LMFDB - Embedded newform 2475.4.a.ca.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$146.029727264$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 103 x^{14} + 4248 x^{12} - 89496 x^{10} + 1015487 x^{8} - 5956953 x^{6} + 15313728 x^{4} + \cdots + 861184 $ x^16 - 103x^14 + 4248x^12 - 89496x^10 + 1015487x^8 - 5956953x^6 + 15313728x^4 - 10273600*x^2 + 861184
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$4.59829$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.59829 q^{2} +13.1443 q^{4} -8.59388 q^{7} +23.6550 q^{8} +O(q^{10})$	$q+4.59829 q^{2} +13.1443 q^{4} -8.59388 q^{7} +23.6550 q^{8} +11.0000 q^{11} +55.0695 q^{13} -39.5172 q^{14} +3.61828 q^{16} -103.250 q^{17} +149.207 q^{19} +50.5812 q^{22} +113.973 q^{23} +253.226 q^{26} -112.961 q^{28} +299.026 q^{29} -202.176 q^{31} -172.602 q^{32} -474.772 q^{34} -241.641 q^{37} +686.097 q^{38} -60.1794 q^{41} +325.474 q^{43} +144.587 q^{44} +524.083 q^{46} +48.0825 q^{47} -269.145 q^{49} +723.850 q^{52} -328.740 q^{53} -203.288 q^{56} +1375.01 q^{58} +362.188 q^{59} +455.354 q^{61} -929.666 q^{62} -822.622 q^{64} +749.885 q^{67} -1357.14 q^{68} +501.761 q^{71} -1045.59 q^{73} -1111.14 q^{74} +1961.22 q^{76} -94.5327 q^{77} +466.940 q^{79} -276.723 q^{82} +1157.65 q^{83} +1496.63 q^{86} +260.205 q^{88} +1100.03 q^{89} -473.261 q^{91} +1498.10 q^{92} +221.098 q^{94} +1167.99 q^{97} -1237.61 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 78 q^{4}+O(q^{10}) $ 16 * q + 78 * q^4	$ 16 q + 78 q^{4} + 176 q^{11} + 124 q^{14} + 306 q^{16} + 84 q^{19} + 532 q^{26} + 652 q^{29} + 80 q^{31} + 360 q^{34} + 1204 q^{41} + 858 q^{44} - 672 q^{46} + 1016 q^{49} + 3332 q^{56} + 712 q^{59} + 880 q^{61} - 962 q^{64} + 1968 q^{71} + 4152 q^{74} - 1048 q^{76} - 1636 q^{79} + 7284 q^{86} + 2776 q^{89} + 144 q^{91} + 1400 q^{94}+O(q^{100}) $ 16 * q + 78 * q^4 + 176 * q^11 + 124 * q^14 + 306 * q^16 + 84 * q^19 + 532 * q^26 + 652 * q^29 + 80 * q^31 + 360 * q^34 + 1204 * q^41 + 858 * q^44 - 672 * q^46 + 1016 * q^49 + 3332 * q^56 + 712 * q^59 + 880 * q^61 - 962 * q^64 + 1968 * q^71 + 4152 * q^74 - 1048 * q^76 - 1636 * q^79 + 7284 * q^86 + 2776 * q^89 + 144 * q^91 + 1400 * q^94

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.59829	1.62574	0.812871	−	0.582444i	$-0.197903\pi$
$2$	4.59829	1.62574	0.812871	+	0.582444i	$0.197903\pi$
$3$	0	0
$3$	0	0
$4$	13.1443	1.64304
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−8.59388	−0.464026	−0.232013	−	0.972713i	$-0.574531\pi$
$7$	−8.59388	−0.464026	−0.232013	+	0.972713i	$0.574531\pi$
$8$	23.6550	1.04541
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	55.0695	1.17489	0.587444	−	0.809265i	$-0.300134\pi$
$13$	55.0695	1.17489	0.587444	+	0.809265i	$0.300134\pi$
$14$	−39.5172	−0.754387
$15$	0	0
$16$	3.61828	0.0565356
$17$	−103.250	−1.47304	−0.736521	−	0.676415i	$-0.763533\pi$
$17$	−103.250	−1.47304	−0.736521	+	0.676415i	$0.763533\pi$
$18$	0	0
$19$	149.207	1.80160	0.900801	−	0.434233i	$-0.142981\pi$
$19$	149.207	1.80160	0.900801	+	0.434233i	$0.142981\pi$
$20$	0	0
$21$	0	0
$22$	50.5812	0.490180
$23$	113.973	1.03326	0.516632	−	0.856207i	$-0.327185\pi$
$23$	113.973	1.03326	0.516632	+	0.856207i	$0.327185\pi$
$24$	0	0
$25$	0	0
$26$	253.226	1.91006
$27$	0	0
$28$	−112.961	−0.762412
$29$	299.026	1.91475	0.957374	−	0.288852i	$-0.0932734\pi$
$29$	299.026	1.91475	0.957374	+	0.288852i	$0.0932734\pi$
$30$	0	0
$31$	−202.176	−1.17135	−0.585676	−	0.810545i	$-0.699171\pi$
$31$	−202.176	−1.17135	−0.585676	+	0.810545i	$0.699171\pi$
$32$	−172.602	−0.953502
$33$	0	0
$34$	−474.772	−2.39479
$35$	0	0
$36$	0	0
$37$	−241.641	−1.07366	−0.536832	−	0.843689i	$-0.680379\pi$
$37$	−241.641	−1.07366	−0.536832	+	0.843689i	$0.680379\pi$
$38$	686.097	2.92894
$39$	0	0
$40$	0	0
$41$	−60.1794	−0.229230	−0.114615	−	0.993410i	$-0.536563\pi$
$41$	−60.1794	−0.229230	−0.114615	+	0.993410i	$0.536563\pi$
$42$	0	0
$43$	325.474	1.15429	0.577143	−	0.816643i	$-0.304167\pi$
$43$	325.474	1.15429	0.577143	+	0.816643i	$0.304167\pi$
$44$	144.587	0.495395
$45$	0	0
$46$	524.083	1.67982
$47$	48.0825	0.149225	0.0746123	−	0.997213i	$-0.476228\pi$
$47$	48.0825	0.149225	0.0746123	+	0.997213i	$0.476228\pi$
$48$	0	0
$49$	−269.145	−0.784680
$50$	0	0
$51$	0	0
$52$	723.850	1.93038
$53$	−328.740	−0.851998	−0.425999	−	0.904724i	$-0.640077\pi$
$53$	−328.740	−0.851998	−0.425999	+	0.904724i	$0.640077\pi$
$54$	0	0
$55$	0	0
$56$	−203.288	−0.485099
$57$	0	0
$58$	1375.01	3.11289
$59$	362.188	0.799202	0.399601	−	0.916689i	$-0.369149\pi$
$59$	362.188	0.799202	0.399601	+	0.916689i	$0.369149\pi$
$60$	0	0
$61$	455.354	0.955773	0.477887	−	0.878422i	$-0.341403\pi$
$61$	455.354	0.955773	0.477887	+	0.878422i	$0.341403\pi$
$62$	−929.666	−1.90432
$63$	0	0
$64$	−822.622	−1.60668
$65$	0	0
$66$	0	0
$67$	749.885	1.36736	0.683680	−	0.729782i	$-0.260379\pi$
$67$	749.885	1.36736	0.683680	+	0.729782i	$0.260379\pi$
$68$	−1357.14	−2.42026
$69$	0	0
$70$	0	0
$71$	501.761	0.838706	0.419353	−	0.907823i	$-0.362257\pi$
$71$	501.761	0.838706	0.419353	+	0.907823i	$0.362257\pi$
$72$	0	0
$73$	−1045.59	−1.67640	−0.838201	−	0.545361i	$-0.816393\pi$
$73$	−1045.59	−1.67640	−0.838201	+	0.545361i	$0.816393\pi$
$74$	−1111.14	−1.74550
$75$	0	0
$76$	1961.22	2.96010
$77$	−94.5327	−0.139909
$78$	0	0
$79$	466.940	0.664998	0.332499	−	0.943104i	$-0.392108\pi$
$79$	466.940	0.664998	0.332499	+	0.943104i	$0.392108\pi$
$80$	0	0
$81$	0	0
$82$	−276.723	−0.372669
$83$	1157.65	1.53095	0.765473	−	0.643468i	$-0.222505\pi$
$83$	1157.65	1.53095	0.765473	+	0.643468i	$0.222505\pi$
$84$	0	0
$85$	0	0
$86$	1496.63	1.87657
$87$	0	0
$88$	260.205	0.315204
$89$	1100.03	1.31015	0.655073	−	0.755566i	$-0.272638\pi$
$89$	1100.03	1.31015	0.655073	+	0.755566i	$0.272638\pi$
$90$	0	0
$91$	−473.261	−0.545178
$92$	1498.10	1.69769
$93$	0	0
$94$	221.098	0.242601
$95$	0	0
$96$	0	0
$97$	1167.99	1.22260	0.611299	−	0.791400i	$-0.290647\pi$
$97$	1167.99	1.22260	0.611299	+	0.791400i	$0.290647\pi$
$98$	−1237.61	−1.27569
$99$	0	0
$100$	0	0
$101$	1535.37	1.51263	0.756314	−	0.654209i	$-0.226998\pi$
$101$	1535.37	1.51263	0.756314	+	0.654209i	$0.226998\pi$
$102$	0	0
$103$	629.531	0.602229	0.301114	−	0.953588i	$-0.402641\pi$
$103$	629.531	0.602229	0.301114	+	0.953588i	$0.402641\pi$
$104$	1302.67	1.22824
$105$	0	0
$106$	−1511.64	−1.38513
$107$	795.146	0.718408	0.359204	−	0.933259i	$-0.383048\pi$
$107$	795.146	0.718408	0.359204	+	0.933259i	$0.383048\pi$
$108$	0	0
$109$	1123.20	0.987005	0.493502	−	0.869745i	$-0.335716\pi$
$109$	1123.20	0.987005	0.493502	+	0.869745i	$0.335716\pi$
$110$	0	0
$111$	0	0
$112$	−31.0950	−0.0262340
$113$	−613.965	−0.511124	−0.255562	−	0.966793i	$-0.582260\pi$
$113$	−613.965	−0.511124	−0.255562	+	0.966793i	$0.582260\pi$
$114$	0	0
$115$	0	0
$116$	3930.48	3.14600
$117$	0	0
$118$	1665.45	1.29930
$119$	887.315	0.683529
$120$	0	0
$121$	121.000	0.0909091
$122$	2093.85	1.55384
$123$	0	0
$124$	−2657.47	−1.92458
$125$	0	0
$126$	0	0
$127$	1656.76	1.15759	0.578793	−	0.815475i	$-0.303524\pi$
$127$	1656.76	1.15759	0.578793	+	0.815475i	$0.303524\pi$
$128$	−2401.84	−1.65855
$129$	0	0
$130$	0	0
$131$	1243.45	0.829316	0.414658	−	0.909977i	$-0.363901\pi$
$131$	1243.45	0.829316	0.414658	+	0.909977i	$0.363901\pi$
$132$	0	0
$133$	−1282.27	−0.835990
$134$	3448.19	2.22297
$135$	0	0
$136$	−2442.37	−1.53994
$137$	−2275.02	−1.41875	−0.709374	−	0.704832i	$-0.751022\pi$
$137$	−2275.02	−1.41875	−0.709374	+	0.704832i	$0.751022\pi$
$138$	0	0
$139$	−662.925	−0.404522	−0.202261	−	0.979332i	$-0.564829\pi$
$139$	−662.925	−0.404522	−0.202261	+	0.979332i	$0.564829\pi$
$140$	0	0
$141$	0	0
$142$	2307.25	1.36352
$143$	605.765	0.354242
$144$	0	0
$145$	0	0
$146$	−4807.94	−2.72540
$147$	0	0
$148$	−3176.20	−1.76407
$149$	−2647.71	−1.45577	−0.727883	−	0.685701i	$-0.759496\pi$
$149$	−2647.71	−1.45577	−0.727883	+	0.685701i	$0.759496\pi$
$150$	0	0
$151$	−503.541	−0.271375	−0.135687	−	0.990752i	$-0.543324\pi$
$151$	−503.541	−0.271375	−0.135687	+	0.990752i	$0.543324\pi$
$152$	3529.49	1.88342
$153$	0	0
$154$	−434.689	−0.227456
$155$	0	0
$156$	0	0
$157$	−1501.10	−0.763062	−0.381531	−	0.924356i	$-0.624603\pi$
$157$	−1501.10	−0.763062	−0.381531	+	0.924356i	$0.624603\pi$
$158$	2147.13	1.08112
$159$	0	0
$160$	0	0
$161$	−979.473	−0.479462
$162$	0	0
$163$	−836.617	−0.402018	−0.201009	−	0.979589i	$-0.564422\pi$
$163$	−836.617	−0.402018	−0.201009	+	0.979589i	$0.564422\pi$
$164$	−791.016	−0.376634
$165$	0	0
$166$	5323.21	2.48892
$167$	−1096.26	−0.507969	−0.253984	−	0.967208i	$-0.581741\pi$
$167$	−1096.26	−0.507969	−0.253984	+	0.967208i	$0.581741\pi$
$168$	0	0
$169$	835.651	0.380360
$170$	0	0
$171$	0	0
$172$	4278.13	1.89654
$173$	4151.11	1.82430	0.912149	−	0.409860i	$-0.134422\pi$
$173$	4151.11	1.82430	0.912149	+	0.409860i	$0.134422\pi$
$174$	0	0
$175$	0	0
$176$	39.8010	0.0170461
$177$	0	0
$178$	5058.26	2.12996
$179$	−44.1421	−0.0184320	−0.00921602	−	0.999958i	$-0.502934\pi$
$179$	−44.1421	−0.0184320	−0.00921602	+	0.999958i	$0.502934\pi$
$180$	0	0
$181$	−404.454	−0.166093	−0.0830465	−	0.996546i	$-0.526465\pi$
$181$	−404.454	−0.166093	−0.0830465	+	0.996546i	$0.526465\pi$
$182$	−2176.19	−0.886319
$183$	0	0
$184$	2696.04	1.08019
$185$	0	0
$186$	0	0
$187$	−1135.75	−0.444139
$188$	632.011	0.245182
$189$	0	0
$190$	0	0
$191$	136.732	0.0517990	0.0258995	−	0.999665i	$-0.491755\pi$
$191$	136.732	0.0517990	0.0258995	+	0.999665i	$0.491755\pi$
$192$	0	0
$193$	1219.61	0.454868	0.227434	−	0.973793i	$-0.426966\pi$
$193$	1219.61	0.454868	0.227434	+	0.973793i	$0.426966\pi$
$194$	5370.78	1.98763
$195$	0	0
$196$	−3537.73	−1.28926
$197$	−297.888	−0.107734	−0.0538671	−	0.998548i	$-0.517155\pi$
$197$	−297.888	−0.107734	−0.0538671	+	0.998548i	$0.517155\pi$
$198$	0	0
$199$	1854.42	0.660583	0.330292	−	0.943879i	$-0.392853\pi$
$199$	1854.42	0.660583	0.330292	+	0.943879i	$0.392853\pi$
$200$	0	0
$201$	0	0
$202$	7060.10	2.45914
$203$	−2569.79	−0.888493
$204$	0	0
$205$	0	0
$206$	2894.77	0.979068
$207$	0	0
$208$	199.257	0.0664229
$209$	1641.28	0.543203
$210$	0	0
$211$	389.567	0.127104	0.0635519	−	0.997979i	$-0.479757\pi$
$211$	389.567	0.127104	0.0635519	+	0.997979i	$0.479757\pi$
$212$	−4321.06	−1.39987
$213$	0	0
$214$	3656.31	1.16795
$215$	0	0
$216$	0	0
$217$	1737.48	0.543538
$218$	5164.82	1.60461
$219$	0	0
$220$	0	0
$221$	−5685.90	−1.73066
$222$	0	0
$223$	−148.027	−0.0444513	−0.0222257	−	0.999753i	$-0.507075\pi$
$223$	−148.027	−0.0444513	−0.0222257	+	0.999753i	$0.507075\pi$
$224$	1483.32	0.442450
$225$	0	0
$226$	−2823.19	−0.830956
$227$	−274.330	−0.0802112	−0.0401056	−	0.999195i	$-0.512769\pi$
$227$	−274.330	−0.0802112	−0.0401056	+	0.999195i	$0.512769\pi$
$228$	0	0
$229$	−290.157	−0.0837297	−0.0418648	−	0.999123i	$-0.513330\pi$
$229$	−290.157	−0.0837297	−0.0418648	+	0.999123i	$0.513330\pi$
$230$	0	0
$231$	0	0
$232$	7073.46	2.00170
$233$	4433.10	1.24645	0.623224	−	0.782044i	$-0.285823\pi$
$233$	4433.10	1.24645	0.623224	+	0.782044i	$0.285823\pi$
$234$	0	0
$235$	0	0
$236$	4760.71	1.31312
$237$	0	0
$238$	4080.13	1.11124
$239$	−2795.84	−0.756685	−0.378343	−	0.925666i	$-0.623506\pi$
$239$	−2795.84	−0.756685	−0.378343	+	0.925666i	$0.623506\pi$
$240$	0	0
$241$	−76.6901	−0.0204981	−0.0102491	−	0.999947i	$-0.503262\pi$
$241$	−76.6901	−0.0204981	−0.0102491	+	0.999947i	$0.503262\pi$
$242$	556.394	0.147795
$243$	0	0
$244$	5985.32	1.57037
$245$	0	0
$246$	0	0
$247$	8216.75	2.11668
$248$	−4782.48	−1.22455
$249$	0	0
$250$	0	0
$251$	−3880.97	−0.975955	−0.487977	−	0.872856i	$-0.662265\pi$
$251$	−3880.97	−0.975955	−0.487977	+	0.872856i	$0.662265\pi$
$252$	0	0
$253$	1253.71	0.311541
$254$	7618.25	1.88194
$255$	0	0
$256$	−4463.39	−1.08969
$257$	−2695.47	−0.654236	−0.327118	−	0.944983i	$-0.606078\pi$
$257$	−2695.47	−0.654236	−0.327118	+	0.944983i	$0.606078\pi$
$258$	0	0
$259$	2076.63	0.498208
$260$	0	0
$261$	0	0
$262$	5717.73	1.34825
$263$	493.066	0.115604	0.0578018	−	0.998328i	$-0.481591\pi$
$263$	493.066	0.115604	0.0578018	+	0.998328i	$0.481591\pi$
$264$	0	0
$265$	0	0
$266$	−5896.24	−1.35910
$267$	0	0
$268$	9856.72	2.24662
$269$	−3210.92	−0.727782	−0.363891	−	0.931442i	$-0.618552\pi$
$269$	−3210.92	−0.727782	−0.363891	+	0.931442i	$0.618552\pi$
$270$	0	0
$271$	−8659.91	−1.94115	−0.970576	−	0.240796i	$-0.922592\pi$
$271$	−8659.91	−1.94115	−0.970576	+	0.240796i	$0.922592\pi$
$272$	−373.585	−0.0832792
$273$	0	0
$274$	−10461.2	−2.30652
$275$	0	0
$276$	0	0
$277$	−7272.40	−1.57746	−0.788730	−	0.614740i	$-0.789261\pi$
$277$	−7272.40	−1.57746	−0.788730	+	0.614740i	$0.789261\pi$
$278$	−3048.33	−0.657649
$279$	0	0
$280$	0	0
$281$	3598.41	0.763926	0.381963	−	0.924178i	$-0.375248\pi$
$281$	3598.41	0.763926	0.381963	+	0.924178i	$0.375248\pi$
$282$	0	0
$283$	506.127	0.106311	0.0531557	−	0.998586i	$-0.483072\pi$
$283$	506.127	0.106311	0.0531557	+	0.998586i	$0.483072\pi$
$284$	6595.30	1.37803
$285$	0	0
$286$	2785.48	0.575906
$287$	517.175	0.106369
$288$	0	0
$289$	5747.48	1.16985
$290$	0	0
$291$	0	0
$292$	−13743.6	−2.75439
$293$	−3069.59	−0.612038	−0.306019	−	0.952025i	$-0.598997\pi$
$293$	−3069.59	−0.612038	−0.306019	+	0.952025i	$0.598997\pi$
$294$	0	0
$295$	0	0
$296$	−5716.02	−1.12242
$297$	0	0
$298$	−12175.0	−2.36670
$299$	6276.46	1.21397
$300$	0	0
$301$	−2797.09	−0.535619
$302$	−2315.43	−0.441185
$303$	0	0
$304$	539.872	0.101855
$305$	0	0
$306$	0	0
$307$	10587.6	1.96830	0.984148	−	0.177350i	$-0.0567526\pi$
$307$	10587.6	1.96830	0.984148	+	0.177350i	$0.0567526\pi$
$308$	−1242.57	−0.229876
$309$	0	0
$310$	0	0
$311$	−295.782	−0.0539300	−0.0269650	−	0.999636i	$-0.508584\pi$
$311$	−295.782	−0.0539300	−0.0269650	+	0.999636i	$0.508584\pi$
$312$	0	0
$313$	−5835.72	−1.05385	−0.526924	−	0.849912i	$-0.676655\pi$
$313$	−5835.72	−1.05385	−0.526924	+	0.849912i	$0.676655\pi$
$314$	−6902.49	−1.24054
$315$	0	0
$316$	6137.60	1.09262
$317$	−1217.65	−0.215741	−0.107871	−	0.994165i	$-0.534403\pi$
$317$	−1217.65	−0.215741	−0.107871	+	0.994165i	$0.534403\pi$
$318$	0	0
$319$	3289.28	0.577318
$320$	0	0
$321$	0	0
$322$	−4503.91	−0.779481
$323$	−15405.6	−2.65383
$324$	0	0
$325$	0	0
$326$	−3847.01	−0.653577
$327$	0	0
$328$	−1423.54	−0.239641
$329$	−413.215	−0.0692441
$330$	0	0
$331$	−4236.59	−0.703516	−0.351758	−	0.936091i	$-0.614416\pi$
$331$	−4236.59	−0.703516	−0.351758	+	0.936091i	$0.614416\pi$
$332$	15216.5	2.51540
$333$	0	0
$334$	−5040.90	−0.825826
$335$	0	0
$336$	0	0
$337$	−4119.34	−0.665859	−0.332930	−	0.942952i	$-0.608037\pi$
$337$	−4119.34	−0.665859	−0.332930	+	0.942952i	$0.608037\pi$
$338$	3842.57	0.618367
$339$	0	0
$340$	0	0
$341$	−2223.94	−0.353176
$342$	0	0
$343$	5260.70	0.828138
$344$	7699.10	1.20671
$345$	0	0
$346$	19088.0	2.96584
$347$	−4702.13	−0.727446	−0.363723	−	0.931507i	$-0.618494\pi$
$347$	−4702.13	−0.727446	−0.363723	+	0.931507i	$0.618494\pi$
$348$	0	0
$349$	−2988.78	−0.458412	−0.229206	−	0.973378i	$-0.573613\pi$
$349$	−2988.78	−0.458412	−0.229206	+	0.973378i	$0.573613\pi$
$350$	0	0
$351$	0	0
$352$	−1898.62	−0.287492
$353$	3152.46	0.475321	0.237661	−	0.971348i	$-0.423619\pi$
$353$	3152.46	0.475321	0.237661	+	0.971348i	$0.423619\pi$
$354$	0	0
$355$	0	0
$356$	14459.1	2.15262
$357$	0	0
$358$	−202.978	−0.0299657
$359$	6882.82	1.01187	0.505935	−	0.862572i	$-0.331148\pi$
$359$	6882.82	1.01187	0.505935	+	0.862572i	$0.331148\pi$
$360$	0	0
$361$	15403.7	2.24577
$362$	−1859.80	−0.270024
$363$	0	0
$364$	−6220.68	−0.895748
$365$	0	0
$366$	0	0
$367$	−2632.89	−0.374484	−0.187242	−	0.982314i	$-0.559955\pi$
$367$	−2632.89	−0.374484	−0.187242	+	0.982314i	$0.559955\pi$
$368$	412.387	0.0584162
$369$	0	0
$370$	0	0
$371$	2825.15	0.395349
$372$	0	0
$373$	3867.85	0.536916	0.268458	−	0.963291i	$-0.413486\pi$
$373$	3867.85	0.536916	0.268458	+	0.963291i	$0.413486\pi$
$374$	−5222.49	−0.722055
$375$	0	0
$376$	1137.39	0.156001
$377$	16467.2	2.24961
$378$	0	0
$379$	−7265.17	−0.984661	−0.492331	−	0.870408i	$-0.663855\pi$
$379$	−7265.17	−0.984661	−0.492331	+	0.870408i	$0.663855\pi$
$380$	0	0
$381$	0	0
$382$	628.736	0.0842119
$383$	−7232.37	−0.964900	−0.482450	−	0.875924i	$-0.660253\pi$
$383$	−7232.37	−0.964900	−0.482450	+	0.875924i	$0.660253\pi$
$384$	0	0
$385$	0	0
$386$	5608.13	0.739499
$387$	0	0
$388$	15352.5	2.00877
$389$	−2093.47	−0.272861	−0.136431	−	0.990650i	$-0.543563\pi$
$389$	−2093.47	−0.272861	−0.136431	+	0.990650i	$0.543563\pi$
$390$	0	0
$391$	−11767.7	−1.52204
$392$	−6366.63	−0.820315
$393$	0	0
$394$	−1369.78	−0.175148
$395$	0	0
$396$	0	0
$397$	3727.71	0.471256	0.235628	−	0.971843i	$-0.424285\pi$
$397$	3727.71	0.471256	0.235628	+	0.971843i	$0.424285\pi$
$398$	8527.15	1.07394
$399$	0	0
$400$	0	0
$401$	−29.2658	−0.00364455	−0.00182228	−	0.999998i	$-0.500580\pi$
$401$	−29.2658	−0.00364455	−0.00182228	+	0.999998i	$0.500580\pi$
$402$	0	0
$403$	−11133.7	−1.37621
$404$	20181.4	2.48531
$405$	0	0
$406$	−11816.7	−1.44446
$407$	−2658.05	−0.323722
$408$	0	0
$409$	−4944.65	−0.597793	−0.298896	−	0.954286i	$-0.596619\pi$
$409$	−4944.65	−0.597793	−0.298896	+	0.954286i	$0.596619\pi$
$410$	0	0
$411$	0	0
$412$	8274.75	0.989484
$413$	−3112.60	−0.370850
$414$	0	0
$415$	0	0
$416$	−9505.12	−1.12026
$417$	0	0
$418$	7547.07	0.883108
$419$	−7421.27	−0.865281	−0.432640	−	0.901567i	$-0.642418\pi$
$419$	−7421.27	−0.865281	−0.432640	+	0.901567i	$0.642418\pi$
$420$	0	0
$421$	−9484.63	−1.09799	−0.548994	−	0.835826i	$-0.684989\pi$
$421$	−9484.63	−1.09799	−0.548994	+	0.835826i	$0.684989\pi$
$422$	1791.34	0.206638
$423$	0	0
$424$	−7776.35	−0.890691
$425$	0	0
$426$	0	0
$427$	−3913.26	−0.443504
$428$	10451.6	1.18037
$429$	0	0
$430$	0	0
$431$	5121.20	0.572342	0.286171	−	0.958179i	$-0.407617\pi$
$431$	5121.20	0.572342	0.286171	+	0.958179i	$0.407617\pi$
$432$	0	0
$433$	−17259.9	−1.91561	−0.957805	−	0.287419i	$-0.907203\pi$
$433$	−17259.9	−1.91561	−0.957805	+	0.287419i	$0.907203\pi$
$434$	7989.44	0.883653
$435$	0	0
$436$	14763.7	1.62169
$437$	17005.6	1.86153
$438$	0	0
$439$	11840.4	1.28727	0.643635	−	0.765333i	$-0.277426\pi$
$439$	11840.4	1.28727	0.643635	+	0.765333i	$0.277426\pi$
$440$	0	0
$441$	0	0
$442$	−26145.5	−2.81360
$443$	−2019.82	−0.216624	−0.108312	−	0.994117i	$-0.534545\pi$
$443$	−2019.82	−0.216624	−0.108312	+	0.994117i	$0.534545\pi$
$444$	0	0
$445$	0	0
$446$	−680.673	−0.0722664
$447$	0	0
$448$	7069.52	0.745543
$449$	−8942.85	−0.939953	−0.469977	−	0.882679i	$-0.655738\pi$
$449$	−8942.85	−0.939953	−0.469977	+	0.882679i	$0.655738\pi$
$450$	0	0
$451$	−661.973	−0.0691155
$452$	−8070.15	−0.839796
$453$	0	0
$454$	−1261.45	−0.130403
$455$	0	0
$456$	0	0
$457$	−11548.3	−1.18207	−0.591035	−	0.806646i	$-0.701281\pi$
$457$	−11548.3	−1.18207	−0.591035	+	0.806646i	$0.701281\pi$
$458$	−1334.23	−0.136123
$459$	0	0
$460$	0	0
$461$	939.830	0.0949507	0.0474753	−	0.998872i	$-0.484882\pi$
$461$	939.830	0.0949507	0.0474753	+	0.998872i	$0.484882\pi$
$462$	0	0
$463$	−14914.6	−1.49706	−0.748532	−	0.663099i	$-0.769241\pi$
$463$	−14914.6	−1.49706	−0.748532	+	0.663099i	$0.769241\pi$
$464$	1081.96	0.108251
$465$	0	0
$466$	20384.7	2.02640
$467$	−15742.2	−1.55988	−0.779939	−	0.625855i	$-0.784750\pi$
$467$	−15742.2	−1.55988	−0.779939	+	0.625855i	$0.784750\pi$
$468$	0	0
$469$	−6444.43	−0.634490
$470$	0	0
$471$	0	0
$472$	8567.57	0.835497
$473$	3580.22	0.348031
$474$	0	0
$475$	0	0
$476$	11663.1	1.12306
$477$	0	0
$478$	−12856.1	−1.23018
$479$	19078.8	1.81990	0.909948	−	0.414721i	$-0.136121\pi$
$479$	19078.8	1.81990	0.909948	+	0.414721i	$0.136121\pi$
$480$	0	0
$481$	−13307.1	−1.26143
$482$	−352.644	−0.0333246
$483$	0	0
$484$	1590.46	0.149367
$485$	0	0
$486$	0	0
$487$	−16536.5	−1.53869	−0.769344	−	0.638835i	$-0.779417\pi$
$487$	−16536.5	−1.53869	−0.769344	+	0.638835i	$0.779417\pi$
$488$	10771.4	0.999178
$489$	0	0
$490$	0	0
$491$	5603.03	0.514992	0.257496	−	0.966279i	$-0.417103\pi$
$491$	5603.03	0.514992	0.257496	+	0.966279i	$0.417103\pi$
$492$	0	0
$493$	−30874.3	−2.82050
$494$	37783.0	3.44117
$495$	0	0
$496$	−731.529	−0.0662231
$497$	−4312.08	−0.389181
$498$	0	0
$499$	5679.93	0.509557	0.254778	−	0.966999i	$-0.417998\pi$
$499$	5679.93	0.509557	0.254778	+	0.966999i	$0.417998\pi$
$500$	0	0
$501$	0	0
$502$	−17845.8	−1.58665
$503$	15590.1	1.38196	0.690982	−	0.722872i	$-0.257178\pi$
$503$	15590.1	1.38196	0.690982	+	0.722872i	$0.257178\pi$
$504$	0	0
$505$	0	0
$506$	5764.91	0.506485
$507$	0	0
$508$	21776.9	1.90196
$509$	12605.1	1.09766	0.548831	−	0.835933i	$-0.315073\pi$
$509$	12605.1	1.09766	0.548831	+	0.835933i	$0.315073\pi$
$510$	0	0
$511$	8985.70	0.777894
$512$	−1309.25	−0.113010
$513$	0	0
$514$	−12394.6	−1.06362
$515$	0	0
$516$	0	0
$517$	528.908	0.0449929
$518$	9548.98	0.809957
$519$	0	0
$520$	0	0
$521$	18743.7	1.57615	0.788075	−	0.615579i	$-0.211078\pi$
$521$	18743.7	1.57615	0.788075	+	0.615579i	$0.211078\pi$
$522$	0	0
$523$	2374.83	0.198555	0.0992773	−	0.995060i	$-0.468347\pi$
$523$	2374.83	0.198555	0.0992773	+	0.995060i	$0.468347\pi$
$524$	16344.2	1.36260
$525$	0	0
$526$	2267.26	0.187942
$527$	20874.6	1.72545
$528$	0	0
$529$	822.919	0.0676353
$530$	0	0
$531$	0	0
$532$	−16854.5	−1.37356
$533$	−3314.05	−0.269320
$534$	0	0
$535$	0	0
$536$	17738.5	1.42946
$537$	0	0
$538$	−14764.8	−1.18319
$539$	−2960.60	−0.236590
$540$	0	0
$541$	16031.8	1.27405	0.637025	−	0.770843i	$-0.280165\pi$
$541$	16031.8	1.27405	0.637025	+	0.770843i	$0.280165\pi$
$542$	−39820.8	−3.15581
$543$	0	0
$544$	17821.1	1.40455
$545$	0	0
$546$	0	0
$547$	−18781.1	−1.46805	−0.734023	−	0.679125i	$-0.762360\pi$
$547$	−18781.1	−1.46805	−0.734023	+	0.679125i	$0.762360\pi$
$548$	−29903.6	−2.33106
$549$	0	0
$550$	0	0
$551$	44616.7	3.44961
$552$	0	0
$553$	−4012.83	−0.308576
$554$	−33440.6	−2.56454
$555$	0	0
$556$	−8713.69	−0.664645
$557$	−7859.50	−0.597877	−0.298939	−	0.954272i	$-0.596633\pi$
$557$	−7859.50	−0.597877	−0.298939	+	0.954272i	$0.596633\pi$
$558$	0	0
$559$	17923.7	1.35616
$560$	0	0
$561$	0	0
$562$	16546.6	1.24195
$563$	−16139.1	−1.20814	−0.604071	−	0.796931i	$-0.706456\pi$
$563$	−16139.1	−1.20814	−0.604071	+	0.796931i	$0.706456\pi$
$564$	0	0
$565$	0	0
$566$	2327.32	0.172835
$567$	0	0
$568$	11869.2	0.876795
$569$	13515.4	0.995776	0.497888	−	0.867241i	$-0.334109\pi$
$569$	13515.4	0.995776	0.497888	+	0.867241i	$0.334109\pi$
$570$	0	0
$571$	21350.5	1.56479	0.782393	−	0.622785i	$-0.213999\pi$
$571$	21350.5	1.56479	0.782393	+	0.622785i	$0.213999\pi$
$572$	7962.35	0.582033
$573$	0	0
$574$	2378.12	0.172928
$575$	0	0
$576$	0	0
$577$	−21993.3	−1.58681	−0.793407	−	0.608692i	$-0.791695\pi$
$577$	−21993.3	−1.58681	−0.793407	+	0.608692i	$0.791695\pi$
$578$	26428.6	1.90188
$579$	0	0
$580$	0	0
$581$	−9948.70	−0.710399
$582$	0	0
$583$	−3616.14	−0.256887
$584$	−24733.5	−1.75253
$585$	0	0
$586$	−14114.9	−0.995017
$587$	4013.68	0.282218	0.141109	−	0.989994i	$-0.454933\pi$
$587$	4013.68	0.282218	0.141109	+	0.989994i	$0.454933\pi$
$588$	0	0
$589$	−30166.1	−2.11031
$590$	0	0
$591$	0	0
$592$	−874.324	−0.0607002
$593$	9422.79	0.652525	0.326263	−	0.945279i	$-0.394211\pi$
$593$	9422.79	0.652525	0.326263	+	0.945279i	$0.394211\pi$
$594$	0	0
$595$	0	0
$596$	−34802.4	−2.39188
$597$	0	0
$598$	28861.0	1.97360
$599$	−6148.99	−0.419434	−0.209717	−	0.977762i	$-0.567254\pi$
$599$	−6148.99	−0.419434	−0.209717	+	0.977762i	$0.567254\pi$
$600$	0	0
$601$	−3087.16	−0.209531	−0.104765	−	0.994497i	$-0.533409\pi$
$601$	−3087.16	−0.209531	−0.104765	+	0.994497i	$0.533409\pi$
$602$	−12861.8	−0.870779
$603$	0	0
$604$	−6618.70	−0.445879
$605$	0	0
$606$	0	0
$607$	−12295.7	−0.822186	−0.411093	−	0.911593i	$-0.634853\pi$
$607$	−12295.7	−0.822186	−0.411093	+	0.911593i	$0.634853\pi$
$608$	−25753.5	−1.71783
$609$	0	0
$610$	0	0
$611$	2647.88	0.175322
$612$	0	0
$613$	4949.57	0.326119	0.163060	−	0.986616i	$-0.447864\pi$
$613$	4949.57	0.326119	0.163060	+	0.986616i	$0.447864\pi$
$614$	48684.9	3.19994
$615$	0	0
$616$	−2236.17	−0.146263
$617$	6590.92	0.430049	0.215025	−	0.976609i	$-0.431017\pi$
$617$	6590.92	0.430049	0.215025	+	0.976609i	$0.431017\pi$
$618$	0	0
$619$	8097.85	0.525816	0.262908	−	0.964821i	$-0.415318\pi$
$619$	8097.85	0.525816	0.262908	+	0.964821i	$0.415318\pi$
$620$	0	0
$621$	0	0
$622$	−1360.09	−0.0876764
$623$	−9453.53	−0.607942
$624$	0	0
$625$	0	0
$626$	−26834.4	−1.71329
$627$	0	0
$628$	−19730.9	−1.25374
$629$	24949.3	1.58155
$630$	0	0
$631$	20435.6	1.28927	0.644633	−	0.764492i	$-0.277010\pi$
$631$	20435.6	1.28927	0.644633	+	0.764492i	$0.277010\pi$
$632$	11045.5	0.695198
$633$	0	0
$634$	−5599.10	−0.350739
$635$	0	0
$636$	0	0
$637$	−14821.7	−0.921910
$638$	15125.1	0.938571
$639$	0	0
$640$	0	0
$641$	23616.9	1.45525	0.727623	−	0.685977i	$-0.240625\pi$
$641$	23616.9	1.45525	0.727623	+	0.685977i	$0.240625\pi$
$642$	0	0
$643$	−1914.53	−0.117421	−0.0587104	−	0.998275i	$-0.518699\pi$
$643$	−1914.53	−0.117421	−0.0587104	+	0.998275i	$0.518699\pi$
$644$	−12874.5	−0.787773
$645$	0	0
$646$	−70839.3	−4.31445
$647$	−7220.11	−0.438720	−0.219360	−	0.975644i	$-0.570397\pi$
$647$	−7220.11	−0.438720	−0.219360	+	0.975644i	$0.570397\pi$
$648$	0	0
$649$	3984.07	0.240968
$650$	0	0
$651$	0	0
$652$	−10996.7	−0.660530
$653$	19852.5	1.18972	0.594861	−	0.803828i	$-0.297207\pi$
$653$	19852.5	1.18972	0.594861	+	0.803828i	$0.297207\pi$
$654$	0	0
$655$	0	0
$656$	−217.746	−0.0129597
$657$	0	0
$658$	−1900.09	−0.112573
$659$	−13246.8	−0.783038	−0.391519	−	0.920170i	$-0.628050\pi$
$659$	−13246.8	−0.783038	−0.391519	+	0.920170i	$0.628050\pi$
$660$	0	0
$661$	−13195.7	−0.776481	−0.388241	−	0.921558i	$-0.626917\pi$
$661$	−13195.7	−0.776481	−0.388241	+	0.921558i	$0.626917\pi$
$662$	−19481.1	−1.14374
$663$	0	0
$664$	27384.2	1.60047
$665$	0	0
$666$	0	0
$667$	34081.0	1.97844
$668$	−14409.5	−0.834612
$669$	0	0
$670$	0	0
$671$	5008.90	0.288176
$672$	0	0
$673$	5500.31	0.315039	0.157520	−	0.987516i	$-0.449650\pi$
$673$	5500.31	0.315039	0.157520	+	0.987516i	$0.449650\pi$
$674$	−18941.9	−1.08252
$675$	0	0
$676$	10984.0	0.624946
$677$	2690.87	0.152760	0.0763801	−	0.997079i	$-0.475664\pi$
$677$	2690.87	0.152760	0.0763801	+	0.997079i	$0.475664\pi$
$678$	0	0
$679$	−10037.6	−0.567317
$680$	0	0
$681$	0	0
$682$	−10226.3	−0.574173
$683$	30673.0	1.71840	0.859202	−	0.511636i	$-0.170960\pi$
$683$	30673.0	1.71840	0.859202	+	0.511636i	$0.170960\pi$
$684$	0	0
$685$	0	0
$686$	24190.3	1.34634
$687$	0	0
$688$	1177.66	0.0652583
$689$	−18103.5	−1.00100
$690$	0	0
$691$	−13853.8	−0.762694	−0.381347	−	0.924432i	$-0.624540\pi$
$691$	−13853.8	−0.762694	−0.381347	+	0.924432i	$0.624540\pi$
$692$	54563.5	2.99739
$693$	0	0
$694$	−21621.8	−1.18264
$695$	0	0
$696$	0	0
$697$	6213.50	0.337666
$698$	−13743.3	−0.745259
$699$	0	0
$700$	0	0
$701$	−27677.6	−1.49125	−0.745627	−	0.666364i	$-0.767850\pi$
$701$	−27677.6	−1.49125	−0.745627	+	0.666364i	$0.767850\pi$
$702$	0	0
$703$	−36054.5	−1.93431
$704$	−9048.84	−0.484433
$705$	0	0
$706$	14495.9	0.772750
$707$	−13194.8	−0.701899
$708$	0	0
$709$	−11530.3	−0.610760	−0.305380	−	0.952231i	$-0.598783\pi$
$709$	−11530.3	−0.610760	−0.305380	+	0.952231i	$0.598783\pi$
$710$	0	0
$711$	0	0
$712$	26021.2	1.36964
$713$	−23042.7	−1.21032
$714$	0	0
$715$	0	0
$716$	−580.217	−0.0302845
$717$	0	0
$718$	31649.2	1.64504
$719$	4960.75	0.257309	0.128654	−	0.991690i	$-0.458934\pi$
$719$	4960.75	0.257309	0.128654	+	0.991690i	$0.458934\pi$
$720$	0	0
$721$	−5410.12	−0.279450
$722$	70830.8	3.65104
$723$	0	0
$724$	−5316.27	−0.272897
$725$	0	0
$726$	0	0
$727$	−25475.4	−1.29963	−0.649814	−	0.760093i	$-0.725153\pi$
$727$	−25475.4	−1.29963	−0.649814	+	0.760093i	$0.725153\pi$
$728$	−11195.0	−0.569937
$729$	0	0
$730$	0	0
$731$	−33605.1	−1.70031
$732$	0	0
$733$	1368.63	0.0689652	0.0344826	−	0.999405i	$-0.489022\pi$
$733$	1368.63	0.0689652	0.0344826	+	0.999405i	$0.489022\pi$
$734$	−12106.8	−0.608815
$735$	0	0
$736$	−19672.0	−0.985219
$737$	8248.74	0.412274
$738$	0	0
$739$	39022.4	1.94244	0.971220	−	0.238185i	$-0.0765524\pi$
$739$	39022.4	1.94244	0.971220	+	0.238185i	$0.0765524\pi$
$740$	0	0
$741$	0	0
$742$	12990.9	0.642736
$743$	−20152.1	−0.995034	−0.497517	−	0.867454i	$-0.665755\pi$
$743$	−20152.1	−0.995034	−0.497517	+	0.867454i	$0.665755\pi$
$744$	0	0
$745$	0	0
$746$	17785.5	0.872886
$747$	0	0
$748$	−14928.6	−0.729737
$749$	−6833.39	−0.333360
$750$	0	0
$751$	−10292.2	−0.500088	−0.250044	−	0.968234i	$-0.580445\pi$
$751$	−10292.2	−0.500088	−0.250044	+	0.968234i	$0.580445\pi$
$752$	173.976	0.00843650
$753$	0	0
$754$	75721.0	3.65729
$755$	0	0
$756$	0	0
$757$	−7740.15	−0.371625	−0.185813	−	0.982585i	$-0.559492\pi$
$757$	−7740.15	−0.371625	−0.185813	+	0.982585i	$0.559492\pi$
$758$	−33407.4	−1.60081
$759$	0	0
$760$	0	0
$761$	−2028.01	−0.0966036	−0.0483018	−	0.998833i	$-0.515381\pi$
$761$	−2028.01	−0.0966036	−0.0483018	+	0.998833i	$0.515381\pi$
$762$	0	0
$763$	−9652.69	−0.457996
$764$	1797.25	0.0851078
$765$	0	0
$766$	−33256.5	−1.56868
$767$	19945.5	0.938972
$768$	0	0
$769$	15647.3	0.733753	0.366877	−	0.930270i	$-0.380427\pi$
$769$	15647.3	0.733753	0.366877	+	0.930270i	$0.380427\pi$
$770$	0	0
$771$	0	0
$772$	16031.0	0.747366
$773$	20997.5	0.977006	0.488503	−	0.872562i	$-0.337543\pi$
$773$	20997.5	0.977006	0.488503	+	0.872562i	$0.337543\pi$
$774$	0	0
$775$	0	0
$776$	27628.9	1.27812
$777$	0	0
$778$	−9626.38	−0.443602
$779$	−8979.18	−0.412982
$780$	0	0
$781$	5519.38	0.252879
$782$	−54111.3	−2.47445
$783$	0	0
$784$	−973.842	−0.0443623
$785$	0	0
$786$	0	0
$787$	−6367.55	−0.288410	−0.144205	−	0.989548i	$-0.546062\pi$
$787$	−6367.55	−0.288410	−0.144205	+	0.989548i	$0.546062\pi$
$788$	−3915.53	−0.177011
$789$	0	0
$790$	0	0
$791$	5276.35	0.237175
$792$	0	0
$793$	25076.1	1.12293
$794$	17141.1	0.766140
$795$	0	0
$796$	24375.0	1.08536
$797$	4774.74	0.212208	0.106104	−	0.994355i	$-0.466162\pi$
$797$	4774.74	0.212208	0.106104	+	0.994355i	$0.466162\pi$
$798$	0	0
$799$	−4964.50	−0.219814
$800$	0	0
$801$	0	0
$802$	−134.573	−0.00592510
$803$	−11501.5	−0.505454
$804$	0	0
$805$	0	0
$806$	−51196.2	−2.23736
$807$	0	0
$808$	36319.3	1.58132
$809$	−14702.8	−0.638966	−0.319483	−	0.947592i	$-0.603509\pi$
$809$	−14702.8	−0.638966	−0.319483	+	0.947592i	$0.603509\pi$
$810$	0	0
$811$	−37891.2	−1.64062	−0.820308	−	0.571922i	$-0.806198\pi$
$811$	−37891.2	−1.64062	−0.820308	+	0.571922i	$0.806198\pi$
$812$	−33778.1	−1.45983
$813$	0	0
$814$	−12222.5	−0.526288
$815$	0	0
$816$	0	0
$817$	48563.0	2.07956
$818$	−22737.0	−0.971857
$819$	0	0
$820$	0	0
$821$	13176.1	0.560107	0.280054	−	0.959984i	$-0.409648\pi$
$821$	13176.1	0.560107	0.280054	+	0.959984i	$0.409648\pi$
$822$	0	0
$823$	6003.58	0.254279	0.127140	−	0.991885i	$-0.459420\pi$
$823$	6003.58	0.254279	0.127140	+	0.991885i	$0.459420\pi$
$824$	14891.6	0.629578
$825$	0	0
$826$	−14312.7	−0.602907
$827$	−23525.9	−0.989209	−0.494605	−	0.869118i	$-0.664687\pi$
$827$	−23525.9	−0.989209	−0.494605	+	0.869118i	$0.664687\pi$
$828$	0	0
$829$	−17994.7	−0.753900	−0.376950	−	0.926234i	$-0.623027\pi$
$829$	−17994.7	−0.753900	−0.376950	+	0.926234i	$0.623027\pi$
$830$	0	0
$831$	0	0
$832$	−45301.4	−1.88767
$833$	27789.1	1.15587
$834$	0	0
$835$	0	0
$836$	21573.4	0.892503
$837$	0	0
$838$	−34125.2	−1.40672
$839$	2877.80	0.118418	0.0592090	−	0.998246i	$-0.481142\pi$
$839$	2877.80	0.118418	0.0592090	+	0.998246i	$0.481142\pi$
$840$	0	0
$841$	65027.4	2.66626
$842$	−43613.1	−1.78504
$843$	0	0
$844$	5120.59	0.208836
$845$	0	0
$846$	0	0
$847$	−1039.86	−0.0421842
$848$	−1189.47	−0.0481682
$849$	0	0
$850$	0	0
$851$	−27540.6	−1.10938
$852$	0	0
$853$	−24054.4	−0.965540	−0.482770	−	0.875747i	$-0.660369\pi$
$853$	−24054.4	−0.965540	−0.482770	+	0.875747i	$0.660369\pi$
$854$	−17994.3	−0.721023
$855$	0	0
$856$	18809.2	0.751033
$857$	9782.84	0.389936	0.194968	−	0.980810i	$-0.437540\pi$
$857$	9782.84	0.389936	0.194968	+	0.980810i	$0.437540\pi$
$858$	0	0
$859$	−16020.2	−0.636323	−0.318162	−	0.948036i	$-0.603066\pi$
$859$	−16020.2	−0.636323	−0.318162	+	0.948036i	$0.603066\pi$
$860$	0	0
$861$	0	0
$862$	23548.8	0.930481
$863$	26720.0	1.05395	0.526975	−	0.849881i	$-0.323326\pi$
$863$	26720.0	1.05395	0.526975	+	0.849881i	$0.323326\pi$
$864$	0	0
$865$	0	0
$866$	−79366.2	−3.11429
$867$	0	0
$868$	22837.9	0.893053
$869$	5136.34	0.200504
$870$	0	0
$871$	41295.8	1.60649
$872$	26569.4	1.03183
$873$	0	0
$874$	78196.8	3.02637
$875$	0	0
$876$	0	0
$877$	−31468.7	−1.21166	−0.605828	−	0.795596i	$-0.707158\pi$
$877$	−31468.7	−1.21166	−0.605828	+	0.795596i	$0.707158\pi$
$878$	54445.6	2.09277
$879$	0	0
$880$	0	0
$881$	−19563.6	−0.748142	−0.374071	−	0.927400i	$-0.622038\pi$
$881$	−19563.6	−0.748142	−0.374071	+	0.927400i	$0.622038\pi$
$882$	0	0
$883$	−45986.4	−1.75262	−0.876311	−	0.481746i	$-0.840003\pi$
$883$	−45986.4	−1.75262	−0.876311	+	0.481746i	$0.840003\pi$
$884$	−74737.2	−2.84354
$885$	0	0
$886$	−9287.73	−0.352175
$887$	−22600.3	−0.855516	−0.427758	−	0.903893i	$-0.640696\pi$
$887$	−22600.3	−0.855516	−0.427758	+	0.903893i	$0.640696\pi$
$888$	0	0
$889$	−14238.0	−0.537150
$890$	0	0
$891$	0	0
$892$	−1945.72	−0.0730352
$893$	7174.25	0.268843
$894$	0	0
$895$	0	0
$896$	20641.1	0.769611
$897$	0	0
$898$	−41121.8	−1.52812
$899$	−60455.9	−2.24284
$900$	0	0
$901$	33942.3	1.25503
$902$	−3043.95	−0.112364
$903$	0	0
$904$	−14523.4	−0.534336
$905$	0	0
$906$	0	0
$907$	28320.7	1.03680	0.518398	−	0.855140i	$-0.326529\pi$
$907$	28320.7	1.03680	0.518398	+	0.855140i	$0.326529\pi$
$908$	−3605.88	−0.131790
$909$	0	0
$910$	0	0
$911$	−34373.6	−1.25011	−0.625054	−	0.780582i	$-0.714923\pi$
$911$	−34373.6	−1.25011	−0.625054	+	0.780582i	$0.714923\pi$
$912$	0	0
$913$	12734.1	0.461598
$914$	−53102.4	−1.92174
$915$	0	0
$916$	−3813.91	−0.137571
$917$	−10686.0	−0.384824
$918$	0	0
$919$	−51419.2	−1.84566	−0.922831	−	0.385205i	$-0.874131\pi$
$919$	−51419.2	−1.84566	−0.922831	+	0.385205i	$0.874131\pi$
$920$	0	0
$921$	0	0
$922$	4321.62	0.154365
$923$	27631.8	0.985385
$924$	0	0
$925$	0	0
$926$	−68581.7	−2.43384
$927$	0	0
$928$	−51612.5	−1.82572
$929$	36062.2	1.27359	0.636793	−	0.771035i	$-0.280260\pi$
$929$	36062.2	1.27359	0.636793	+	0.771035i	$0.280260\pi$
$930$	0	0
$931$	−40158.3	−1.41368
$932$	58270.1	2.04796
$933$	0	0
$934$	−72387.4	−2.53596
$935$	0	0
$936$	0	0
$937$	−15270.1	−0.532393	−0.266196	−	0.963919i	$-0.585767\pi$
$937$	−15270.1	−0.532393	−0.266196	+	0.963919i	$0.585767\pi$
$938$	−29633.4	−1.03152
$939$	0	0
$940$	0	0
$941$	54312.6	1.88155	0.940777	−	0.339027i	$-0.110098\pi$
$941$	54312.6	1.88155	0.940777	+	0.339027i	$0.110098\pi$
$942$	0	0
$943$	−6858.85	−0.236855
$944$	1310.50	0.0451833
$945$	0	0
$946$	16462.9	0.565808
$947$	−14308.3	−0.490980	−0.245490	−	0.969399i	$-0.578949\pi$
$947$	−14308.3	−0.490980	−0.245490	+	0.969399i	$0.578949\pi$
$948$	0	0
$949$	−57580.3	−1.96958
$950$	0	0
$951$	0	0
$952$	20989.4	0.714571
$953$	−3405.22	−0.115746	−0.0578730	−	0.998324i	$-0.518432\pi$
$953$	−3405.22	−0.115746	−0.0578730	+	0.998324i	$0.518432\pi$
$954$	0	0
$955$	0	0
$956$	−36749.3	−1.24326
$957$	0	0
$958$	87729.7	2.95868
$959$	19551.3	0.658336
$960$	0	0
$961$	11084.2	0.372066
$962$	−61189.7	−2.05077
$963$	0	0
$964$	−1008.04	−0.0336792
$965$	0	0
$966$	0	0
$967$	−37008.2	−1.23072	−0.615358	−	0.788248i	$-0.710988\pi$
$967$	−37008.2	−1.23072	−0.615358	+	0.788248i	$0.710988\pi$
$968$	2862.26	0.0950376
$969$	0	0
$970$	0	0
$971$	−14892.3	−0.492190	−0.246095	−	0.969246i	$-0.579148\pi$
$971$	−14892.3	−0.492190	−0.246095	+	0.969246i	$0.579148\pi$
$972$	0	0
$973$	5697.10	0.187709
$974$	−76039.8	−2.50151
$975$	0	0
$976$	1647.60	0.0540352
$977$	10306.1	0.337483	0.168742	−	0.985660i	$-0.446030\pi$
$977$	10306.1	0.337483	0.168742	+	0.985660i	$0.446030\pi$
$978$	0	0
$979$	12100.3	0.395024
$980$	0	0
$981$	0	0
$982$	25764.4	0.837245
$983$	16103.5	0.522503	0.261251	−	0.965271i	$-0.415865\pi$
$983$	16103.5	0.522503	0.261251	+	0.965271i	$0.415865\pi$
$984$	0	0
$985$	0	0
$986$	−141969.	−4.58541
$987$	0	0
$988$	108004.	3.47778
$989$	37095.4	1.19268
$990$	0	0
$991$	−37230.7	−1.19341	−0.596707	−	0.802459i	$-0.703525\pi$
$991$	−37230.7	−1.19341	−0.596707	+	0.802459i	$0.703525\pi$
$992$	34896.1	1.11689
$993$	0	0
$994$	−19828.2	−0.632709
$995$	0	0
$996$	0	0
$997$	40064.4	1.27267	0.636335	−	0.771412i	$-0.280449\pi$
$997$	40064.4	1.27267	0.636335	+	0.771412i	$0.280449\pi$
$998$	26118.0	0.828408
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.ca.1.14		16
3.2	odd	2		2475.4.a.bz.1.3		16
5.2	odd	4		495.4.c.f.199.14	yes	16
5.3	odd	4		495.4.c.f.199.3	yes	16
5.4	even	2	inner	2475.4.a.ca.1.3		16
15.2	even	4		495.4.c.e.199.3	✓	16
15.8	even	4		495.4.c.e.199.14	yes	16
15.14	odd	2		2475.4.a.bz.1.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.c.e.199.3	✓	16	15.2	even	4
495.4.c.e.199.14	yes	16	15.8	even	4
495.4.c.f.199.3	yes	16	5.3	odd	4
495.4.c.f.199.14	yes	16	5.2	odd	4
2475.4.a.bz.1.3		16	3.2	odd	2
2475.4.a.bz.1.14		16	15.14	odd	2
2475.4.a.ca.1.3		16	5.4	even	2	inner
2475.4.a.ca.1.14		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+4.59829 q^{2} +13.1443 q^{4} -8.59388 q^{7} +23.6550 q^{8} +O(q^{10})\)	\(q+4.59829 q^{2} +13.1443 q^{4} -8.59388 q^{7} +23.6550 q^{8} +11.0000 q^{11} +55.0695 q^{13} -39.5172 q^{14} +3.61828 q^{16} -103.250 q^{17} +149.207 q^{19} +50.5812 q^{22} +113.973 q^{23} +253.226 q^{26} -112.961 q^{28} +299.026 q^{29} -202.176 q^{31} -172.602 q^{32} -474.772 q^{34} -241.641 q^{37} +686.097 q^{38} -60.1794 q^{41} +325.474 q^{43} +144.587 q^{44} +524.083 q^{46} +48.0825 q^{47} -269.145 q^{49} +723.850 q^{52} -328.740 q^{53} -203.288 q^{56} +1375.01 q^{58} +362.188 q^{59} +455.354 q^{61} -929.666 q^{62} -822.622 q^{64} +749.885 q^{67} -1357.14 q^{68} +501.761 q^{71} -1045.59 q^{73} -1111.14 q^{74} +1961.22 q^{76} -94.5327 q^{77} +466.940 q^{79} -276.723 q^{82} +1157.65 q^{83} +1496.63 q^{86} +260.205 q^{88} +1100.03 q^{89} -473.261 q^{91} +1498.10 q^{92} +221.098 q^{94} +1167.99 q^{97} -1237.61 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 78 q^{4}+O(q^{10}) \) 16 * q + 78 * q^4	\( 16 q + 78 q^{4} + 176 q^{11} + 124 q^{14} + 306 q^{16} + 84 q^{19} + 532 q^{26} + 652 q^{29} + 80 q^{31} + 360 q^{34} + 1204 q^{41} + 858 q^{44} - 672 q^{46} + 1016 q^{49} + 3332 q^{56} + 712 q^{59} + 880 q^{61} - 962 q^{64} + 1968 q^{71} + 4152 q^{74} - 1048 q^{76} - 1636 q^{79} + 7284 q^{86} + 2776 q^{89} + 144 q^{91} + 1400 q^{94}+O(q^{100}) \) 16 * q + 78 * q^4 + 176 * q^11 + 124 * q^14 + 306 * q^16 + 84 * q^19 + 532 * q^26 + 652 * q^29 + 80 * q^31 + 360 * q^34 + 1204 * q^41 + 858 * q^44 - 672 * q^46 + 1016 * q^49 + 3332 * q^56 + 712 * q^59 + 880 * q^61 - 962 * q^64 + 1968 * q^71 + 4152 * q^74 - 1048 * q^76 - 1636 * q^79 + 7284 * q^86 + 2776 * q^89 + 144 * q^91 + 1400 * q^94

Introduction

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