0.625436
$239$	3983.20	1.07804	0.539020	−	0.842293i	$-0.318795\pi$
$239$	3983.20	1.07804	0.539020	+	0.842293i	$0.318795\pi$
$240$	0	0
$241$	2282.78	0.610152	0.305076	−	0.952328i	$-0.401318\pi$
$241$	2282.78	0.610152	0.305076	+	0.952328i	$0.401318\pi$
$242$	4268.12	1.13374
$243$	0	0
$244$	−1910.84	−0.501347
$245$	0	0
$246$	0	0
$247$	11583.2	2.98390
$248$	5517.83	1.41283
$249$	0	0
$250$	0	0
$251$	855.255	0.215073	0.107536	−	0.994201i	$-0.465704\pi$
$251$	855.255	0.215073	0.107536	+	0.994201i	$0.465704\pi$
$252$	0	0
$253$	243.921	0.0606134
$254$	−3017.45	−0.745400
$255$	0	0
$256$	3330.17	0.813029
$257$	2892.91	0.702158	0.351079	−	0.936346i	$-0.385815\pi$
$257$	2892.91	0.702158	0.351079	+	0.936346i	$0.385815\pi$
$258$	0	0
$259$	−1108.35	−0.265906
$260$	0	0
$261$	0	0
$262$	2401.15	0.566198
$263$	4167.10	0.977014	0.488507	−	0.872560i	$-0.337542\pi$
$263$	4167.10	0.977014	0.488507	+	0.872560i	$0.337542\pi$
$264$	0	0
$265$	0	0
$266$	3227.74	0.744005
$267$	0	0
$268$	−1002.18	−0.228425
$269$	7519.40	1.70433	0.852167	−	0.523270i	$-0.175288\pi$
$269$	7519.40	1.70433	0.852167	+	0.523270i	$0.175288\pi$
$270$	0	0
$271$	7793.64	1.74697	0.873487	−	0.486848i	$-0.161853\pi$
$271$	7793.64	1.74697	0.873487	+	0.486848i	$0.161853\pi$
$272$	7880.07	1.75661
$273$	0	0
$274$	1584.59	0.349373
$275$	0	0
$276$	0	0
$277$	3192.46	0.692477	0.346238	−	0.938147i	$-0.387459\pi$
$277$	3192.46	0.692477	0.346238	+	0.938147i	$0.387459\pi$
$278$	490.586	0.105840
$279$	0	0
$280$	0	0
$281$	2566.33	0.544819	0.272410	−	0.962181i	$-0.412179\pi$
$281$	2566.33	0.544819	0.272410	+	0.962181i	$0.412179\pi$
$282$	0	0
$283$	−6114.58	−1.28436	−0.642181	−	0.766553i	$-0.721970\pi$
$283$	−6114.58	−1.28436	−0.642181	+	0.766553i	$0.721970\pi$
$284$	2432.43	0.508234
$285$	0	0
$286$	−545.571	−0.112798
$287$	−2655.60	−0.546186
$288$	0	0
$289$	5483.94	1.11621
$290$	0	0
$291$	0	0
$292$	−1109.20	−0.222298
$293$	3546.49	0.707127	0.353563	−	0.935411i	$-0.384970\pi$
$293$	3546.49	0.707127	0.353563	+	0.935411i	$0.384970\pi$
$294$	0	0
$295$	0	0
$296$	2877.57	0.565053
$297$	0	0
$298$	−4611.02	−0.896339
$299$	9396.13	1.81737
$300$	0	0
$301$	1672.86	0.320339
$302$	−2947.87	−0.561691
$303$	0	0
$304$	−11075.9	−2.08963
$305$	0	0
$306$	0	0
$307$	1705.97	0.317150	0.158575	−	0.987347i	$-0.449310\pi$
$307$	1705.97	0.317150	0.158575	+	0.987347i	$0.449310\pi$
$308$	−34.5328	−0.00638860
$309$	0	0
$310$	0	0
$311$	10323.6	1.88230	0.941152	−	0.337983i	$-0.109745\pi$
$311$	10323.6	1.88230	0.941152	+	0.337983i	$0.109745\pi$
$312$	0	0
$313$	7118.85	1.28556	0.642781	−	0.766050i	$-0.277780\pi$
$313$	7118.85	1.28556	0.642781	+	0.766050i	$0.277780\pi$
$314$	−546.193	−0.0981640
$315$	0	0
$316$	2826.06	0.503095
$317$	10755.2	1.90559	0.952797	−	0.303607i	$-0.0981909\pi$
$317$	10755.2	1.90559	0.952797	+	0.303607i	$0.0981909\pi$
$318$	0	0
$319$	−380.164	−0.0667245
$320$	0	0
$321$	0	0
$322$	2618.29	0.453141
$323$	−14613.6	−2.51740
$324$	0	0
$325$	0	0
$326$	−3383.46	−0.574823
$327$	0	0
$328$	6894.64	1.16065
$329$	881.694	0.147749
$330$	0	0
$331$	−494.875	−0.0821776	−0.0410888	−	0.999155i	$-0.513083\pi$
$331$	−494.875	−0.0821776	−0.0410888	+	0.999155i	$0.513083\pi$
$332$	−3117.15	−0.515289
$333$	0	0
$334$	9302.35	1.52396
$335$	0	0
$336$	0	0
$337$	409.916	0.0662598	0.0331299	−	0.999451i	$-0.489452\pi$
$337$	409.916	0.0662598	0.0331299	+	0.999451i	$0.489452\pi$
$338$	−13947.6	−2.24452
$339$	0	0
$340$	0	0
$341$	637.013	0.101162
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−4343.18	−0.680723
$345$	0	0
$346$	−2795.18	−0.434306
$347$	7377.44	1.14133	0.570666	−	0.821182i	$-0.306685\pi$
$347$	7377.44	1.14133	0.570666	+	0.821182i	$0.306685\pi$
$348$	0	0
$349$	−11284.7	−1.73083	−0.865413	−	0.501059i	$-0.832944\pi$
$349$	−11284.7	−1.73083	−0.865413	+	0.501059i	$0.832944\pi$
$350$	0	0
$351$	0	0
$352$	216.632	0.0328026
$353$	3910.15	0.589565	0.294783	−	0.955564i	$-0.404753\pi$
$353$	3910.15	0.589565	0.294783	+	0.955564i	$0.404753\pi$
$354$	0	0
$355$	0	0
$356$	2083.78	0.310224
$357$	0	0
$358$	−6671.08	−0.984854
$359$	−12865.9	−1.89147	−0.945733	−	0.324945i	$-0.894654\pi$
$359$	−12865.9	−1.89147	−0.945733	+	0.324945i	$0.894654\pi$
$360$	0	0
$361$	13681.3	1.99464
$362$	6924.50	1.00537
$363$	0	0
$364$	−1330.24	−0.191549
$365$	0	0
$366$	0	0
$367$	892.357	0.126923	0.0634614	−	0.997984i	$-0.479786\pi$
$367$	892.357	0.126923	0.0634614	+	0.997984i	$0.479786\pi$
$368$	−8984.61	−1.27270
$369$	0	0
$370$	0	0
$371$	−304.363	−0.0425923
$372$	0	0
$373$	8666.00	1.20297	0.601486	−	0.798884i	$-0.294576\pi$
$373$	8666.00	1.20297	0.601486	+	0.798884i	$0.294576\pi$
$374$	688.299	0.0951634
$375$	0	0
$376$	−2289.11	−0.313967
$377$	−14644.4	−2.00059
$378$	0	0
$379$	−7111.00	−0.963767	−0.481883	−	0.876235i	$-0.660047\pi$
$379$	−7111.00	−0.963767	−0.481883	+	0.876235i	$0.660047\pi$
$380$	0	0
$381$	0	0
$382$	−13349.4	−1.78799
$383$	14017.9	1.87019	0.935094	−	0.354401i	$-0.115315\pi$
$383$	14017.9	1.87019	0.935094	+	0.354401i	$0.115315\pi$
$384$	0	0
$385$	0	0
$386$	5985.90	0.789312
$387$	0	0
$388$	−317.209	−0.0415048
$389$	4848.24	0.631917	0.315959	−	0.948773i	$-0.397674\pi$
$389$	4848.24	0.631917	0.315959	+	0.948773i	$0.397674\pi$
$390$	0	0
$391$	−11854.3	−1.53324
$392$	890.518	0.114740
$393$	0	0
$394$	−312.245	−0.0399256
$395$	0	0
$396$	0	0
$397$	4088.76	0.516899	0.258449	−	0.966025i	$-0.416788\pi$
$397$	4088.76	0.516899	0.258449	+	0.966025i	$0.416788\pi$
$398$	3014.55	0.379663
$399$	0	0
$400$	0	0
$401$	−7187.01	−0.895018	−0.447509	−	0.894279i	$-0.647689\pi$
$401$	−7187.01	−0.895018	−0.447509	+	0.894279i	$0.647689\pi$
$402$	0	0
$403$	24538.5	3.03313
$404$	−1376.96	−0.169570
$405$	0	0
$406$	−4080.75	−0.498828
$407$	332.206	0.0404590
$408$	0	0
$409$	924.965	0.111825	0.0559127	−	0.998436i	$-0.482193\pi$
$409$	924.965	0.111825	0.0559127	+	0.998436i	$0.482193\pi$
$410$	0	0
$411$	0	0
$412$	2000.53	0.239221
$413$	217.661	0.0259332
$414$	0	0
$415$	0	0
$416$	8344.92	0.983518
$417$	0	0
$418$	−967.448	−0.113204
$419$	−4460.85	−0.520112	−0.260056	−	0.965594i	$-0.583741\pi$
$419$	−4460.85	−0.520112	−0.260056	+	0.965594i	$0.583741\pi$
$420$	0	0
$421$	−3443.27	−0.398610	−0.199305	−	0.979938i	$-0.563868\pi$
$421$	−3443.27	−0.398610	−0.199305	+	0.979938i	$0.563868\pi$
$422$	4318.28	0.498130
$423$	0	0
$424$	790.206	0.0905090
$425$	0	0
$426$	0	0
$427$	5688.73	0.644723
$428$	−1332.71	−0.150512
$429$	0	0
$430$	0	0
$431$	6214.55	0.694535	0.347267	−	0.937766i	$-0.387110\pi$
$431$	6214.55	0.694535	0.347267	+	0.937766i	$0.387110\pi$
$432$	0	0
$433$	13129.1	1.45714	0.728571	−	0.684970i	$-0.240185\pi$
$433$	13129.1	1.45714	0.728571	+	0.684970i	$0.240185\pi$
$434$	6837.81	0.756279
$435$	0	0
$436$	−2613.45	−0.287067
$437$	16661.9	1.82391
$438$	0	0
$439$	6830.51	0.742602	0.371301	−	0.928512i	$-0.378912\pi$
$439$	6830.51	0.742602	0.371301	+	0.928512i	$0.378912\pi$
$440$	0	0
$441$	0	0
$442$	26514.1	2.85328
$443$	−12837.6	−1.37682	−0.688410	−	0.725322i	$-0.741691\pi$
$443$	−12837.6	−1.37682	−0.688410	+	0.725322i	$0.741691\pi$
$444$	0	0
$445$	0	0
$446$	−2608.60	−0.276953
$447$	0	0
$448$	−2002.42	−0.211172
$449$	16800.5	1.76585	0.882923	−	0.469518i	$-0.155572\pi$
$449$	16800.5	1.76585	0.882923	+	0.469518i	$0.155572\pi$
$450$	0	0
$451$	795.962	0.0831051
$452$	−453.187	−0.0471596
$453$	0	0
$454$	8315.52	0.859618
$455$	0	0
$456$	0	0
$457$	9601.60	0.982809	0.491405	−	0.870931i	$-0.336484\pi$
$457$	9601.60	0.982809	0.491405	+	0.870931i	$0.336484\pi$
$458$	5267.96	0.537457
$459$	0	0
$460$	0	0
$461$	−3506.77	−0.354287	−0.177144	−	0.984185i	$-0.556686\pi$
$461$	−3506.77	−0.354287	−0.177144	+	0.984185i	$0.556686\pi$
$462$	0	0
$463$	−4212.28	−0.422810	−0.211405	−	0.977399i	$-0.567804\pi$
$463$	−4212.28	−0.422810	−0.211405	+	0.977399i	$0.567804\pi$
$464$	14003.0	1.40102
$465$	0	0
$466$	10734.6	1.06711
$467$	−560.881	−0.0555771	−0.0277885	−	0.999614i	$-0.508847\pi$
$467$	−560.881	−0.0555771	−0.0277885	+	0.999614i	$0.508847\pi$
$468$	0	0
$469$	2983.57	0.293750
$470$	0	0
$471$	0	0
$472$	−565.105	−0.0551082
$473$	−501.405	−0.0487413
$474$	0	0
$475$	0	0
$476$	1678.25	0.161602
$477$	0	0
$478$	−12815.3	−1.22627
$479$	−13267.8	−1.26559	−0.632796	−	0.774318i	$-0.718093\pi$
$479$	−13267.8	−1.26559	−0.632796	+	0.774318i	$0.718093\pi$
$480$	0	0
$481$	12797.0	1.21308
$482$	−7344.48	−0.694049
$483$	0	0
$484$	−3119.22	−0.292939
$485$	0	0
$486$	0	0
$487$	−16137.0	−1.50151	−0.750755	−	0.660581i	$-0.770310\pi$
$487$	−16137.0	−1.50151	−0.750755	+	0.660581i	$0.770310\pi$
$488$	−14769.4	−1.37004
$489$	0	0
$490$	0	0
$491$	−5186.66	−0.476722	−0.238361	−	0.971177i	$-0.576610\pi$
$491$	−5186.66	−0.476722	−0.238361	+	0.971177i	$0.576610\pi$
$492$	0	0
$493$	18475.5	1.68782
$494$	−37267.2	−3.39420
$495$	0	0
$496$	−23463.8	−2.12411
$497$	−7241.56	−0.653579
$498$	0	0
$499$	14822.5	1.32975	0.664877	−	0.746953i	$-0.268484\pi$
$499$	14822.5	1.32975	0.664877	+	0.746953i	$0.268484\pi$
$500$	0	0
$501$	0	0
$502$	−2751.65	−0.244646
$503$	1690.53	0.149855	0.0749273	−	0.997189i	$-0.476128\pi$
$503$	1690.53	0.149855	0.0749273	+	0.997189i	$0.476128\pi$
$504$	0	0
$505$	0	0
$506$	−784.778	−0.0689479
$507$	0	0
$508$	2205.21	0.192599
$509$	−5355.09	−0.466326	−0.233163	−	0.972438i	$-0.574908\pi$
$509$	−5355.09	−0.466326	−0.233163	+	0.972438i	$0.574908\pi$
$510$	0	0
$511$	3302.19	0.285871
$512$	3256.60	0.281100
$513$	0	0
$514$	−9307.47	−0.798706
$515$	0	0
$516$	0	0
$517$	−264.269	−0.0224808
$518$	3565.95	0.302469
$519$	0	0
$520$	0	0
$521$	15355.6	1.29125	0.645624	−	0.763655i	$-0.276597\pi$
$521$	15355.6	1.29125	0.645624	+	0.763655i	$0.276597\pi$
$522$	0	0
$523$	6142.95	0.513599	0.256800	−	0.966465i	$-0.417332\pi$
$523$	6142.95	0.513599	0.256800	+	0.966465i	$0.417332\pi$
$524$	−1754.81	−0.146296
$525$	0	0
$526$	−13407.0	−1.11136
$527$	−30958.1	−2.55893
$528$	0	0
$529$	1348.89	0.110864
$530$	0	0
$531$	0	0
$532$	−2358.89	−0.192238
$533$	30661.4	2.49173
$534$	0	0
$535$	0	0
$536$	−7746.14	−0.624221
$537$	0	0
$538$	−24192.5	−1.93868
$539$	102.807	0.00821561
$540$	0	0
$541$	4877.07	0.387582	0.193791	−	0.981043i	$-0.437922\pi$
$541$	4877.07	0.387582	0.193791	+	0.981043i	$0.437922\pi$
$542$	−25074.8	−1.98719
$543$	0	0
$544$	−10528.1	−0.829755
$545$	0	0
$546$	0	0
$547$	3823.80	0.298892	0.149446	−	0.988770i	$-0.452251\pi$
$547$	3823.80	0.298892	0.149446	+	0.988770i	$0.452251\pi$
$548$	−1158.04	−0.0902722
$549$	0	0
$550$	0	0
$551$	−25968.5	−2.00780
$552$	0	0
$553$	−8413.41	−0.646970
$554$	−10271.2	−0.787694
$555$	0	0
$556$	−358.529	−0.0273472
$557$	−10514.6	−0.799849	−0.399925	−	0.916548i	$-0.630964\pi$
$557$	−10514.6	−0.799849	−0.399925	+	0.916548i	$0.630964\pi$
$558$	0	0
$559$	−19314.7	−1.46141
$560$	0	0
$561$	0	0
$562$	−8256.75	−0.619733
$563$	−1679.55	−0.125728	−0.0628639	−	0.998022i	$-0.520023\pi$
$563$	−1679.55	−0.125728	−0.0628639	+	0.998022i	$0.520023\pi$
$564$	0	0
$565$	0	0
$566$	19672.7	1.46096
$567$	0	0
$568$	18801.0	1.38886
$569$	−7067.14	−0.520686	−0.260343	−	0.965516i	$-0.583836\pi$
$569$	−7067.14	−0.520686	−0.260343	+	0.965516i	$0.583836\pi$
$570$	0	0
$571$	−10609.6	−0.777577	−0.388788	−	0.921327i	$-0.627106\pi$
$571$	−10609.6	−0.777577	−0.388788	+	0.921327i	$0.627106\pi$
$572$	398.713	0.0291452
$573$	0	0
$574$	8543.99	0.621288
$575$	0	0
$576$	0	0
$577$	4002.39	0.288773	0.144386	−	0.989521i	$-0.453879\pi$
$577$	4002.39	0.288773	0.144386	+	0.989521i	$0.453879\pi$
$578$	−17643.7	−1.26969
$579$	0	0
$580$	0	0
$581$	9280.03	0.662652
$582$	0	0
$583$	91.2266	0.00648065
$584$	−8573.34	−0.607478
$585$	0	0
$586$	−11410.3	−0.804358
$587$	−2334.57	−0.164154	−0.0820768	−	0.996626i	$-0.526155\pi$
$587$	−2334.57	−0.164154	−0.0820768	+	0.996626i	$0.526155\pi$
$588$	0	0
$589$	43513.6	3.04405
$590$	0	0
$591$	0	0
$592$	−12236.5	−0.849522
$593$	−12401.0	−0.858768	−0.429384	−	0.903122i	$-0.641269\pi$
$593$	−12401.0	−0.858768	−0.429384	+	0.903122i	$0.641269\pi$
$594$	0	0
$595$	0	0
$596$	3369.81	0.231599
$597$	0	0
$598$	−30230.6	−2.06726
$599$	−15848.8	−1.08108	−0.540538	−	0.841320i	$-0.681779\pi$
$599$	−15848.8	−1.08108	−0.540538	+	0.841320i	$0.681779\pi$
$600$	0	0
$601$	298.912	0.0202877	0.0101438	−	0.999949i	$-0.496771\pi$
$601$	298.912	0.0202877	0.0101438	+	0.999949i	$0.496771\pi$
$602$	−5382.16	−0.364386
$603$	0	0
$604$	2154.36	0.145132
$605$	0	0
$606$	0	0
$607$	8731.31	0.583843	0.291922	−	0.956442i	$-0.405705\pi$
$607$	8731.31	0.583843	0.291922	+	0.956442i	$0.405705\pi$
$608$	14797.8	0.987059
$609$	0	0
$610$	0	0
$611$	−10180.0	−0.674039
$612$	0	0
$613$	−10037.7	−0.661368	−0.330684	−	0.943742i	$-0.607279\pi$
$613$	−10037.7	−0.661368	−0.330684	+	0.943742i	$0.607279\pi$
$614$	−5488.70	−0.360759
$615$	0	0
$616$	−266.914	−0.0174582
$617$	20567.7	1.34202	0.671008	−	0.741450i	$-0.265862\pi$
$617$	20567.7	1.34202	0.671008	+	0.741450i	$0.265862\pi$
$618$	0	0
$619$	15714.5	1.02039	0.510193	−	0.860060i	$-0.329574\pi$
$619$	15714.5	1.02039	0.510193	+	0.860060i	$0.329574\pi$
$620$	0	0
$621$	0	0
$622$	−33214.5	−2.14113
$623$	−6203.58	−0.398943
$624$	0	0
$625$	0	0
$626$	−22903.8	−1.46233
$627$	0	0
$628$	399.168	0.0253639
$629$	−16144.8	−1.02343
$630$	0	0
$631$	−8962.39	−0.565431	−0.282716	−	0.959204i	$-0.591235\pi$
$631$	−8962.39	−0.565431	−0.282716	+	0.959204i	$0.591235\pi$
$632$	21843.4	1.37482
$633$	0	0
$634$	−34603.2	−2.16762
$635$	0	0
$636$	0	0
$637$	3960.25	0.246328
$638$	1223.12	0.0758993
$639$	0	0
$640$	0	0
$641$	−26919.8	−1.65877	−0.829383	−	0.558681i	$-0.811308\pi$
$641$	−26919.8	−1.65877	−0.829383	+	0.558681i	$0.811308\pi$
$642$	0	0
$643$	−5675.19	−0.348068	−0.174034	−	0.984740i	$-0.555680\pi$
$643$	−5675.19	−0.348068	−0.174034	+	0.984740i	$0.555680\pi$
$644$	−1913.49	−0.117084
$645$	0	0
$646$	47016.8	2.86355
$647$	−8444.54	−0.513121	−0.256560	−	0.966528i	$-0.582589\pi$
$647$	−8444.54	−0.513121	−0.256560	+	0.966528i	$0.582589\pi$
$648$	0	0
$649$	−65.2393	−0.00394587
$650$	0	0
$651$	0	0
$652$	2472.69	0.148525
$653$	25968.1	1.55622	0.778108	−	0.628130i	$-0.216180\pi$
$653$	25968.1	1.55622	0.778108	+	0.628130i	$0.216180\pi$
$654$	0	0
$655$	0	0
$656$	−29318.5	−1.74496
$657$	0	0
$658$	−2836.71	−0.168065
$659$	21195.4	1.25289	0.626446	−	0.779465i	$-0.284509\pi$
$659$	21195.4	1.25289	0.626446	+	0.779465i	$0.284509\pi$
$660$	0	0
$661$	−9283.40	−0.546267	−0.273134	−	0.961976i	$-0.588060\pi$
$661$	−9283.40	−0.546267	−0.273134	+	0.961976i	$0.588060\pi$
$662$	1592.18	0.0934773
$663$	0	0
$664$	−24093.4	−1.40814
$665$	0	0
$666$	0	0
$667$	−21065.2	−1.22286
$668$	−6798.33	−0.393765
$669$	0	0
$670$	0	0
$671$	−1705.08	−0.0980980
$672$	0	0
$673$	−25358.3	−1.45244	−0.726219	−	0.687463i	$-0.758724\pi$
$673$	−25358.3	−1.45244	−0.726219	+	0.687463i	$0.758724\pi$
$674$	−1318.84	−0.0753707
$675$	0	0
$676$	10193.1	0.579945
$677$	−7140.67	−0.405374	−0.202687	−	0.979244i	$-0.564967\pi$
$677$	−7140.67	−0.405374	−0.202687	+	0.979244i	$0.564967\pi$
$678$	0	0
$679$	944.359	0.0533743
$680$	0	0
$681$	0	0
$682$	−2049.49	−0.115072
$683$	−10666.3	−0.597561	−0.298780	−	0.954322i	$-0.596580\pi$
$683$	−10666.3	−0.597561	−0.298780	+	0.954322i	$0.596580\pi$
$684$	0	0
$685$	0	0
$686$	1103.55	0.0614194
$687$	0	0
$688$	18468.8	1.02342
$689$	3514.16	0.194309
$690$	0	0
$691$	−19360.0	−1.06583	−0.532915	−	0.846169i	$-0.678903\pi$
$691$	−19360.0	−1.06583	−0.532915	+	0.846169i	$0.678903\pi$
$692$	2042.77	0.112217
$693$	0	0
$694$	−23735.8	−1.29827
$695$	0	0
$696$	0	0
$697$	−38682.8	−2.10218
$698$	36306.9	1.96882
$699$	0	0
$700$	0	0
$701$	19252.3	1.03730	0.518651	−	0.854986i	$-0.326434\pi$
$701$	19252.3	1.03730	0.518651	+	0.854986i	$0.326434\pi$
$702$	0	0
$703$	22692.5	1.21745
$704$	600.183	0.0321310
$705$	0	0
$706$	−12580.3	−0.670632
$707$	4099.32	0.218063
$708$	0	0
$709$	−19553.0	−1.03572	−0.517861	−	0.855465i	$-0.673272\pi$
$709$	−19553.0	−1.03572	−0.517861	+	0.855465i	$0.673272\pi$
$710$	0	0
$711$	0	0
$712$	16106.1	0.847756
$713$	35297.5	1.85400
$714$	0	0
$715$	0	0
$716$	4875.35	0.254470
$717$	0	0
$718$	41394.0	2.15155
$719$	16334.8	0.847265	0.423633	−	0.905834i	$-0.360755\pi$
$719$	16334.8	0.847265	0.423633	+	0.905834i	$0.360755\pi$
$720$	0	0
$721$	−5955.76	−0.307634
$722$	−44017.3	−2.26891
$723$	0	0
$724$	−5060.55	−0.259770
$725$	0	0
$726$	0	0
$727$	−30698.8	−1.56610	−0.783051	−	0.621957i	$-0.786338\pi$
$727$	−30698.8	−1.56610	−0.783051	+	0.621957i	$0.786338\pi$
$728$	−10281.9	−0.523449
$729$	0	0
$730$	0	0
$731$	24367.7	1.23293
$732$	0	0
$733$	7388.56	0.372309	0.186155	−	0.982520i	$-0.440397\pi$
$733$	7388.56	0.372309	0.186155	+	0.982520i	$0.440397\pi$
$734$	−2871.02	−0.144375
$735$	0	0
$736$	12003.8	0.601175
$737$	−894.264	−0.0446956
$738$	0	0
$739$	22650.8	1.12750	0.563751	−	0.825945i	$-0.309358\pi$
$739$	22650.8	1.12750	0.563751	+	0.825945i	$0.309358\pi$
$740$	0	0
$741$	0	0
$742$	979.241	0.0484489
$743$	17656.4	0.871806	0.435903	−	0.899994i	$-0.356429\pi$
$743$	17656.4	0.871806	0.435903	+	0.899994i	$0.356429\pi$
$744$	0	0
$745$	0	0
$746$	−27881.5	−1.36838
$747$	0	0
$748$	−503.021	−0.0245886
$749$	3967.59	0.193555
$750$	0	0
$751$	22109.0	1.07426	0.537130	−	0.843500i	$-0.319508\pi$
$751$	22109.0	1.07426	0.537130	+	0.843500i	$0.319508\pi$
$752$	9734.12	0.472031
$753$	0	0
$754$	47116.0	2.27568
$755$	0	0
$756$	0	0
$757$	−11034.2	−0.529784	−0.264892	−	0.964278i	$-0.585336\pi$
$757$	−11034.2	−0.529784	−0.264892	+	0.964278i	$0.585336\pi$
$758$	22878.5	1.09629
$759$	0	0
$760$	0	0
$761$	6357.12	0.302819	0.151410	−	0.988471i	$-0.451619\pi$
$761$	6357.12	0.302819	0.151410	+	0.988471i	$0.451619\pi$
$762$	0	0
$763$	7780.46	0.369163
$764$	9755.96	0.461987
$765$	0	0
$766$	−45100.4	−2.12734
$767$	−2513.10	−0.118309
$768$	0	0
$769$	−35810.1	−1.67925	−0.839627	−	0.543163i	$-0.817226\pi$
$769$	−35810.1	−1.67925	−0.839627	+	0.543163i	$0.817226\pi$
$770$	0	0
$771$	0	0
$772$	−4374.60	−0.203945
$773$	−3158.57	−0.146967	−0.0734837	−	0.997296i	$-0.523412\pi$
$773$	−3158.57	−0.146967	−0.0734837	+	0.997296i	$0.523412\pi$
$774$	0	0
$775$	0	0
$776$	−2451.80	−0.113421
$777$	0	0
$778$	−15598.5	−0.718807
$779$	54371.1	2.50070
$780$	0	0
$781$	2170.51	0.0994455
$782$	38139.3	1.74406
$783$	0	0
$784$	−3786.81	−0.172504
$785$	0	0
$786$	0	0
$787$	−20371.2	−0.922687	−0.461344	−	0.887222i	$-0.652632\pi$
$787$	−20371.2	−0.922687	−0.461344	+	0.887222i	$0.652632\pi$
$788$	228.195	0.0103161
$789$	0	0
$790$	0	0
$791$	1349.18	0.0606463
$792$	0	0
$793$	−65681.6	−2.94127
$794$	−13154.9	−0.587973
$795$	0	0
$796$	−2203.09	−0.0980986
$797$	−31056.5	−1.38027	−0.690136	−	0.723680i	$-0.742449\pi$
$797$	−31056.5	−1.38027	−0.690136	+	0.723680i	$0.742449\pi$
$798$	0	0
$799$	12843.2	0.568660
$800$	0	0
$801$	0	0
$802$	23123.1	1.01809
$803$	−989.761	−0.0434968
$804$	0	0
$805$	0	0
$806$	−78948.8	−3.45019
$807$	0	0
$808$	−10642.9	−0.463386
$809$	−9549.52	−0.415010	−0.207505	−	0.978234i	$-0.566534\pi$
$809$	−9549.52	−0.415010	−0.207505	+	0.978234i	$0.566534\pi$
$810$	0	0
$811$	−26730.9	−1.15740	−0.578698	−	0.815542i	$-0.696439\pi$
$811$	−26730.9	−1.15740	−0.578698	+	0.815542i	$0.696439\pi$
$812$	2982.28	0.128889
$813$	0	0
$814$	−1068.82	−0.0460222
$815$	0	0
$816$	0	0
$817$	−34250.3	−1.46667
$818$	−2975.93	−0.127202
$819$	0	0
$820$	0	0
$821$	30878.1	1.31261	0.656305	−	0.754496i	$-0.272118\pi$
$821$	30878.1	1.31261	0.656305	+	0.754496i	$0.272118\pi$
$822$	0	0
$823$	20618.3	0.873280	0.436640	−	0.899636i	$-0.356168\pi$
$823$	20618.3	0.873280	0.436640	+	0.899636i	$0.356168\pi$
$824$	15462.7	0.653724
$825$	0	0
$826$	−700.290	−0.0294990
$827$	−12493.2	−0.525311	−0.262656	−	0.964890i	$-0.584598\pi$
$827$	−12493.2	−0.525311	−0.262656	+	0.964890i	$0.584598\pi$
$828$	0	0
$829$	−36795.9	−1.54158	−0.770792	−	0.637087i	$-0.780139\pi$
$829$	−36795.9	−1.54158	−0.770792	+	0.637087i	$0.780139\pi$
$830$	0	0
$831$	0	0
$832$	23119.8	0.963381
$833$	−4996.31	−0.207817
$834$	0	0
$835$	0	0
$836$	707.028	0.0292501
$837$	0	0
$838$	14352.1	0.591628
$839$	−6904.41	−0.284108	−0.142054	−	0.989859i	$-0.545371\pi$
$839$	−6904.41	−0.284108	−0.142054	+	0.989859i	$0.545371\pi$
$840$	0	0
$841$	8442.32	0.346153
$842$	11078.2	0.453419
$843$	0	0
$844$	−3155.88	−0.128708
$845$	0	0
$846$	0	0
$847$	9286.19	0.376714
$848$	−3360.25	−0.136075
$849$	0	0
$850$	0	0
$851$	18407.8	0.741495
$852$	0	0
$853$	34285.8	1.37623	0.688115	−	0.725602i	$-0.258439\pi$
$853$	34285.8	1.37623	0.688115	+	0.725602i	$0.258439\pi$
$854$	−18302.6	−0.733374
$855$	0	0
$856$	−10300.9	−0.411306
$857$	−34301.1	−1.36722	−0.683608	−	0.729849i	$-0.739590\pi$
$857$	−34301.1	−1.36722	−0.683608	+	0.729849i	$0.739590\pi$
$858$	0	0
$859$	6823.12	0.271015	0.135507	−	0.990776i	$-0.456734\pi$
$859$	6823.12	0.271015	0.135507	+	0.990776i	$0.456734\pi$
$860$	0	0
$861$	0	0
$862$	−19994.3	−0.790035
$863$	8059.34	0.317895	0.158947	−	0.987287i	$-0.449190\pi$
$863$	8059.34	0.317895	0.158947	+	0.987287i	$0.449190\pi$
$864$	0	0
$865$	0	0
$866$	−42240.7	−1.65750
$867$	0	0
$868$	−4997.19	−0.195410
$869$	2521.75	0.0984400
$870$	0	0
$871$	−34448.1	−1.34010
$872$	−20200.1	−0.784474
$873$	0	0
$874$	−53607.1	−2.07470
$875$	0	0
$876$	0	0
$877$	−162.992	−0.00627577	−0.00313789	−	0.999995i	$-0.500999\pi$
$877$	−162.992	−0.00627577	−0.00313789	+	0.999995i	$0.500999\pi$
$878$	−21976.1	−0.844712
$879$	0	0
$880$	0	0
$881$	−10633.9	−0.406658	−0.203329	−	0.979110i	$-0.565176\pi$
$881$	−10633.9	−0.406658	−0.203329	+	0.979110i	$0.565176\pi$
$882$	0	0
$883$	−42208.3	−1.60863	−0.804317	−	0.594201i	$-0.797468\pi$
$883$	−42208.3	−1.60863	−0.804317	+	0.594201i	$0.797468\pi$
$884$	−19377.0	−0.737238
$885$	0	0
$886$	41302.8	1.56614
$887$	15696.0	0.594161	0.297080	−	0.954852i	$-0.403987\pi$
$887$	15696.0	0.594161	0.297080	+	0.954852i	$0.403987\pi$
$888$	0	0
$889$	−6565.09	−0.247678
$890$	0	0
$891$	0	0
$892$	1906.41	0.0715599
$893$	−18051.9	−0.676466
$894$	0	0
$895$	0	0
$896$	12224.5	0.455796
$897$	0	0
$898$	−54053.0	−2.00865
$899$	−55013.0	−2.04092
$900$	0	0
$901$	−4433.50	−0.163931
$902$	−2560.88	−0.0945322
$903$	0	0
$904$	−3502.82	−0.128874
$905$	0	0
$906$	0	0
$907$	−24251.5	−0.887827	−0.443913	−	0.896070i	$-0.646410\pi$
$907$	−24251.5	−0.887827	−0.443913	+	0.896070i	$0.646410\pi$
$908$	−6077.13	−0.222111
$909$	0	0
$910$	0	0
$911$	34110.8	1.24055	0.620276	−	0.784384i	$-0.287021\pi$
$911$	34110.8	1.24055	0.620276	+	0.784384i	$0.287021\pi$
$912$	0	0
$913$	−2781.50	−0.100826
$914$	−30891.6	−1.11795
$915$	0	0
$916$	−3849.92	−0.138870
$917$	5224.21	0.188134
$918$	0	0
$919$	55153.2	1.97969	0.989846	−	0.142146i	$-0.0454002\pi$
$919$	55153.2	1.97969	0.989846	+	0.142146i	$0.0454002\pi$
$920$	0	0
$921$	0	0
$922$	11282.5	0.403003
$923$	83610.6	2.98166
$924$	0	0
$925$	0	0
$926$	13552.3	0.480947
$927$	0	0
$928$	−18708.5	−0.661786
$929$	−14429.8	−0.509607	−0.254804	−	0.966993i	$-0.582011\pi$
$929$	−14429.8	−0.509607	−0.254804	+	0.966993i	$0.582011\pi$
$930$	0	0
$931$	7022.62	0.247215
$932$	−7845.06	−0.275723
$933$	0	0
$934$	1804.55	0.0632190
$935$	0	0
$936$	0	0
$937$	3625.32	0.126397	0.0631985	−	0.998001i	$-0.479870\pi$
$937$	3625.32	0.126397	0.0631985	+	0.998001i	$0.479870\pi$
$938$	−9599.18	−0.334141
$939$	0	0
$940$	0	0
$941$	15634.7	0.541632	0.270816	−	0.962631i	$-0.412707\pi$
$941$	15634.7	0.541632	0.270816	+	0.962631i	$0.412707\pi$
$942$	0	0
$943$	44105.0	1.52307
$944$	2403.03	0.0828517
$945$	0	0
$946$	1613.19	0.0554433
$947$	−32438.0	−1.11309	−0.556544	−	0.830818i	$-0.687873\pi$
$947$	−32438.0	−1.11309	−0.556544	+	0.830818i	$0.687873\pi$
$948$	0	0
$949$	−38126.8	−1.30416
$950$	0	0
$951$	0	0
$952$	12971.7	0.441613
$953$	10940.1	0.371862	0.185931	−	0.982563i	$-0.440470\pi$
$953$	10940.1	0.371862	0.185931	+	0.982563i	$0.440470\pi$
$954$	0	0
$955$	0	0
$956$	9365.66	0.316849
$957$	0	0
$958$	42686.9	1.43961
$959$	3447.60	0.116088
$960$	0	0
$961$	62390.3	2.09427
$962$	−41172.2	−1.37988
$963$	0	0
$964$	5367.48	0.179331
$965$	0	0
$966$	0	0
$967$	27827.5	0.925412	0.462706	−	0.886512i	$-0.346879\pi$
$967$	27827.5	0.925412	0.462706	+	0.886512i	$0.346879\pi$
$968$	−24109.4	−0.800521
$969$	0	0
$970$	0	0
$971$	58041.1	1.91826	0.959129	−	0.282970i	$-0.0913198\pi$
$971$	58041.1	1.91826	0.959129	+	0.282970i	$0.0913198\pi$
$972$	0	0
$973$	1067.37	0.0351679
$974$	51918.1	1.70797
$975$	0	0
$976$	62805.0	2.05977
$977$	28430.6	0.930988	0.465494	−	0.885051i	$-0.345877\pi$
$977$	28430.6	0.930988	0.465494	+	0.885051i	$0.345877\pi$
$978$	0	0
$979$	1859.39	0.0607012
$980$	0	0
$981$	0	0
$982$	16687.2	0.542272
$983$	9356.47	0.303586	0.151793	−	0.988412i	$-0.451495\pi$
$983$	9356.47	0.303586	0.151793	+	0.988412i	$0.451495\pi$
$984$	0	0
$985$	0	0
$986$	−59442.1	−1.91990
$987$	0	0
$988$	27235.6	0.877003
$989$	−27783.3	−0.893284
$990$	0	0
$991$	−9199.90	−0.294898	−0.147449	−	0.989070i	$-0.547106\pi$
$991$	−9199.90	−0.294898	−0.147449	+	0.989070i	$0.547106\pi$
$992$	31348.5	1.00334
$993$	0	0
$994$	23298.6	0.743447
$995$	0	0
$996$	0	0
$997$	5396.25	0.171415	0.0857076	−	0.996320i	$-0.472685\pi$
$997$	5396.25	0.171415	0.0857076	+	0.996320i	$0.472685\pi$
$998$	−47689.1	−1.51260
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.bf.1.2		4
3.2	odd	2		525.4.a.v.1.3	yes	4
5.4	even	2		1575.4.a.bm.1.3		4
15.2	even	4		525.4.d.o.274.7		8
15.8	even	4		525.4.d.o.274.2		8
15.14	odd	2		525.4.a.s.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.s.1.2	✓	4	15.14	odd	2
525.4.a.v.1.3	yes	4	3.2	odd	2
525.4.d.o.274.2		8	15.8	even	4
525.4.d.o.274.7		8	15.2	even	4
1575.4.a.bf.1.2		4	1.1	even	1	trivial
1575.4.a.bm.1.3		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)

\(f(q)\)	\(=\)	\(q-3.21734 q^{2} +2.35129 q^{4} -7.00000 q^{7} +18.1738 q^{8} +2.09810 q^{11} +80.8215 q^{13} +22.5214 q^{14} -77.2818 q^{16} -101.965 q^{17} +143.319 q^{19} -6.75032 q^{22} +116.258 q^{23} -260.030 q^{26} -16.4590 q^{28} -181.194 q^{29} +303.614 q^{31} +103.251 q^{32} +328.058 q^{34} +158.336 q^{37} -461.106 q^{38} +379.372 q^{41} -238.980 q^{43} +4.93326 q^{44} -374.041 q^{46} -125.956 q^{47} +49.0000 q^{49} +190.035 q^{52} +43.4805 q^{53} -127.217 q^{56} +582.964 q^{58} -31.0944 q^{59} -812.675 q^{61} -976.830 q^{62} +286.059 q^{64} -426.225 q^{67} -239.750 q^{68} +1034.51 q^{71} -471.741 q^{73} -509.422 q^{74} +336.984 q^{76} -14.6867 q^{77} +1201.92 q^{79} -1220.57 q^{82} -1325.72 q^{83} +768.880 q^{86} +38.1306 q^{88} +886.226 q^{89} -565.751 q^{91} +273.356 q^{92} +405.245 q^{94} -134.908 q^{97} -157.650 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} + 16 q^{4} - 28 q^{7} - 93 q^{8} - 57 q^{11} + 43 q^{13} + 42 q^{14} + 216 q^{16} - 99 q^{17} - 12 q^{19} - 41 q^{22} - 156 q^{23} + 81 q^{26} - 112 q^{28} - 378 q^{29} - 93 q^{31} - 690 q^{32}+ \cdots - 294 q^{98}+O(q^{100}) \) 4 * q - 6 * q^2 + 16 * q^4 - 28 * q^7 - 93 * q^8 - 57 * q^11 + 43 * q^13 + 42 * q^14 + 216 * q^16 - 99 * q^17 - 12 * q^19 - 41 * q^22 - 156 * q^23 + 81 * q^26 - 112 * q^28 - 378 * q^29 - 93 * q^31 - 690 * q^32 + 783 * q^34 + 81 * q^37 - 216 * q^38 + 465 * q^41 - 64 * q^43 - 681 * q^44 + 310 * q^46 - 744 * q^47 + 196 * q^49 - 727 * q^52 - 729 * q^53 + 651 * q^56 + 1172 * q^58 - 231 * q^59 - 1353 * q^61 + 165 * q^62 + 3107 * q^64 + 1487 * q^67 - 2577 * q^68 + 1725 * q^71 + 512 * q^73 + 1953 * q^74 - 3046 * q^76 + 399 * q^77 + 1629 * q^79 - 693 * q^82 - 321 * q^83 + 4542 * q^86 + 3482 * q^88 + 978 * q^89 - 301 * q^91 - 852 * q^92 + 2480 * q^94 + 2616 * q^97 - 294 * q^98

Introduction

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