LMFDB - Embedded newform 2475.4.a.y.1.3

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:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.03932$ 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)

$f(q)$	$=$	$q+4.03932 q^{2} +8.31608 q^{4} +6.49923 q^{7} +1.27677 q^{8} -11.0000 q^{11} -30.5385 q^{13} +26.2524 q^{14} -61.3714 q^{16} -43.1314 q^{17} -6.46044 q^{19} -44.4325 q^{22} +108.104 q^{23} -123.355 q^{26} +54.0481 q^{28} +274.857 q^{29} -68.5519 q^{31} -258.113 q^{32} -174.221 q^{34} +402.200 q^{37} -26.0958 q^{38} +268.450 q^{41} +30.5334 q^{43} -91.4769 q^{44} +436.666 q^{46} +31.5850 q^{47} -300.760 q^{49} -253.961 q^{52} +252.497 q^{53} +8.29800 q^{56} +1110.24 q^{58} +558.331 q^{59} +335.270 q^{61} -276.903 q^{62} -551.628 q^{64} -28.7521 q^{67} -358.684 q^{68} -89.0081 q^{71} +717.622 q^{73} +1624.61 q^{74} -53.7256 q^{76} -71.4915 q^{77} +200.702 q^{79} +1084.36 q^{82} +243.681 q^{83} +123.334 q^{86} -14.0444 q^{88} +312.832 q^{89} -198.477 q^{91} +899.002 q^{92} +127.582 q^{94} -27.9996 q^{97} -1214.87 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 7 q^{4} - 16 q^{7} - 3 q^{8} - 33 q^{11} - 45 q^{13} + 116 q^{14} - 85 q^{16} - 58 q^{17} - 169 q^{19} - 11 q^{22} + 155 q^{23} - 167 q^{26} - 100 q^{28} + 277 q^{29} - 173 q^{31} + 97 q^{32}+ \cdots - 2273 q^{98}+O(q^{100}) $ 3 * q + q^2 + 7 * q^4 - 16 * q^7 - 3 * q^8 - 33 * q^11 - 45 * q^13 + 116 * q^14 - 85 * q^16 - 58 * q^17 - 169 * q^19 - 11 * q^22 + 155 * q^23 - 167 * q^26 - 100 * q^28 + 277 * q^29 - 173 * q^31 + 97 * q^32 - 146 * q^34 - 60 * q^37 + 169 * q^38 - 44 * q^41 + 109 * q^43 - 77 * q^44 + 425 * q^46 + 270 * q^47 - 427 * q^49 - 45 * q^52 - 148 * q^53 - 168 * q^56 + 783 * q^58 + 684 * q^59 - 1038 * q^61 + 953 * q^62 - 1129 * q^64 - 314 * q^67 - 366 * q^68 + 1459 * q^71 + 1170 * q^73 + 1764 * q^74 + 211 * q^76 + 176 * q^77 - 506 * q^79 + 1040 * q^82 - 347 * q^83 + 527 * q^86 + 33 * q^88 + 607 * q^89 - 398 * q^91 + 687 * q^92 - 1822 * q^94 - 1263 * q^97 - 2273 * 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 $

$2$	4.03932	1.42811	0.714057	−	0.700087i	$-0.246856\pi$
$2$	4.03932	1.42811	0.714057	+	0.700087i	$0.246856\pi$
$3$	0	0
$3$	0	0
$4$	8.31608	1.03951
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.49923	0.350925	0.175463	−	0.984486i	$-0.443858\pi$
$7$	6.49923	0.350925	0.175463	+	0.984486i	$0.443858\pi$
$8$	1.27677	0.0564257
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−30.5385	−0.651528	−0.325764	−	0.945451i	$-0.605622\pi$
$13$	−30.5385	−0.651528	−0.325764	+	0.945451i	$0.605622\pi$
$14$	26.2524	0.501161
$15$	0	0
$16$	−61.3714	−0.958928
$17$	−43.1314	−0.615347	−0.307674	−	0.951492i	$-0.599550\pi$
$17$	−43.1314	−0.615347	−0.307674	+	0.951492i	$0.599550\pi$
$18$	0	0
$19$	−6.46044	−0.0780066	−0.0390033	−	0.999239i	$-0.512418\pi$
$19$	−6.46044	−0.0780066	−0.0390033	+	0.999239i	$0.512418\pi$
$20$	0	0
$21$	0	0
$22$	−44.4325	−0.430593
$23$	108.104	0.980054	0.490027	−	0.871707i	$-0.336987\pi$
$23$	108.104	0.980054	0.490027	+	0.871707i	$0.336987\pi$
$24$	0	0
$25$	0	0
$26$	−123.355	−0.930457
$27$	0	0
$28$	54.0481	0.364791
$29$	274.857	1.75999	0.879995	−	0.474984i	$-0.157546\pi$
$29$	274.857	1.75999	0.879995	+	0.474984i	$0.157546\pi$
$30$	0	0
$31$	−68.5519	−0.397170	−0.198585	−	0.980084i	$-0.563635\pi$
$31$	−68.5519	−0.397170	−0.198585	+	0.980084i	$0.563635\pi$
$32$	−258.113	−1.42588
$33$	0	0
$34$	−174.221	−0.878786
$35$	0	0
$36$	0	0
$37$	402.200	1.78706	0.893530	−	0.449004i	$-0.148221\pi$
$37$	402.200	1.78706	0.893530	+	0.449004i	$0.148221\pi$
$38$	−26.0958	−0.111402
$39$	0	0
$40$	0	0
$41$	268.450	1.02256	0.511280	−	0.859414i	$-0.329172\pi$
$41$	268.450	1.02256	0.511280	+	0.859414i	$0.329172\pi$
$42$	0	0
$43$	30.5334	0.108286	0.0541431	−	0.998533i	$-0.482757\pi$
$43$	30.5334	0.108286	0.0541431	+	0.998533i	$0.482757\pi$
$44$	−91.4769	−0.313424
$45$	0	0
$46$	436.666	1.39963
$47$	31.5850	0.0980243	0.0490121	−	0.998798i	$-0.484393\pi$
$47$	31.5850	0.0980243	0.0490121	+	0.998798i	$0.484393\pi$
$48$	0	0
$49$	−300.760	−0.876851
$50$	0	0
$51$	0	0
$52$	−253.961	−0.677271
$53$	252.497	0.654398	0.327199	−	0.944956i	$-0.393895\pi$
$53$	252.497	0.654398	0.327199	+	0.944956i	$0.393895\pi$
$54$	0	0
$55$	0	0
$56$	8.29800	0.0198012
$57$	0	0
$58$	1110.24	2.51347
$59$	558.331	1.23201	0.616004	−	0.787743i	$-0.288750\pi$
$59$	558.331	1.23201	0.616004	+	0.787743i	$0.288750\pi$
$60$	0	0
$61$	335.270	0.703719	0.351860	−	0.936053i	$-0.385549\pi$
$61$	335.270	0.703719	0.351860	+	0.936053i	$0.385549\pi$
$62$	−276.903	−0.567205
$63$	0	0
$64$	−551.628	−1.07740
$65$	0	0
$66$	0	0
$67$	−28.7521	−0.0524273	−0.0262136	−	0.999656i	$-0.508345\pi$
$67$	−28.7521	−0.0524273	−0.0262136	+	0.999656i	$0.508345\pi$
$68$	−358.684	−0.639660
$69$	0	0
$70$	0	0
$71$	−89.0081	−0.148779	−0.0743895	−	0.997229i	$-0.523701\pi$
$71$	−89.0081	−0.148779	−0.0743895	+	0.997229i	$0.523701\pi$
$72$	0	0
$73$	717.622	1.15057	0.575283	−	0.817954i	$-0.304892\pi$
$73$	717.622	1.15057	0.575283	+	0.817954i	$0.304892\pi$
$74$	1624.61	2.55213
$75$	0	0
$76$	−53.7256	−0.0810887
$77$	−71.4915	−0.105808
$78$	0	0
$79$	200.702	0.285832	0.142916	−	0.989735i	$-0.454352\pi$
$79$	200.702	0.285832	0.142916	+	0.989735i	$0.454352\pi$
$80$	0	0
$81$	0	0
$82$	1084.36	1.46033
$83$	243.681	0.322258	0.161129	−	0.986933i	$-0.448486\pi$
$83$	243.681	0.322258	0.161129	+	0.986933i	$0.448486\pi$
$84$	0	0
$85$	0	0
$86$	123.334	0.154645
$87$	0	0
$88$	−14.0444	−0.0170130
$89$	312.832	0.372585	0.186293	−	0.982494i	$-0.440353\pi$
$89$	312.832	0.372585	0.186293	+	0.982494i	$0.440353\pi$
$90$	0	0
$91$	−198.477	−0.228638
$92$	899.002	1.01878
$93$	0	0
$94$	127.582	0.139990
$95$	0	0
$96$	0	0
$97$	−27.9996	−0.0293085	−0.0146543	−	0.999893i	$-0.504665\pi$
$97$	−27.9996	−0.0293085	−0.0146543	+	0.999893i	$0.504665\pi$
$98$	−1214.87	−1.25224
$99$	0	0
$100$	0	0
$101$	228.278	0.224896	0.112448	−	0.993658i	$-0.464131\pi$
$101$	228.278	0.224896	0.112448	+	0.993658i	$0.464131\pi$
$102$	0	0
$103$	800.328	0.765618	0.382809	−	0.923827i	$-0.374957\pi$
$103$	800.328	0.765618	0.382809	+	0.923827i	$0.374957\pi$
$104$	−38.9906	−0.0367629
$105$	0	0
$106$	1019.91	0.934555
$107$	−1046.89	−0.945857	−0.472929	−	0.881101i	$-0.656803\pi$
$107$	−1046.89	−0.945857	−0.472929	+	0.881101i	$0.656803\pi$
$108$	0	0
$109$	317.023	0.278581	0.139291	−	0.990252i	$-0.455518\pi$
$109$	317.023	0.278581	0.139291	+	0.990252i	$0.455518\pi$
$110$	0	0
$111$	0	0
$112$	−398.867	−0.336512
$113$	1333.82	1.11040	0.555198	−	0.831718i	$-0.312642\pi$
$113$	1333.82	1.11040	0.555198	+	0.831718i	$0.312642\pi$
$114$	0	0
$115$	0	0
$116$	2285.74	1.82953
$117$	0	0
$118$	2255.28	1.75945
$119$	−280.321	−0.215941
$120$	0	0
$121$	121.000	0.0909091
$122$	1354.26	1.00499
$123$	0	0
$124$	−570.083	−0.412863
$125$	0	0
$126$	0	0
$127$	−564.888	−0.394691	−0.197345	−	0.980334i	$-0.563232\pi$
$127$	−564.888	−0.394691	−0.197345	+	0.980334i	$0.563232\pi$
$128$	−163.299	−0.112763
$129$	0	0
$130$	0	0
$131$	214.082	0.142782	0.0713908	−	0.997448i	$-0.477256\pi$
$131$	214.082	0.142782	0.0713908	+	0.997448i	$0.477256\pi$
$132$	0	0
$133$	−41.9879	−0.0273745
$134$	−116.139	−0.0748721
$135$	0	0
$136$	−55.0688	−0.0347214
$137$	−65.2288	−0.0406779	−0.0203389	−	0.999793i	$-0.506475\pi$
$137$	−65.2288	−0.0406779	−0.0203389	+	0.999793i	$0.506475\pi$
$138$	0	0
$139$	−407.084	−0.248406	−0.124203	−	0.992257i	$-0.539637\pi$
$139$	−407.084	−0.248406	−0.124203	+	0.992257i	$0.539637\pi$
$140$	0	0
$141$	0	0
$142$	−359.532	−0.212474
$143$	335.924	0.196443
$144$	0	0
$145$	0	0
$146$	2898.70	1.64314
$147$	0	0
$148$	3344.73	1.85767
$149$	−792.092	−0.435508	−0.217754	−	0.976004i	$-0.569873\pi$
$149$	−792.092	−0.435508	−0.217754	+	0.976004i	$0.569873\pi$
$150$	0	0
$151$	−2861.80	−1.54232	−0.771159	−	0.636642i	$-0.780323\pi$
$151$	−2861.80	−1.54232	−0.771159	+	0.636642i	$0.780323\pi$
$152$	−8.24848	−0.00440158
$153$	0	0
$154$	−288.777	−0.151106
$155$	0	0
$156$	0	0
$157$	−2991.55	−1.52071	−0.760355	−	0.649507i	$-0.774975\pi$
$157$	−2991.55	−1.52071	−0.760355	+	0.649507i	$0.774975\pi$
$158$	810.698	0.408201
$159$	0	0
$160$	0	0
$161$	702.592	0.343926
$162$	0	0
$163$	−2907.60	−1.39718	−0.698592	−	0.715520i	$-0.746190\pi$
$163$	−2907.60	−1.39718	−0.698592	+	0.715520i	$0.746190\pi$
$164$	2232.46	1.06296
$165$	0	0
$166$	984.304	0.460222
$167$	1100.39	0.509885	0.254942	−	0.966956i	$-0.417944\pi$
$167$	1100.39	0.509885	0.254942	+	0.966956i	$0.417944\pi$
$168$	0	0
$169$	−1264.40	−0.575511
$170$	0	0
$171$	0	0
$172$	253.919	0.112565
$173$	3754.75	1.65011	0.825054	−	0.565054i	$-0.191145\pi$
$173$	3754.75	1.65011	0.825054	+	0.565054i	$0.191145\pi$
$174$	0	0
$175$	0	0
$176$	675.086	0.289128
$177$	0	0
$178$	1263.63	0.532094
$179$	3218.79	1.34404	0.672021	−	0.740532i	$-0.265426\pi$
$179$	3218.79	1.34404	0.672021	+	0.740532i	$0.265426\pi$
$180$	0	0
$181$	−456.885	−0.187624	−0.0938121	−	0.995590i	$-0.529905\pi$
$181$	−456.885	−0.187624	−0.0938121	+	0.995590i	$0.529905\pi$
$182$	−801.711	−0.326521
$183$	0	0
$184$	138.024	0.0553002
$185$	0	0
$186$	0	0
$187$	474.445	0.185534
$188$	262.663	0.101897
$189$	0	0
$190$	0	0
$191$	4907.11	1.85899	0.929493	−	0.368841i	$-0.120245\pi$
$191$	4907.11	1.85899	0.929493	+	0.368841i	$0.120245\pi$
$192$	0	0
$193$	−1462.17	−0.545333	−0.272667	−	0.962109i	$-0.587906\pi$
$193$	−1462.17	−0.545333	−0.272667	+	0.962109i	$0.587906\pi$
$194$	−113.099	−0.0418559
$195$	0	0
$196$	−2501.15	−0.911496
$197$	2210.10	0.799307	0.399653	−	0.916666i	$-0.369131\pi$
$197$	2210.10	0.799307	0.399653	+	0.916666i	$0.369131\pi$
$198$	0	0
$199$	−395.272	−0.140804	−0.0704022	−	0.997519i	$-0.522428\pi$
$199$	−395.272	−0.140804	−0.0704022	+	0.997519i	$0.522428\pi$
$200$	0	0
$201$	0	0
$202$	922.086	0.321177
$203$	1786.36	0.617625
$204$	0	0
$205$	0	0
$206$	3232.78	1.09339
$207$	0	0
$208$	1874.19	0.624769
$209$	71.0648	0.0235199
$210$	0	0
$211$	1559.98	0.508974	0.254487	−	0.967076i	$-0.418093\pi$
$211$	1559.98	0.508974	0.254487	+	0.967076i	$0.418093\pi$
$212$	2099.78	0.680253
$213$	0	0
$214$	−4228.72	−1.35079
$215$	0	0
$216$	0	0
$217$	−445.534	−0.139377
$218$	1280.56	0.397846
$219$	0	0
$220$	0	0
$221$	1317.17	0.400916
$222$	0	0
$223$	−756.215	−0.227085	−0.113542	−	0.993533i	$-0.536220\pi$
$223$	−756.215	−0.227085	−0.113542	+	0.993533i	$0.536220\pi$
$224$	−1677.53	−0.500379
$225$	0	0
$226$	5387.70	1.58577
$227$	−5758.69	−1.68378	−0.841890	−	0.539649i	$-0.818557\pi$
$227$	−5758.69	−1.68378	−0.841890	+	0.539649i	$0.818557\pi$
$228$	0	0
$229$	−1912.63	−0.551921	−0.275961	−	0.961169i	$-0.588996\pi$
$229$	−1912.63	−0.551921	−0.275961	+	0.961169i	$0.588996\pi$
$230$	0	0
$231$	0	0
$232$	350.929	0.0993086
$233$	1034.86	0.290970	0.145485	−	0.989360i	$-0.453526\pi$
$233$	1034.86	0.290970	0.145485	+	0.989360i	$0.453526\pi$
$234$	0	0
$235$	0	0
$236$	4643.13	1.28069
$237$	0	0
$238$	−1132.30	−0.308388
$239$	−2647.23	−0.716465	−0.358232	−	0.933632i	$-0.616620\pi$
$239$	−2647.23	−0.716465	−0.358232	+	0.933632i	$0.616620\pi$
$240$	0	0
$241$	6734.00	1.79990	0.899948	−	0.435996i	$-0.143604\pi$
$241$	6734.00	1.79990	0.899948	+	0.435996i	$0.143604\pi$
$242$	488.757	0.129829
$243$	0	0
$244$	2788.13	0.731523
$245$	0	0
$246$	0	0
$247$	197.292	0.0508235
$248$	−87.5248	−0.0224106
$249$	0	0
$250$	0	0
$251$	5377.57	1.35231	0.676154	−	0.736761i	$-0.263646\pi$
$251$	5377.57	1.35231	0.676154	+	0.736761i	$0.263646\pi$
$252$	0	0
$253$	−1189.14	−0.295497
$254$	−2281.76	−0.563663
$255$	0	0
$256$	3753.41	0.916360
$257$	5882.33	1.42774	0.713871	−	0.700278i	$-0.246940\pi$
$257$	5882.33	1.42774	0.713871	+	0.700278i	$0.246940\pi$
$258$	0	0
$259$	2613.99	0.627124
$260$	0	0
$261$	0	0
$262$	864.744	0.203909
$263$	−4001.67	−0.938227	−0.469113	−	0.883138i	$-0.655426\pi$
$263$	−4001.67	−0.938227	−0.469113	+	0.883138i	$0.655426\pi$
$264$	0	0
$265$	0	0
$266$	−169.602	−0.0390939
$267$	0	0
$268$	−239.105	−0.0544987
$269$	1758.29	0.398530	0.199265	−	0.979946i	$-0.436144\pi$
$269$	1758.29	0.398530	0.199265	+	0.979946i	$0.436144\pi$
$270$	0	0
$271$	6018.44	1.34906	0.674528	−	0.738249i	$-0.264347\pi$
$271$	6018.44	1.34906	0.674528	+	0.738249i	$0.264347\pi$
$272$	2647.03	0.590074
$273$	0	0
$274$	−263.480	−0.0580927
$275$	0	0
$276$	0	0
$277$	−2910.30	−0.631275	−0.315637	−	0.948880i	$-0.602218\pi$
$277$	−2910.30	−0.631275	−0.315637	+	0.948880i	$0.602218\pi$
$278$	−1644.34	−0.354752
$279$	0	0
$280$	0	0
$281$	−6893.19	−1.46339	−0.731697	−	0.681631i	$-0.761271\pi$
$281$	−6893.19	−1.46339	−0.731697	+	0.681631i	$0.761271\pi$
$282$	0	0
$283$	−3120.16	−0.655385	−0.327693	−	0.944784i	$-0.606271\pi$
$283$	−3120.16	−0.655385	−0.327693	+	0.944784i	$0.606271\pi$
$284$	−740.199	−0.154657
$285$	0	0
$286$	1356.90	0.280543
$287$	1744.72	0.358842
$288$	0	0
$289$	−3052.68	−0.621348
$290$	0	0
$291$	0	0
$292$	5967.81	1.19603
$293$	8869.62	1.76849	0.884247	−	0.467020i	$-0.154672\pi$
$293$	8869.62	1.76849	0.884247	+	0.467020i	$0.154672\pi$
$294$	0	0
$295$	0	0
$296$	513.515	0.100836
$297$	0	0
$298$	−3199.51	−0.621955
$299$	−3301.34	−0.638533
$300$	0	0
$301$	198.444	0.0380003
$302$	−11559.7	−2.20261
$303$	0	0
$304$	396.486	0.0748028
$305$	0	0
$306$	0	0
$307$	5713.42	1.06216	0.531078	−	0.847323i	$-0.321787\pi$
$307$	5713.42	1.06216	0.531078	+	0.847323i	$0.321787\pi$
$308$	−594.529	−0.109988
$309$	0	0
$310$	0	0
$311$	−1984.01	−0.361746	−0.180873	−	0.983506i	$-0.557892\pi$
$311$	−1984.01	−0.361746	−0.180873	+	0.983506i	$0.557892\pi$
$312$	0	0
$313$	2675.50	0.483156	0.241578	−	0.970381i	$-0.422335\pi$
$313$	2675.50	0.483156	0.241578	+	0.970381i	$0.422335\pi$
$314$	−12083.8	−2.17175
$315$	0	0
$316$	1669.05	0.297125
$317$	−7083.76	−1.25509	−0.627545	−	0.778580i	$-0.715940\pi$
$317$	−7083.76	−1.25509	−0.627545	+	0.778580i	$0.715940\pi$
$318$	0	0
$319$	−3023.43	−0.530657
$320$	0	0
$321$	0	0
$322$	2837.99	0.491165
$323$	278.648	0.0480012
$324$	0	0
$325$	0	0
$326$	−11744.7	−1.99534
$327$	0	0
$328$	342.749	0.0576986
$329$	205.278	0.0343992
$330$	0	0
$331$	7812.39	1.29730	0.648652	−	0.761085i	$-0.275333\pi$
$331$	7812.39	1.29730	0.648652	+	0.761085i	$0.275333\pi$
$332$	2026.47	0.334991
$333$	0	0
$334$	4444.82	0.728173
$335$	0	0
$336$	0	0
$337$	11866.4	1.91811	0.959056	−	0.283218i	$-0.0914019\pi$
$337$	11866.4	1.91811	0.959056	+	0.283218i	$0.0914019\pi$
$338$	−5107.30	−0.821895
$339$	0	0
$340$	0	0
$341$	754.071	0.119751
$342$	0	0
$343$	−4183.94	−0.658635
$344$	38.9841	0.00611012
$345$	0	0
$346$	15166.6	2.35654
$347$	3485.50	0.539225	0.269613	−	0.962969i	$-0.413104\pi$
$347$	3485.50	0.539225	0.269613	+	0.962969i	$0.413104\pi$
$348$	0	0
$349$	−6366.76	−0.976518	−0.488259	−	0.872699i	$-0.662368\pi$
$349$	−6366.76	−0.976518	−0.488259	+	0.872699i	$0.662368\pi$
$350$	0	0
$351$	0	0
$352$	2839.24	0.429920
$353$	8223.15	1.23987	0.619935	−	0.784653i	$-0.287159\pi$
$353$	8223.15	1.23987	0.619935	+	0.784653i	$0.287159\pi$
$354$	0	0
$355$	0	0
$356$	2601.54	0.387306
$357$	0	0
$358$	13001.7	1.91945
$359$	9377.54	1.37863	0.689314	−	0.724462i	$-0.257912\pi$
$359$	9377.54	1.37863	0.689314	+	0.724462i	$0.257912\pi$
$360$	0	0
$361$	−6817.26	−0.993915
$362$	−1845.50	−0.267949
$363$	0	0
$364$	−1650.55	−0.237671
$365$	0	0
$366$	0	0
$367$	310.895	0.0442196	0.0221098	−	0.999756i	$-0.492962\pi$
$367$	310.895	0.0442196	0.0221098	+	0.999756i	$0.492962\pi$
$368$	−6634.49	−0.939801
$369$	0	0
$370$	0	0
$371$	1641.03	0.229645
$372$	0	0
$373$	4738.51	0.657777	0.328889	−	0.944369i	$-0.393326\pi$
$373$	4738.51	0.657777	0.328889	+	0.944369i	$0.393326\pi$
$374$	1916.44	0.264964
$375$	0	0
$376$	40.3267	0.00553109
$377$	−8393.74	−1.14668
$378$	0	0
$379$	−4735.47	−0.641806	−0.320903	−	0.947112i	$-0.603986\pi$
$379$	−4735.47	−0.641806	−0.320903	+	0.947112i	$0.603986\pi$
$380$	0	0
$381$	0	0
$382$	19821.4	2.65484
$383$	3632.14	0.484579	0.242289	−	0.970204i	$-0.422102\pi$
$383$	3632.14	0.484579	0.242289	+	0.970204i	$0.422102\pi$
$384$	0	0
$385$	0	0
$386$	−5906.17	−0.778798
$387$	0	0
$388$	−232.847	−0.0304665
$389$	−6557.50	−0.854701	−0.427350	−	0.904086i	$-0.640553\pi$
$389$	−6557.50	−0.854701	−0.427350	+	0.904086i	$0.640553\pi$
$390$	0	0
$391$	−4662.68	−0.603073
$392$	−384.001	−0.0494769
$393$	0	0
$394$	8927.31	1.14150
$395$	0	0
$396$	0	0
$397$	−4084.97	−0.516420	−0.258210	−	0.966089i	$-0.583133\pi$
$397$	−4084.97	−0.516420	−0.258210	+	0.966089i	$0.583133\pi$
$398$	−1596.63	−0.201085
$399$	0	0
$400$	0	0
$401$	6676.04	0.831386	0.415693	−	0.909505i	$-0.363539\pi$
$401$	6676.04	0.831386	0.415693	+	0.909505i	$0.363539\pi$
$402$	0	0
$403$	2093.47	0.258768
$404$	1898.38	0.233782
$405$	0	0
$406$	7215.67	0.882039
$407$	−4424.20	−0.538819
$408$	0	0
$409$	1718.56	0.207769	0.103884	−	0.994589i	$-0.466873\pi$
$409$	1718.56	0.207769	0.103884	+	0.994589i	$0.466873\pi$
$410$	0	0
$411$	0	0
$412$	6655.60	0.795868
$413$	3628.72	0.432343
$414$	0	0
$415$	0	0
$416$	7882.39	0.929004
$417$	0	0
$418$	287.053	0.0335891
$419$	−5732.23	−0.668348	−0.334174	−	0.942511i	$-0.608457\pi$
$419$	−5732.23	−0.668348	−0.334174	+	0.942511i	$0.608457\pi$
$420$	0	0
$421$	−9647.38	−1.11683	−0.558414	−	0.829563i	$-0.688590\pi$
$421$	−9647.38	−1.11683	−0.558414	+	0.829563i	$0.688590\pi$
$422$	6301.26	0.726873
$423$	0	0
$424$	322.379	0.0369248
$425$	0	0
$426$	0	0
$427$	2178.99	0.246953
$428$	−8706.03	−0.983229
$429$	0	0
$430$	0	0
$431$	3054.57	0.341376	0.170688	−	0.985325i	$-0.445401\pi$
$431$	3054.57	0.341376	0.170688	+	0.985325i	$0.445401\pi$
$432$	0	0
$433$	−11022.8	−1.22337	−0.611686	−	0.791100i	$-0.709509\pi$
$433$	−11022.8	−1.22337	−0.611686	+	0.791100i	$0.709509\pi$
$434$	−1799.65	−0.199046
$435$	0	0
$436$	2636.39	0.289588
$437$	−698.399	−0.0764507
$438$	0	0
$439$	−7589.30	−0.825097	−0.412548	−	0.910936i	$-0.635361\pi$
$439$	−7589.30	−0.825097	−0.412548	+	0.910936i	$0.635361\pi$
$440$	0	0
$441$	0	0
$442$	5320.47	0.572554
$443$	4043.30	0.433641	0.216821	−	0.976211i	$-0.430431\pi$
$443$	4043.30	0.433641	0.216821	+	0.976211i	$0.430431\pi$
$444$	0	0
$445$	0	0
$446$	−3054.59	−0.324303
$447$	0	0
$448$	−3585.16	−0.378086
$449$	−14619.3	−1.53658	−0.768291	−	0.640101i	$-0.778893\pi$
$449$	−14619.3	−1.53658	−0.768291	+	0.640101i	$0.778893\pi$
$450$	0	0
$451$	−2952.96	−0.308313
$452$	11092.1	1.15427
$453$	0	0
$454$	−23261.2	−2.40463
$455$	0	0
$456$	0	0
$457$	10910.8	1.11682	0.558411	−	0.829565i	$-0.311411\pi$
$457$	10910.8	1.11682	0.558411	+	0.829565i	$0.311411\pi$
$458$	−7725.71	−0.788207
$459$	0	0
$460$	0	0
$461$	−16371.0	−1.65395	−0.826977	−	0.562236i	$-0.809941\pi$
$461$	−16371.0	−1.65395	−0.826977	+	0.562236i	$0.809941\pi$
$462$	0	0
$463$	16023.5	1.60837	0.804186	−	0.594377i	$-0.202601\pi$
$463$	16023.5	1.60837	0.804186	+	0.594377i	$0.202601\pi$
$464$	−16868.4	−1.68770
$465$	0	0
$466$	4180.13	0.415538
$467$	−15097.3	−1.49597	−0.747985	−	0.663716i	$-0.768978\pi$
$467$	−15097.3	−1.49597	−0.747985	+	0.663716i	$0.768978\pi$
$468$	0	0
$469$	−186.866	−0.0183981
$470$	0	0
$471$	0	0
$472$	712.859	0.0695170
$473$	−335.868	−0.0326495
$474$	0	0
$475$	0	0
$476$	−2331.17	−0.224473
$477$	0	0
$478$	−10693.0	−1.02319
$479$	10459.0	0.997667	0.498834	−	0.866698i	$-0.333762\pi$
$479$	10459.0	0.997667	0.498834	+	0.866698i	$0.333762\pi$
$480$	0	0
$481$	−12282.6	−1.16432
$482$	27200.8	2.57046
$483$	0	0
$484$	1006.25	0.0945010
$485$	0	0
$486$	0	0
$487$	7939.86	0.738788	0.369394	−	0.929273i	$-0.379565\pi$
$487$	7939.86	0.738788	0.369394	+	0.929273i	$0.379565\pi$
$488$	428.061	0.0397078
$489$	0	0
$490$	0	0
$491$	−11492.7	−1.05633	−0.528167	−	0.849141i	$-0.677120\pi$
$491$	−11492.7	−1.05633	−0.528167	+	0.849141i	$0.677120\pi$
$492$	0	0
$493$	−11855.0	−1.08300
$494$	796.927	0.0725818
$495$	0	0
$496$	4207.13	0.380858
$497$	−578.484	−0.0522103
$498$	0	0
$499$	4813.54	0.431831	0.215915	−	0.976412i	$-0.430726\pi$
$499$	4813.54	0.431831	0.215915	+	0.976412i	$0.430726\pi$
$500$	0	0
$501$	0	0
$502$	21721.7	1.93125
$503$	17756.3	1.57399	0.786993	−	0.616962i	$-0.211637\pi$
$503$	17756.3	1.57399	0.786993	+	0.616962i	$0.211637\pi$
$504$	0	0
$505$	0	0
$506$	−4803.33	−0.422004
$507$	0	0
$508$	−4697.66	−0.410285
$509$	−3183.05	−0.277183	−0.138592	−	0.990350i	$-0.544258\pi$
$509$	−3183.05	−0.277183	−0.138592	+	0.990350i	$0.544258\pi$
$510$	0	0
$511$	4663.99	0.403763
$512$	16467.6	1.42143
$513$	0	0
$514$	23760.6	2.03898
$515$	0	0
$516$	0	0
$517$	−347.435	−0.0295554
$518$	10558.7	0.895605
$519$	0	0
$520$	0	0
$521$	−7246.60	−0.609365	−0.304683	−	0.952454i	$-0.598550\pi$
$521$	−7246.60	−0.609365	−0.304683	+	0.952454i	$0.598550\pi$
$522$	0	0
$523$	17985.0	1.50369	0.751845	−	0.659340i	$-0.229164\pi$
$523$	17985.0	1.50369	0.751845	+	0.659340i	$0.229164\pi$
$524$	1780.32	0.148423
$525$	0	0
$526$	−16164.0	−1.33990
$527$	2956.74	0.244398
$528$	0	0
$529$	−480.530	−0.0394945
$530$	0	0
$531$	0	0
$532$	−349.175	−0.0284561
$533$	−8198.09	−0.666226
$534$	0	0
$535$	0	0
$536$	−36.7097	−0.00295825
$537$	0	0
$538$	7102.28	0.569147
$539$	3308.36	0.264381
$540$	0	0
$541$	−14549.3	−1.15624	−0.578119	−	0.815952i	$-0.696213\pi$
$541$	−14549.3	−1.15624	−0.578119	+	0.815952i	$0.696213\pi$
$542$	24310.4	1.92661
$543$	0	0
$544$	11132.8	0.877414
$545$	0	0
$546$	0	0
$547$	−5487.29	−0.428920	−0.214460	−	0.976733i	$-0.568799\pi$
$547$	−5487.29	−0.428920	−0.214460	+	0.976733i	$0.568799\pi$
$548$	−542.448	−0.0422851
$549$	0	0
$550$	0	0
$551$	−1775.70	−0.137291
$552$	0	0
$553$	1304.41	0.100306
$554$	−11755.6	−0.901532
$555$	0	0
$556$	−3385.35	−0.258221
$557$	−7366.60	−0.560382	−0.280191	−	0.959944i	$-0.590398\pi$
$557$	−7366.60	−0.560382	−0.280191	+	0.959944i	$0.590398\pi$
$558$	0	0
$559$	−932.446	−0.0705515
$560$	0	0
$561$	0	0
$562$	−27843.8	−2.08989
$563$	−4516.14	−0.338068	−0.169034	−	0.985610i	$-0.554065\pi$
$563$	−4516.14	−0.338068	−0.169034	+	0.985610i	$0.554065\pi$
$564$	0	0
$565$	0	0
$566$	−12603.3	−0.935965
$567$	0	0
$568$	−113.643	−0.00839496
$569$	−2174.35	−0.160200	−0.0800999	−	0.996787i	$-0.525524\pi$
$569$	−2174.35	−0.160200	−0.0800999	+	0.996787i	$0.525524\pi$
$570$	0	0
$571$	−25339.7	−1.85715	−0.928575	−	0.371145i	$-0.878965\pi$
$571$	−25339.7	−1.85715	−0.928575	+	0.371145i	$0.878965\pi$
$572$	2793.57	0.204205
$573$	0	0
$574$	7047.48	0.512467
$575$	0	0
$576$	0	0
$577$	21909.1	1.58074	0.790370	−	0.612630i	$-0.209888\pi$
$577$	21909.1	1.58074	0.790370	+	0.612630i	$0.209888\pi$
$578$	−12330.8	−0.887356
$579$	0	0
$580$	0	0
$581$	1583.74	0.113089
$582$	0	0
$583$	−2777.46	−0.197308
$584$	916.237	0.0649215
$585$	0	0
$586$	35827.2	2.52561
$587$	−922.235	−0.0648462	−0.0324231	−	0.999474i	$-0.510322\pi$
$587$	−922.235	−0.0648462	−0.0324231	+	0.999474i	$0.510322\pi$
$588$	0	0
$589$	442.875	0.0309819
$590$	0	0
$591$	0	0
$592$	−24683.6	−1.71366
$593$	−9075.86	−0.628501	−0.314250	−	0.949340i	$-0.601753\pi$
$593$	−9075.86	−0.628501	−0.314250	+	0.949340i	$0.601753\pi$
$594$	0	0
$595$	0	0
$596$	−6587.10	−0.452715
$597$	0	0
$598$	−13335.2	−0.911898
$599$	−19987.0	−1.36335	−0.681675	−	0.731655i	$-0.738748\pi$
$599$	−19987.0	−1.36335	−0.681675	+	0.731655i	$0.738748\pi$
$600$	0	0
$601$	11116.7	0.754509	0.377255	−	0.926110i	$-0.376868\pi$
$601$	11116.7	0.754509	0.377255	+	0.926110i	$0.376868\pi$
$602$	801.577	0.0542688
$603$	0	0
$604$	−23799.0	−1.60326
$605$	0	0
$606$	0	0
$607$	−19355.0	−1.29423	−0.647114	−	0.762393i	$-0.724024\pi$
$607$	−19355.0	−1.29423	−0.647114	+	0.762393i	$0.724024\pi$
$608$	1667.52	0.111228
$609$	0	0
$610$	0	0
$611$	−964.559	−0.0638656
$612$	0	0
$613$	12325.5	0.812111	0.406056	−	0.913848i	$-0.366904\pi$
$613$	12325.5	0.812111	0.406056	+	0.913848i	$0.366904\pi$
$614$	23078.3	1.51688
$615$	0	0
$616$	−91.2780	−0.00597029
$617$	28207.0	1.84047	0.920236	−	0.391363i	$-0.127996\pi$
$617$	28207.0	1.84047	0.920236	+	0.391363i	$0.127996\pi$
$618$	0	0
$619$	17236.9	1.11924	0.559621	−	0.828748i	$-0.310947\pi$
$619$	17236.9	1.11924	0.559621	+	0.828748i	$0.310947\pi$
$620$	0	0
$621$	0	0
$622$	−8014.06	−0.516615
$623$	2033.16	0.130750
$624$	0	0
$625$	0	0
$626$	10807.2	0.690003
$627$	0	0
$628$	−24878.0	−1.58080
$629$	−17347.4	−1.09966
$630$	0	0
$631$	−23582.7	−1.48782	−0.743908	−	0.668282i	$-0.767030\pi$
$631$	−23582.7	−1.48782	−0.743908	+	0.668282i	$0.767030\pi$
$632$	256.250	0.0161283
$633$	0	0
$634$	−28613.6	−1.79241
$635$	0	0
$636$	0	0
$637$	9184.77	0.571294
$638$	−12212.6	−0.757839
$639$	0	0
$640$	0	0
$641$	13003.1	0.801233	0.400617	−	0.916246i	$-0.368796\pi$
$641$	13003.1	0.801233	0.400617	+	0.916246i	$0.368796\pi$
$642$	0	0
$643$	21269.1	1.30446	0.652232	−	0.758019i	$-0.273833\pi$
$643$	21269.1	1.30446	0.652232	+	0.758019i	$0.273833\pi$
$644$	5842.82	0.357514
$645$	0	0
$646$	1125.55	0.0685511
$647$	29247.2	1.77716	0.888582	−	0.458717i	$-0.151691\pi$
$647$	29247.2	1.77716	0.888582	+	0.458717i	$0.151691\pi$
$648$	0	0
$649$	−6141.64	−0.371465
$650$	0	0
$651$	0	0
$652$	−24179.9	−1.45239
$653$	−1678.30	−0.100577	−0.0502885	−	0.998735i	$-0.516014\pi$
$653$	−1678.30	−0.100577	−0.0502885	+	0.998735i	$0.516014\pi$
$654$	0	0
$655$	0	0
$656$	−16475.2	−0.980561
$657$	0	0
$658$	829.182	0.0491260
$659$	7620.41	0.450454	0.225227	−	0.974306i	$-0.427688\pi$
$659$	7620.41	0.450454	0.225227	+	0.974306i	$0.427688\pi$
$660$	0	0
$661$	−22762.2	−1.33941	−0.669703	−	0.742629i	$-0.733579\pi$
$661$	−22762.2	−1.33941	−0.669703	+	0.742629i	$0.733579\pi$
$662$	31556.7	1.85270
$663$	0	0
$664$	311.124	0.0181837
$665$	0	0
$666$	0	0
$667$	29713.2	1.72488
$668$	9150.93	0.530030
$669$	0	0
$670$	0	0
$671$	−3687.96	−0.212179
$672$	0	0
$673$	−698.528	−0.0400093	−0.0200047	−	0.999800i	$-0.506368\pi$
$673$	−698.528	−0.0400093	−0.0200047	+	0.999800i	$0.506368\pi$
$674$	47932.1	2.73928
$675$	0	0
$676$	−10514.8	−0.598250
$677$	−10860.1	−0.616524	−0.308262	−	0.951302i	$-0.599747\pi$
$677$	−10860.1	−0.616524	−0.308262	+	0.951302i	$0.599747\pi$
$678$	0	0
$679$	−181.976	−0.0102851
$680$	0	0
$681$	0	0
$682$	3045.93	0.171019
$683$	−6534.95	−0.366110	−0.183055	−	0.983103i	$-0.558599\pi$
$683$	−6534.95	−0.366110	−0.183055	+	0.983103i	$0.558599\pi$
$684$	0	0
$685$	0	0
$686$	−16900.3	−0.940606
$687$	0	0
$688$	−1873.88	−0.103839
$689$	−7710.88	−0.426359
$690$	0	0
$691$	−10915.3	−0.600924	−0.300462	−	0.953794i	$-0.597141\pi$
$691$	−10915.3	−0.600924	−0.300462	+	0.953794i	$0.597141\pi$
$692$	31224.9	1.71530
$693$	0	0
$694$	14079.0	0.770076
$695$	0	0
$696$	0	0
$697$	−11578.6	−0.629229
$698$	−25717.3	−1.39458
$699$	0	0
$700$	0	0
$701$	−18405.4	−0.991674	−0.495837	−	0.868416i	$-0.665139\pi$
$701$	−18405.4	−0.991674	−0.495837	+	0.868416i	$0.665139\pi$
$702$	0	0
$703$	−2598.39	−0.139403
$704$	6067.91	0.324848
$705$	0	0
$706$	33215.9	1.77068
$707$	1483.63	0.0789217
$708$	0	0
$709$	−21436.2	−1.13547	−0.567737	−	0.823210i	$-0.692181\pi$
$709$	−21436.2	−1.13547	−0.567737	+	0.823210i	$0.692181\pi$
$710$	0	0
$711$	0	0
$712$	399.413	0.0210234
$713$	−7410.73	−0.389248
$714$	0	0
$715$	0	0
$716$	26767.7	1.39715
$717$	0	0
$718$	37878.9	1.96884
$719$	−28641.7	−1.48561	−0.742807	−	0.669506i	$-0.766506\pi$
$719$	−28641.7	−1.48561	−0.742807	+	0.669506i	$0.766506\pi$
$720$	0	0
$721$	5201.51	0.268675
$722$	−27537.1	−1.41942
$723$	0	0
$724$	−3799.49	−0.195037
$725$	0	0
$726$	0	0
$727$	1723.13	0.0879055	0.0439527	−	0.999034i	$-0.486005\pi$
$727$	1723.13	0.0879055	0.0439527	+	0.999034i	$0.486005\pi$
$728$	−253.409	−0.0129010
$729$	0	0
$730$	0	0
$731$	−1316.95	−0.0666335
$732$	0	0
$733$	18813.4	0.948005	0.474003	−	0.880523i	$-0.342809\pi$
$733$	18813.4	0.948005	0.474003	+	0.880523i	$0.342809\pi$
$734$	1255.80	0.0631506
$735$	0	0
$736$	−27903.0	−1.39744
$737$	316.273	0.0158074
$738$	0	0
$739$	−10893.3	−0.542242	−0.271121	−	0.962545i	$-0.587394\pi$
$739$	−10893.3	−0.542242	−0.271121	+	0.962545i	$0.587394\pi$
$740$	0	0
$741$	0	0
$742$	6628.65	0.327959
$743$	11888.6	0.587013	0.293507	−	0.955957i	$-0.405178\pi$
$743$	11888.6	0.587013	0.293507	+	0.955957i	$0.405178\pi$
$744$	0	0
$745$	0	0
$746$	19140.4	0.939381
$747$	0	0
$748$	3945.53	0.192865
$749$	−6803.98	−0.331925
$750$	0	0
$751$	−32134.8	−1.56140	−0.780702	−	0.624903i	$-0.785138\pi$
$751$	−32134.8	−1.56140	−0.780702	+	0.624903i	$0.785138\pi$
$752$	−1938.41	−0.0939982
$753$	0	0
$754$	−33905.0	−1.63759
$755$	0	0
$756$	0	0
$757$	−17366.1	−0.833794	−0.416897	−	0.908954i	$-0.636882\pi$
$757$	−17366.1	−0.833794	−0.416897	+	0.908954i	$0.636882\pi$
$758$	−19128.1	−0.916573
$759$	0	0
$760$	0	0
$761$	27393.9	1.30490	0.652450	−	0.757832i	$-0.273741\pi$
$761$	27393.9	1.30490	0.652450	+	0.757832i	$0.273741\pi$
$762$	0	0
$763$	2060.41	0.0977611
$764$	40808.0	1.93243
$765$	0	0
$766$	14671.4	0.692034
$767$	−17050.6	−0.802689
$768$	0	0
$769$	−12091.3	−0.567000	−0.283500	−	0.958972i	$-0.591496\pi$
$769$	−12091.3	−0.567000	−0.283500	+	0.958972i	$0.591496\pi$
$770$	0	0
$771$	0	0
$772$	−12159.5	−0.566879
$773$	18895.8	0.879216	0.439608	−	0.898190i	$-0.355117\pi$
$773$	18895.8	0.879216	0.439608	+	0.898190i	$0.355117\pi$
$774$	0	0
$775$	0	0
$776$	−35.7490	−0.00165375
$777$	0	0
$778$	−26487.8	−1.22061
$779$	−1734.31	−0.0797664
$780$	0	0
$781$	979.089	0.0448586
$782$	−18834.0	−0.861258
$783$	0	0
$784$	18458.1	0.840838
$785$	0	0
$786$	0	0
$787$	−23705.9	−1.07373	−0.536863	−	0.843669i	$-0.680391\pi$
$787$	−23705.9	−1.07373	−0.536863	+	0.843669i	$0.680391\pi$
$788$	18379.4	0.830888
$789$	0	0
$790$	0	0
$791$	8668.77	0.389666
$792$	0	0
$793$	−10238.6	−0.458493
$794$	−16500.5	−0.737507
$795$	0	0
$796$	−3287.11	−0.146368
$797$	−37248.0	−1.65545	−0.827725	−	0.561135i	$-0.810365\pi$
$797$	−37248.0	−1.65545	−0.827725	+	0.561135i	$0.810365\pi$
$798$	0	0
$799$	−1362.30	−0.0603189
$800$	0	0
$801$	0	0
$802$	26966.7	1.18731
$803$	−7893.84	−0.346909
$804$	0	0
$805$	0	0
$806$	8456.21	0.369550
$807$	0	0
$808$	291.458	0.0126899
$809$	24861.0	1.08043	0.540214	−	0.841528i	$-0.318343\pi$
$809$	24861.0	1.08043	0.540214	+	0.841528i	$0.318343\pi$
$810$	0	0
$811$	4131.24	0.178875	0.0894375	−	0.995992i	$-0.471493\pi$
$811$	4131.24	0.178875	0.0894375	+	0.995992i	$0.471493\pi$
$812$	14855.5	0.642027
$813$	0	0
$814$	−17870.7	−0.769495
$815$	0	0
$816$	0	0
$817$	−197.259	−0.00844704
$818$	6941.81	0.296717
$819$	0	0
$820$	0	0
$821$	−25297.3	−1.07537	−0.537687	−	0.843145i	$-0.680702\pi$
$821$	−25297.3	−1.07537	−0.537687	+	0.843145i	$0.680702\pi$
$822$	0	0
$823$	18155.5	0.768969	0.384484	−	0.923131i	$-0.374379\pi$
$823$	18155.5	0.768969	0.384484	+	0.923131i	$0.374379\pi$
$824$	1021.83	0.0432005
$825$	0	0
$826$	14657.6	0.617435
$827$	−29566.8	−1.24321	−0.621607	−	0.783329i	$-0.713520\pi$
$827$	−29566.8	−1.24321	−0.621607	+	0.783329i	$0.713520\pi$
$828$	0	0
$829$	27249.4	1.14163	0.570814	−	0.821079i	$-0.306628\pi$
$829$	27249.4	1.14163	0.570814	+	0.821079i	$0.306628\pi$
$830$	0	0
$831$	0	0
$832$	16845.9	0.701956
$833$	12972.2	0.539568
$834$	0	0
$835$	0	0
$836$	590.981	0.0244492
$837$	0	0
$838$	−23154.3	−0.954478
$839$	15506.1	0.638058	0.319029	−	0.947745i	$-0.396643\pi$
$839$	15506.1	0.638058	0.319029	+	0.947745i	$0.396643\pi$
$840$	0	0
$841$	51157.5	2.09756
$842$	−38968.8	−1.59496
$843$	0	0
$844$	12972.9	0.529084
$845$	0	0
$846$	0	0
$847$	786.406	0.0319023
$848$	−15496.1	−0.627520
$849$	0	0
$850$	0	0
$851$	43479.4	1.75141
$852$	0	0
$853$	−8698.16	−0.349143	−0.174572	−	0.984644i	$-0.555854\pi$
$853$	−8698.16	−0.349143	−0.174572	+	0.984644i	$0.555854\pi$
$854$	8801.64	0.352677
$855$	0	0
$856$	−1336.64	−0.0533707
$857$	−28468.8	−1.13474	−0.567372	−	0.823462i	$-0.692040\pi$
$857$	−28468.8	−1.13474	−0.567372	+	0.823462i	$0.692040\pi$
$858$	0	0
$859$	−21154.3	−0.840251	−0.420125	−	0.907466i	$-0.638014\pi$
$859$	−21154.3	−0.840251	−0.420125	+	0.907466i	$0.638014\pi$
$860$	0	0
$861$	0	0
$862$	12338.4	0.487525
$863$	−45959.5	−1.81284	−0.906420	−	0.422378i	$-0.861195\pi$
$863$	−45959.5	−1.81284	−0.906420	+	0.422378i	$0.861195\pi$
$864$	0	0
$865$	0	0
$866$	−44524.5	−1.74712
$867$	0	0
$868$	−3705.10	−0.144884
$869$	−2207.72	−0.0861815
$870$	0	0
$871$	878.047	0.0341579
$872$	404.765	0.0157191
$873$	0	0
$874$	−2821.06	−0.109180
$875$	0	0
$876$	0	0
$877$	7315.10	0.281657	0.140829	−	0.990034i	$-0.455023\pi$
$877$	7315.10	0.281657	0.140829	+	0.990034i	$0.455023\pi$
$878$	−30655.6	−1.17833
$879$	0	0
$880$	0	0
$881$	−4652.19	−0.177907	−0.0889536	−	0.996036i	$-0.528352\pi$
$881$	−4652.19	−0.177907	−0.0889536	+	0.996036i	$0.528352\pi$
$882$	0	0
$883$	38954.3	1.48462	0.742309	−	0.670058i	$-0.233731\pi$
$883$	38954.3	1.48462	0.742309	+	0.670058i	$0.233731\pi$
$884$	10953.7	0.416756
$885$	0	0
$886$	16332.2	0.619289
$887$	−16917.5	−0.640398	−0.320199	−	0.947350i	$-0.603750\pi$
$887$	−16917.5	−0.640398	−0.320199	+	0.947350i	$0.603750\pi$
$888$	0	0
$889$	−3671.33	−0.138507
$890$	0	0
$891$	0	0
$892$	−6288.75	−0.236057
$893$	−204.053	−0.00764654
$894$	0	0
$895$	0	0
$896$	−1061.32	−0.0395715
$897$	0	0
$898$	−59051.8	−2.19441
$899$	−18842.0	−0.699016
$900$	0	0
$901$	−10890.5	−0.402682
$902$	−11927.9	−0.440306
$903$	0	0
$904$	1702.97	0.0626549
$905$	0	0
$906$	0	0
$907$	−38985.4	−1.42722	−0.713610	−	0.700543i	$-0.752941\pi$
$907$	−38985.4	−1.42722	−0.713610	+	0.700543i	$0.752941\pi$
$908$	−47889.8	−1.75031
$909$	0	0
$910$	0	0
$911$	−27759.8	−1.00957	−0.504787	−	0.863244i	$-0.668429\pi$
$911$	−27759.8	−1.00957	−0.504787	+	0.863244i	$0.668429\pi$
$912$	0	0
$913$	−2680.49	−0.0971646
$914$	44072.3	1.59495
$915$	0	0
$916$	−15905.6	−0.573728
$917$	1391.37	0.0501057
$918$	0	0
$919$	29705.5	1.06626	0.533130	−	0.846033i	$-0.321016\pi$
$919$	29705.5	1.06626	0.533130	+	0.846033i	$0.321016\pi$
$920$	0	0
$921$	0	0
$922$	−66127.6	−2.36204
$923$	2718.18	0.0969338
$924$	0	0
$925$	0	0
$926$	64724.1	2.29694
$927$	0	0
$928$	−70944.1	−2.50954
$929$	−2940.01	−0.103831	−0.0519153	−	0.998651i	$-0.516533\pi$
$929$	−2940.01	−0.103831	−0.0519153	+	0.998651i	$0.516533\pi$
$930$	0	0
$931$	1943.04	0.0684002
$932$	8605.99	0.302466
$933$	0	0
$934$	−60982.6	−2.13642
$935$	0	0
$936$	0	0
$937$	20041.9	0.698762	0.349381	−	0.936981i	$-0.386392\pi$
$937$	20041.9	0.698762	0.349381	+	0.936981i	$0.386392\pi$
$938$	−754.813	−0.0262745
$939$	0	0
$940$	0	0
$941$	14446.1	0.500457	0.250229	−	0.968187i	$-0.419494\pi$
$941$	14446.1	0.500457	0.250229	+	0.968187i	$0.419494\pi$
$942$	0	0
$943$	29020.6	1.00216
$944$	−34265.6	−1.18141
$945$	0	0
$946$	−1356.68	−0.0466272
$947$	3901.33	0.133871	0.0669357	−	0.997757i	$-0.478678\pi$
$947$	3901.33	0.133871	0.0669357	+	0.997757i	$0.478678\pi$
$948$	0	0
$949$	−21915.1	−0.749626
$950$	0	0
$951$	0	0
$952$	−357.904	−0.0121846
$953$	613.963	0.0208691	0.0104345	−	0.999946i	$-0.496679\pi$
$953$	613.963	0.0208691	0.0104345	+	0.999946i	$0.496679\pi$
$954$	0	0
$955$	0	0
$956$	−22014.6	−0.744773
$957$	0	0
$958$	42247.1	1.42478
$959$	−423.937	−0.0142749
$960$	0	0
$961$	−25091.6	−0.842256
$962$	−49613.3	−1.66278
$963$	0	0
$964$	56000.5	1.87101
$965$	0	0
$966$	0	0
$967$	20617.4	0.685637	0.342819	−	0.939402i	$-0.388618\pi$
$967$	20617.4	0.685637	0.342819	+	0.939402i	$0.388618\pi$
$968$	154.489	0.00512961
$969$	0	0
$970$	0	0
$971$	−22816.0	−0.754069	−0.377035	−	0.926199i	$-0.623056\pi$
$971$	−22816.0	−0.754069	−0.377035	+	0.926199i	$0.623056\pi$
$972$	0	0
$973$	−2645.73	−0.0871719
$974$	32071.6	1.05507
$975$	0	0
$976$	−20576.0	−0.674816
$977$	22650.2	0.741702	0.370851	−	0.928692i	$-0.379066\pi$
$977$	22650.2	0.741702	0.370851	+	0.928692i	$0.379066\pi$
$978$	0	0
$979$	−3441.15	−0.112339
$980$	0	0
$981$	0	0
$982$	−46422.8	−1.50857
$983$	−8963.01	−0.290820	−0.145410	−	0.989372i	$-0.546450\pi$
$983$	−8963.01	−0.290820	−0.145410	+	0.989372i	$0.546450\pi$
$984$	0	0
$985$	0	0
$986$	−47886.0	−1.54665
$987$	0	0
$988$	1640.70	0.0528316
$989$	3300.78	0.106126
$990$	0	0
$991$	422.204	0.0135336	0.00676678	−	0.999977i	$-0.497846\pi$
$991$	422.204	0.0135336	0.00676678	+	0.999977i	$0.497846\pi$
$992$	17694.1	0.566319
$993$	0	0
$994$	−2336.68	−0.0745623
$995$	0	0
$996$	0	0
$997$	12577.3	0.399524	0.199762	−	0.979844i	$-0.435983\pi$
$997$	12577.3	0.399524	0.199762	+	0.979844i	$0.435983\pi$
$998$	19443.4	0.616704
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.y.1.3		3
3.2	odd	2		825.4.a.o.1.1	✓	3
5.4	even	2		2475.4.a.v.1.1		3
15.2	even	4		825.4.c.n.199.1		6
15.8	even	4		825.4.c.n.199.6		6
15.14	odd	2		825.4.a.q.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.o.1.1	✓	3	3.2	odd	2
825.4.a.q.1.3	yes	3	15.14	odd	2
825.4.c.n.199.1		6	15.2	even	4
825.4.c.n.199.6		6	15.8	even	4
2475.4.a.v.1.1		3	5.4	even	2
2475.4.a.y.1.3		3	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)

\(f(q)\)	\(=\)	\(q+4.03932 q^{2} +8.31608 q^{4} +6.49923 q^{7} +1.27677 q^{8} -11.0000 q^{11} -30.5385 q^{13} +26.2524 q^{14} -61.3714 q^{16} -43.1314 q^{17} -6.46044 q^{19} -44.4325 q^{22} +108.104 q^{23} -123.355 q^{26} +54.0481 q^{28} +274.857 q^{29} -68.5519 q^{31} -258.113 q^{32} -174.221 q^{34} +402.200 q^{37} -26.0958 q^{38} +268.450 q^{41} +30.5334 q^{43} -91.4769 q^{44} +436.666 q^{46} +31.5850 q^{47} -300.760 q^{49} -253.961 q^{52} +252.497 q^{53} +8.29800 q^{56} +1110.24 q^{58} +558.331 q^{59} +335.270 q^{61} -276.903 q^{62} -551.628 q^{64} -28.7521 q^{67} -358.684 q^{68} -89.0081 q^{71} +717.622 q^{73} +1624.61 q^{74} -53.7256 q^{76} -71.4915 q^{77} +200.702 q^{79} +1084.36 q^{82} +243.681 q^{83} +123.334 q^{86} -14.0444 q^{88} +312.832 q^{89} -198.477 q^{91} +899.002 q^{92} +127.582 q^{94} -27.9996 q^{97} -1214.87 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 7 q^{4} - 16 q^{7} - 3 q^{8} - 33 q^{11} - 45 q^{13} + 116 q^{14} - 85 q^{16} - 58 q^{17} - 169 q^{19} - 11 q^{22} + 155 q^{23} - 167 q^{26} - 100 q^{28} + 277 q^{29} - 173 q^{31} + 97 q^{32}+ \cdots - 2273 q^{98}+O(q^{100}) \) 3 * q + q^2 + 7 * q^4 - 16 * q^7 - 3 * q^8 - 33 * q^11 - 45 * q^13 + 116 * q^14 - 85 * q^16 - 58 * q^17 - 169 * q^19 - 11 * q^22 + 155 * q^23 - 167 * q^26 - 100 * q^28 + 277 * q^29 - 173 * q^31 + 97 * q^32 - 146 * q^34 - 60 * q^37 + 169 * q^38 - 44 * q^41 + 109 * q^43 - 77 * q^44 + 425 * q^46 + 270 * q^47 - 427 * q^49 - 45 * q^52 - 148 * q^53 - 168 * q^56 + 783 * q^58 + 684 * q^59 - 1038 * q^61 + 953 * q^62 - 1129 * q^64 - 314 * q^67 - 366 * q^68 + 1459 * q^71 + 1170 * q^73 + 1764 * q^74 + 211 * q^76 + 176 * q^77 - 506 * q^79 + 1040 * q^82 - 347 * q^83 + 527 * q^86 + 33 * q^88 + 607 * q^89 - 398 * q^91 + 687 * q^92 - 1822 * q^94 - 1263 * q^97 - 2273 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more