LMFDB - Embedded newform 575.4.a.r.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.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:	$33.9260982533$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 4 x^{16} - 96 x^{15} + 368 x^{14} + 3705 x^{13} - 13440 x^{12} - 73933 x^{11} + 248806 x^{10} + \cdots - 2150912 $ x^17 - 4x^16 - 96x^15 + 368x^14 + 3705x^13 - 13440x^12 - 73933x^11 + 248806x^10 + 820568x^9 - 2474496x^8 - 5096659x^7 + 12861046x^6 + 16846350x^5 - 30611400x^4 - 25094976x^3 + 22642432x^2 + 5724640x - 2150912
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$4.94561$ of defining polynomial
Character	$\chi$	$=$	575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.94561 q^{2} +5.42743 q^{3} +16.4591 q^{4} +26.8419 q^{6} +28.7748 q^{7} +41.8353 q^{8} +2.45696 q^{9} +16.7210 q^{11} +89.3304 q^{12} -65.8600 q^{13} +142.309 q^{14} +75.2286 q^{16} +73.6189 q^{17} +12.1512 q^{18} -76.3128 q^{19} +156.173 q^{21} +82.6955 q^{22} +23.0000 q^{23} +227.058 q^{24} -325.718 q^{26} -133.206 q^{27} +473.606 q^{28} +36.8512 q^{29} +25.8215 q^{31} +37.3689 q^{32} +90.7519 q^{33} +364.090 q^{34} +40.4393 q^{36} +95.0278 q^{37} -377.413 q^{38} -357.450 q^{39} +159.298 q^{41} +772.371 q^{42} -492.347 q^{43} +275.212 q^{44} +113.749 q^{46} -421.614 q^{47} +408.298 q^{48} +484.988 q^{49} +399.561 q^{51} -1084.00 q^{52} +40.8194 q^{53} -658.783 q^{54} +1203.80 q^{56} -414.182 q^{57} +182.252 q^{58} +495.047 q^{59} -653.531 q^{61} +127.703 q^{62} +70.6985 q^{63} -417.017 q^{64} +448.824 q^{66} -859.860 q^{67} +1211.70 q^{68} +124.831 q^{69} +1171.37 q^{71} +102.788 q^{72} +842.104 q^{73} +469.970 q^{74} -1256.04 q^{76} +481.143 q^{77} -1767.81 q^{78} +497.161 q^{79} -789.301 q^{81} +787.825 q^{82} +1187.34 q^{83} +2570.46 q^{84} -2434.96 q^{86} +200.007 q^{87} +699.528 q^{88} -540.814 q^{89} -1895.11 q^{91} +378.559 q^{92} +140.144 q^{93} -2085.14 q^{94} +202.817 q^{96} +641.210 q^{97} +2398.56 q^{98} +41.0828 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 4 q^{2} + 12 q^{3} + 72 q^{4} + 12 q^{6} + 72 q^{7} + 48 q^{8} + 155 q^{9} - 4 q^{11} + 342 q^{12} + 208 q^{13} - 118 q^{14} + 220 q^{16} + 268 q^{17} + 180 q^{18} - 72 q^{19} - 16 q^{21} + 318 q^{22}+ \cdots + 1908 q^{99}+O(q^{100}) $ 17 * q + 4 * q^2 + 12 * q^3 + 72 * q^4 + 12 * q^6 + 72 * q^7 + 48 * q^8 + 155 * q^9 - 4 * q^11 + 342 * q^12 + 208 * q^13 - 118 * q^14 + 220 * q^16 + 268 * q^17 + 180 * q^18 - 72 * q^19 - 16 * q^21 + 318 * q^22 + 391 * q^23 - 54 * q^24 - 72 * q^26 + 408 * q^27 + 928 * q^28 - 28 * q^29 - 40 * q^31 + 268 * q^32 + 1382 * q^33 - 132 * q^34 + 932 * q^36 + 586 * q^37 + 596 * q^38 + 600 * q^39 + 134 * q^41 - 104 * q^42 + 1264 * q^43 + 806 * q^44 + 92 * q^46 + 184 * q^47 + 2928 * q^48 + 737 * q^49 - 552 * q^51 + 1558 * q^52 + 1102 * q^53 - 782 * q^54 - 1150 * q^56 + 1188 * q^57 + 3882 * q^58 + 354 * q^59 + 550 * q^61 - 1114 * q^62 + 3102 * q^63 - 50 * q^64 - 708 * q^66 + 898 * q^67 + 3774 * q^68 + 276 * q^69 + 680 * q^71 + 102 * q^72 + 4980 * q^73 - 794 * q^74 + 54 * q^76 + 584 * q^77 + 4108 * q^78 - 1984 * q^79 + 2457 * q^81 + 1798 * q^82 + 2634 * q^83 + 974 * q^84 - 3074 * q^86 - 472 * q^87 + 7584 * q^88 - 598 * q^89 - 272 * q^91 + 1656 * q^92 + 2384 * q^93 + 1170 * q^94 + 1480 * q^96 + 6266 * q^97 + 3486 * q^98 + 1908 * q^99

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 $

$2$	4.94561	1.74854	0.874269	−	0.485442i	$-0.161341\pi$
$2$	4.94561	1.74854	0.874269	+	0.485442i	$0.161341\pi$
$3$	5.42743	1.04451	0.522254	−	0.852790i	$-0.325091\pi$
$3$	5.42743	1.04451	0.522254	+	0.852790i	$0.325091\pi$
$4$	16.4591	2.05738
$5$	0	0
$5$	0	0
$6$	26.8419	1.82636
$7$	28.7748	1.55369	0.776846	−	0.629691i	$-0.216818\pi$
$7$	28.7748	1.55369	0.776846	+	0.629691i	$0.216818\pi$
$8$	41.8353	1.84888
$9$	2.45696	0.0909985
$10$	0	0
$11$	16.7210	0.458324	0.229162	−	0.973388i	$-0.426401\pi$
$11$	16.7210	0.458324	0.229162	+	0.973388i	$0.426401\pi$
$12$	89.3304	2.14896
$13$	−65.8600	−1.40510	−0.702549	−	0.711635i	$-0.747955\pi$
$13$	−65.8600	−1.40510	−0.702549	+	0.711635i	$0.747955\pi$
$14$	142.309	2.71669
$15$	0	0
$16$	75.2286	1.17545
$17$	73.6189	1.05031	0.525153	−	0.851008i	$-0.324008\pi$
$17$	73.6189	1.05031	0.525153	+	0.851008i	$0.324008\pi$
$18$	12.1512	0.159114
$19$	−76.3128	−0.921439	−0.460720	−	0.887546i	$-0.652409\pi$
$19$	−76.3128	−0.921439	−0.460720	+	0.887546i	$0.652409\pi$
$20$	0	0
$21$	156.173	1.62284
$22$	82.6955	0.801397
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	227.058	1.93117
$25$	0	0
$26$	−325.718	−2.45687
$27$	−133.206	−0.949460
$28$	473.606	3.19654
$29$	36.8512	0.235969	0.117984	−	0.993015i	$-0.462357\pi$
$29$	36.8512	0.235969	0.117984	+	0.993015i	$0.462357\pi$
$30$	0	0
$31$	25.8215	0.149602	0.0748012	−	0.997198i	$-0.476168\pi$
$31$	25.8215	0.149602	0.0748012	+	0.997198i	$0.476168\pi$
$32$	37.3689	0.206436
$33$	90.7519	0.478724
$34$	364.090	1.83650
$35$	0	0
$36$	40.4393	0.187219
$37$	95.0278	0.422229	0.211114	−	0.977461i	$-0.432291\pi$
$37$	95.0278	0.422229	0.211114	+	0.977461i	$0.432291\pi$
$38$	−377.413	−1.61117
$39$	−357.450	−1.46764
$40$	0	0
$41$	159.298	0.606783	0.303392	−	0.952866i	$-0.401881\pi$
$41$	159.298	0.606783	0.303392	+	0.952866i	$0.401881\pi$
$42$	772.371	2.83761
$43$	−492.347	−1.74610	−0.873049	−	0.487633i	$-0.837861\pi$
$43$	−492.347	−1.74610	−0.873049	+	0.487633i	$0.837861\pi$
$44$	275.212	0.942949
$45$	0	0
$46$	113.749	0.364595
$47$	−421.614	−1.30848	−0.654241	−	0.756286i	$-0.727012\pi$
$47$	−421.614	−1.30848	−0.654241	+	0.756286i	$0.727012\pi$
$48$	408.298	1.22776
$49$	484.988	1.41396
$50$	0	0
$51$	399.561	1.09705
$52$	−1084.00	−2.89083
$53$	40.8194	0.105792	0.0528959	−	0.998600i	$-0.483155\pi$
$53$	40.8194	0.105792	0.0528959	+	0.998600i	$0.483155\pi$
$54$	−658.783	−1.66017
$55$	0	0
$56$	1203.80	2.87259
$57$	−414.182	−0.962451
$58$	182.252	0.412601
$59$	495.047	1.09237	0.546183	−	0.837666i	$-0.316080\pi$
$59$	495.047	1.09237	0.546183	+	0.837666i	$0.316080\pi$
$60$	0	0
$61$	−653.531	−1.37174	−0.685870	−	0.727724i	$-0.740578\pi$
$61$	−653.531	−1.37174	−0.685870	+	0.727724i	$0.740578\pi$
$62$	127.703	0.261585
$63$	70.6985	0.141384
$64$	−417.017	−0.814486
$65$	0	0
$66$	448.824	0.837066
$67$	−859.860	−1.56789	−0.783945	−	0.620831i	$-0.786795\pi$
$67$	−859.860	−1.56789	−0.783945	+	0.620831i	$0.786795\pi$
$68$	1211.70	2.16088
$69$	124.831	0.217795
$70$	0	0
$71$	1171.37	1.95798	0.978990	−	0.203906i	$-0.0653638\pi$
$71$	1171.37	1.95798	0.978990	+	0.203906i	$0.0653638\pi$
$72$	102.788	0.168245
$73$	842.104	1.35015	0.675074	−	0.737750i	$-0.264112\pi$
$73$	842.104	1.35015	0.675074	+	0.737750i	$0.264112\pi$
$74$	469.970	0.738283
$75$	0	0
$76$	−1256.04	−1.89575
$77$	481.143	0.712095
$78$	−1767.81	−2.56622
$79$	497.161	0.708037	0.354019	−	0.935238i	$-0.384815\pi$
$79$	497.161	0.708037	0.354019	+	0.935238i	$0.384815\pi$
$80$	0	0
$81$	−789.301	−1.08272
$82$	787.825	1.06098
$83$	1187.34	1.57021	0.785106	−	0.619362i	$-0.212609\pi$
$83$	1187.34	1.57021	0.785106	+	0.619362i	$0.212609\pi$
$84$	2570.46	3.33882
$85$	0	0
$86$	−2434.96	−3.05312
$87$	200.007	0.246472
$88$	699.528	0.847385
$89$	−540.814	−0.644114	−0.322057	−	0.946720i	$-0.604374\pi$
$89$	−540.814	−0.644114	−0.322057	+	0.946720i	$0.604374\pi$
$90$	0	0
$91$	−1895.11	−2.18309
$92$	378.559	0.428994
$93$	140.144	0.156261
$94$	−2085.14	−2.28793
$95$	0	0
$96$	202.817	0.215624
$97$	641.210	0.671186	0.335593	−	0.942007i	$-0.391063\pi$
$97$	641.210	0.671186	0.335593	+	0.942007i	$0.391063\pi$
$98$	2398.56	2.47236
$99$	41.0828	0.0417068
$100$	0	0
$101$	205.740	0.202692	0.101346	−	0.994851i	$-0.467685\pi$
$101$	205.740	0.202692	0.101346	+	0.994851i	$0.467685\pi$
$102$	1976.07	1.91824
$103$	790.392	0.756113	0.378056	−	0.925783i	$-0.376593\pi$
$103$	790.392	0.756113	0.378056	+	0.925783i	$0.376593\pi$
$104$	−2755.27	−2.59785
$105$	0	0
$106$	201.877	0.184981
$107$	−192.007	−0.173476	−0.0867382	−	0.996231i	$-0.527644\pi$
$107$	−192.007	−0.173476	−0.0867382	+	0.996231i	$0.527644\pi$
$108$	−2192.44	−1.95340
$109$	−1238.31	−1.08815	−0.544074	−	0.839037i	$-0.683119\pi$
$109$	−1238.31	−1.08815	−0.544074	+	0.839037i	$0.683119\pi$
$110$	0	0
$111$	515.756	0.441022
$112$	2164.69	1.82628
$113$	−1829.76	−1.52327	−0.761633	−	0.648009i	$-0.775602\pi$
$113$	−1829.76	−1.52327	−0.761633	+	0.648009i	$0.775602\pi$
$114$	−2048.38	−1.68288
$115$	0	0
$116$	606.537	0.485479
$117$	−161.815	−0.127862
$118$	2448.31	1.91004
$119$	2118.37	1.63185
$120$	0	0
$121$	−1051.41	−0.789939
$122$	−3232.11	−2.39854
$123$	864.577	0.633791
$124$	424.998	0.307790
$125$	0	0
$126$	349.647	0.247215
$127$	−1177.03	−0.822399	−0.411199	−	0.911545i	$-0.634890\pi$
$127$	−1177.03	−0.822399	−0.411199	+	0.911545i	$0.634890\pi$
$128$	−2361.35	−1.63059
$129$	−2672.18	−1.82381
$130$	0	0
$131$	−1118.17	−0.745763	−0.372881	−	0.927879i	$-0.621630\pi$
$131$	−1118.17	−0.745763	−0.372881	+	0.927879i	$0.621630\pi$
$132$	1493.69	0.984919
$133$	−2195.88	−1.43163
$134$	−4252.53	−2.74151
$135$	0	0
$136$	3079.87	1.94189
$137$	174.022	0.108523	0.0542616	−	0.998527i	$-0.482720\pi$
$137$	174.022	0.108523	0.0542616	+	0.998527i	$0.482720\pi$
$138$	617.365	0.380823
$139$	871.055	0.531524	0.265762	−	0.964039i	$-0.414376\pi$
$139$	871.055	0.531524	0.265762	+	0.964039i	$0.414376\pi$
$140$	0	0
$141$	−2288.28	−1.36672
$142$	5793.17	3.42360
$143$	−1101.24	−0.643991
$144$	184.834	0.106964
$145$	0	0
$146$	4164.72	2.36078
$147$	2632.24	1.47689
$148$	1564.07	0.868687
$149$	−1163.23	−0.639565	−0.319782	−	0.947491i	$-0.603610\pi$
$149$	−1163.23	−0.639565	−0.319782	+	0.947491i	$0.603610\pi$
$150$	0	0
$151$	−133.547	−0.0719730	−0.0359865	−	0.999352i	$-0.511457\pi$
$151$	−133.547	−0.0719730	−0.0359865	+	0.999352i	$0.511457\pi$
$152$	−3192.57	−1.70363
$153$	180.879	0.0955762
$154$	2379.54	1.24512
$155$	0	0
$156$	−5883.30	−3.01950
$157$	1356.69	0.689653	0.344826	−	0.938666i	$-0.387938\pi$
$157$	1356.69	0.689653	0.344826	+	0.938666i	$0.387938\pi$
$158$	2458.76	1.23803
$159$	221.544	0.110501
$160$	0	0
$161$	661.820	0.323967
$162$	−3903.58	−1.89317
$163$	3579.07	1.71984	0.859922	−	0.510425i	$-0.170512\pi$
$163$	3579.07	1.71984	0.859922	+	0.510425i	$0.170512\pi$
$164$	2621.89	1.24839
$165$	0	0
$166$	5872.12	2.74557
$167$	1903.72	0.882123	0.441062	−	0.897477i	$-0.354602\pi$
$167$	1903.72	0.882123	0.441062	+	0.897477i	$0.354602\pi$
$168$	6533.55	3.00044
$169$	2140.54	0.974302
$170$	0	0
$171$	−187.497	−0.0838496
$172$	−8103.58	−3.59239
$173$	−2700.03	−1.18659	−0.593293	−	0.804986i	$-0.702173\pi$
$173$	−2700.03	−1.18659	−0.593293	+	0.804986i	$0.702173\pi$
$174$	989.158	0.430965
$175$	0	0
$176$	1257.90	0.538736
$177$	2686.83	1.14099
$178$	−2674.66	−1.12626
$179$	−754.312	−0.314972	−0.157486	−	0.987521i	$-0.550339\pi$
$179$	−754.312	−0.314972	−0.157486	+	0.987521i	$0.550339\pi$
$180$	0	0
$181$	−3006.53	−1.23466	−0.617330	−	0.786705i	$-0.711786\pi$
$181$	−3006.53	−1.23466	−0.617330	+	0.786705i	$0.711786\pi$
$182$	−9372.47	−3.81722
$183$	−3546.99	−1.43279
$184$	962.212	0.385517
$185$	0	0
$186$	693.099	0.273228
$187$	1230.98	0.481381
$188$	−6939.37	−2.69205
$189$	−3832.96	−1.47517
$190$	0	0
$191$	2145.54	0.812806	0.406403	−	0.913694i	$-0.366783\pi$
$191$	2145.54	0.812806	0.406403	+	0.913694i	$0.366783\pi$
$192$	−2263.33	−0.850737
$193$	3612.08	1.34717	0.673584	−	0.739111i	$-0.264754\pi$
$193$	3612.08	1.34717	0.673584	+	0.739111i	$0.264754\pi$
$194$	3171.18	1.17359
$195$	0	0
$196$	7982.45	2.90906
$197$	2155.29	0.779483	0.389742	−	0.920924i	$-0.372564\pi$
$197$	2155.29	0.779483	0.389742	+	0.920924i	$0.372564\pi$
$198$	203.179	0.0729260
$199$	−4366.37	−1.55539	−0.777697	−	0.628639i	$-0.783612\pi$
$199$	−4366.37	−1.55539	−0.777697	+	0.628639i	$0.783612\pi$
$200$	0	0
$201$	−4666.83	−1.63767
$202$	1017.51	0.354414
$203$	1060.39	0.366623
$204$	6576.40	2.25706
$205$	0	0
$206$	3908.97	1.32209
$207$	56.5101	0.0189745
$208$	−4954.56	−1.65162
$209$	−1276.02	−0.422318
$210$	0	0
$211$	−1373.27	−0.448057	−0.224028	−	0.974583i	$-0.571921\pi$
$211$	−1373.27	−0.448057	−0.224028	+	0.974583i	$0.571921\pi$
$212$	671.849	0.217655
$213$	6357.55	2.04513
$214$	−949.590	−0.303330
$215$	0	0
$216$	−5572.69	−1.75543
$217$	743.007	0.232436
$218$	−6124.18	−1.90267
$219$	4570.46	1.41024
$220$	0	0
$221$	−4848.54	−1.47578
$222$	2550.73	0.771143
$223$	−1005.22	−0.301859	−0.150929	−	0.988545i	$-0.548227\pi$
$223$	−1005.22	−0.301859	−0.150929	+	0.988545i	$0.548227\pi$
$224$	1075.28	0.320738
$225$	0	0
$226$	−9049.26	−2.66349
$227$	−2971.43	−0.868813	−0.434407	−	0.900717i	$-0.643042\pi$
$227$	−2971.43	−0.868813	−0.434407	+	0.900717i	$0.643042\pi$
$228$	−6817.05	−1.98013
$229$	291.005	0.0839744	0.0419872	−	0.999118i	$-0.486631\pi$
$229$	291.005	0.0839744	0.0419872	+	0.999118i	$0.486631\pi$
$230$	0	0
$231$	2611.37	0.743789
$232$	1541.68	0.436277
$233$	−4215.49	−1.18526	−0.592631	−	0.805474i	$-0.701911\pi$
$233$	−4215.49	−1.18526	−0.592631	+	0.805474i	$0.701911\pi$
$234$	−800.276	−0.223571
$235$	0	0
$236$	8148.02	2.24742
$237$	2698.30	0.739551
$238$	10476.6	2.85335
$239$	−234.318	−0.0634176	−0.0317088	−	0.999497i	$-0.510095\pi$
$239$	−234.318	−0.0634176	−0.0317088	+	0.999497i	$0.510095\pi$
$240$	0	0
$241$	4644.28	1.24135	0.620673	−	0.784070i	$-0.286860\pi$
$241$	4644.28	1.24135	0.620673	+	0.784070i	$0.286860\pi$
$242$	−5199.86	−1.38124
$243$	−687.325	−0.181448
$244$	−10756.5	−2.82220
$245$	0	0
$246$	4275.86	1.10821
$247$	5025.96	1.29471
$248$	1080.25	0.276596
$249$	6444.20	1.64010
$250$	0	0
$251$	−961.417	−0.241769	−0.120885	−	0.992667i	$-0.538573\pi$
$251$	−961.417	−0.241769	−0.120885	+	0.992667i	$0.538573\pi$
$252$	1163.63	0.290881
$253$	384.583	0.0955672
$254$	−5821.14	−1.43800
$255$	0	0
$256$	−8342.20	−2.03667
$257$	−2800.34	−0.679689	−0.339845	−	0.940482i	$-0.610375\pi$
$257$	−2800.34	−0.679689	−0.339845	+	0.940482i	$0.610375\pi$
$258$	−13215.5	−3.18901
$259$	2734.40	0.656014
$260$	0	0
$261$	90.5419	0.0214728
$262$	−5530.03	−1.30399
$263$	7316.97	1.71553	0.857763	−	0.514045i	$-0.171853\pi$
$263$	7316.97	1.71553	0.857763	+	0.514045i	$0.171853\pi$
$264$	3796.63	0.885101
$265$	0	0
$266$	−10860.0	−2.50326
$267$	−2935.23	−0.672783
$268$	−14152.5	−3.22575
$269$	3429.15	0.777244	0.388622	−	0.921397i	$-0.372951\pi$
$269$	3429.15	0.777244	0.388622	+	0.921397i	$0.372951\pi$
$270$	0	0
$271$	7608.53	1.70548	0.852740	−	0.522336i	$-0.174939\pi$
$271$	7608.53	1.70548	0.852740	+	0.522336i	$0.174939\pi$
$272$	5538.24	1.23458
$273$	−10285.6	−2.28026
$274$	860.644	0.189757
$275$	0	0
$276$	2054.60	0.448088
$277$	3156.98	0.684782	0.342391	−	0.939558i	$-0.388763\pi$
$277$	3156.98	0.684782	0.342391	+	0.939558i	$0.388763\pi$
$278$	4307.90	0.929391
$279$	63.4423	0.0136136
$280$	0	0
$281$	2217.02	0.470663	0.235331	−	0.971915i	$-0.424382\pi$
$281$	2217.02	0.470663	0.235331	+	0.971915i	$0.424382\pi$
$282$	−11316.9	−2.38976
$283$	2162.68	0.454268	0.227134	−	0.973864i	$-0.427065\pi$
$283$	2162.68	0.454268	0.227134	+	0.973864i	$0.427065\pi$
$284$	19279.7	4.02832
$285$	0	0
$286$	−5446.33	−1.12604
$287$	4583.76	0.942755
$288$	91.8138	0.0187853
$289$	506.736	0.103142
$290$	0	0
$291$	3480.12	0.701059
$292$	13860.3	2.77777
$293$	−2576.23	−0.513669	−0.256835	−	0.966455i	$-0.582680\pi$
$293$	−2576.23	−0.513669	−0.256835	+	0.966455i	$0.582680\pi$
$294$	13018.0	2.58240
$295$	0	0
$296$	3975.51	0.780649
$297$	−2227.33	−0.435161
$298$	−5752.86	−1.11830
$299$	−1514.78	−0.292983
$300$	0	0
$301$	−14167.2	−2.71290
$302$	−660.473	−0.125848
$303$	1116.64	0.211713
$304$	−5740.90	−1.08310
$305$	0	0
$306$	894.555	0.167119
$307$	3252.03	0.604571	0.302286	−	0.953217i	$-0.402250\pi$
$307$	3252.03	0.604571	0.302286	+	0.953217i	$0.402250\pi$
$308$	7919.16	1.46505
$309$	4289.79	0.789766
$310$	0	0
$311$	3954.42	0.721012	0.360506	−	0.932757i	$-0.382604\pi$
$311$	3954.42	0.721012	0.360506	+	0.932757i	$0.382604\pi$
$312$	−14954.0	−2.71348
$313$	6423.16	1.15993	0.579966	−	0.814641i	$-0.303066\pi$
$313$	6423.16	1.15993	0.579966	+	0.814641i	$0.303066\pi$
$314$	6709.65	1.20588
$315$	0	0
$316$	8182.80	1.45670
$317$	2582.98	0.457649	0.228825	−	0.973468i	$-0.426512\pi$
$317$	2582.98	0.457649	0.228825	+	0.973468i	$0.426512\pi$
$318$	1095.67	0.193214
$319$	616.189	0.108150
$320$	0	0
$321$	−1042.10	−0.181198
$322$	3273.10	0.566469
$323$	−5618.06	−0.967793
$324$	−12991.2	−2.22757
$325$	0	0
$326$	17700.7	3.00721
$327$	−6720.81	−1.13658
$328$	6664.27	1.12187
$329$	−12131.8	−2.03298
$330$	0	0
$331$	−7154.87	−1.18812	−0.594059	−	0.804421i	$-0.702475\pi$
$331$	−7154.87	−1.18812	−0.594059	+	0.804421i	$0.702475\pi$
$332$	19542.5	3.23053
$333$	233.479	0.0384222
$334$	9415.08	1.54243
$335$	0	0
$336$	11748.7	1.90757
$337$	−3151.02	−0.509338	−0.254669	−	0.967028i	$-0.581967\pi$
$337$	−3151.02	−0.509338	−0.254669	+	0.967028i	$0.581967\pi$
$338$	10586.3	1.70360
$339$	−9930.87	−1.59106
$340$	0	0
$341$	431.760	0.0685664
$342$	−927.289	−0.146614
$343$	4085.67	0.643165
$344$	−20597.5	−3.22832
$345$	0	0
$346$	−13353.3	−2.07479
$347$	−8874.67	−1.37296	−0.686480	−	0.727148i	$-0.740845\pi$
$347$	−8874.67	−1.37296	−0.686480	+	0.727148i	$0.740845\pi$
$348$	3291.93	0.507087
$349$	−3484.70	−0.534475	−0.267238	−	0.963631i	$-0.586111\pi$
$349$	−3484.70	−0.534475	−0.267238	+	0.963631i	$0.586111\pi$
$350$	0	0
$351$	8772.92	1.33409
$352$	624.844	0.0946145
$353$	3409.79	0.514121	0.257060	−	0.966395i	$-0.417246\pi$
$353$	3409.79	0.514121	0.257060	+	0.966395i	$0.417246\pi$
$354$	13288.0	1.99506
$355$	0	0
$356$	−8901.30	−1.32519
$357$	11497.3	1.70448
$358$	−3730.54	−0.550740
$359$	−2191.51	−0.322183	−0.161091	−	0.986940i	$-0.551501\pi$
$359$	−2191.51	−0.322183	−0.161091	+	0.986940i	$0.551501\pi$
$360$	0	0
$361$	−1035.36	−0.150950
$362$	−14869.1	−2.15885
$363$	−5706.44	−0.825098
$364$	−31191.7	−4.49146
$365$	0	0
$366$	−17542.1	−2.50529
$367$	−1446.01	−0.205670	−0.102835	−	0.994698i	$-0.532791\pi$
$367$	−1446.01	−0.205670	−0.102835	+	0.994698i	$0.532791\pi$
$368$	1730.26	0.245098
$369$	391.388	0.0552164
$370$	0	0
$371$	1174.57	0.164368
$372$	2306.64	0.321489
$373$	9909.59	1.37560	0.687801	−	0.725900i	$-0.258576\pi$
$373$	9909.59	1.37560	0.687801	+	0.725900i	$0.258576\pi$
$374$	6087.95	0.841712
$375$	0	0
$376$	−17638.3	−2.41922
$377$	−2427.02	−0.331560
$378$	−18956.3	−2.57939
$379$	6430.45	0.871531	0.435765	−	0.900060i	$-0.356478\pi$
$379$	6430.45	0.871531	0.435765	+	0.900060i	$0.356478\pi$
$380$	0	0
$381$	−6388.25	−0.859003
$382$	10611.0	1.42122
$383$	881.749	0.117638	0.0588189	−	0.998269i	$-0.481267\pi$
$383$	881.749	0.117638	0.0588189	+	0.998269i	$0.481267\pi$
$384$	−12816.1	−1.70317
$385$	0	0
$386$	17864.0	2.35557
$387$	−1209.68	−0.158892
$388$	10553.7	1.38089
$389$	−8893.90	−1.15923	−0.579613	−	0.814892i	$-0.696796\pi$
$389$	−8893.90	−1.15923	−0.579613	+	0.814892i	$0.696796\pi$
$390$	0	0
$391$	1693.23	0.219004
$392$	20289.6	2.61424
$393$	−6068.78	−0.778956
$394$	10659.2	1.36296
$395$	0	0
$396$	676.185	0.0858070
$397$	−3537.94	−0.447265	−0.223633	−	0.974674i	$-0.571792\pi$
$397$	−3537.94	−0.447265	−0.223633	+	0.974674i	$0.571792\pi$
$398$	−21594.4	−2.71967
$399$	−11918.0	−1.49535
$400$	0	0
$401$	8308.81	1.03472	0.517360	−	0.855768i	$-0.326915\pi$
$401$	8308.81	1.03472	0.517360	+	0.855768i	$0.326915\pi$
$402$	−23080.3	−2.86354
$403$	−1700.60	−0.210206
$404$	3386.28	0.417015
$405$	0	0
$406$	5244.26	0.641054
$407$	1588.96	0.193518
$408$	16715.8	2.02832
$409$	15235.6	1.84194	0.920969	−	0.389637i	$-0.127400\pi$
$409$	15235.6	1.84194	0.920969	+	0.389637i	$0.127400\pi$
$410$	0	0
$411$	944.490	0.113353
$412$	13009.1	1.55561
$413$	14244.9	1.69720
$414$	279.477	0.0331776
$415$	0	0
$416$	−2461.11	−0.290063
$417$	4727.59	0.555182
$418$	−6310.72	−0.738439
$419$	2610.62	0.304385	0.152193	−	0.988351i	$-0.451367\pi$
$419$	2610.62	0.304385	0.152193	+	0.988351i	$0.451367\pi$
$420$	0	0
$421$	5889.61	0.681810	0.340905	−	0.940098i	$-0.389267\pi$
$421$	5889.61	0.681810	0.340905	+	0.940098i	$0.389267\pi$
$422$	−6791.67	−0.783444
$423$	−1035.89	−0.119070
$424$	1707.69	0.195596
$425$	0	0
$426$	31442.0	3.57598
$427$	−18805.2	−2.13126
$428$	−3160.25	−0.356908
$429$	−5976.92	−0.672654
$430$	0	0
$431$	−15351.4	−1.71567	−0.857833	−	0.513929i	$-0.828189\pi$
$431$	−15351.4	−1.71567	−0.857833	+	0.513929i	$0.828189\pi$
$432$	−10020.9	−1.11604
$433$	2875.39	0.319128	0.159564	−	0.987188i	$-0.448991\pi$
$433$	2875.39	0.319128	0.159564	+	0.987188i	$0.448991\pi$
$434$	3674.63	0.406423
$435$	0	0
$436$	−20381.4	−2.23874
$437$	−1755.19	−0.192133
$438$	22603.7	2.46586
$439$	13824.2	1.50294	0.751471	−	0.659766i	$-0.229345\pi$
$439$	13824.2	1.50294	0.751471	+	0.659766i	$0.229345\pi$
$440$	0	0
$441$	1191.60	0.128668
$442$	−23979.0	−2.58046
$443$	−10229.0	−1.09705	−0.548526	−	0.836133i	$-0.684811\pi$
$443$	−10229.0	−1.09705	−0.548526	+	0.836133i	$0.684811\pi$
$444$	8488.87	0.907351
$445$	0	0
$446$	−4971.43	−0.527812
$447$	−6313.32	−0.668031
$448$	−11999.6	−1.26546
$449$	8117.62	0.853216	0.426608	−	0.904437i	$-0.359708\pi$
$449$	8117.62	0.853216	0.426608	+	0.904437i	$0.359708\pi$
$450$	0	0
$451$	2663.61	0.278104
$452$	−30116.1	−3.13394
$453$	−724.818	−0.0751764
$454$	−14695.5	−1.51915
$455$	0	0
$456$	−17327.4	−1.77945
$457$	11728.0	1.20047	0.600235	−	0.799824i	$-0.295074\pi$
$457$	11728.0	1.20047	0.600235	+	0.799824i	$0.295074\pi$
$458$	1439.20	0.146832
$459$	−9806.44	−0.997223
$460$	0	0
$461$	1557.75	0.157379	0.0786897	−	0.996899i	$-0.474926\pi$
$461$	1557.75	0.157379	0.0786897	+	0.996899i	$0.474926\pi$
$462$	12914.8	1.30054
$463$	1824.04	0.183089	0.0915446	−	0.995801i	$-0.470820\pi$
$463$	1824.04	0.183089	0.0915446	+	0.995801i	$0.470820\pi$
$464$	2772.26	0.277369
$465$	0	0
$466$	−20848.2	−2.07247
$467$	13059.8	1.29408	0.647040	−	0.762456i	$-0.276007\pi$
$467$	13059.8	1.29408	0.647040	+	0.762456i	$0.276007\pi$
$468$	−2663.33	−0.263061
$469$	−24742.3	−2.43602
$470$	0	0
$471$	7363.32	0.720348
$472$	20710.4	2.01965
$473$	−8232.53	−0.800279
$474$	13344.8	1.29313
$475$	0	0
$476$	34866.4	3.35735
$477$	100.291	0.00962690
$478$	−1158.85	−0.110888
$479$	−18106.1	−1.72712	−0.863559	−	0.504247i	$-0.831770\pi$
$479$	−18106.1	−1.72712	−0.863559	+	0.504247i	$0.831770\pi$
$480$	0	0
$481$	−6258.53	−0.593273
$482$	22968.8	2.17054
$483$	3591.98	0.338387
$484$	−17305.2	−1.62521
$485$	0	0
$486$	−3399.24	−0.317269
$487$	5675.51	0.528095	0.264047	−	0.964510i	$-0.414942\pi$
$487$	5675.51	0.528095	0.264047	+	0.964510i	$0.414942\pi$
$488$	−27340.7	−2.53618
$489$	19425.2	1.79639
$490$	0	0
$491$	−2.76101	−0.000253773 0	−0.000126887	−	1.00000i	$-0.500040\pi$
$491$	−2.76101	−0.000253773 0	−0.000126887		1.00000i	$0.500040\pi$
$492$	14230.1	1.30395
$493$	2712.94	0.247839
$494$	24856.4	2.26385
$495$	0	0
$496$	1942.51	0.175850
$497$	33706.1	3.04210
$498$	31870.5	2.86778
$499$	1240.62	0.111299	0.0556493	−	0.998450i	$-0.482277\pi$
$499$	1240.62	0.111299	0.0556493	+	0.998450i	$0.482277\pi$
$500$	0	0
$501$	10332.3	0.921385
$502$	−4754.80	−0.422743
$503$	11807.7	1.04668	0.523340	−	0.852124i	$-0.324686\pi$
$503$	11807.7	1.04668	0.523340	+	0.852124i	$0.324686\pi$
$504$	2957.69	0.261401
$505$	0	0
$506$	1902.00	0.167103
$507$	11617.6	1.01767
$508$	−19372.8	−1.69199
$509$	8573.55	0.746593	0.373296	−	0.927712i	$-0.378227\pi$
$509$	8573.55	0.746593	0.373296	+	0.927712i	$0.378227\pi$
$510$	0	0
$511$	24231.4	2.09771
$512$	−22366.5	−1.93060
$513$	10165.3	0.874870
$514$	−13849.4	−1.18846
$515$	0	0
$516$	−43981.6	−3.75229
$517$	−7049.80	−0.599709
$518$	13523.3	1.14706
$519$	−14654.2	−1.23940
$520$	0	0
$521$	1036.56	0.0871638	0.0435819	−	0.999050i	$-0.486123\pi$
$521$	1036.56	0.0871638	0.0435819	+	0.999050i	$0.486123\pi$
$522$	447.785	0.0375460
$523$	4294.28	0.359036	0.179518	−	0.983755i	$-0.442546\pi$
$523$	4294.28	0.359036	0.179518	+	0.983755i	$0.442546\pi$
$524$	−18404.0	−1.53432
$525$	0	0
$526$	36186.9	2.99966
$527$	1900.95	0.157128
$528$	6827.14	0.562714
$529$	529.000	0.0434783
$530$	0	0
$531$	1216.31	0.0994037
$532$	−36142.2	−2.94542
$533$	−10491.4	−0.852591
$534$	−14516.5	−1.17639
$535$	0	0
$536$	−35972.5	−2.89883
$537$	−4093.97	−0.328991
$538$	16959.2	1.35904
$539$	8109.48	0.648052
$540$	0	0
$541$	5428.80	0.431427	0.215714	−	0.976457i	$-0.430792\pi$
$541$	5428.80	0.431427	0.215714	+	0.976457i	$0.430792\pi$
$542$	37628.8	2.98210
$543$	−16317.7	−1.28961
$544$	2751.05	0.216821
$545$	0	0
$546$	−50868.4	−3.98712
$547$	10768.0	0.841694	0.420847	−	0.907132i	$-0.361733\pi$
$547$	10768.0	0.841694	0.420847	+	0.907132i	$0.361733\pi$
$548$	2864.24	0.223274
$549$	−1605.70	−0.124826
$550$	0	0
$551$	−2812.22	−0.217431
$552$	5222.34	0.402676
$553$	14305.7	1.10007
$554$	15613.2	1.19737
$555$	0	0
$556$	14336.8	1.09355
$557$	16061.8	1.22183	0.610916	−	0.791695i	$-0.290801\pi$
$557$	16061.8	1.22183	0.610916	+	0.791695i	$0.290801\pi$
$558$	313.761	0.0238039
$559$	32426.0	2.45344
$560$	0	0
$561$	6681.05	0.502806
$562$	10964.5	0.822972
$563$	7755.76	0.580580	0.290290	−	0.956939i	$-0.406248\pi$
$563$	7755.76	0.580580	0.290290	+	0.956939i	$0.406248\pi$
$564$	−37662.9	−2.81187
$565$	0	0
$566$	10695.8	0.794304
$567$	−22712.0	−1.68221
$568$	49004.8	3.62007
$569$	3993.21	0.294207	0.147104	−	0.989121i	$-0.453005\pi$
$569$	3993.21	0.294207	0.147104	+	0.989121i	$0.453005\pi$
$570$	0	0
$571$	−16239.2	−1.19017	−0.595087	−	0.803662i	$-0.702882\pi$
$571$	−16239.2	−1.19017	−0.595087	+	0.803662i	$0.702882\pi$
$572$	−18125.5	−1.32494
$573$	11644.8	0.848983
$574$	22669.5	1.64844
$575$	0	0
$576$	−1024.59	−0.0741170
$577$	−6260.37	−0.451685	−0.225843	−	0.974164i	$-0.572514\pi$
$577$	−6260.37	−0.451685	−0.225843	+	0.974164i	$0.572514\pi$
$578$	2506.12	0.180347
$579$	19604.3	1.40713
$580$	0	0
$581$	34165.5	2.43962
$582$	17211.3	1.22583
$583$	682.540	0.0484870
$584$	35229.7	2.49626
$585$	0	0
$586$	−12741.0	−0.898170
$587$	24332.7	1.71093	0.855466	−	0.517859i	$-0.173271\pi$
$587$	24332.7	1.71093	0.855466	+	0.517859i	$0.173271\pi$
$588$	43324.2	3.03854
$589$	−1970.51	−0.137849
$590$	0	0
$591$	11697.7	0.814177
$592$	7148.80	0.496307
$593$	6503.42	0.450360	0.225180	−	0.974317i	$-0.427703\pi$
$593$	6503.42	0.450360	0.225180	+	0.974317i	$0.427703\pi$
$594$	−11015.5	−0.760895
$595$	0	0
$596$	−19145.6	−1.31583
$597$	−23698.1	−1.62462
$598$	−7491.52	−0.512292
$599$	8996.68	0.613680	0.306840	−	0.951761i	$-0.400728\pi$
$599$	8996.68	0.613680	0.306840	+	0.951761i	$0.400728\pi$
$600$	0	0
$601$	−24207.6	−1.64301	−0.821504	−	0.570203i	$-0.806864\pi$
$601$	−24207.6	−1.64301	−0.821504	+	0.570203i	$0.806864\pi$
$602$	−70065.3	−4.74361
$603$	−2112.64	−0.142676
$604$	−2198.06	−0.148076
$605$	0	0
$606$	5522.45	0.370189
$607$	24601.9	1.64507	0.822536	−	0.568712i	$-0.192558\pi$
$607$	24601.9	1.64507	0.822536	+	0.568712i	$0.192558\pi$
$608$	−2851.72	−0.190218
$609$	5755.17	0.382941
$610$	0	0
$611$	27767.5	1.83855
$612$	2977.09	0.196637
$613$	−17927.6	−1.18122	−0.590612	−	0.806956i	$-0.701114\pi$
$613$	−17927.6	−1.18122	−0.590612	+	0.806956i	$0.701114\pi$
$614$	16083.3	1.05712
$615$	0	0
$616$	20128.7	1.31658
$617$	11105.0	0.724589	0.362295	−	0.932064i	$-0.381993\pi$
$617$	11105.0	0.724589	0.362295	+	0.932064i	$0.381993\pi$
$618$	21215.7	1.38094
$619$	−9846.80	−0.639380	−0.319690	−	0.947522i	$-0.603579\pi$
$619$	−9846.80	−0.639380	−0.319690	+	0.947522i	$0.603579\pi$
$620$	0	0
$621$	−3063.73	−0.197976
$622$	19557.0	1.26072
$623$	−15561.8	−1.00076
$624$	−26890.5	−1.72513
$625$	0	0
$626$	31766.5	2.02818
$627$	−6925.53	−0.441115
$628$	22329.8	1.41888
$629$	6995.83	0.443469
$630$	0	0
$631$	5127.98	0.323521	0.161760	−	0.986830i	$-0.448283\pi$
$631$	5127.98	0.323521	0.161760	+	0.986830i	$0.448283\pi$
$632$	20798.9	1.30907
$633$	−7453.34	−0.467999
$634$	12774.4	0.800217
$635$	0	0
$636$	3646.41	0.227342
$637$	−31941.3	−1.98675
$638$	3047.43	0.189105
$639$	2878.02	0.178173
$640$	0	0
$641$	−16730.1	−1.03089	−0.515445	−	0.856922i	$-0.672373\pi$
$641$	−16730.1	−1.03089	−0.515445	+	0.856922i	$0.672373\pi$
$642$	−5153.83	−0.316831
$643$	−8215.70	−0.503881	−0.251941	−	0.967743i	$-0.581069\pi$
$643$	−8215.70	−0.503881	−0.251941	+	0.967743i	$0.581069\pi$
$644$	10892.9	0.666525
$645$	0	0
$646$	−27784.7	−1.69222
$647$	2755.97	0.167462	0.0837312	−	0.996488i	$-0.473316\pi$
$647$	2755.97	0.167462	0.0837312	+	0.996488i	$0.473316\pi$
$648$	−33020.7	−2.00181
$649$	8277.67	0.500658
$650$	0	0
$651$	4032.62	0.242781
$652$	58908.2	3.53838
$653$	17413.8	1.04357	0.521787	−	0.853076i	$-0.325265\pi$
$653$	17413.8	1.04357	0.521787	+	0.853076i	$0.325265\pi$
$654$	−33238.5	−1.98735
$655$	0	0
$656$	11983.7	0.713241
$657$	2069.02	0.122861
$658$	−59999.4	−3.55474
$659$	10288.6	0.608175	0.304088	−	0.952644i	$-0.401648\pi$
$659$	10288.6	0.608175	0.304088	+	0.952644i	$0.401648\pi$
$660$	0	0
$661$	−6418.58	−0.377691	−0.188846	−	0.982007i	$-0.560475\pi$
$661$	−6418.58	−0.377691	−0.188846	+	0.982007i	$0.560475\pi$
$662$	−35385.2	−2.07747
$663$	−26315.1	−1.54147
$664$	49672.7	2.90313
$665$	0	0
$666$	1154.70	0.0671827
$667$	847.578	0.0492029
$668$	31333.5	1.81487
$669$	−5455.76	−0.315294
$670$	0	0
$671$	−10927.7	−0.628702
$672$	5836.01	0.335013
$673$	−27817.4	−1.59329	−0.796645	−	0.604448i	$-0.793394\pi$
$673$	−27817.4	−1.59329	−0.796645	+	0.604448i	$0.793394\pi$
$674$	−15583.7	−0.890597
$675$	0	0
$676$	35231.3	2.00451
$677$	−12716.6	−0.721921	−0.360960	−	0.932581i	$-0.617551\pi$
$677$	−12716.6	−0.721921	−0.360960	+	0.932581i	$0.617551\pi$
$678$	−49114.2	−2.78204
$679$	18450.7	1.04282
$680$	0	0
$681$	−16127.2	−0.907483
$682$	2135.32	0.119891
$683$	14735.9	0.825557	0.412778	−	0.910831i	$-0.364558\pi$
$683$	14735.9	0.825557	0.412778	+	0.910831i	$0.364558\pi$
$684$	−3086.03	−0.172511
$685$	0	0
$686$	20206.1	1.12460
$687$	1579.41	0.0877120
$688$	−37038.6	−2.05244
$689$	−2688.36	−0.148648
$690$	0	0
$691$	20890.4	1.15008	0.575041	−	0.818125i	$-0.304986\pi$
$691$	20890.4	1.15008	0.575041	+	0.818125i	$0.304986\pi$
$692$	−44440.0	−2.44127
$693$	1182.15	0.0647996
$694$	−43890.7	−2.40067
$695$	0	0
$696$	8367.37	0.455696
$697$	11727.3	0.637308
$698$	−17234.0	−0.934551
$699$	−22879.3	−1.23802
$700$	0	0
$701$	−31086.0	−1.67490	−0.837448	−	0.546517i	$-0.815953\pi$
$701$	−31086.0	−1.67490	−0.837448	+	0.546517i	$0.815953\pi$
$702$	43387.5	2.33270
$703$	−7251.83	−0.389058
$704$	−6972.93	−0.373299
$705$	0	0
$706$	16863.5	0.898960
$707$	5920.11	0.314920
$708$	44222.8	2.34745
$709$	4701.78	0.249054	0.124527	−	0.992216i	$-0.460259\pi$
$709$	4701.78	0.249054	0.124527	+	0.992216i	$0.460259\pi$
$710$	0	0
$711$	1221.50	0.0644303
$712$	−22625.1	−1.19089
$713$	593.894	0.0311942
$714$	56861.1	2.98035
$715$	0	0
$716$	−12415.3	−0.648018
$717$	−1271.75	−0.0662402
$718$	−10838.4	−0.563348
$719$	−631.878	−0.0327748	−0.0163874	−	0.999866i	$-0.505217\pi$
$719$	−631.878	−0.0327748	−0.0163874	+	0.999866i	$0.505217\pi$
$720$	0	0
$721$	22743.4	1.17477
$722$	−5120.51	−0.263941
$723$	25206.5	1.29660
$724$	−49484.6	−2.54017
$725$	0	0
$726$	−28221.9	−1.44272
$727$	−22570.8	−1.15145	−0.575724	−	0.817644i	$-0.695280\pi$
$727$	−22570.8	−1.15145	−0.575724	+	0.817644i	$0.695280\pi$
$728$	−79282.4	−4.03627
$729$	17580.7	0.893194
$730$	0	0
$731$	−36246.0	−1.83394
$732$	−58380.2	−2.94781
$733$	−4267.53	−0.215041	−0.107520	−	0.994203i	$-0.534291\pi$
$733$	−4267.53	−0.215041	−0.107520	+	0.994203i	$0.534291\pi$
$734$	−7151.39	−0.359622
$735$	0	0
$736$	859.484	0.0430448
$737$	−14377.7	−0.718602
$738$	1935.65	0.0965479
$739$	35877.7	1.78590	0.892951	−	0.450154i	$-0.148631\pi$
$739$	35877.7	1.78590	0.892951	+	0.450154i	$0.148631\pi$
$740$	0	0
$741$	27278.0	1.35234
$742$	5808.96	0.287404
$743$	6006.44	0.296575	0.148287	−	0.988944i	$-0.452624\pi$
$743$	6006.44	0.296575	0.148287	+	0.988944i	$0.452624\pi$
$744$	5862.97	0.288907
$745$	0	0
$746$	49009.0	2.40529
$747$	2917.25	0.142887
$748$	20260.8	0.990385
$749$	−5524.95	−0.269529
$750$	0	0
$751$	29115.9	1.41472	0.707359	−	0.706855i	$-0.249887\pi$
$751$	29115.9	1.41472	0.707359	+	0.706855i	$0.249887\pi$
$752$	−31717.4	−1.53805
$753$	−5218.02	−0.252530
$754$	−12003.1	−0.579745
$755$	0	0
$756$	−63087.0	−3.03499
$757$	−16513.9	−0.792875	−0.396438	−	0.918062i	$-0.629754\pi$
$757$	−16513.9	−0.792875	−0.396438	+	0.918062i	$0.629754\pi$
$758$	31802.5	1.52390
$759$	2087.29	0.0998208
$760$	0	0
$761$	7596.54	0.361859	0.180929	−	0.983496i	$-0.442089\pi$
$761$	7596.54	0.361859	0.180929	+	0.983496i	$0.442089\pi$
$762$	−31593.8	−1.50200
$763$	−35632.0	−1.69065
$764$	35313.6	1.67225
$765$	0	0
$766$	4360.79	0.205694
$767$	−32603.8	−1.53488
$768$	−45276.7	−2.12732
$769$	10360.9	0.485858	0.242929	−	0.970044i	$-0.421892\pi$
$769$	10360.9	0.485858	0.242929	+	0.970044i	$0.421892\pi$
$770$	0	0
$771$	−15198.6	−0.709942
$772$	59451.5	2.77164
$773$	−28121.4	−1.30848	−0.654241	−	0.756286i	$-0.727012\pi$
$773$	−28121.4	−1.30848	−0.654241	+	0.756286i	$0.727012\pi$
$774$	−5982.59	−0.277829
$775$	0	0
$776$	26825.2	1.24094
$777$	14840.8	0.685212
$778$	−43985.8	−2.02695
$779$	−12156.4	−0.559114
$780$	0	0
$781$	19586.5	0.897390
$782$	8374.08	0.382937
$783$	−4908.79	−0.224043
$784$	36484.9	1.66203
$785$	0	0
$786$	−30013.8	−1.36203
$787$	−23907.1	−1.08284	−0.541421	−	0.840751i	$-0.682114\pi$
$787$	−23907.1	−1.08284	−0.541421	+	0.840751i	$0.682114\pi$
$788$	35474.1	1.60370
$789$	39712.3	1.79188
$790$	0	0
$791$	−52650.8	−2.36668
$792$	1718.71	0.0771108
$793$	43041.6	1.92743
$794$	−17497.3	−0.782060
$795$	0	0
$796$	−71866.4	−3.20004
$797$	−25272.2	−1.12320	−0.561598	−	0.827410i	$-0.689813\pi$
$797$	−25272.2	−1.12320	−0.561598	+	0.827410i	$0.689813\pi$
$798$	−58941.8	−2.61468
$799$	−31038.7	−1.37431
$800$	0	0
$801$	−1328.76	−0.0586134
$802$	41092.2	1.80925
$803$	14080.8	0.618806
$804$	−76811.6	−3.36933
$805$	0	0
$806$	−8410.52	−0.367553
$807$	18611.4	0.811838
$808$	8607.18	0.374752
$809$	−36235.2	−1.57474	−0.787368	−	0.616484i	$-0.788557\pi$
$809$	−36235.2	−1.57474	−0.787368	+	0.616484i	$0.788557\pi$
$810$	0	0
$811$	−4007.44	−0.173514	−0.0867572	−	0.996229i	$-0.527650\pi$
$811$	−4007.44	−0.173514	−0.0867572	+	0.996229i	$0.527650\pi$
$812$	17453.0	0.754285
$813$	41294.7	1.78139
$814$	7858.37	0.338373
$815$	0	0
$816$	30058.4	1.28953
$817$	37572.4	1.60892
$818$	75349.4	3.22070
$819$	−4656.20	−0.198658
$820$	0	0
$821$	−18085.0	−0.768785	−0.384392	−	0.923170i	$-0.625589\pi$
$821$	−18085.0	−0.768785	−0.384392	+	0.923170i	$0.625589\pi$
$822$	4671.08	0.198203
$823$	−5589.80	−0.236753	−0.118377	−	0.992969i	$-0.537769\pi$
$823$	−5589.80	−0.236753	−0.118377	+	0.992969i	$0.537769\pi$
$824$	33066.3	1.39796
$825$	0	0
$826$	70449.6	2.96762
$827$	9001.00	0.378471	0.189235	−	0.981932i	$-0.439399\pi$
$827$	9001.00	0.378471	0.189235	+	0.981932i	$0.439399\pi$
$828$	930.104	0.0390378
$829$	−35846.4	−1.50181	−0.750904	−	0.660412i	$-0.770382\pi$
$829$	−35846.4	−1.50181	−0.750904	+	0.660412i	$0.770382\pi$
$830$	0	0
$831$	17134.3	0.715261
$832$	27464.7	1.14443
$833$	35704.3	1.48509
$834$	23380.8	0.970757
$835$	0	0
$836$	−21002.2	−0.868870
$837$	−3439.56	−0.142041
$838$	12911.1	0.532229
$839$	−24753.9	−1.01859	−0.509296	−	0.860592i	$-0.670094\pi$
$839$	−24753.9	−1.01859	−0.509296	+	0.860592i	$0.670094\pi$
$840$	0	0
$841$	−23031.0	−0.944319
$842$	29127.7	1.19217
$843$	12032.7	0.491612
$844$	−22602.8	−0.921825
$845$	0	0
$846$	−5123.10	−0.208198
$847$	−30254.1	−1.22732
$848$	3070.78	0.124353
$849$	11737.8	0.474486
$850$	0	0
$851$	2185.64	0.0880408
$852$	104639.	4.20761
$853$	17445.9	0.700278	0.350139	−	0.936698i	$-0.386134\pi$
$853$	17445.9	0.700278	0.350139	+	0.936698i	$0.386134\pi$
$854$	−93003.3	−3.72659
$855$	0	0
$856$	−8032.65	−0.320737
$857$	−10455.3	−0.416738	−0.208369	−	0.978050i	$-0.566816\pi$
$857$	−10455.3	−0.416738	−0.208369	+	0.978050i	$0.566816\pi$
$858$	−29559.5	−1.17616
$859$	−22362.5	−0.888240	−0.444120	−	0.895967i	$-0.646484\pi$
$859$	−22362.5	−0.888240	−0.444120	+	0.895967i	$0.646484\pi$
$860$	0	0
$861$	24878.0	0.984715
$862$	−75922.1	−2.99991
$863$	−17633.2	−0.695527	−0.347763	−	0.937582i	$-0.613059\pi$
$863$	−17633.2	−0.695527	−0.347763	+	0.937582i	$0.613059\pi$
$864$	−4977.74	−0.196002
$865$	0	0
$866$	14220.5	0.558007
$867$	2750.27	0.107733
$868$	12229.2	0.478210
$869$	8313.01	0.324511
$870$	0	0
$871$	56630.4	2.20304
$872$	−51804.9	−2.01185
$873$	1575.43	0.0610769
$874$	−8680.50	−0.335952
$875$	0	0
$876$	75225.5	2.90141
$877$	3700.88	0.142497	0.0712486	−	0.997459i	$-0.477302\pi$
$877$	3700.88	0.142497	0.0712486	+	0.997459i	$0.477302\pi$
$878$	68369.0	2.62795
$879$	−13982.3	−0.536532
$880$	0	0
$881$	7648.64	0.292496	0.146248	−	0.989248i	$-0.453280\pi$
$881$	7648.64	0.292496	0.146248	+	0.989248i	$0.453280\pi$
$882$	5893.17	0.224981
$883$	−40610.2	−1.54772	−0.773862	−	0.633354i	$-0.781678\pi$
$883$	−40610.2	−1.54772	−0.773862	+	0.633354i	$0.781678\pi$
$884$	−79802.5	−3.03625
$885$	0	0
$886$	−50588.6	−1.91824
$887$	27976.1	1.05901	0.529507	−	0.848305i	$-0.322377\pi$
$887$	27976.1	1.05901	0.529507	+	0.848305i	$0.322377\pi$
$888$	21576.8	0.815395
$889$	−33868.8	−1.27775
$890$	0	0
$891$	−13197.9	−0.496236
$892$	−16545.0	−0.621040
$893$	32174.5	1.20569
$894$	−31223.2	−1.16808
$895$	0	0
$896$	−67947.4	−2.53344
$897$	−8221.36	−0.306024
$898$	40146.6	1.49188
$899$	951.553	0.0353015
$900$	0	0
$901$	3005.07	0.111114
$902$	13173.2	0.486275
$903$	−76891.3	−2.83365
$904$	−76548.4	−2.81633
$905$	0	0
$906$	−3584.67	−0.131449
$907$	−4.67623	−0.000171192 0	−8.55962e−5	−	1.00000i	$-0.500027\pi$
$907$	−4.67623	−0.000171192 0	−8.55962e−5		1.00000i	$0.500027\pi$
$908$	−48907.0	−1.78748
$909$	505.494	0.0184446
$910$	0	0
$911$	−27794.1	−1.01082	−0.505411	−	0.862879i	$-0.668659\pi$
$911$	−27794.1	−1.01082	−0.505411	+	0.862879i	$0.668659\pi$
$912$	−31158.3	−1.13131
$913$	19853.5	0.719666
$914$	58002.4	2.09907
$915$	0	0
$916$	4789.67	0.172768
$917$	−32175.1	−1.15869
$918$	−48498.8	−1.74368
$919$	29644.0	1.06405	0.532027	−	0.846727i	$-0.321430\pi$
$919$	29644.0	1.06405	0.532027	+	0.846727i	$0.321430\pi$
$920$	0	0
$921$	17650.2	0.631480
$922$	7704.05	0.275184
$923$	−77146.8	−2.75116
$924$	42980.7	1.53026
$925$	0	0
$926$	9020.99	0.320139
$927$	1941.96	0.0688051
$928$	1377.09	0.0487124
$929$	−10032.8	−0.354324	−0.177162	−	0.984182i	$-0.556692\pi$
$929$	−10032.8	−0.354324	−0.177162	+	0.984182i	$0.556692\pi$
$930$	0	0
$931$	−37010.8	−1.30288
$932$	−69383.0	−2.43854
$933$	21462.3	0.753103
$934$	64588.7	2.26275
$935$	0	0
$936$	−6769.60	−0.236401
$937$	50301.8	1.75377	0.876887	−	0.480696i	$-0.159616\pi$
$937$	50301.8	1.75377	0.876887	+	0.480696i	$0.159616\pi$
$938$	−122366.	−4.25947
$939$	34861.2	1.21156
$940$	0	0
$941$	−35772.6	−1.23927	−0.619635	−	0.784890i	$-0.712720\pi$
$941$	−35772.6	−1.23927	−0.619635	+	0.784890i	$0.712720\pi$
$942$	36416.1	1.25956
$943$	3663.85	0.126523
$944$	37241.7	1.28402
$945$	0	0
$946$	−40714.9	−1.39932
$947$	22482.0	0.771456	0.385728	−	0.922613i	$-0.373950\pi$
$947$	22482.0	0.771456	0.385728	+	0.922613i	$0.373950\pi$
$948$	44411.6	1.52154
$949$	−55461.0	−1.89709
$950$	0	0
$951$	14018.9	0.478018
$952$	88622.5	3.01709
$953$	−43503.4	−1.47871	−0.739357	−	0.673314i	$-0.764870\pi$
$953$	−43503.4	−1.47871	−0.739357	+	0.673314i	$0.764870\pi$
$954$	496.003	0.0168330
$955$	0	0
$956$	−3856.66	−0.130474
$957$	3344.32	0.112964
$958$	−89545.8	−3.01993
$959$	5007.44	0.168612
$960$	0	0
$961$	−29124.3	−0.977619
$962$	−30952.3	−1.03736
$963$	−471.752	−0.0157861
$964$	76440.5	2.55392
$965$	0	0
$966$	17764.5	0.591682
$967$	22419.0	0.745548	0.372774	−	0.927922i	$-0.378407\pi$
$967$	22419.0	0.745548	0.372774	+	0.927922i	$0.378407\pi$
$968$	−43986.0	−1.46050
$969$	−30491.6	−1.01087
$970$	0	0
$971$	14978.2	0.495031	0.247515	−	0.968884i	$-0.420386\pi$
$971$	14978.2	0.495031	0.247515	+	0.968884i	$0.420386\pi$
$972$	−11312.7	−0.373309
$973$	25064.4	0.825825
$974$	28068.9	0.923393
$975$	0	0
$976$	−49164.2	−1.61241
$977$	−31907.0	−1.04483	−0.522413	−	0.852692i	$-0.674968\pi$
$977$	−31907.0	−1.04483	−0.522413	+	0.852692i	$0.674968\pi$
$978$	96069.3	3.14106
$979$	−9042.94	−0.295213
$980$	0	0
$981$	−3042.47	−0.0990198
$982$	−13.6549	−0.000443732 0
$983$	11156.9	0.362005	0.181002	−	0.983483i	$-0.442066\pi$
$983$	11156.9	0.362005	0.181002	+	0.983483i	$0.442066\pi$
$984$	36169.8	1.17180
$985$	0	0
$986$	13417.2	0.433357
$987$	−65844.7	−2.12346
$988$	82722.6	2.66372
$989$	−11324.0	−0.364087
$990$	0	0
$991$	−18389.7	−0.589473	−0.294737	−	0.955579i	$-0.595232\pi$
$991$	−18389.7	−0.589473	−0.294737	+	0.955579i	$0.595232\pi$
$992$	964.919	0.0308833
$993$	−38832.5	−1.24100
$994$	166697.	5.31923
$995$	0	0
$996$	106066.	3.37432
$997$	−49492.0	−1.57214	−0.786072	−	0.618135i	$-0.787889\pi$
$997$	−49492.0	−1.57214	−0.786072	+	0.618135i	$0.787889\pi$
$998$	6135.64	0.194610
$999$	−12658.2	−0.400889

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.a.r.1.15		17
5.2	odd	4		115.4.b.a.24.31	yes	34
5.3	odd	4		115.4.b.a.24.4	✓	34
5.4	even	2		575.4.a.q.1.3		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.b.a.24.4	✓	34	5.3	odd	4
115.4.b.a.24.31	yes	34	5.2	odd	4
575.4.a.q.1.3		17	5.4	even	2
575.4.a.r.1.15		17	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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.a (trivial)

\(f(q)\)	\(=\)	\(q+4.94561 q^{2} +5.42743 q^{3} +16.4591 q^{4} +26.8419 q^{6} +28.7748 q^{7} +41.8353 q^{8} +2.45696 q^{9} +16.7210 q^{11} +89.3304 q^{12} -65.8600 q^{13} +142.309 q^{14} +75.2286 q^{16} +73.6189 q^{17} +12.1512 q^{18} -76.3128 q^{19} +156.173 q^{21} +82.6955 q^{22} +23.0000 q^{23} +227.058 q^{24} -325.718 q^{26} -133.206 q^{27} +473.606 q^{28} +36.8512 q^{29} +25.8215 q^{31} +37.3689 q^{32} +90.7519 q^{33} +364.090 q^{34} +40.4393 q^{36} +95.0278 q^{37} -377.413 q^{38} -357.450 q^{39} +159.298 q^{41} +772.371 q^{42} -492.347 q^{43} +275.212 q^{44} +113.749 q^{46} -421.614 q^{47} +408.298 q^{48} +484.988 q^{49} +399.561 q^{51} -1084.00 q^{52} +40.8194 q^{53} -658.783 q^{54} +1203.80 q^{56} -414.182 q^{57} +182.252 q^{58} +495.047 q^{59} -653.531 q^{61} +127.703 q^{62} +70.6985 q^{63} -417.017 q^{64} +448.824 q^{66} -859.860 q^{67} +1211.70 q^{68} +124.831 q^{69} +1171.37 q^{71} +102.788 q^{72} +842.104 q^{73} +469.970 q^{74} -1256.04 q^{76} +481.143 q^{77} -1767.81 q^{78} +497.161 q^{79} -789.301 q^{81} +787.825 q^{82} +1187.34 q^{83} +2570.46 q^{84} -2434.96 q^{86} +200.007 q^{87} +699.528 q^{88} -540.814 q^{89} -1895.11 q^{91} +378.559 q^{92} +140.144 q^{93} -2085.14 q^{94} +202.817 q^{96} +641.210 q^{97} +2398.56 q^{98} +41.0828 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 4 q^{2} + 12 q^{3} + 72 q^{4} + 12 q^{6} + 72 q^{7} + 48 q^{8} + 155 q^{9} - 4 q^{11} + 342 q^{12} + 208 q^{13} - 118 q^{14} + 220 q^{16} + 268 q^{17} + 180 q^{18} - 72 q^{19} - 16 q^{21} + 318 q^{22}+ \cdots + 1908 q^{99}+O(q^{100}) \) 17 * q + 4 * q^2 + 12 * q^3 + 72 * q^4 + 12 * q^6 + 72 * q^7 + 48 * q^8 + 155 * q^9 - 4 * q^11 + 342 * q^12 + 208 * q^13 - 118 * q^14 + 220 * q^16 + 268 * q^17 + 180 * q^18 - 72 * q^19 - 16 * q^21 + 318 * q^22 + 391 * q^23 - 54 * q^24 - 72 * q^26 + 408 * q^27 + 928 * q^28 - 28 * q^29 - 40 * q^31 + 268 * q^32 + 1382 * q^33 - 132 * q^34 + 932 * q^36 + 586 * q^37 + 596 * q^38 + 600 * q^39 + 134 * q^41 - 104 * q^42 + 1264 * q^43 + 806 * q^44 + 92 * q^46 + 184 * q^47 + 2928 * q^48 + 737 * q^49 - 552 * q^51 + 1558 * q^52 + 1102 * q^53 - 782 * q^54 - 1150 * q^56 + 1188 * q^57 + 3882 * q^58 + 354 * q^59 + 550 * q^61 - 1114 * q^62 + 3102 * q^63 - 50 * q^64 - 708 * q^66 + 898 * q^67 + 3774 * q^68 + 276 * q^69 + 680 * q^71 + 102 * q^72 + 4980 * q^73 - 794 * q^74 + 54 * q^76 + 584 * q^77 + 4108 * q^78 - 1984 * q^79 + 2457 * q^81 + 1798 * q^82 + 2634 * q^83 + 974 * q^84 - 3074 * q^86 - 472 * q^87 + 7584 * q^88 - 598 * q^89 - 272 * q^91 + 1656 * q^92 + 2384 * q^93 + 1170 * q^94 + 1480 * q^96 + 6266 * q^97 + 3486 * q^98 + 1908 * q^99

Introduction

L-functions

Modular forms

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Database

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