LMFDB - Embedded newform 539.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,4,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, 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("539.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	539.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:	$31.8020294931$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{2} -5.92820 q^{3} -7.46410 q^{4} +12.8564 q^{5} +4.33975 q^{6} +11.3205 q^{8} +8.14359 q^{9} +O(q^{10})$	\(q-0.732051 q^{2} -5.92820 q^{3} -7.46410 q^{4} +12.8564 q^{5} +4.33975 q^{6} +11.3205 q^{8} +8.14359 q^{9} -9.41154 q^{10} -11.0000 q^{11} +44.2487 q^{12} -74.6410 q^{13} -76.2154 q^{15} +51.4256 q^{16} +82.7846 q^{17} -5.96152 q^{18} +67.9230 q^{19} -95.9615 q^{20} +8.05256 q^{22} +13.3538 q^{23} -67.1103 q^{24} +40.2872 q^{25} +54.6410 q^{26} +111.785 q^{27} +168.995 q^{29} +55.7935 q^{30} +65.4974 q^{31} -128.210 q^{32} +65.2102 q^{33} -60.6025 q^{34} -60.7846 q^{36} +40.8564 q^{37} -49.7231 q^{38} +442.487 q^{39} +145.541 q^{40} -274.928 q^{41} -2.28719 q^{43} +82.1051 q^{44} +104.697 q^{45} -9.77568 q^{46} -71.8461 q^{47} -304.862 q^{48} -29.4923 q^{50} -490.764 q^{51} +557.128 q^{52} -149.005 q^{53} -81.8320 q^{54} -141.420 q^{55} -402.662 q^{57} -123.713 q^{58} -545.631 q^{59} +568.879 q^{60} -101.303 q^{61} -47.9474 q^{62} -317.549 q^{64} -959.615 q^{65} -47.7372 q^{66} +411.641 q^{67} -617.913 q^{68} -79.1642 q^{69} -470.636 q^{71} +92.1896 q^{72} -610.600 q^{73} -29.9090 q^{74} -238.831 q^{75} -506.985 q^{76} -323.923 q^{78} -978.225 q^{79} +661.149 q^{80} -882.559 q^{81} +201.261 q^{82} -26.1539 q^{83} +1064.31 q^{85} +1.67434 q^{86} -1001.84 q^{87} -124.526 q^{88} +352.887 q^{89} -76.6438 q^{90} -99.6743 q^{92} -388.282 q^{93} +52.5950 q^{94} +873.246 q^{95} +760.056 q^{96} -847.585 q^{97} -89.5795 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 - 8 * q^4 - 2 * q^5 + 26 * q^6 - 12 * q^8 + 44 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9} - 50 q^{10} - 22 q^{11} + 40 q^{12} - 80 q^{13} - 194 q^{15} - 8 q^{16} + 124 q^{17} + 92 q^{18} - 72 q^{19} - 88 q^{20} - 22 q^{22} - 98 q^{23} - 252 q^{24} + 136 q^{25} + 40 q^{26} + 182 q^{27} + 144 q^{29} - 266 q^{30} + 34 q^{31} - 104 q^{32} - 22 q^{33} + 52 q^{34} - 80 q^{36} + 54 q^{37} - 432 q^{38} + 400 q^{39} + 492 q^{40} - 536 q^{41} - 60 q^{43} + 88 q^{44} - 428 q^{45} - 314 q^{46} + 272 q^{47} - 776 q^{48} + 232 q^{50} - 164 q^{51} + 560 q^{52} - 492 q^{53} + 110 q^{54} + 22 q^{55} - 1512 q^{57} - 192 q^{58} - 634 q^{59} + 632 q^{60} - 840 q^{61} - 134 q^{62} + 224 q^{64} - 880 q^{65} - 286 q^{66} + 754 q^{67} - 640 q^{68} - 962 q^{69} - 678 q^{71} - 744 q^{72} + 400 q^{73} + 6 q^{74} + 520 q^{75} - 432 q^{76} - 440 q^{78} + 316 q^{79} + 1544 q^{80} - 1294 q^{81} - 512 q^{82} - 468 q^{83} + 452 q^{85} - 156 q^{86} - 1200 q^{87} + 132 q^{88} + 1842 q^{89} - 1532 q^{90} - 40 q^{92} - 638 q^{93} + 992 q^{94} + 2952 q^{95} + 952 q^{96} - 2194 q^{97} - 484 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 - 8 * q^4 - 2 * q^5 + 26 * q^6 - 12 * q^8 + 44 * q^9 - 50 * q^10 - 22 * q^11 + 40 * q^12 - 80 * q^13 - 194 * q^15 - 8 * q^16 + 124 * q^17 + 92 * q^18 - 72 * q^19 - 88 * q^20 - 22 * q^22 - 98 * q^23 - 252 * q^24 + 136 * q^25 + 40 * q^26 + 182 * q^27 + 144 * q^29 - 266 * q^30 + 34 * q^31 - 104 * q^32 - 22 * q^33 + 52 * q^34 - 80 * q^36 + 54 * q^37 - 432 * q^38 + 400 * q^39 + 492 * q^40 - 536 * q^41 - 60 * q^43 + 88 * q^44 - 428 * q^45 - 314 * q^46 + 272 * q^47 - 776 * q^48 + 232 * q^50 - 164 * q^51 + 560 * q^52 - 492 * q^53 + 110 * q^54 + 22 * q^55 - 1512 * q^57 - 192 * q^58 - 634 * q^59 + 632 * q^60 - 840 * q^61 - 134 * q^62 + 224 * q^64 - 880 * q^65 - 286 * q^66 + 754 * q^67 - 640 * q^68 - 962 * q^69 - 678 * q^71 - 744 * q^72 + 400 * q^73 + 6 * q^74 + 520 * q^75 - 432 * q^76 - 440 * q^78 + 316 * q^79 + 1544 * q^80 - 1294 * q^81 - 512 * q^82 - 468 * q^83 + 452 * q^85 - 156 * q^86 - 1200 * q^87 + 132 * q^88 + 1842 * q^89 - 1532 * q^90 - 40 * q^92 - 638 * q^93 + 992 * q^94 + 2952 * q^95 + 952 * q^96 - 2194 * q^97 - 484 * q^99

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 $

$2$	−0.732051	−0.258819	−0.129410	−	0.991591i	$-0.541308\pi$
$2$	−0.732051	−0.258819	−0.129410	+	0.991591i	$0.541308\pi$
$3$	−5.92820	−1.14088	−0.570442	−	0.821338i	$-0.693228\pi$
$3$	−5.92820	−1.14088	−0.570442	+	0.821338i	$0.693228\pi$
$4$	−7.46410	−0.933013
$5$	12.8564	1.14991	0.574956	−	0.818184i	$-0.305019\pi$
$5$	12.8564	1.14991	0.574956	+	0.818184i	$0.305019\pi$
$6$	4.33975	0.295282
$7$	0	0
$7$	0	0
$8$	11.3205	0.500301
$9$	8.14359	0.301615
$10$	−9.41154	−0.297619
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	44.2487	1.06446
$13$	−74.6410	−1.59244	−0.796219	−	0.605009i	$-0.793170\pi$
$13$	−74.6410	−1.59244	−0.796219	+	0.605009i	$0.793170\pi$
$14$	0	0
$15$	−76.2154	−1.31192
$16$	51.4256	0.803525
$17$	82.7846	1.18107	0.590536	−	0.807011i	$-0.298916\pi$
$17$	82.7846	1.18107	0.590536	+	0.807011i	$0.298916\pi$
$18$	−5.96152	−0.0780636
$19$	67.9230	0.820138	0.410069	−	0.912055i	$-0.365505\pi$
$19$	67.9230	0.820138	0.410069	+	0.912055i	$0.365505\pi$
$20$	−95.9615	−1.07288
$21$	0	0
$22$	8.05256	0.0780369
$23$	13.3538	0.121064	0.0605319	−	0.998166i	$-0.480720\pi$
$23$	13.3538	0.121064	0.0605319	+	0.998166i	$0.480720\pi$
$24$	−67.1103	−0.570784
$25$	40.2872	0.322297
$26$	54.6410	0.412153
$27$	111.785	0.796776
$28$	0	0
$29$	168.995	1.08212	0.541061	−	0.840983i	$-0.318023\pi$
$29$	168.995	1.08212	0.541061	+	0.840983i	$0.318023\pi$
$30$	55.7935	0.339549
$31$	65.4974	0.379474	0.189737	−	0.981835i	$-0.439237\pi$
$31$	65.4974	0.379474	0.189737	+	0.981835i	$0.439237\pi$
$32$	−128.210	−0.708268
$33$	65.2102	0.343989
$34$	−60.6025	−0.305684
$35$	0	0
$36$	−60.7846	−0.281410
$37$	40.8564	0.181534	0.0907669	−	0.995872i	$-0.471068\pi$
$37$	40.8564	0.181534	0.0907669	+	0.995872i	$0.471068\pi$
$38$	−49.7231	−0.212267
$39$	442.487	1.81679
$40$	145.541	0.575302
$41$	−274.928	−1.04723	−0.523617	−	0.851954i	$-0.675418\pi$
$41$	−274.928	−1.04723	−0.523617	+	0.851954i	$0.675418\pi$
$42$	0	0
$43$	−2.28719	−0.00811146	−0.00405573	−	0.999992i	$-0.501291\pi$
$43$	−2.28719	−0.00811146	−0.00405573	+	0.999992i	$0.501291\pi$
$44$	82.1051	0.281314
$45$	104.697	0.346830
$46$	−9.77568	−0.0313336
$47$	−71.8461	−0.222975	−0.111488	−	0.993766i	$-0.535562\pi$
$47$	−71.8461	−0.222975	−0.111488	+	0.993766i	$0.535562\pi$
$48$	−304.862	−0.916729
$49$	0	0
$50$	−29.4923	−0.0834167
$51$	−490.764	−1.34746
$52$	557.128	1.48576
$53$	−149.005	−0.386178	−0.193089	−	0.981181i	$-0.561851\pi$
$53$	−149.005	−0.386178	−0.193089	+	0.981181i	$0.561851\pi$
$54$	−81.8320	−0.206221
$55$	−141.420	−0.346711
$56$	0	0
$57$	−402.662	−0.935681
$58$	−123.713	−0.280074
$59$	−545.631	−1.20398	−0.601992	−	0.798502i	$-0.705626\pi$
$59$	−545.631	−1.20398	−0.601992	+	0.798502i	$0.705626\pi$
$60$	568.879	1.22403
$61$	−101.303	−0.212631	−0.106315	−	0.994332i	$-0.533905\pi$
$61$	−101.303	−0.212631	−0.106315	+	0.994332i	$0.533905\pi$
$62$	−47.9474	−0.0982150
$63$	0	0
$64$	−317.549	−0.620212
$65$	−959.615	−1.83116
$66$	−47.7372	−0.0890310
$67$	411.641	0.750596	0.375298	−	0.926904i	$-0.377540\pi$
$67$	411.641	0.750596	0.375298	+	0.926904i	$0.377540\pi$
$68$	−617.913	−1.10195
$69$	−79.1642	−0.138120
$70$	0	0
$71$	−470.636	−0.786679	−0.393339	−	0.919393i	$-0.628680\pi$
$71$	−470.636	−0.786679	−0.393339	+	0.919393i	$0.628680\pi$
$72$	92.1896	0.150898
$73$	−610.600	−0.978977	−0.489488	−	0.872010i	$-0.662816\pi$
$73$	−610.600	−0.978977	−0.489488	+	0.872010i	$0.662816\pi$
$74$	−29.9090	−0.0469844
$75$	−238.831	−0.367704
$76$	−506.985	−0.765199
$77$	0	0
$78$	−323.923	−0.470219
$79$	−978.225	−1.39315	−0.696576	−	0.717483i	$-0.745294\pi$
$79$	−978.225	−1.39315	−0.696576	+	0.717483i	$0.745294\pi$
$80$	661.149	0.923983
$81$	−882.559	−1.21064
$82$	201.261	0.271044
$83$	−26.1539	−0.0345875	−0.0172938	−	0.999850i	$-0.505505\pi$
$83$	−26.1539	−0.0345875	−0.0172938	+	0.999850i	$0.505505\pi$
$84$	0	0
$85$	1064.31	1.35813
$86$	1.67434	0.00209940
$87$	−1001.84	−1.23458
$88$	−124.526	−0.150846
$89$	352.887	0.420292	0.210146	−	0.977670i	$-0.432606\pi$
$89$	352.887	0.420292	0.210146	+	0.977670i	$0.432606\pi$
$90$	−76.6438	−0.0897663
$91$	0	0
$92$	−99.6743	−0.112954
$93$	−388.282	−0.432935
$94$	52.5950	0.0577102
$95$	873.246	0.943086
$96$	760.056	0.808051
$97$	−847.585	−0.887208	−0.443604	−	0.896223i	$-0.646300\pi$
$97$	−847.585	−0.887208	−0.443604	+	0.896223i	$0.646300\pi$
$98$	0	0
$99$	−89.5795	−0.0909402
$100$	−300.708	−0.300708
$101$	−1293.46	−1.27430	−0.637150	−	0.770740i	$-0.719887\pi$
$101$	−1293.46	−1.27430	−0.637150	+	0.770740i	$0.719887\pi$
$102$	359.264	0.348750
$103$	1725.24	1.65042	0.825209	−	0.564828i	$-0.191057\pi$
$103$	1725.24	1.65042	0.825209	+	0.564828i	$0.191057\pi$
$104$	−844.974	−0.796697
$105$	0	0
$106$	109.079	0.0999502
$107$	−484.179	−0.437452	−0.218726	−	0.975786i	$-0.570190\pi$
$107$	−484.179	−0.437452	−0.218726	+	0.975786i	$0.570190\pi$
$108$	−834.372	−0.743402
$109$	−64.2563	−0.0564645	−0.0282323	−	0.999601i	$-0.508988\pi$
$109$	−64.2563	−0.0564645	−0.0282323	+	0.999601i	$0.508988\pi$
$110$	103.527	0.0897355
$111$	−242.205	−0.207109
$112$	0	0
$113$	−2005.08	−1.66922	−0.834612	−	0.550839i	$-0.814308\pi$
$113$	−2005.08	−1.66922	−0.834612	+	0.550839i	$0.814308\pi$
$114$	294.769	0.242172
$115$	171.682	0.139213
$116$	−1261.39	−1.00963
$117$	−607.846	−0.480302
$118$	399.429	0.311614
$119$	0	0
$120$	−862.797	−0.656352
$121$	121.000	0.0909091
$122$	74.1587	0.0550329
$123$	1629.83	1.19477
$124$	−488.879	−0.354054
$125$	−1089.10	−0.779298
$126$	0	0
$127$	109.605	0.0765816	0.0382908	−	0.999267i	$-0.487809\pi$
$127$	109.605	0.0765816	0.0382908	+	0.999267i	$0.487809\pi$
$128$	1258.14	0.868791
$129$	13.5589	0.00925423
$130$	702.487	0.473940
$131$	−1156.71	−0.771469	−0.385734	−	0.922610i	$-0.626052\pi$
$131$	−1156.71	−0.771469	−0.385734	+	0.922610i	$0.626052\pi$
$132$	−486.736	−0.320946
$133$	0	0
$134$	−301.342	−0.194269
$135$	1437.15	0.916223
$136$	937.164	0.590891
$137$	198.323	0.123678	0.0618391	−	0.998086i	$-0.480303\pi$
$137$	198.323	0.123678	0.0618391	+	0.998086i	$0.480303\pi$
$138$	57.9522	0.0357480
$139$	2900.14	1.76969	0.884844	−	0.465888i	$-0.154265\pi$
$139$	2900.14	1.76969	0.884844	+	0.465888i	$0.154265\pi$
$140$	0	0
$141$	425.918	0.254389
$142$	344.529	0.203607
$143$	821.051	0.480138
$144$	418.789	0.242355
$145$	2172.67	1.24435
$146$	446.990	0.253378
$147$	0	0
$148$	−304.956	−0.169373
$149$	3488.34	1.91796	0.958980	−	0.283472i	$-0.0914864\pi$
$149$	3488.34	1.91796	0.958980	+	0.283472i	$0.0914864\pi$
$150$	174.836	0.0951687
$151$	−1163.32	−0.626953	−0.313477	−	0.949596i	$-0.601494\pi$
$151$	−1163.32	−0.626953	−0.313477	+	0.949596i	$0.601494\pi$
$152$	768.923	0.410315
$153$	674.164	0.356228
$154$	0	0
$155$	842.061	0.436361
$156$	−3302.77	−1.69508
$157$	−342.057	−0.173880	−0.0869398	−	0.996214i	$-0.527709\pi$
$157$	−342.057	−0.173880	−0.0869398	+	0.996214i	$0.527709\pi$
$158$	716.111	0.360574
$159$	883.333	0.440584
$160$	−1648.32	−0.814446
$161$	0	0
$162$	646.078	0.313338
$163$	−1394.89	−0.670285	−0.335142	−	0.942167i	$-0.608784\pi$
$163$	−1394.89	−0.670285	−0.335142	+	0.942167i	$0.608784\pi$
$164$	2052.09	0.977082
$165$	838.369	0.395557
$166$	19.1460	0.00895191
$167$	−478.703	−0.221815	−0.110908	−	0.993831i	$-0.535376\pi$
$167$	−478.703	−0.221815	−0.110908	+	0.993831i	$0.535376\pi$
$168$	0	0
$169$	3374.28	1.53586
$170$	−779.131	−0.351509
$171$	553.138	0.247365
$172$	17.0718	0.00756809
$173$	−1808.58	−0.794822	−0.397411	−	0.917641i	$-0.630091\pi$
$173$	−1808.58	−0.794822	−0.397411	+	0.917641i	$0.630091\pi$
$174$	733.395	0.319532
$175$	0	0
$176$	−565.682	−0.242272
$177$	3234.61	1.37361
$178$	−258.331	−0.108780
$179$	−4429.85	−1.84973	−0.924867	−	0.380292i	$-0.875824\pi$
$179$	−4429.85	−1.84973	−0.924867	+	0.380292i	$0.875824\pi$
$180$	−781.472	−0.323597
$181$	−3409.17	−1.40001	−0.700005	−	0.714138i	$-0.746819\pi$
$181$	−3409.17	−1.40001	−0.700005	+	0.714138i	$0.746819\pi$
$182$	0	0
$183$	600.543	0.242587
$184$	151.172	0.0605682
$185$	525.267	0.208748
$186$	284.242	0.112052
$187$	−910.631	−0.356106
$188$	536.267	0.208039
$189$	0	0
$190$	−639.261	−0.244089
$191$	2923.75	1.10762	0.553810	−	0.832643i	$-0.313173\pi$
$191$	2923.75	1.10762	0.553810	+	0.832643i	$0.313173\pi$
$192$	1882.49	0.707590
$193$	−2484.18	−0.926505	−0.463253	−	0.886226i	$-0.653318\pi$
$193$	−2484.18	−0.926505	−0.463253	+	0.886226i	$0.653318\pi$
$194$	620.475	0.229626
$195$	5688.79	2.08914
$196$	0	0
$197$	−5125.67	−1.85375	−0.926876	−	0.375369i	$-0.877516\pi$
$197$	−5125.67	−1.85375	−0.926876	+	0.375369i	$0.877516\pi$
$198$	65.5768	0.0235371
$199$	7.69219	0.00274013	0.00137006	−	0.999999i	$-0.499564\pi$
$199$	7.69219	0.00274013	0.00137006	+	0.999999i	$0.499564\pi$
$200$	456.071	0.161246
$201$	−2440.29	−0.856343
$202$	946.879	0.329813
$203$	0	0
$204$	3663.11	1.25720
$205$	−3534.59	−1.20423
$206$	−1262.96	−0.427160
$207$	108.748	0.0365146
$208$	−3838.46	−1.27956
$209$	−747.154	−0.247281
$210$	0	0
$211$	3107.34	1.01383	0.506915	−	0.861996i	$-0.330786\pi$
$211$	3107.34	1.01383	0.506915	+	0.861996i	$0.330786\pi$
$212$	1112.19	0.360309
$213$	2790.03	0.897509
$214$	354.444	0.113221
$215$	−29.4050	−0.00932746
$216$	1265.46	0.398628
$217$	0	0
$218$	47.0388	0.0146141
$219$	3619.76	1.11690
$220$	1055.58	0.323486
$221$	−6179.13	−1.88078
$222$	177.306	0.0536037
$223$	12.3185	0.00369913	0.00184957	−	0.999998i	$-0.499411\pi$
$223$	12.3185	0.00369913	0.00184957	+	0.999998i	$0.499411\pi$
$224$	0	0
$225$	328.082	0.0972096
$226$	1467.82	0.432027
$227$	−4615.90	−1.34964	−0.674820	−	0.737983i	$-0.735779\pi$
$227$	−4615.90	−1.34964	−0.674820	+	0.737983i	$0.735779\pi$
$228$	3005.51	0.873003
$229$	−5074.63	−1.46437	−0.732186	−	0.681105i	$-0.761500\pi$
$229$	−5074.63	−1.46437	−0.732186	+	0.681105i	$0.761500\pi$
$230$	−125.680	−0.0360309
$231$	0	0
$232$	1913.11	0.541386
$233$	211.683	0.0595184	0.0297592	−	0.999557i	$-0.490526\pi$
$233$	211.683	0.0595184	0.0297592	+	0.999557i	$0.490526\pi$
$234$	444.974	0.124311
$235$	−923.683	−0.256402
$236$	4072.64	1.12333
$237$	5799.12	1.58942
$238$	0	0
$239$	4312.49	1.16716	0.583581	−	0.812055i	$-0.301651\pi$
$239$	4312.49	1.16716	0.583581	+	0.812055i	$0.301651\pi$
$240$	−3919.42	−1.05416
$241$	996.584	0.266372	0.133186	−	0.991091i	$-0.457479\pi$
$241$	996.584	0.266372	0.133186	+	0.991091i	$0.457479\pi$
$242$	−88.5781	−0.0235290
$243$	2213.80	0.584426
$244$	756.133	0.198387
$245$	0	0
$246$	−1193.12	−0.309230
$247$	−5069.85	−1.30602
$248$	741.464	0.189851
$249$	155.046	0.0394603
$250$	797.278	0.201697
$251$	276.892	0.0696306	0.0348153	−	0.999394i	$-0.488916\pi$
$251$	276.892	0.0696306	0.0348153	+	0.999394i	$0.488916\pi$
$252$	0	0
$253$	−146.892	−0.0365021
$254$	−80.2364	−0.0198208
$255$	−6309.46	−1.54947
$256$	1619.36	0.395352
$257$	3235.18	0.785233	0.392617	−	0.919702i	$-0.371570\pi$
$257$	3235.18	0.785233	0.392617	+	0.919702i	$0.371570\pi$
$258$	−9.92581	−0.00239517
$259$	0	0
$260$	7162.67	1.70850
$261$	1376.23	0.326384
$262$	846.772	0.199671
$263$	207.944	0.0487544	0.0243772	−	0.999703i	$-0.492240\pi$
$263$	207.944	0.0487544	0.0243772	+	0.999703i	$0.492240\pi$
$264$	738.213	0.172098
$265$	−1915.67	−0.444071
$266$	0	0
$267$	−2091.99	−0.479504
$268$	−3072.53	−0.700316
$269$	−5033.04	−1.14078	−0.570390	−	0.821374i	$-0.693208\pi$
$269$	−5033.04	−1.14078	−0.570390	+	0.821374i	$0.693208\pi$
$270$	−1052.07	−0.237136
$271$	−1487.01	−0.333319	−0.166660	−	0.986015i	$-0.553298\pi$
$271$	−1487.01	−0.333319	−0.166660	+	0.986015i	$0.553298\pi$
$272$	4257.25	0.949021
$273$	0	0
$274$	−145.183	−0.0320102
$275$	−443.159	−0.0971764
$276$	590.890	0.128867
$277$	−235.836	−0.0511552	−0.0255776	−	0.999673i	$-0.508142\pi$
$277$	−235.836	−0.0511552	−0.0255776	+	0.999673i	$0.508142\pi$
$278$	−2123.05	−0.458029
$279$	533.384	0.114455
$280$	0	0
$281$	−4915.01	−1.04343	−0.521717	−	0.853118i	$-0.674708\pi$
$281$	−4915.01	−1.04343	−0.521717	+	0.853118i	$0.674708\pi$
$282$	−311.794	−0.0658406
$283$	5199.56	1.09216	0.546081	−	0.837733i	$-0.316119\pi$
$283$	5199.56	1.09216	0.546081	+	0.837733i	$0.316119\pi$
$284$	3512.87	0.733981
$285$	−5176.78	−1.07595
$286$	−601.051	−0.124269
$287$	0	0
$288$	−1044.09	−0.213624
$289$	1940.29	0.394930
$290$	−1590.50	−0.322060
$291$	5024.65	1.01220
$292$	4557.58	0.913398
$293$	8880.92	1.77075	0.885373	−	0.464881i	$-0.153903\pi$
$293$	8880.92	1.77075	0.885373	+	0.464881i	$0.153903\pi$
$294$	0	0
$295$	−7014.85	−1.38448
$296$	462.515	0.0908215
$297$	−1229.63	−0.240237
$298$	−2553.64	−0.496405
$299$	−996.743	−0.192786
$300$	1782.66	0.343072
$301$	0	0
$302$	851.612	0.162267
$303$	7667.90	1.45383
$304$	3492.99	0.659001
$305$	−1302.39	−0.244507
$306$	−493.522	−0.0921987
$307$	1497.93	0.278474	0.139237	−	0.990259i	$-0.455535\pi$
$307$	1497.93	0.278474	0.139237	+	0.990259i	$0.455535\pi$
$308$	0	0
$309$	−10227.6	−1.88293
$310$	−616.432	−0.112939
$311$	7484.71	1.36469	0.682345	−	0.731030i	$-0.260960\pi$
$311$	7484.71	1.36469	0.682345	+	0.731030i	$0.260960\pi$
$312$	5009.18	0.908939
$313$	658.363	0.118891	0.0594455	−	0.998232i	$-0.481067\pi$
$313$	658.363	0.118891	0.0594455	+	0.998232i	$0.481067\pi$
$314$	250.403	0.0450034
$315$	0	0
$316$	7301.57	1.29983
$317$	233.708	0.0414080	0.0207040	−	0.999786i	$-0.493409\pi$
$317$	233.708	0.0414080	0.0207040	+	0.999786i	$0.493409\pi$
$318$	−646.645	−0.114032
$319$	−1858.94	−0.326272
$320$	−4082.53	−0.713189
$321$	2870.31	0.499082
$322$	0	0
$323$	5622.98	0.968641
$324$	6587.51	1.12955
$325$	−3007.08	−0.513239
$326$	1021.13	0.173482
$327$	380.924	0.0644194
$328$	−3112.33	−0.523931
$329$	0	0
$330$	−613.729	−0.102378
$331$	8532.95	1.41696	0.708480	−	0.705731i	$-0.249381\pi$
$331$	8532.95	1.41696	0.708480	+	0.705731i	$0.249381\pi$
$332$	195.215	0.0322706
$333$	332.718	0.0547533
$334$	350.435	0.0574100
$335$	5292.22	0.863120
$336$	0	0
$337$	11691.2	1.88979	0.944895	−	0.327373i	$-0.106163\pi$
$337$	11691.2	1.88979	0.944895	+	0.327373i	$0.106163\pi$
$338$	−2470.15	−0.397509
$339$	11886.5	1.90439
$340$	−7944.14	−1.26715
$341$	−720.472	−0.114416
$342$	−404.925	−0.0640229
$343$	0	0
$344$	−25.8921	−0.00405817
$345$	−1017.77	−0.158825
$346$	1323.98	0.205715
$347$	4598.79	0.711459	0.355729	−	0.934589i	$-0.384232\pi$
$347$	4598.79	0.711459	0.355729	+	0.934589i	$0.384232\pi$
$348$	7477.80	1.15187
$349$	−6720.27	−1.03074	−0.515369	−	0.856968i	$-0.672345\pi$
$349$	−6720.27	−1.03074	−0.515369	+	0.856968i	$0.672345\pi$
$350$	0	0
$351$	−8343.72	−1.26882
$352$	1410.31	0.213551
$353$	−5738.70	−0.865270	−0.432635	−	0.901569i	$-0.642416\pi$
$353$	−5738.70	−0.865270	−0.432635	+	0.901569i	$0.642416\pi$
$354$	−2367.90	−0.355515
$355$	−6050.69	−0.904611
$356$	−2633.99	−0.392138
$357$	0	0
$358$	3242.87	0.478746
$359$	−4115.27	−0.605001	−0.302501	−	0.953149i	$-0.597821\pi$
$359$	−4115.27	−0.605001	−0.302501	+	0.953149i	$0.597821\pi$
$360$	1185.23	0.173519
$361$	−2245.46	−0.327374
$362$	2495.69	0.362349
$363$	−717.313	−0.103717
$364$	0	0
$365$	−7850.12	−1.12574
$366$	−439.628	−0.0627861
$367$	−9662.99	−1.37440	−0.687199	−	0.726469i	$-0.741160\pi$
$367$	−9662.99	−1.37440	−0.687199	+	0.726469i	$0.741160\pi$
$368$	686.729	0.0972778
$369$	−2238.90	−0.315861
$370$	−384.522	−0.0540279
$371$	0	0
$372$	2898.18	0.403934
$373$	−141.780	−0.0196812	−0.00984062	−	0.999952i	$-0.503132\pi$
$373$	−141.780	−0.0196812	−0.00984062	+	0.999952i	$0.503132\pi$
$374$	666.628	0.0921671
$375$	6456.42	0.889088
$376$	−813.334	−0.111555
$377$	−12613.9	−1.72321
$378$	0	0
$379$	−2819.73	−0.382163	−0.191082	−	0.981574i	$-0.561200\pi$
$379$	−2819.73	−0.382163	−0.191082	+	0.981574i	$0.561200\pi$
$380$	−6518.00	−0.879911
$381$	−649.760	−0.0873707
$382$	−2140.34	−0.286673
$383$	6337.84	0.845557	0.422778	−	0.906233i	$-0.361055\pi$
$383$	6337.84	0.845557	0.422778	+	0.906233i	$0.361055\pi$
$384$	−7458.53	−0.991189
$385$	0	0
$386$	1818.55	0.239797
$387$	−18.6259	−0.00244653
$388$	6326.46	0.827776
$389$	−8805.25	−1.14767	−0.573836	−	0.818970i	$-0.694545\pi$
$389$	−8805.25	−1.14767	−0.573836	+	0.818970i	$0.694545\pi$
$390$	−4164.49	−0.540710
$391$	1105.49	0.142985
$392$	0	0
$393$	6857.23	0.880156
$394$	3752.25	0.479786
$395$	−12576.5	−1.60200
$396$	668.631	0.0848484
$397$	−4315.26	−0.545534	−0.272767	−	0.962080i	$-0.587939\pi$
$397$	−4315.26	−0.545534	−0.272767	+	0.962080i	$0.587939\pi$
$398$	−5.63108	−0.000709197 0
$399$	0	0
$400$	2071.79	0.258974
$401$	361.681	0.0450411	0.0225206	−	0.999746i	$-0.492831\pi$
$401$	361.681	0.0450411	0.0225206	+	0.999746i	$0.492831\pi$
$402$	1786.42	0.221638
$403$	−4888.79	−0.604288
$404$	9654.53	1.18894
$405$	−11346.5	−1.39213
$406$	0	0
$407$	−449.420	−0.0547345
$408$	−5555.70	−0.674137
$409$	−9220.50	−1.11473	−0.557365	−	0.830268i	$-0.688188\pi$
$409$	−9220.50	−1.11473	−0.557365	+	0.830268i	$0.688188\pi$
$410$	2587.50	0.311677
$411$	−1175.70	−0.141102
$412$	−12877.4	−1.53986
$413$	0	0
$414$	−79.6092	−0.00945067
$415$	−336.245	−0.0397726
$416$	9569.74	1.12787
$417$	−17192.6	−2.01901
$418$	546.954	0.0640010
$419$	14912.9	1.73876	0.869380	−	0.494144i	$-0.164519\pi$
$419$	14912.9	1.73876	0.869380	+	0.494144i	$0.164519\pi$
$420$	0	0
$421$	−13486.0	−1.56121	−0.780603	−	0.625027i	$-0.785088\pi$
$421$	−13486.0	−1.56121	−0.780603	+	0.625027i	$0.785088\pi$
$422$	−2274.73	−0.262399
$423$	−585.085	−0.0672525
$424$	−1686.81	−0.193205
$425$	3335.16	0.380656
$426$	−2042.44	−0.232292
$427$	0	0
$428$	3613.96	0.408148
$429$	−4867.36	−0.547782
$430$	21.5260	0.00241413
$431$	406.334	0.0454116	0.0227058	−	0.999742i	$-0.492772\pi$
$431$	406.334	0.0454116	0.0227058	+	0.999742i	$0.492772\pi$
$432$	5748.59	0.640230
$433$	1766.69	0.196078	0.0980391	−	0.995183i	$-0.468743\pi$
$433$	1766.69	0.196078	0.0980391	+	0.995183i	$0.468743\pi$
$434$	0	0
$435$	−12880.0	−1.41965
$436$	479.615	0.0526821
$437$	907.033	0.0992889
$438$	−2649.85	−0.289075
$439$	−7824.19	−0.850634	−0.425317	−	0.905044i	$-0.639837\pi$
$439$	−7824.19	−0.850634	−0.425317	+	0.905044i	$0.639837\pi$
$440$	−1600.95	−0.173460
$441$	0	0
$442$	4523.44	0.486783
$443$	11667.9	1.25137	0.625686	−	0.780075i	$-0.284819\pi$
$443$	11667.9	1.25137	0.625686	+	0.780075i	$0.284819\pi$
$444$	1807.84	0.193235
$445$	4536.86	0.483299
$446$	−9.01776	−0.000957406 0
$447$	−20679.6	−2.18817
$448$	0	0
$449$	16975.3	1.78421	0.892107	−	0.451825i	$-0.149227\pi$
$449$	16975.3	1.78421	0.892107	+	0.451825i	$0.149227\pi$
$450$	−240.173	−0.0251597
$451$	3024.21	0.315753
$452$	14966.1	1.55741
$453$	6896.42	0.715280
$454$	3379.07	0.349312
$455$	0	0
$456$	−4558.33	−0.468122
$457$	−16192.9	−1.65748	−0.828741	−	0.559632i	$-0.810943\pi$
$457$	−16192.9	−1.65748	−0.828741	+	0.559632i	$0.810943\pi$
$458$	3714.89	0.379007
$459$	9254.05	0.941050
$460$	−1281.45	−0.129887
$461$	−8586.04	−0.867444	−0.433722	−	0.901047i	$-0.642800\pi$
$461$	−8586.04	−0.867444	−0.433722	+	0.901047i	$0.642800\pi$
$462$	0	0
$463$	−7917.20	−0.794694	−0.397347	−	0.917668i	$-0.630069\pi$
$463$	−7917.20	−0.794694	−0.397347	+	0.917668i	$0.630069\pi$
$464$	8690.67	0.869513
$465$	−4991.91	−0.497837
$466$	−154.962	−0.0154045
$467$	15155.0	1.50169	0.750844	−	0.660480i	$-0.229647\pi$
$467$	15155.0	1.50169	0.750844	+	0.660480i	$0.229647\pi$
$468$	4537.03	0.448128
$469$	0	0
$470$	676.183	0.0663617
$471$	2027.78	0.198376
$472$	−6176.82	−0.602354
$473$	25.1591	0.00244570
$474$	−4245.25	−0.411373
$475$	2736.43	0.264328
$476$	0	0
$477$	−1213.44	−0.116477
$478$	−3156.96	−0.302084
$479$	−10001.1	−0.953993	−0.476996	−	0.878905i	$-0.658275\pi$
$479$	−10001.1	−0.953993	−0.476996	+	0.878905i	$0.658275\pi$
$480$	9771.59	0.929188
$481$	−3049.56	−0.289081
$482$	−729.550	−0.0689421
$483$	0	0
$484$	−903.156	−0.0848193
$485$	−10896.9	−1.02021
$486$	−1620.62	−0.151261
$487$	7044.54	0.655480	0.327740	−	0.944768i	$-0.393713\pi$
$487$	7044.54	0.655480	0.327740	+	0.944768i	$0.393713\pi$
$488$	−1146.80	−0.106379
$489$	8269.21	0.764717
$490$	0	0
$491$	−13326.4	−1.22487	−0.612437	−	0.790520i	$-0.709811\pi$
$491$	−13326.4	−1.22487	−0.612437	+	0.790520i	$0.709811\pi$
$492$	−12165.2	−1.11474
$493$	13990.2	1.27806
$494$	3711.38	0.338022
$495$	−1151.67	−0.104573
$496$	3368.25	0.304917
$497$	0	0
$498$	−113.501	−0.0102131
$499$	−20069.1	−1.80044	−0.900218	−	0.435440i	$-0.856593\pi$
$499$	−20069.1	−1.80044	−0.900218	+	0.435440i	$0.856593\pi$
$500$	8129.17	0.727095
$501$	2837.85	0.253065
$502$	−202.699	−0.0180217
$503$	−7782.35	−0.689856	−0.344928	−	0.938629i	$-0.612097\pi$
$503$	−7782.35	−0.689856	−0.344928	+	0.938629i	$0.612097\pi$
$504$	0	0
$505$	−16629.3	−1.46533
$506$	107.532	0.00944744
$507$	−20003.4	−1.75224
$508$	−818.102	−0.0714516
$509$	1475.93	0.128526	0.0642628	−	0.997933i	$-0.479530\pi$
$509$	1475.93	0.128526	0.0642628	+	0.997933i	$0.479530\pi$
$510$	4618.85	0.401031
$511$	0	0
$512$	−11250.6	−0.971116
$513$	7592.75	0.653466
$514$	−2368.32	−0.203233
$515$	22180.4	1.89784
$516$	−101.205	−0.00863431
$517$	790.307	0.0672295
$518$	0	0
$519$	10721.7	0.906799
$520$	−10863.3	−0.916132
$521$	−7609.43	−0.639875	−0.319938	−	0.947439i	$-0.603662\pi$
$521$	−7609.43	−0.639875	−0.319938	+	0.947439i	$0.603662\pi$
$522$	−1007.47	−0.0844744
$523$	−12452.9	−1.04116	−0.520581	−	0.853812i	$-0.674285\pi$
$523$	−12452.9	−1.04116	−0.520581	+	0.853812i	$0.674285\pi$
$524$	8633.82	0.719790
$525$	0	0
$526$	−152.226	−0.0126186
$527$	5422.18	0.448186
$528$	3353.48	0.276404
$529$	−11988.7	−0.985344
$530$	1402.37	0.114934
$531$	−4443.39	−0.363139
$532$	0	0
$533$	20520.9	1.66765
$534$	1531.44	0.124105
$535$	−6224.81	−0.503031
$536$	4659.99	0.375524
$537$	26261.0	2.11033
$538$	3684.44	0.295255
$539$	0	0
$540$	−10727.0	−0.854847
$541$	9312.17	0.740039	0.370020	−	0.929024i	$-0.379351\pi$
$541$	9312.17	0.740039	0.370020	+	0.929024i	$0.379351\pi$
$542$	1088.57	0.0862693
$543$	20210.3	1.59725
$544$	−10613.8	−0.836515
$545$	−826.105	−0.0649292
$546$	0	0
$547$	−11018.6	−0.861278	−0.430639	−	0.902524i	$-0.641712\pi$
$547$	−11018.6	−0.861278	−0.430639	+	0.902524i	$0.641712\pi$
$548$	−1480.31	−0.115393
$549$	−824.968	−0.0641325
$550$	324.415	0.0251511
$551$	11478.6	0.887490
$552$	−896.179	−0.0691013
$553$	0	0
$554$	172.644	0.0132399
$555$	−3113.89	−0.238157
$556$	−21646.9	−1.65114
$557$	−12018.4	−0.914250	−0.457125	−	0.889403i	$-0.651121\pi$
$557$	−12018.4	−0.914250	−0.457125	+	0.889403i	$0.651121\pi$
$558$	−390.464	−0.0296231
$559$	170.718	0.0129170
$560$	0	0
$561$	5398.40	0.406276
$562$	3598.04	0.270061
$563$	8763.89	0.656046	0.328023	−	0.944670i	$-0.393618\pi$
$563$	8763.89	0.656046	0.328023	+	0.944670i	$0.393618\pi$
$564$	−3179.10	−0.237348
$565$	−25778.1	−1.91946
$566$	−3806.34	−0.282672
$567$	0	0
$568$	−5327.84	−0.393576
$569$	−10273.2	−0.756895	−0.378447	−	0.925623i	$-0.623542\pi$
$569$	−10273.2	−0.756895	−0.378447	+	0.925623i	$0.623542\pi$
$570$	3789.67	0.278477
$571$	2602.62	0.190747	0.0953734	−	0.995442i	$-0.469596\pi$
$571$	2602.62	0.190747	0.0953734	+	0.995442i	$0.469596\pi$
$572$	−6128.41	−0.447975
$573$	−17332.6	−1.26366
$574$	0	0
$575$	537.988	0.0390185
$576$	−2585.99	−0.187065
$577$	19727.0	1.42331	0.711653	−	0.702532i	$-0.247947\pi$
$577$	19727.0	1.42331	0.711653	+	0.702532i	$0.247947\pi$
$578$	−1420.39	−0.102215
$579$	14726.7	1.05703
$580$	−16217.0	−1.16099
$581$	0	0
$582$	−3678.30	−0.261977
$583$	1639.06	0.116437
$584$	−6912.30	−0.489783
$585$	−7814.72	−0.552306
$586$	−6501.28	−0.458303
$587$	10116.2	0.711309	0.355654	−	0.934618i	$-0.384258\pi$
$587$	10116.2	0.711309	0.355654	+	0.934618i	$0.384258\pi$
$588$	0	0
$589$	4448.78	0.311221
$590$	5135.23	0.358329
$591$	30386.0	2.11491
$592$	2101.07	0.145867
$593$	−3130.32	−0.216774	−0.108387	−	0.994109i	$-0.534569\pi$
$593$	−3130.32	−0.216774	−0.108387	+	0.994109i	$0.534569\pi$
$594$	900.152	0.0621779
$595$	0	0
$596$	−26037.3	−1.78948
$597$	−45.6009	−0.00312616
$598$	729.667	0.0498968
$599$	10080.1	0.687581	0.343790	−	0.939046i	$-0.388289\pi$
$599$	10080.1	0.687581	0.343790	+	0.939046i	$0.388289\pi$
$600$	−2703.68	−0.183962
$601$	−4777.02	−0.324224	−0.162112	−	0.986772i	$-0.551831\pi$
$601$	−4777.02	−0.324224	−0.162112	+	0.986772i	$0.551831\pi$
$602$	0	0
$603$	3352.24	0.226391
$604$	8683.16	0.584955
$605$	1555.63	0.104537
$606$	−5613.29	−0.376278
$607$	2571.35	0.171941	0.0859703	−	0.996298i	$-0.472601\pi$
$607$	2571.35	0.171941	0.0859703	+	0.996298i	$0.472601\pi$
$608$	−8708.43	−0.580877
$609$	0	0
$610$	953.414	0.0632830
$611$	5362.67	0.355074
$612$	−5032.03	−0.332366
$613$	12711.9	0.837564	0.418782	−	0.908087i	$-0.362457\pi$
$613$	12711.9	0.837564	0.418782	+	0.908087i	$0.362457\pi$
$614$	−1096.56	−0.0720744
$615$	20953.8	1.37388
$616$	0	0
$617$	16236.1	1.05939	0.529693	−	0.848189i	$-0.322307\pi$
$617$	16236.1	1.05939	0.529693	+	0.848189i	$0.322307\pi$
$618$	7487.11	0.487339
$619$	−12657.3	−0.821874	−0.410937	−	0.911664i	$-0.634798\pi$
$619$	−12657.3	−0.821874	−0.410937	+	0.911664i	$0.634798\pi$
$620$	−6285.23	−0.407131
$621$	1492.75	0.0964607
$622$	−5479.19	−0.353208
$623$	0	0
$624$	22755.2	1.45983
$625$	−19037.8	−1.21842
$626$	−481.955	−0.0307713
$627$	4429.28	0.282119
$628$	2553.15	0.162232
$629$	3382.28	0.214404
$630$	0	0
$631$	−3949.97	−0.249201	−0.124600	−	0.992207i	$-0.539765\pi$
$631$	−3949.97	−0.249201	−0.124600	+	0.992207i	$0.539765\pi$
$632$	−11074.0	−0.696994
$633$	−18421.0	−1.15666
$634$	−171.086	−0.0107172
$635$	1409.13	0.0880621
$636$	−6593.29	−0.411070
$637$	0	0
$638$	1360.84	0.0844455
$639$	−3832.67	−0.237274
$640$	16175.2	0.999033
$641$	−7398.27	−0.455872	−0.227936	−	0.973676i	$-0.573198\pi$
$641$	−7398.27	−0.455872	−0.227936	+	0.973676i	$0.573198\pi$
$642$	−2101.22	−0.129172
$643$	12491.7	0.766134	0.383067	−	0.923721i	$-0.374868\pi$
$643$	12491.7	0.766134	0.383067	+	0.923721i	$0.374868\pi$
$644$	0	0
$645$	174.319	0.0106415
$646$	−4116.31	−0.250703
$647$	10472.0	0.636315	0.318158	−	0.948038i	$-0.396936\pi$
$647$	10472.0	0.636315	0.318158	+	0.948038i	$0.396936\pi$
$648$	−9991.02	−0.605685
$649$	6001.94	0.363015
$650$	2201.33	0.132836
$651$	0	0
$652$	10411.6	0.625384
$653$	6337.94	0.379820	0.189910	−	0.981801i	$-0.439180\pi$
$653$	6337.94	0.379820	0.189910	+	0.981801i	$0.439180\pi$
$654$	−278.856	−0.0166730
$655$	−14871.2	−0.887121
$656$	−14138.4	−0.841479
$657$	−4972.48	−0.295274
$658$	0	0
$659$	15196.7	0.898302	0.449151	−	0.893456i	$-0.351726\pi$
$659$	15196.7	0.898302	0.449151	+	0.893456i	$0.351726\pi$
$660$	−6257.67	−0.369060
$661$	−2298.17	−0.135232	−0.0676161	−	0.997711i	$-0.521539\pi$
$661$	−2298.17	−0.135232	−0.0676161	+	0.997711i	$0.521539\pi$
$662$	−6246.55	−0.366736
$663$	36631.1	2.14575
$664$	−296.075	−0.0173042
$665$	0	0
$666$	−243.566	−0.0141712
$667$	2256.73	0.131006
$668$	3573.09	0.206956
$669$	−73.0265	−0.00422028
$670$	−3874.18	−0.223392
$671$	1114.33	0.0641106
$672$	0	0
$673$	23199.6	1.32880	0.664398	−	0.747379i	$-0.268688\pi$
$673$	23199.6	1.32880	0.664398	+	0.747379i	$0.268688\pi$
$674$	−8558.54	−0.489114
$675$	4503.49	0.256799
$676$	−25186.0	−1.43298
$677$	2145.38	0.121793	0.0608963	−	0.998144i	$-0.480604\pi$
$677$	2145.38	0.121793	0.0608963	+	0.998144i	$0.480604\pi$
$678$	−8701.55	−0.492892
$679$	0	0
$680$	12048.6	0.679472
$681$	27364.0	1.53978
$682$	527.422	0.0296129
$683$	−29544.6	−1.65519	−0.827593	−	0.561329i	$-0.810290\pi$
$683$	−29544.6	−1.65519	−0.827593	+	0.561329i	$0.810290\pi$
$684$	−4128.68	−0.230795
$685$	2549.72	0.142219
$686$	0	0
$687$	30083.4	1.67068
$688$	−117.620	−0.00651776
$689$	11121.9	0.614964
$690$	745.057	0.0411070
$691$	−27803.1	−1.53065	−0.765325	−	0.643644i	$-0.777422\pi$
$691$	−27803.1	−1.53065	−0.765325	+	0.643644i	$0.777422\pi$
$692$	13499.5	0.741579
$693$	0	0
$694$	−3366.55	−0.184139
$695$	37285.4	2.03498
$696$	−11341.3	−0.617659
$697$	−22759.8	−1.23686
$698$	4919.58	0.266775
$699$	−1254.90	−0.0679036
$700$	0	0
$701$	−19697.8	−1.06130	−0.530652	−	0.847590i	$-0.678053\pi$
$701$	−19697.8	−1.06130	−0.530652	+	0.847590i	$0.678053\pi$
$702$	6108.02	0.328394
$703$	2775.09	0.148883
$704$	3493.03	0.187001
$705$	5475.78	0.292524
$706$	4201.02	0.223948
$707$	0	0
$708$	−24143.5	−1.28159
$709$	19122.5	1.01292	0.506460	−	0.862263i	$-0.330954\pi$
$709$	19122.5	1.01292	0.506460	+	0.862263i	$0.330954\pi$
$710$	4429.41	0.234131
$711$	−7966.27	−0.420195
$712$	3994.86	0.210272
$713$	874.641	0.0459405
$714$	0	0
$715$	10555.8	0.552117
$716$	33064.8	1.72582
$717$	−25565.3	−1.33160
$718$	3012.59	0.156586
$719$	−1837.44	−0.0953060	−0.0476530	−	0.998864i	$-0.515174\pi$
$719$	−1837.44	−0.0953060	−0.0476530	+	0.998864i	$0.515174\pi$
$720$	5384.13	0.278687
$721$	0	0
$722$	1643.79	0.0847307
$723$	−5907.95	−0.303899
$724$	25446.4	1.30623
$725$	6808.33	0.348765
$726$	525.109	0.0268438
$727$	7555.46	0.385442	0.192721	−	0.981254i	$-0.438269\pi$
$727$	7555.46	0.385442	0.192721	+	0.981254i	$0.438269\pi$
$728$	0	0
$729$	10705.2	0.543881
$730$	5746.69	0.291362
$731$	−189.344	−0.00958021
$732$	−4482.51	−0.226337
$733$	−11984.6	−0.603905	−0.301952	−	0.953323i	$-0.597638\pi$
$733$	−11984.6	−0.603905	−0.301952	+	0.953323i	$0.597638\pi$
$734$	7073.80	0.355720
$735$	0	0
$736$	−1712.10	−0.0857456
$737$	−4528.05	−0.226313
$738$	1638.99	0.0817508
$739$	−27142.5	−1.35109	−0.675543	−	0.737321i	$-0.736091\pi$
$739$	−27142.5	−1.35109	−0.675543	+	0.737321i	$0.736091\pi$
$740$	−3920.64	−0.194764
$741$	30055.1	1.49001
$742$	0	0
$743$	−29222.6	−1.44290	−0.721450	−	0.692467i	$-0.756524\pi$
$743$	−29222.6	−1.44290	−0.721450	+	0.692467i	$0.756524\pi$
$744$	−4395.55	−0.216598
$745$	44847.6	2.20549
$746$	103.790	0.00509388
$747$	−212.987	−0.0104321
$748$	6797.04	0.332252
$749$	0	0
$750$	−4726.43	−0.230113
$751$	−8859.39	−0.430471	−0.215236	−	0.976562i	$-0.569052\pi$
$751$	−8859.39	−0.430471	−0.215236	+	0.976562i	$0.569052\pi$
$752$	−3694.73	−0.179166
$753$	−1641.47	−0.0794404
$754$	9234.05	0.446000
$755$	−14956.2	−0.720941
$756$	0	0
$757$	35734.4	1.71571	0.857853	−	0.513896i	$-0.171798\pi$
$757$	35734.4	1.71571	0.857853	+	0.513896i	$0.171798\pi$
$758$	2064.19	0.0989112
$759$	870.806	0.0416446
$760$	9885.59	0.471826
$761$	−34394.7	−1.63838	−0.819189	−	0.573524i	$-0.805576\pi$
$761$	−34394.7	−1.63838	−0.819189	+	0.573524i	$0.805576\pi$
$762$	475.658	0.0226132
$763$	0	0
$764$	−21823.2	−1.03342
$765$	8667.33	0.409631
$766$	−4639.62	−0.218846
$767$	40726.4	1.91727
$768$	−9599.92	−0.451051
$769$	11602.7	0.544091	0.272045	−	0.962284i	$-0.412300\pi$
$769$	11602.7	0.544091	0.272045	+	0.962284i	$0.412300\pi$
$770$	0	0
$771$	−19178.8	−0.895859
$772$	18542.2	0.864441
$773$	12680.6	0.590026	0.295013	−	0.955493i	$-0.404676\pi$
$773$	12680.6	0.590026	0.295013	+	0.955493i	$0.404676\pi$
$774$	13.6351	0.000633210 0
$775$	2638.71	0.122303
$776$	−9595.09	−0.443871
$777$	0	0
$778$	6445.89	0.297039
$779$	−18674.0	−0.858876
$780$	−42461.7	−1.94920
$781$	5176.99	0.237193
$782$	−809.276	−0.0370072
$783$	18891.0	0.862210
$784$	0	0
$785$	−4397.62	−0.199946
$786$	−5019.84	−0.227801
$787$	4417.61	0.200090	0.100045	−	0.994983i	$-0.468101\pi$
$787$	4417.61	0.200090	0.100045	+	0.994983i	$0.468101\pi$
$788$	38258.5	1.72957
$789$	−1232.74	−0.0556231
$790$	9206.61	0.414628
$791$	0	0
$792$	−1014.09	−0.0454974
$793$	7561.33	0.338601
$794$	3158.99	0.141194
$795$	11356.5	0.506633
$796$	−57.4153	−0.00255657
$797$	27030.1	1.20132	0.600661	−	0.799504i	$-0.294904\pi$
$797$	27030.1	1.20132	0.600661	+	0.799504i	$0.294904\pi$
$798$	0	0
$799$	−5947.75	−0.263350
$800$	−5165.23	−0.228273
$801$	2873.77	0.126766
$802$	−264.769	−0.0116575
$803$	6716.60	0.295173
$804$	18214.6	0.798979
$805$	0	0
$806$	3578.85	0.156401
$807$	29836.9	1.30150
$808$	−14642.6	−0.637533
$809$	23647.0	1.02767	0.513835	−	0.857889i	$-0.328224\pi$
$809$	23647.0	1.02767	0.513835	+	0.857889i	$0.328224\pi$
$810$	8306.24	0.360311
$811$	−33486.1	−1.44988	−0.724941	−	0.688811i	$-0.758133\pi$
$811$	−33486.1	−1.44988	−0.724941	+	0.688811i	$0.758133\pi$
$812$	0	0
$813$	8815.30	0.380278
$814$	328.999	0.0141663
$815$	−17933.3	−0.770768
$816$	−25237.8	−1.08272
$817$	−155.353	−0.00665251
$818$	6749.88	0.288513
$819$	0	0
$820$	26382.5	1.12356
$821$	2605.69	0.110766	0.0553832	−	0.998465i	$-0.482362\pi$
$821$	2605.69	0.110766	0.0553832	+	0.998465i	$0.482362\pi$
$822$	860.673	0.0365200
$823$	31976.2	1.35434	0.677169	−	0.735828i	$-0.263207\pi$
$823$	31976.2	1.35434	0.677169	+	0.735828i	$0.263207\pi$
$824$	19530.6	0.825705
$825$	2627.14	0.110867
$826$	0	0
$827$	−37759.0	−1.58768	−0.793839	−	0.608128i	$-0.791921\pi$
$827$	−37759.0	−1.58768	−0.793839	+	0.608128i	$0.791921\pi$
$828$	−811.707	−0.0340686
$829$	1137.55	0.0476584	0.0238292	−	0.999716i	$-0.492414\pi$
$829$	1137.55	0.0476584	0.0238292	+	0.999716i	$0.492414\pi$
$830$	246.149	0.0102939
$831$	1398.08	0.0583621
$832$	23702.2	0.987649
$833$	0	0
$834$	12585.9	0.522557
$835$	−6154.40	−0.255068
$836$	5576.83	0.230716
$837$	7321.60	0.302356
$838$	−10917.0	−0.450024
$839$	37372.2	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$839$	37372.2	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$840$	0	0
$841$	4170.26	0.170989
$842$	9872.44	0.404070
$843$	29137.2	1.19044
$844$	−23193.5	−0.945917
$845$	43381.1	1.76610
$846$	428.312	0.0174062
$847$	0	0
$848$	−7662.68	−0.310304
$849$	−30824.0	−1.24603
$850$	−2441.51	−0.0985211
$851$	545.589	0.0219772
$852$	−20825.0	−0.837387
$853$	22490.8	0.902780	0.451390	−	0.892327i	$-0.350928\pi$
$853$	22490.8	0.902780	0.451390	+	0.892327i	$0.350928\pi$
$854$	0	0
$855$	7111.36	0.284449
$856$	−5481.16	−0.218858
$857$	−43409.5	−1.73027	−0.865135	−	0.501539i	$-0.832767\pi$
$857$	−43409.5	−1.73027	−0.865135	+	0.501539i	$0.832767\pi$
$858$	3563.15	0.141776
$859$	−29533.2	−1.17306	−0.586532	−	0.809926i	$-0.699507\pi$
$859$	−29533.2	−1.17306	−0.586532	+	0.809926i	$0.699507\pi$
$860$	219.482	0.00870264
$861$	0	0
$862$	−297.457	−0.0117534
$863$	14351.6	0.566090	0.283045	−	0.959107i	$-0.408655\pi$
$863$	14351.6	0.566090	0.283045	+	0.959107i	$0.408655\pi$
$864$	−14331.9	−0.564331
$865$	−23251.9	−0.913975
$866$	−1293.31	−0.0507488
$867$	−11502.4	−0.450569
$868$	0	0
$869$	10760.5	0.420051
$870$	9428.82	0.367433
$871$	−30725.3	−1.19528
$872$	−727.413	−0.0282492
$873$	−6902.39	−0.267595
$874$	−663.994	−0.0256979
$875$	0	0
$876$	−27018.3	−1.04208
$877$	−43248.7	−1.66523	−0.832614	−	0.553854i	$-0.813156\pi$
$877$	−43248.7	−1.66523	−0.832614	+	0.553854i	$0.813156\pi$
$878$	5727.71	0.220160
$879$	−52647.9	−2.02021
$880$	−7272.64	−0.278591
$881$	−3816.13	−0.145935	−0.0729675	−	0.997334i	$-0.523247\pi$
$881$	−3816.13	−0.145935	−0.0729675	+	0.997334i	$0.523247\pi$
$882$	0	0
$883$	48787.6	1.85938	0.929690	−	0.368343i	$-0.120075\pi$
$883$	48787.6	1.85938	0.929690	+	0.368343i	$0.120075\pi$
$884$	46121.6	1.75479
$885$	41585.5	1.57953
$886$	−8541.49	−0.323879
$887$	−41495.1	−1.57077	−0.785384	−	0.619009i	$-0.787534\pi$
$887$	−41495.1	−1.57077	−0.785384	+	0.619009i	$0.787534\pi$
$888$	−2741.88	−0.103617
$889$	0	0
$890$	−3321.21	−0.125087
$891$	9708.15	0.365023
$892$	−91.9464	−0.00345134
$893$	−4880.01	−0.182870
$894$	15138.5	0.566340
$895$	−56951.9	−2.12703
$896$	0	0
$897$	5908.90	0.219947
$898$	−12426.7	−0.461788
$899$	11068.7	0.410637
$900$	−2448.84	−0.0906978
$901$	−12335.3	−0.456104
$902$	−2213.88	−0.0817228
$903$	0	0
$904$	−22698.5	−0.835113
$905$	−43829.7	−1.60989
$906$	−5048.53	−0.185128
$907$	21615.3	0.791316	0.395658	−	0.918398i	$-0.370517\pi$
$907$	21615.3	0.791316	0.395658	+	0.918398i	$0.370517\pi$
$908$	34453.6	1.25923
$909$	−10533.4	−0.384347
$910$	0	0
$911$	3646.35	0.132611	0.0663057	−	0.997799i	$-0.478879\pi$
$911$	3646.35	0.132611	0.0663057	+	0.997799i	$0.478879\pi$
$912$	−20707.1	−0.751844
$913$	287.693	0.0104285
$914$	11854.0	0.428988
$915$	7720.82	0.278954
$916$	37877.6	1.36628
$917$	0	0
$918$	−6774.43	−0.243562
$919$	31280.0	1.12278	0.561388	−	0.827553i	$-0.310267\pi$
$919$	31280.0	1.12278	0.561388	+	0.827553i	$0.310267\pi$
$920$	1943.53	0.0696482
$921$	−8880.05	−0.317707
$922$	6285.42	0.224511
$923$	35128.7	1.25274
$924$	0	0
$925$	1645.99	0.0585079
$926$	5795.79	0.205682
$927$	14049.7	0.497790
$928$	−21666.9	−0.766433
$929$	−6557.92	−0.231602	−0.115801	−	0.993272i	$-0.536944\pi$
$929$	−6557.92	−0.231602	−0.115801	+	0.993272i	$0.536944\pi$
$930$	3654.33	0.128850
$931$	0	0
$932$	−1580.02	−0.0555314
$933$	−44370.9	−1.55695
$934$	−11094.2	−0.388665
$935$	−11707.4	−0.409491
$936$	−6881.13	−0.240296
$937$	24473.3	0.853265	0.426632	−	0.904425i	$-0.359700\pi$
$937$	24473.3	0.853265	0.426632	+	0.904425i	$0.359700\pi$
$938$	0	0
$939$	−3902.91	−0.135641
$940$	6894.46	0.239226
$941$	−15420.8	−0.534224	−0.267112	−	0.963665i	$-0.586069\pi$
$941$	−15420.8	−0.534224	−0.267112	+	0.963665i	$0.586069\pi$
$942$	−1484.44	−0.0513436
$943$	−3671.34	−0.126782
$944$	−28059.4	−0.967432
$945$	0	0
$946$	−18.4177	−0.000632993 0
$947$	−33141.2	−1.13722	−0.568608	−	0.822609i	$-0.692518\pi$
$947$	−33141.2	−1.13722	−0.568608	+	0.822609i	$0.692518\pi$
$948$	−43285.2	−1.48295
$949$	45575.8	1.55896
$950$	−2003.20	−0.0684132
$951$	−1385.47	−0.0472417
$952$	0	0
$953$	−20735.4	−0.704813	−0.352406	−	0.935847i	$-0.614637\pi$
$953$	−20735.4	−0.704813	−0.352406	+	0.935847i	$0.614637\pi$
$954$	888.298	0.0301464
$955$	37589.0	1.27367
$956$	−32188.9	−1.08898
$957$	11020.2	0.372239
$958$	7321.32	0.246911
$959$	0	0
$960$	24202.1	0.813666
$961$	−25501.1	−0.856000
$962$	2232.44	0.0748198
$963$	−3942.96	−0.131942
$964$	−7438.60	−0.248528
$965$	−31937.7	−1.06540
$966$	0	0
$967$	−8178.87	−0.271990	−0.135995	−	0.990710i	$-0.543423\pi$
$967$	−8178.87	−0.271990	−0.135995	+	0.990710i	$0.543423\pi$
$968$	1369.78	0.0454819
$969$	−33334.2	−1.10511
$970$	7977.08	0.264050
$971$	20576.1	0.680039	0.340020	−	0.940418i	$-0.389566\pi$
$971$	20576.1	0.680039	0.340020	+	0.940418i	$0.389566\pi$
$972$	−16524.1	−0.545277
$973$	0	0
$974$	−5156.96	−0.169651
$975$	17826.6	0.585546
$976$	−5209.55	−0.170854
$977$	14541.9	0.476188	0.238094	−	0.971242i	$-0.423477\pi$
$977$	14541.9	0.476188	0.238094	+	0.971242i	$0.423477\pi$
$978$	−6053.48	−0.197923
$979$	−3881.76	−0.126723
$980$	0	0
$981$	−523.277	−0.0170305
$982$	9755.62	0.317021
$983$	29285.7	0.950223	0.475111	−	0.879926i	$-0.342408\pi$
$983$	29285.7	0.950223	0.475111	+	0.879926i	$0.342408\pi$
$984$	18450.5	0.597745
$985$	−65897.7	−2.13165
$986$	−10241.5	−0.330787
$987$	0	0
$988$	37841.8	1.21853
$989$	−30.5427	−0.000982004 0
$990$	843.082	0.0270655
$991$	−38085.9	−1.22083	−0.610413	−	0.792083i	$-0.708997\pi$
$991$	−38085.9	−1.22083	−0.610413	+	0.792083i	$0.708997\pi$
$992$	−8397.44	−0.268769
$993$	−50585.1	−1.61658
$994$	0	0
$995$	98.8940	0.00315090
$996$	−1157.28	−0.0368170
$997$	26803.6	0.851434	0.425717	−	0.904856i	$-0.360022\pi$
$997$	26803.6	0.851434	0.425717	+	0.904856i	$0.360022\pi$
$998$	14691.6	0.465987
$999$	4567.12	0.144642

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.4.a.e.1.1		2
7.6	odd	2		11.4.a.a.1.1	✓	2
21.20	even	2		99.4.a.c.1.2		2
28.27	even	2		176.4.a.i.1.1		2
35.13	even	4		275.4.b.c.199.3		4
35.27	even	4		275.4.b.c.199.2		4
35.34	odd	2		275.4.a.b.1.2		2
56.13	odd	2		704.4.a.p.1.1		2
56.27	even	2		704.4.a.n.1.2		2
77.6	even	10		121.4.c.f.3.1		8
77.13	even	10		121.4.c.f.81.1		8
77.20	odd	10		121.4.c.c.81.2		8
77.27	odd	10		121.4.c.c.3.2		8
77.41	even	10		121.4.c.f.9.2		8
77.48	odd	10		121.4.c.c.27.1		8
77.62	even	10		121.4.c.f.27.2		8
77.69	odd	10		121.4.c.c.9.1		8
77.76	even	2		121.4.a.c.1.2		2
84.83	odd	2		1584.4.a.bc.1.2		2
91.90	odd	2		1859.4.a.a.1.2		2
105.104	even	2		2475.4.a.q.1.1		2
231.230	odd	2		1089.4.a.v.1.1		2
308.307	odd	2		1936.4.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.1	✓	2	7.6	odd	2
99.4.a.c.1.2		2	21.20	even	2
121.4.a.c.1.2		2	77.76	even	2
121.4.c.c.3.2		8	77.27	odd	10
121.4.c.c.9.1		8	77.69	odd	10
121.4.c.c.27.1		8	77.48	odd	10
121.4.c.c.81.2		8	77.20	odd	10
121.4.c.f.3.1		8	77.6	even	10
121.4.c.f.9.2		8	77.41	even	10
121.4.c.f.27.2		8	77.62	even	10
121.4.c.f.81.1		8	77.13	even	10
176.4.a.i.1.1		2	28.27	even	2
275.4.a.b.1.2		2	35.34	odd	2
275.4.b.c.199.2		4	35.27	even	4
275.4.b.c.199.3		4	35.13	even	4
539.4.a.e.1.1		2	1.1	even	1	trivial
704.4.a.n.1.2		2	56.27	even	2
704.4.a.p.1.1		2	56.13	odd	2
1089.4.a.v.1.1		2	231.230	odd	2
1584.4.a.bc.1.2		2	84.83	odd	2
1859.4.a.a.1.2		2	91.90	odd	2
1936.4.a.w.1.1		2	308.307	odd	2
2475.4.a.q.1.1		2	105.104	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{2} -5.92820 q^{3} -7.46410 q^{4} +12.8564 q^{5} +4.33975 q^{6} +11.3205 q^{8} +8.14359 q^{9} +O(q^{10})\)	\(q-0.732051 q^{2} -5.92820 q^{3} -7.46410 q^{4} +12.8564 q^{5} +4.33975 q^{6} +11.3205 q^{8} +8.14359 q^{9} -9.41154 q^{10} -11.0000 q^{11} +44.2487 q^{12} -74.6410 q^{13} -76.2154 q^{15} +51.4256 q^{16} +82.7846 q^{17} -5.96152 q^{18} +67.9230 q^{19} -95.9615 q^{20} +8.05256 q^{22} +13.3538 q^{23} -67.1103 q^{24} +40.2872 q^{25} +54.6410 q^{26} +111.785 q^{27} +168.995 q^{29} +55.7935 q^{30} +65.4974 q^{31} -128.210 q^{32} +65.2102 q^{33} -60.6025 q^{34} -60.7846 q^{36} +40.8564 q^{37} -49.7231 q^{38} +442.487 q^{39} +145.541 q^{40} -274.928 q^{41} -2.28719 q^{43} +82.1051 q^{44} +104.697 q^{45} -9.77568 q^{46} -71.8461 q^{47} -304.862 q^{48} -29.4923 q^{50} -490.764 q^{51} +557.128 q^{52} -149.005 q^{53} -81.8320 q^{54} -141.420 q^{55} -402.662 q^{57} -123.713 q^{58} -545.631 q^{59} +568.879 q^{60} -101.303 q^{61} -47.9474 q^{62} -317.549 q^{64} -959.615 q^{65} -47.7372 q^{66} +411.641 q^{67} -617.913 q^{68} -79.1642 q^{69} -470.636 q^{71} +92.1896 q^{72} -610.600 q^{73} -29.9090 q^{74} -238.831 q^{75} -506.985 q^{76} -323.923 q^{78} -978.225 q^{79} +661.149 q^{80} -882.559 q^{81} +201.261 q^{82} -26.1539 q^{83} +1064.31 q^{85} +1.67434 q^{86} -1001.84 q^{87} -124.526 q^{88} +352.887 q^{89} -76.6438 q^{90} -99.6743 q^{92} -388.282 q^{93} +52.5950 q^{94} +873.246 q^{95} +760.056 q^{96} -847.585 q^{97} -89.5795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 - 8 * q^4 - 2 * q^5 + 26 * q^6 - 12 * q^8 + 44 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9} - 50 q^{10} - 22 q^{11} + 40 q^{12} - 80 q^{13} - 194 q^{15} - 8 q^{16} + 124 q^{17} + 92 q^{18} - 72 q^{19} - 88 q^{20} - 22 q^{22} - 98 q^{23} - 252 q^{24} + 136 q^{25} + 40 q^{26} + 182 q^{27} + 144 q^{29} - 266 q^{30} + 34 q^{31} - 104 q^{32} - 22 q^{33} + 52 q^{34} - 80 q^{36} + 54 q^{37} - 432 q^{38} + 400 q^{39} + 492 q^{40} - 536 q^{41} - 60 q^{43} + 88 q^{44} - 428 q^{45} - 314 q^{46} + 272 q^{47} - 776 q^{48} + 232 q^{50} - 164 q^{51} + 560 q^{52} - 492 q^{53} + 110 q^{54} + 22 q^{55} - 1512 q^{57} - 192 q^{58} - 634 q^{59} + 632 q^{60} - 840 q^{61} - 134 q^{62} + 224 q^{64} - 880 q^{65} - 286 q^{66} + 754 q^{67} - 640 q^{68} - 962 q^{69} - 678 q^{71} - 744 q^{72} + 400 q^{73} + 6 q^{74} + 520 q^{75} - 432 q^{76} - 440 q^{78} + 316 q^{79} + 1544 q^{80} - 1294 q^{81} - 512 q^{82} - 468 q^{83} + 452 q^{85} - 156 q^{86} - 1200 q^{87} + 132 q^{88} + 1842 q^{89} - 1532 q^{90} - 40 q^{92} - 638 q^{93} + 992 q^{94} + 2952 q^{95} + 952 q^{96} - 2194 q^{97} - 484 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 - 8 * q^4 - 2 * q^5 + 26 * q^6 - 12 * q^8 + 44 * q^9 - 50 * q^10 - 22 * q^11 + 40 * q^12 - 80 * q^13 - 194 * q^15 - 8 * q^16 + 124 * q^17 + 92 * q^18 - 72 * q^19 - 88 * q^20 - 22 * q^22 - 98 * q^23 - 252 * q^24 + 136 * q^25 + 40 * q^26 + 182 * q^27 + 144 * q^29 - 266 * q^30 + 34 * q^31 - 104 * q^32 - 22 * q^33 + 52 * q^34 - 80 * q^36 + 54 * q^37 - 432 * q^38 + 400 * q^39 + 492 * q^40 - 536 * q^41 - 60 * q^43 + 88 * q^44 - 428 * q^45 - 314 * q^46 + 272 * q^47 - 776 * q^48 + 232 * q^50 - 164 * q^51 + 560 * q^52 - 492 * q^53 + 110 * q^54 + 22 * q^55 - 1512 * q^57 - 192 * q^58 - 634 * q^59 + 632 * q^60 - 840 * q^61 - 134 * q^62 + 224 * q^64 - 880 * q^65 - 286 * q^66 + 754 * q^67 - 640 * q^68 - 962 * q^69 - 678 * q^71 - 744 * q^72 + 400 * q^73 + 6 * q^74 + 520 * q^75 - 432 * q^76 - 440 * q^78 + 316 * q^79 + 1544 * q^80 - 1294 * q^81 - 512 * q^82 - 468 * q^83 + 452 * q^85 - 156 * q^86 - 1200 * q^87 + 132 * q^88 + 1842 * q^89 - 1532 * q^90 - 40 * q^92 - 638 * q^93 + 992 * q^94 + 2952 * q^95 + 952 * q^96 - 2194 * q^97 - 484 * q^99

Introduction

L-functions

Modular forms

Varieties

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