LMFDB - Embedded newform 1521.4.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 25x^{2} + 24x + 78 $ x^4 - 2x^3 - 25x^2 + 24*x + 78
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.46610$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46610 q^{2} -5.85055 q^{4} -9.85055 q^{5} +29.9396 q^{7} -20.3063 q^{8} +O(q^{10})$	$q+1.46610 q^{2} -5.85055 q^{4} -9.85055 q^{5} +29.9396 q^{7} -20.3063 q^{8} -14.4419 q^{10} -46.9257 q^{11} +43.8945 q^{14} +17.0333 q^{16} +48.2616 q^{17} +120.300 q^{19} +57.6311 q^{20} -68.7979 q^{22} -130.697 q^{23} -27.9667 q^{25} -175.163 q^{28} +194.946 q^{29} -32.0123 q^{31} +187.423 q^{32} +70.7565 q^{34} -294.921 q^{35} -32.4250 q^{37} +176.372 q^{38} +200.028 q^{40} +241.825 q^{41} +96.4087 q^{43} +274.541 q^{44} -191.615 q^{46} -539.015 q^{47} +553.380 q^{49} -41.0021 q^{50} +152.277 q^{53} +462.244 q^{55} -607.963 q^{56} +285.810 q^{58} -327.792 q^{59} -98.4180 q^{61} -46.9332 q^{62} +138.515 q^{64} -441.151 q^{67} -282.357 q^{68} -432.385 q^{70} -345.049 q^{71} +773.839 q^{73} -47.5383 q^{74} -703.822 q^{76} -1404.94 q^{77} -150.332 q^{79} -167.787 q^{80} +354.540 q^{82} -337.966 q^{83} -475.403 q^{85} +141.345 q^{86} +952.889 q^{88} -169.913 q^{89} +764.649 q^{92} -790.250 q^{94} -1185.02 q^{95} +214.201 q^{97} +811.311 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8	$ 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29} + 26 q^{31} - 654 q^{32} + 1062 q^{34} - 88 q^{35} + 102 q^{37} - 332 q^{38} - 998 q^{40} + 1054 q^{41} + 450 q^{43} + 44 q^{44} - 172 q^{46} + 96 q^{47} + 1070 q^{49} - 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} + 722 q^{58} - 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} - 1134 q^{67} - 1786 q^{68} - 2324 q^{70} - 1064 q^{71} + 952 q^{73} - 1158 q^{74} - 1708 q^{76} - 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 1734 q^{82} + 404 q^{83} - 1394 q^{85} - 3168 q^{86} + 3060 q^{88} - 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} + 2166 q^{97} + 1906 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 - 62 * q^10 - 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 + 124 * q^19 + 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 - 144 * q^28 - 194 * q^29 + 26 * q^31 - 654 * q^32 + 1062 * q^34 - 88 * q^35 + 102 * q^37 - 332 * q^38 - 998 * q^40 + 1054 * q^41 + 450 * q^43 + 44 * q^44 - 172 * q^46 + 96 * q^47 + 1070 * q^49 - 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 + 722 * q^58 - 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 - 1134 * q^67 - 1786 * q^68 - 2324 * q^70 - 1064 * q^71 + 952 * q^73 - 1158 * q^74 - 1708 * q^76 - 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 1734 * q^82 + 404 * q^83 - 1394 * q^85 - 3168 * q^86 + 3060 * q^88 - 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 + 2166 * q^97 + 1906 * q^98

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 $

$2$	1.46610	0.518345	0.259173	−	0.965831i	$-0.416550\pi$
$2$	1.46610	0.518345	0.259173	+	0.965831i	$0.416550\pi$
$3$	0	0
$3$	0	0
$4$	−5.85055	−0.731318
$5$	−9.85055	−0.881060	−0.440530	−	0.897738i	$-0.645209\pi$
$5$	−9.85055	−0.881060	−0.440530	+	0.897738i	$0.645209\pi$
$6$	0	0
$7$	29.9396	1.61659	0.808293	−	0.588780i	$-0.200392\pi$
$7$	29.9396	1.61659	0.808293	+	0.588780i	$0.200392\pi$
$8$	−20.3063	−0.897421
$9$	0	0
$10$	−14.4419	−0.456693
$11$	−46.9257	−1.28624	−0.643120	−	0.765765i	$-0.722360\pi$
$11$	−46.9257	−1.28624	−0.643120	+	0.765765i	$0.722360\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	43.8945	0.837950
$15$	0	0
$16$	17.0333	0.266145
$17$	48.2616	0.688539	0.344270	−	0.938871i	$-0.388127\pi$
$17$	48.2616	0.688539	0.344270	+	0.938871i	$0.388127\pi$
$18$	0	0
$19$	120.300	1.45257	0.726283	−	0.687396i	$-0.241246\pi$
$19$	120.300	1.45257	0.726283	+	0.687396i	$0.241246\pi$
$20$	57.6311	0.644335
$21$	0	0
$22$	−68.7979	−0.666717
$23$	−130.697	−1.18488	−0.592440	−	0.805615i	$-0.701835\pi$
$23$	−130.697	−1.18488	−0.592440	+	0.805615i	$0.701835\pi$
$24$	0	0
$25$	−27.9667	−0.223734
$26$	0	0
$27$	0	0
$28$	−175.163	−1.18224
$29$	194.946	1.24829	0.624147	−	0.781307i	$-0.285447\pi$
$29$	194.946	1.24829	0.624147	+	0.781307i	$0.285447\pi$
$30$	0	0
$31$	−32.0123	−0.185470	−0.0927351	−	0.995691i	$-0.529561\pi$
$31$	−32.0123	−0.185470	−0.0927351	+	0.995691i	$0.529561\pi$
$32$	187.423	1.03538
$33$	0	0
$34$	70.7565	0.356901
$35$	−294.921	−1.42431
$36$	0	0
$37$	−32.4250	−0.144071	−0.0720355	−	0.997402i	$-0.522949\pi$
$37$	−32.4250	−0.144071	−0.0720355	+	0.997402i	$0.522949\pi$
$38$	176.372	0.752931
$39$	0	0
$40$	200.028	0.790681
$41$	241.825	0.921140	0.460570	−	0.887623i	$-0.347645\pi$
$41$	241.825	0.921140	0.460570	+	0.887623i	$0.347645\pi$
$42$	0	0
$43$	96.4087	0.341911	0.170956	−	0.985279i	$-0.445314\pi$
$43$	96.4087	0.341911	0.170956	+	0.985279i	$0.445314\pi$
$44$	274.541	0.940651
$45$	0	0
$46$	−191.615	−0.614177
$47$	−539.015	−1.67284	−0.836419	−	0.548090i	$-0.815355\pi$
$47$	−539.015	−1.67284	−0.836419	+	0.548090i	$0.815355\pi$
$48$	0	0
$49$	553.380	1.61335
$50$	−41.0021	−0.115971
$51$	0	0
$52$	0	0
$53$	152.277	0.394657	0.197328	−	0.980337i	$-0.436774\pi$
$53$	152.277	0.394657	0.197328	+	0.980337i	$0.436774\pi$
$54$	0	0
$55$	462.244	1.13325
$56$	−607.963	−1.45076
$57$	0	0
$58$	285.810	0.647047
$59$	−327.792	−0.723304	−0.361652	−	0.932313i	$-0.617787\pi$
$59$	−327.792	−0.723304	−0.361652	+	0.932313i	$0.617787\pi$
$60$	0	0
$61$	−98.4180	−0.206576	−0.103288	−	0.994651i	$-0.532936\pi$
$61$	−98.4180	−0.206576	−0.103288	+	0.994651i	$0.532936\pi$
$62$	−46.9332	−0.0961376
$63$	0	0
$64$	138.515	0.270537
$65$	0	0
$66$	0	0
$67$	−441.151	−0.804405	−0.402202	−	0.915551i	$-0.631755\pi$
$67$	−441.151	−0.804405	−0.402202	+	0.915551i	$0.631755\pi$
$68$	−282.357	−0.503541
$69$	0	0
$70$	−432.385	−0.738284
$71$	−345.049	−0.576757	−0.288379	−	0.957516i	$-0.593116\pi$
$71$	−345.049	−0.576757	−0.288379	+	0.957516i	$0.593116\pi$
$72$	0	0
$73$	773.839	1.24070	0.620349	−	0.784326i	$-0.286991\pi$
$73$	773.839	1.24070	0.620349	+	0.784326i	$0.286991\pi$
$74$	−47.5383	−0.0746785
$75$	0	0
$76$	−703.822	−1.06229
$77$	−1404.94	−2.07932
$78$	0	0
$79$	−150.332	−0.214097	−0.107049	−	0.994254i	$-0.534140\pi$
$79$	−150.332	−0.214097	−0.107049	+	0.994254i	$0.534140\pi$
$80$	−167.787	−0.234489
$81$	0	0
$82$	354.540	0.477469
$83$	−337.966	−0.446947	−0.223473	−	0.974710i	$-0.571740\pi$
$83$	−337.966	−0.446947	−0.223473	+	0.974710i	$0.571740\pi$
$84$	0	0
$85$	−475.403	−0.606644
$86$	141.345	0.177228
$87$	0	0
$88$	952.889	1.15430
$89$	−169.913	−0.202368	−0.101184	−	0.994868i	$-0.532263\pi$
$89$	−169.913	−0.202368	−0.101184	+	0.994868i	$0.532263\pi$
$90$	0	0
$91$	0	0
$92$	764.649	0.866524
$93$	0	0
$94$	−790.250	−0.867108
$95$	−1185.02	−1.27980
$96$	0	0
$97$	214.201	0.224215	0.112107	−	0.993696i	$-0.464240\pi$
$97$	214.201	0.224215	0.112107	+	0.993696i	$0.464240\pi$
$98$	811.311	0.836273
$99$	0	0
$100$	163.621	0.163621
$101$	−1595.11	−1.57148	−0.785741	−	0.618556i	$-0.787718\pi$
$101$	−1595.11	−1.57148	−0.785741	+	0.618556i	$0.787718\pi$
$102$	0	0
$103$	−1570.30	−1.50219	−0.751096	−	0.660193i	$-0.770475\pi$
$103$	−1570.30	−1.50219	−0.751096	+	0.660193i	$0.770475\pi$
$104$	0	0
$105$	0	0
$106$	223.253	0.204568
$107$	6.73302	0.00608323	0.00304161	−	0.999995i	$-0.499032\pi$
$107$	6.73302	0.00608323	0.00304161	+	0.999995i	$0.499032\pi$
$108$	0	0
$109$	−542.422	−0.476648	−0.238324	−	0.971186i	$-0.576598\pi$
$109$	−542.422	−0.476648	−0.238324	+	0.971186i	$0.576598\pi$
$110$	677.697	0.587417
$111$	0	0
$112$	509.969	0.430246
$113$	1442.82	1.20114	0.600572	−	0.799571i	$-0.294940\pi$
$113$	1442.82	1.20114	0.600572	+	0.799571i	$0.294940\pi$
$114$	0	0
$115$	1287.44	1.04395
$116$	−1140.54	−0.912900
$117$	0	0
$118$	−480.577	−0.374921
$119$	1444.93	1.11308
$120$	0	0
$121$	871.025	0.654414
$122$	−144.291	−0.107078
$123$	0	0
$124$	187.289	0.135638
$125$	1506.81	1.07818
$126$	0	0
$127$	−2492.84	−1.74176	−0.870881	−	0.491494i	$-0.836451\pi$
$127$	−2492.84	−1.74176	−0.870881	+	0.491494i	$0.836451\pi$
$128$	−1296.31	−0.895144
$129$	0	0
$130$	0	0
$131$	−744.561	−0.496585	−0.248292	−	0.968685i	$-0.579869\pi$
$131$	−744.561	−0.496585	−0.248292	+	0.968685i	$0.579869\pi$
$132$	0	0
$133$	3601.74	2.34820
$134$	−646.772	−0.416959
$135$	0	0
$136$	−980.016	−0.617909
$137$	222.167	0.138547	0.0692736	−	0.997598i	$-0.477932\pi$
$137$	222.167	0.138547	0.0692736	+	0.997598i	$0.477932\pi$
$138$	0	0
$139$	−777.145	−0.474220	−0.237110	−	0.971483i	$-0.576200\pi$
$139$	−777.145	−0.474220	−0.237110	+	0.971483i	$0.576200\pi$
$140$	1725.45	1.04162
$141$	0	0
$142$	−505.876	−0.298959
$143$	0	0
$144$	0	0
$145$	−1920.32	−1.09982
$146$	1134.53	0.643110
$147$	0	0
$148$	189.704	0.105362
$149$	−1778.29	−0.977742	−0.488871	−	0.872356i	$-0.662591\pi$
$149$	−1778.29	−0.977742	−0.488871	+	0.872356i	$0.662591\pi$
$150$	0	0
$151$	1166.00	0.628394	0.314197	−	0.949358i	$-0.398265\pi$
$151$	1166.00	0.628394	0.314197	+	0.949358i	$0.398265\pi$
$152$	−2442.85	−1.30356
$153$	0	0
$154$	−2059.78	−1.07780
$155$	315.338	0.163410
$156$	0	0
$157$	517.628	0.263129	0.131564	−	0.991308i	$-0.458000\pi$
$157$	517.628	0.263129	0.131564	+	0.991308i	$0.458000\pi$
$158$	−220.402	−0.110976
$159$	0	0
$160$	−1846.22	−0.912227
$161$	−3913.02	−1.91546
$162$	0	0
$163$	610.188	0.293212	0.146606	−	0.989195i	$-0.453165\pi$
$163$	610.188	0.293212	0.146606	+	0.989195i	$0.453165\pi$
$164$	−1414.81	−0.673647
$165$	0	0
$166$	−495.493	−0.231673
$167$	−2983.01	−1.38223	−0.691115	−	0.722745i	$-0.742880\pi$
$167$	−2983.01	−1.38223	−0.691115	+	0.722745i	$0.742880\pi$
$168$	0	0
$169$	0	0
$170$	−696.990	−0.314451
$171$	0	0
$172$	−564.044	−0.250046
$173$	−978.212	−0.429896	−0.214948	−	0.976625i	$-0.568958\pi$
$173$	−978.212	−0.429896	−0.214948	+	0.976625i	$0.568958\pi$
$174$	0	0
$175$	−837.313	−0.361685
$176$	−799.298	−0.342326
$177$	0	0
$178$	−249.110	−0.104897
$179$	−1852.94	−0.773717	−0.386858	−	0.922139i	$-0.626440\pi$
$179$	−1852.94	−0.773717	−0.386858	+	0.922139i	$0.626440\pi$
$180$	0	0
$181$	852.777	0.350201	0.175101	−	0.984551i	$-0.443975\pi$
$181$	852.777	0.350201	0.175101	+	0.984551i	$0.443975\pi$
$182$	0	0
$183$	0	0
$184$	2653.98	1.06334
$185$	319.403	0.126935
$186$	0	0
$187$	−2264.71	−0.885627
$188$	3153.53	1.22338
$189$	0	0
$190$	−1737.36	−0.663377
$191$	−4441.29	−1.68252	−0.841258	−	0.540633i	$-0.818185\pi$
$191$	−4441.29	−1.68252	−0.841258	+	0.540633i	$0.818185\pi$
$192$	0	0
$193$	−2482.43	−0.925851	−0.462925	−	0.886397i	$-0.653200\pi$
$193$	−2482.43	−0.925851	−0.462925	+	0.886397i	$0.653200\pi$
$194$	314.041	0.116221
$195$	0	0
$196$	−3237.57	−1.17987
$197$	−1260.23	−0.455775	−0.227888	−	0.973687i	$-0.573182\pi$
$197$	−1260.23	−0.455775	−0.227888	+	0.973687i	$0.573182\pi$
$198$	0	0
$199$	−5520.96	−1.96669	−0.983344	−	0.181756i	$-0.941822\pi$
$199$	−5520.96	−1.96669	−0.983344	+	0.181756i	$0.941822\pi$
$200$	567.901	0.200783
$201$	0	0
$202$	−2338.60	−0.814570
$203$	5836.60	2.01798
$204$	0	0
$205$	−2382.11	−0.811580
$206$	−2302.21	−0.778654
$207$	0	0
$208$	0	0
$209$	−5645.18	−1.86835
$210$	0	0
$211$	−4527.79	−1.47728	−0.738640	−	0.674100i	$-0.764532\pi$
$211$	−4527.79	−1.47728	−0.738640	+	0.674100i	$0.764532\pi$
$212$	−890.901	−0.288620
$213$	0	0
$214$	9.87129	0.00315321
$215$	−949.679	−0.301244
$216$	0	0
$217$	−958.435	−0.299829
$218$	−795.245	−0.247068
$219$	0	0
$220$	−2704.38	−0.828770
$221$	0	0
$222$	0	0
$223$	4481.17	1.34566	0.672829	−	0.739798i	$-0.265079\pi$
$223$	4481.17	1.34566	0.672829	+	0.739798i	$0.265079\pi$
$224$	5611.37	1.67377
$225$	0	0
$226$	2115.32	0.622607
$227$	5759.75	1.68409	0.842044	−	0.539408i	$-0.181352\pi$
$227$	5759.75	1.68409	0.842044	+	0.539408i	$0.181352\pi$
$228$	0	0
$229$	−4635.08	−1.33753	−0.668766	−	0.743473i	$-0.733177\pi$
$229$	−4635.08	−1.33753	−0.668766	+	0.743473i	$0.733177\pi$
$230$	1887.51	0.541126
$231$	0	0
$232$	−3958.63	−1.12024
$233$	−5886.33	−1.65505	−0.827524	−	0.561431i	$-0.810251\pi$
$233$	−5886.33	−1.65505	−0.827524	+	0.561431i	$0.810251\pi$
$234$	0	0
$235$	5309.59	1.47387
$236$	1917.76	0.528965
$237$	0	0
$238$	2118.42	0.576961
$239$	−2135.84	−0.578060	−0.289030	−	0.957320i	$-0.593333\pi$
$239$	−2135.84	−0.578060	−0.289030	+	0.957320i	$0.593333\pi$
$240$	0	0
$241$	4692.92	1.25435	0.627173	−	0.778880i	$-0.284212\pi$
$241$	4692.92	1.25435	0.627173	+	0.778880i	$0.284212\pi$
$242$	1277.01	0.339212
$243$	0	0
$244$	575.799	0.151073
$245$	−5451.09	−1.42146
$246$	0	0
$247$	0	0
$248$	650.051	0.166445
$249$	0	0
$250$	2209.13	0.558871
$251$	3902.88	0.981464	0.490732	−	0.871310i	$-0.336729\pi$
$251$	3902.88	0.981464	0.490732	+	0.871310i	$0.336729\pi$
$252$	0	0
$253$	6133.06	1.52404
$254$	−3654.76	−0.902834
$255$	0	0
$256$	−3008.64	−0.734531
$257$	4130.83	1.00262	0.501312	−	0.865267i	$-0.332851\pi$
$257$	4130.83	1.00262	0.501312	+	0.865267i	$0.332851\pi$
$258$	0	0
$259$	−970.790	−0.232903
$260$	0	0
$261$	0	0
$262$	−1091.60	−0.257402
$263$	−6352.17	−1.48932	−0.744661	−	0.667443i	$-0.767389\pi$
$263$	−6352.17	−1.48932	−0.744661	+	0.667443i	$0.767389\pi$
$264$	0	0
$265$	−1500.01	−0.347716
$266$	5280.52	1.21718
$267$	0	0
$268$	2580.97	0.588276
$269$	−181.524	−0.0411439	−0.0205719	−	0.999788i	$-0.506549\pi$
$269$	−181.524	−0.0411439	−0.0205719	+	0.999788i	$0.506549\pi$
$270$	0	0
$271$	3460.35	0.775651	0.387825	−	0.921733i	$-0.373226\pi$
$271$	3460.35	0.775651	0.387825	+	0.921733i	$0.373226\pi$
$272$	822.053	0.183251
$273$	0	0
$274$	325.719	0.0718153
$275$	1312.36	0.287776
$276$	0	0
$277$	−6437.94	−1.39646	−0.698228	−	0.715876i	$-0.746028\pi$
$277$	−6437.94	−1.39646	−0.698228	+	0.715876i	$0.746028\pi$
$278$	−1139.37	−0.245810
$279$	0	0
$280$	5988.77	1.27820
$281$	2974.26	0.631421	0.315711	−	0.948855i	$-0.397757\pi$
$281$	2974.26	0.631421	0.315711	+	0.948855i	$0.397757\pi$
$282$	0	0
$283$	3035.72	0.637649	0.318825	−	0.947814i	$-0.396712\pi$
$283$	3035.72	0.637649	0.318825	+	0.947814i	$0.396712\pi$
$284$	2018.72	0.421793
$285$	0	0
$286$	0	0
$287$	7240.15	1.48910
$288$	0	0
$289$	−2583.81	−0.525914
$290$	−2815.39	−0.570087
$291$	0	0
$292$	−4527.38	−0.907346
$293$	1955.74	0.389952	0.194976	−	0.980808i	$-0.437537\pi$
$293$	1955.74	0.389952	0.194976	+	0.980808i	$0.437537\pi$
$294$	0	0
$295$	3228.93	0.637274
$296$	658.431	0.129292
$297$	0	0
$298$	−2607.16	−0.506808
$299$	0	0
$300$	0	0
$301$	2886.44	0.552729
$302$	1709.47	0.325725
$303$	0	0
$304$	2049.10	0.386593
$305$	969.471	0.182006
$306$	0	0
$307$	−1027.56	−0.191029	−0.0955147	−	0.995428i	$-0.530450\pi$
$307$	−1027.56	−0.191029	−0.0955147	+	0.995428i	$0.530450\pi$
$308$	8219.66	1.52064
$309$	0	0
$310$	462.318	0.0847029
$311$	−3405.61	−0.620947	−0.310474	−	0.950582i	$-0.600488\pi$
$311$	−3405.61	−0.620947	−0.310474	+	0.950582i	$0.600488\pi$
$312$	0	0
$313$	4813.20	0.869196	0.434598	−	0.900625i	$-0.356890\pi$
$313$	4813.20	0.869196	0.434598	+	0.900625i	$0.356890\pi$
$314$	758.895	0.136391
$315$	0	0
$316$	879.525	0.156573
$317$	−1141.33	−0.202219	−0.101110	−	0.994875i	$-0.532239\pi$
$317$	−1141.33	−0.202219	−0.101110	+	0.994875i	$0.532239\pi$
$318$	0	0
$319$	−9147.98	−1.60561
$320$	−1364.45	−0.238359
$321$	0	0
$322$	−5736.88	−0.992870
$323$	5805.88	1.00015
$324$	0	0
$325$	0	0
$326$	894.597	0.151985
$327$	0	0
$328$	−4910.58	−0.826650
$329$	−16137.9	−2.70429
$330$	0	0
$331$	7652.57	1.27076	0.635382	−	0.772198i	$-0.280843\pi$
$331$	7652.57	1.27076	0.635382	+	0.772198i	$0.280843\pi$
$332$	1977.29	0.326860
$333$	0	0
$334$	−4373.40	−0.716472
$335$	4345.57	0.708729
$336$	0	0
$337$	−2503.69	−0.404702	−0.202351	−	0.979313i	$-0.564858\pi$
$337$	−2503.69	−0.404702	−0.202351	+	0.979313i	$0.564858\pi$
$338$	0	0
$339$	0	0
$340$	2781.37	0.443650
$341$	1502.20	0.238559
$342$	0	0
$343$	6298.69	0.991537
$344$	−1957.71	−0.306838
$345$	0	0
$346$	−1434.16	−0.222835
$347$	−4496.12	−0.695574	−0.347787	−	0.937574i	$-0.613067\pi$
$347$	−4496.12	−0.695574	−0.347787	+	0.937574i	$0.613067\pi$
$348$	0	0
$349$	3577.61	0.548726	0.274363	−	0.961626i	$-0.411533\pi$
$349$	3577.61	0.548726	0.274363	+	0.961626i	$0.411533\pi$
$350$	−1227.59	−0.187478
$351$	0	0
$352$	−8794.96	−1.33174
$353$	8045.17	1.21303	0.606517	−	0.795070i	$-0.292566\pi$
$353$	8045.17	1.21303	0.606517	+	0.795070i	$0.292566\pi$
$354$	0	0
$355$	3398.92	0.508157
$356$	994.086	0.147996
$357$	0	0
$358$	−2716.60	−0.401052
$359$	2172.90	0.319447	0.159724	−	0.987162i	$-0.448940\pi$
$359$	2172.90	0.319447	0.159724	+	0.987162i	$0.448940\pi$
$360$	0	0
$361$	7613.14	1.10995
$362$	1250.26	0.181525
$363$	0	0
$364$	0	0
$365$	−7622.74	−1.09313
$366$	0	0
$367$	7662.76	1.08990	0.544949	−	0.838469i	$-0.316549\pi$
$367$	7662.76	1.08990	0.544949	+	0.838469i	$0.316549\pi$
$368$	−2226.20	−0.315349
$369$	0	0
$370$	468.278	0.0657962
$371$	4559.10	0.637996
$372$	0	0
$373$	10542.5	1.46346	0.731732	−	0.681593i	$-0.238712\pi$
$373$	10542.5	1.46346	0.731732	+	0.681593i	$0.238712\pi$
$374$	−3320.30	−0.459060
$375$	0	0
$376$	10945.4	1.50124
$377$	0	0
$378$	0	0
$379$	5475.54	0.742110	0.371055	−	0.928611i	$-0.378996\pi$
$379$	5475.54	0.742110	0.371055	+	0.928611i	$0.378996\pi$
$380$	6933.03	0.935939
$381$	0	0
$382$	−6511.39	−0.872124
$383$	−808.085	−0.107810	−0.0539050	−	0.998546i	$-0.517167\pi$
$383$	−808.085	−0.107810	−0.0539050	+	0.998546i	$0.517167\pi$
$384$	0	0
$385$	13839.4	1.83200
$386$	−3639.49	−0.479910
$387$	0	0
$388$	−1253.19	−0.163972
$389$	7060.26	0.920230	0.460115	−	0.887859i	$-0.347808\pi$
$389$	7060.26	0.920230	0.460115	+	0.887859i	$0.347808\pi$
$390$	0	0
$391$	−6307.65	−0.815836
$392$	−11237.1	−1.44786
$393$	0	0
$394$	−1847.63	−0.236249
$395$	1480.85	0.188633
$396$	0	0
$397$	−1419.89	−0.179502	−0.0897511	−	0.995964i	$-0.528607\pi$
$397$	−1419.89	−0.179502	−0.0897511	+	0.995964i	$0.528607\pi$
$398$	−8094.29	−1.01942
$399$	0	0
$400$	−476.365	−0.0595456
$401$	−10670.7	−1.32886	−0.664428	−	0.747353i	$-0.731325\pi$
$401$	−10670.7	−1.32886	−0.664428	+	0.747353i	$0.731325\pi$
$402$	0	0
$403$	0	0
$404$	9332.28	1.14925
$405$	0	0
$406$	8557.05	1.04601
$407$	1521.56	0.185310
$408$	0	0
$409$	6351.56	0.767883	0.383942	−	0.923357i	$-0.374566\pi$
$409$	6351.56	0.767883	0.383942	+	0.923357i	$0.374566\pi$
$410$	−3492.42	−0.420678
$411$	0	0
$412$	9187.09	1.09858
$413$	−9813.97	−1.16928
$414$	0	0
$415$	3329.15	0.393787
$416$	0	0
$417$	0	0
$418$	−8276.40	−0.968450
$419$	−5617.87	−0.655015	−0.327507	−	0.944849i	$-0.606209\pi$
$419$	−5617.87	−0.655015	−0.327507	+	0.944849i	$0.606209\pi$
$420$	0	0
$421$	−1518.29	−0.175765	−0.0878825	−	0.996131i	$-0.528010\pi$
$421$	−1518.29	−0.175765	−0.0878825	+	0.996131i	$0.528010\pi$
$422$	−6638.20	−0.765741
$423$	0	0
$424$	−3092.17	−0.354173
$425$	−1349.72	−0.154050
$426$	0	0
$427$	−2946.60	−0.333948
$428$	−39.3918	−0.00444878
$429$	0	0
$430$	−1392.33	−0.156149
$431$	4970.13	0.555459	0.277730	−	0.960659i	$-0.410418\pi$
$431$	4970.13	0.555459	0.277730	+	0.960659i	$0.410418\pi$
$432$	0	0
$433$	−3298.71	−0.366111	−0.183055	−	0.983103i	$-0.558599\pi$
$433$	−3298.71	−0.366111	−0.183055	+	0.983103i	$0.558599\pi$
$434$	−1405.16	−0.155415
$435$	0	0
$436$	3173.46	0.348581
$437$	−15722.9	−1.72112
$438$	0	0
$439$	−6048.47	−0.657581	−0.328790	−	0.944403i	$-0.606641\pi$
$439$	−6048.47	−0.657581	−0.328790	+	0.944403i	$0.606641\pi$
$440$	−9386.47	−1.01701
$441$	0	0
$442$	0	0
$443$	−6822.62	−0.731722	−0.365861	−	0.930670i	$-0.619225\pi$
$443$	−6822.62	−0.731722	−0.365861	+	0.930670i	$0.619225\pi$
$444$	0	0
$445$	1673.74	0.178299
$446$	6569.86	0.697515
$447$	0	0
$448$	4147.09	0.437347
$449$	−410.410	−0.0431369	−0.0215684	−	0.999767i	$-0.506866\pi$
$449$	−410.410	−0.0431369	−0.0215684	+	0.999767i	$0.506866\pi$
$450$	0	0
$451$	−11347.8	−1.18481
$452$	−8441.30	−0.878419
$453$	0	0
$454$	8444.38	0.872939
$455$	0	0
$456$	0	0
$457$	−16642.8	−1.70354	−0.851771	−	0.523915i	$-0.824471\pi$
$457$	−16642.8	−1.70354	−0.851771	+	0.523915i	$0.824471\pi$
$458$	−6795.50	−0.693303
$459$	0	0
$460$	−7532.21	−0.763459
$461$	11729.7	1.18505	0.592523	−	0.805553i	$-0.298132\pi$
$461$	11729.7	1.18505	0.592523	+	0.805553i	$0.298132\pi$
$462$	0	0
$463$	−3564.93	−0.357832	−0.178916	−	0.983864i	$-0.557259\pi$
$463$	−3564.93	−0.357832	−0.178916	+	0.983864i	$0.557259\pi$
$464$	3320.56	0.332227
$465$	0	0
$466$	−8629.95	−0.857886
$467$	−1134.81	−0.112447	−0.0562233	−	0.998418i	$-0.517906\pi$
$467$	−1134.81	−0.112447	−0.0562233	+	0.998418i	$0.517906\pi$
$468$	0	0
$469$	−13207.9	−1.30039
$470$	7784.40	0.763974
$471$	0	0
$472$	6656.25	0.649107
$473$	−4524.05	−0.439780
$474$	0	0
$475$	−3364.40	−0.324988
$476$	−8453.65	−0.814018
$477$	0	0
$478$	−3131.36	−0.299635
$479$	19372.6	1.84793	0.923963	−	0.382481i	$-0.124930\pi$
$479$	19372.6	1.84793	0.923963	+	0.382481i	$0.124930\pi$
$480$	0	0
$481$	0	0
$482$	6880.29	0.650184
$483$	0	0
$484$	−5095.97	−0.478585
$485$	−2110.00	−0.197547
$486$	0	0
$487$	−9045.95	−0.841707	−0.420853	−	0.907129i	$-0.638269\pi$
$487$	−9045.95	−0.841707	−0.420853	+	0.907129i	$0.638269\pi$
$488$	1998.51	0.185385
$489$	0	0
$490$	−7991.86	−0.736807
$491$	−14403.3	−1.32385	−0.661927	−	0.749568i	$-0.730261\pi$
$491$	−14403.3	−1.32385	−0.661927	+	0.749568i	$0.730261\pi$
$492$	0	0
$493$	9408.40	0.859499
$494$	0	0
$495$	0	0
$496$	−545.273	−0.0493619
$497$	−10330.6	−0.932378
$498$	0	0
$499$	−9319.75	−0.836091	−0.418045	−	0.908426i	$-0.637285\pi$
$499$	−9319.75	−0.836091	−0.418045	+	0.908426i	$0.637285\pi$
$500$	−8815.64	−0.788495
$501$	0	0
$502$	5722.02	0.508737
$503$	−2745.98	−0.243414	−0.121707	−	0.992566i	$-0.538837\pi$
$503$	−2745.98	−0.243414	−0.121707	+	0.992566i	$0.538837\pi$
$504$	0	0
$505$	15712.7	1.38457
$506$	8991.69	0.789979
$507$	0	0
$508$	14584.5	1.27378
$509$	1182.11	0.102939	0.0514697	−	0.998675i	$-0.483609\pi$
$509$	1182.11	0.102939	0.0514697	+	0.998675i	$0.483609\pi$
$510$	0	0
$511$	23168.4	2.00570
$512$	5959.48	0.514403
$513$	0	0
$514$	6056.22	0.519705
$515$	15468.3	1.32352
$516$	0	0
$517$	25293.7	2.15167
$518$	−1423.28	−0.120724
$519$	0	0
$520$	0	0
$521$	−10858.8	−0.913115	−0.456558	−	0.889694i	$-0.650918\pi$
$521$	−10858.8	−0.913115	−0.456558	+	0.889694i	$0.650918\pi$
$522$	0	0
$523$	−10161.7	−0.849602	−0.424801	−	0.905287i	$-0.639656\pi$
$523$	−10161.7	−0.849602	−0.424801	+	0.905287i	$0.639656\pi$
$524$	4356.09	0.363162
$525$	0	0
$526$	−9312.93	−0.771983
$527$	−1544.96	−0.127703
$528$	0	0
$529$	4914.73	0.403939
$530$	−2199.16	−0.180237
$531$	0	0
$532$	−21072.1	−1.71728
$533$	0	0
$534$	0	0
$535$	−66.3239	−0.00535969
$536$	8958.14	0.721889
$537$	0	0
$538$	−266.132	−0.0213267
$539$	−25967.8	−2.07516
$540$	0	0
$541$	−9573.04	−0.760771	−0.380386	−	0.924828i	$-0.624209\pi$
$541$	−9573.04	−0.760771	−0.380386	+	0.924828i	$0.624209\pi$
$542$	5073.23	0.402055
$543$	0	0
$544$	9045.34	0.712896
$545$	5343.15	0.419955
$546$	0	0
$547$	15958.9	1.24745	0.623724	−	0.781645i	$-0.285619\pi$
$547$	15958.9	1.24745	0.623724	+	0.781645i	$0.285619\pi$
$548$	−1299.80	−0.101322
$549$	0	0
$550$	1924.05	0.149167
$551$	23452.0	1.81323
$552$	0	0
$553$	−4500.89	−0.346107
$554$	−9438.67	−0.723846
$555$	0	0
$556$	4546.72	0.346806
$557$	2145.09	0.163178	0.0815892	−	0.996666i	$-0.474000\pi$
$557$	2145.09	0.163178	0.0815892	+	0.996666i	$0.474000\pi$
$558$	0	0
$559$	0	0
$560$	−5023.47	−0.379072
$561$	0	0
$562$	4360.57	0.327294
$563$	22318.6	1.67072	0.835362	−	0.549701i	$-0.185258\pi$
$563$	22318.6	1.67072	0.835362	+	0.549701i	$0.185258\pi$
$564$	0	0
$565$	−14212.6	−1.05828
$566$	4450.67	0.330523
$567$	0	0
$568$	7006.67	0.517594
$569$	19753.0	1.45534	0.727671	−	0.685927i	$-0.240603\pi$
$569$	19753.0	1.45534	0.727671	+	0.685927i	$0.240603\pi$
$570$	0	0
$571$	−10640.6	−0.779850	−0.389925	−	0.920847i	$-0.627499\pi$
$571$	−10640.6	−0.779850	−0.389925	+	0.920847i	$0.627499\pi$
$572$	0	0
$573$	0	0
$574$	10614.8	0.771870
$575$	3655.17	0.265098
$576$	0	0
$577$	6547.89	0.472430	0.236215	−	0.971701i	$-0.424093\pi$
$577$	6547.89	0.472430	0.236215	+	0.971701i	$0.424093\pi$
$578$	−3788.13	−0.272605
$579$	0	0
$580$	11234.9	0.804319
$581$	−10118.6	−0.722529
$582$	0	0
$583$	−7145.69	−0.507623
$584$	−15713.8	−1.11343
$585$	0	0
$586$	2867.32	0.202130
$587$	−5900.33	−0.414877	−0.207439	−	0.978248i	$-0.566513\pi$
$587$	−5900.33	−0.414877	−0.207439	+	0.978248i	$0.566513\pi$
$588$	0	0
$589$	−3851.08	−0.269408
$590$	4733.94	0.330328
$591$	0	0
$592$	−552.303	−0.0383437
$593$	15261.5	1.05686	0.528428	−	0.848978i	$-0.322782\pi$
$593$	15261.5	1.05686	0.528428	+	0.848978i	$0.322782\pi$
$594$	0	0
$595$	−14233.4	−0.980693
$596$	10404.0	0.715041
$597$	0	0
$598$	0	0
$599$	18900.6	1.28925	0.644623	−	0.764501i	$-0.277014\pi$
$599$	18900.6	1.28925	0.644623	+	0.764501i	$0.277014\pi$
$600$	0	0
$601$	−18507.0	−1.25610	−0.628049	−	0.778174i	$-0.716146\pi$
$601$	−18507.0	−1.25610	−0.628049	+	0.778174i	$0.716146\pi$
$602$	4231.81	0.286505
$603$	0	0
$604$	−6821.72	−0.459556
$605$	−8580.08	−0.576578
$606$	0	0
$607$	−6408.50	−0.428522	−0.214261	−	0.976776i	$-0.568734\pi$
$607$	−6408.50	−0.428522	−0.214261	+	0.976776i	$0.568734\pi$
$608$	22547.0	1.50395
$609$	0	0
$610$	1421.34	0.0943418
$611$	0	0
$612$	0	0
$613$	−3507.26	−0.231088	−0.115544	−	0.993302i	$-0.536861\pi$
$613$	−3507.26	−0.231088	−0.115544	+	0.993302i	$0.536861\pi$
$614$	−1506.51	−0.0990192
$615$	0	0
$616$	28529.1	1.86602
$617$	−14507.8	−0.946614	−0.473307	−	0.880897i	$-0.656940\pi$
$617$	−14507.8	−0.946614	−0.473307	+	0.880897i	$0.656940\pi$
$618$	0	0
$619$	4750.85	0.308486	0.154243	−	0.988033i	$-0.450706\pi$
$619$	4750.85	0.308486	0.154243	+	0.988033i	$0.450706\pi$
$620$	−1844.90	−0.119505
$621$	0	0
$622$	−4992.98	−0.321865
$623$	−5087.14	−0.327146
$624$	0	0
$625$	−11347.0	−0.726209
$626$	7056.65	0.450544
$627$	0	0
$628$	−3028.40	−0.192431
$629$	−1564.88	−0.0991986
$630$	0	0
$631$	4825.19	0.304418	0.152209	−	0.988348i	$-0.451361\pi$
$631$	4825.19	0.304418	0.152209	+	0.988348i	$0.451361\pi$
$632$	3052.69	0.192135
$633$	0	0
$634$	−1673.30	−0.104819
$635$	24555.8	1.53460
$636$	0	0
$637$	0	0
$638$	−13411.9	−0.832258
$639$	0	0
$640$	12769.3	0.788675
$641$	5911.89	0.364284	0.182142	−	0.983272i	$-0.441697\pi$
$641$	5911.89	0.364284	0.182142	+	0.983272i	$0.441697\pi$
$642$	0	0
$643$	23408.3	1.43567	0.717833	−	0.696216i	$-0.245134\pi$
$643$	23408.3	1.43567	0.717833	+	0.696216i	$0.245134\pi$
$644$	22893.3	1.40081
$645$	0	0
$646$	8512.02	0.518422
$647$	16398.7	0.996442	0.498221	−	0.867050i	$-0.333987\pi$
$647$	16398.7	0.996442	0.498221	+	0.867050i	$0.333987\pi$
$648$	0	0
$649$	15381.9	0.930342
$650$	0	0
$651$	0	0
$652$	−3569.93	−0.214431
$653$	−27529.6	−1.64979	−0.824897	−	0.565283i	$-0.808767\pi$
$653$	−27529.6	−1.64979	−0.824897	+	0.565283i	$0.808767\pi$
$654$	0	0
$655$	7334.34	0.437521
$656$	4119.07	0.245157
$657$	0	0
$658$	−23659.8	−1.40175
$659$	24179.7	1.42930	0.714650	−	0.699482i	$-0.246586\pi$
$659$	24179.7	1.42930	0.714650	+	0.699482i	$0.246586\pi$
$660$	0	0
$661$	4525.04	0.266269	0.133134	−	0.991098i	$-0.457496\pi$
$661$	4525.04	0.266269	0.133134	+	0.991098i	$0.457496\pi$
$662$	11219.4	0.658695
$663$	0	0
$664$	6862.84	0.401099
$665$	−35479.1	−2.06890
$666$	0	0
$667$	−25478.8	−1.47908
$668$	17452.2	1.01085
$669$	0	0
$670$	6371.05	0.367366
$671$	4618.34	0.265706
$672$	0	0
$673$	3287.18	0.188279	0.0941393	−	0.995559i	$-0.469990\pi$
$673$	3287.18	0.188279	0.0941393	+	0.995559i	$0.469990\pi$
$674$	−3670.66	−0.209775
$675$	0	0
$676$	0	0
$677$	9724.21	0.552041	0.276020	−	0.961152i	$-0.410984\pi$
$677$	9724.21	0.552041	0.276020	+	0.961152i	$0.410984\pi$
$678$	0	0
$679$	6413.10	0.362463
$680$	9653.69	0.544415
$681$	0	0
$682$	2202.38	0.123656
$683$	−14548.7	−0.815065	−0.407532	−	0.913191i	$-0.633611\pi$
$683$	−14548.7	−0.815065	−0.407532	+	0.913191i	$0.633611\pi$
$684$	0	0
$685$	−2188.46	−0.122068
$686$	9234.52	0.513959
$687$	0	0
$688$	1642.15	0.0909979
$689$	0	0
$690$	0	0
$691$	6728.96	0.370451	0.185226	−	0.982696i	$-0.440698\pi$
$691$	6728.96	0.370451	0.185226	+	0.982696i	$0.440698\pi$
$692$	5723.07	0.314391
$693$	0	0
$694$	−6591.76	−0.360548
$695$	7655.31	0.417816
$696$	0	0
$697$	11670.9	0.634241
$698$	5245.15	0.284429
$699$	0	0
$700$	4898.74	0.264507
$701$	29159.8	1.57111	0.785557	−	0.618789i	$-0.212376\pi$
$701$	29159.8	1.57111	0.785557	+	0.618789i	$0.212376\pi$
$702$	0	0
$703$	−3900.73	−0.209273
$704$	−6499.92	−0.347976
$705$	0	0
$706$	11795.0	0.628770
$707$	−47757.0	−2.54044
$708$	0	0
$709$	−20489.0	−1.08531	−0.542653	−	0.839957i	$-0.682580\pi$
$709$	−20489.0	−1.08531	−0.542653	+	0.839957i	$0.682580\pi$
$710$	4983.16	0.263401
$711$	0	0
$712$	3450.31	0.181609
$713$	4183.91	0.219760
$714$	0	0
$715$	0	0
$716$	10840.7	0.565833
$717$	0	0
$718$	3185.70	0.165584
$719$	17990.1	0.933127	0.466563	−	0.884488i	$-0.345492\pi$
$719$	17990.1	0.933127	0.466563	+	0.884488i	$0.345492\pi$
$720$	0	0
$721$	−47014.0	−2.42842
$722$	11161.6	0.575337
$723$	0	0
$724$	−4989.21	−0.256108
$725$	−5452.00	−0.279286
$726$	0	0
$727$	−37652.7	−1.92086	−0.960428	−	0.278528i	$-0.910153\pi$
$727$	−37652.7	−1.92086	−0.960428	+	0.278528i	$0.910153\pi$
$728$	0	0
$729$	0	0
$730$	−11175.7	−0.566618
$731$	4652.84	0.235419
$732$	0	0
$733$	4524.26	0.227977	0.113989	−	0.993482i	$-0.463637\pi$
$733$	4524.26	0.227977	0.113989	+	0.993482i	$0.463637\pi$
$734$	11234.4	0.564944
$735$	0	0
$736$	−24495.6	−1.22679
$737$	20701.3	1.03466
$738$	0	0
$739$	818.302	0.0407331	0.0203665	−	0.999793i	$-0.493517\pi$
$739$	818.302	0.0407331	0.0203665	+	0.999793i	$0.493517\pi$
$740$	−1868.68	−0.0928300
$741$	0	0
$742$	6684.10	0.330702
$743$	−39002.2	−1.92578	−0.962888	−	0.269901i	$-0.913009\pi$
$743$	−39002.2	−1.92578	−0.962888	+	0.269901i	$0.913009\pi$
$744$	0	0
$745$	17517.2	0.861449
$746$	15456.4	0.758579
$747$	0	0
$748$	13249.8	0.647675
$749$	201.584	0.00983407
$750$	0	0
$751$	−22377.7	−1.08732	−0.543658	−	0.839307i	$-0.682961\pi$
$751$	−22377.7	−1.08732	−0.543658	+	0.839307i	$0.682961\pi$
$752$	−9181.18	−0.445217
$753$	0	0
$754$	0	0
$755$	−11485.7	−0.553653
$756$	0	0
$757$	−34512.6	−1.65704	−0.828521	−	0.559958i	$-0.810817\pi$
$757$	−34512.6	−1.65704	−0.828521	+	0.559958i	$0.810817\pi$
$758$	8027.70	0.384669
$759$	0	0
$760$	24063.4	1.14852
$761$	−19975.6	−0.951529	−0.475764	−	0.879573i	$-0.657828\pi$
$761$	−19975.6	−0.951529	−0.475764	+	0.879573i	$0.657828\pi$
$762$	0	0
$763$	−16239.9	−0.770542
$764$	25984.0	1.23046
$765$	0	0
$766$	−1184.73	−0.0558827
$767$	0	0
$768$	0	0
$769$	−33064.9	−1.55052	−0.775260	−	0.631642i	$-0.782381\pi$
$769$	−33064.9	−1.55052	−0.775260	+	0.631642i	$0.782381\pi$
$770$	20290.0	0.949610
$771$	0	0
$772$	14523.6	0.677091
$773$	18564.0	0.863779	0.431890	−	0.901927i	$-0.357847\pi$
$773$	18564.0	0.863779	0.431890	+	0.901927i	$0.357847\pi$
$774$	0	0
$775$	895.279	0.0414960
$776$	−4349.64	−0.201215
$777$	0	0
$778$	10351.1	0.476997
$779$	29091.6	1.33802
$780$	0	0
$781$	16191.7	0.741848
$782$	−9247.66	−0.422885
$783$	0	0
$784$	9425.86	0.429385
$785$	−5098.91	−0.231832
$786$	0	0
$787$	36436.8	1.65036	0.825179	−	0.564871i	$-0.191074\pi$
$787$	36436.8	1.65036	0.825179	+	0.564871i	$0.191074\pi$
$788$	7373.04	0.333317
$789$	0	0
$790$	2171.08	0.0977768
$791$	43197.5	1.94175
$792$	0	0
$793$	0	0
$794$	−2081.71	−0.0930441
$795$	0	0
$796$	32300.7	1.43827
$797$	−12365.6	−0.549576	−0.274788	−	0.961505i	$-0.588608\pi$
$797$	−12365.6	−0.549576	−0.274788	+	0.961505i	$0.588608\pi$
$798$	0	0
$799$	−26013.7	−1.15181
$800$	−5241.61	−0.231649
$801$	0	0
$802$	−15644.4	−0.688806
$803$	−36313.0	−1.59584
$804$	0	0
$805$	38545.4	1.68763
$806$	0	0
$807$	0	0
$808$	32390.9	1.41028
$809$	8093.75	0.351744	0.175872	−	0.984413i	$-0.443726\pi$
$809$	8093.75	0.351744	0.175872	+	0.984413i	$0.443726\pi$
$810$	0	0
$811$	15984.7	0.692105	0.346052	−	0.938215i	$-0.387522\pi$
$811$	15984.7	0.692105	0.346052	+	0.938215i	$0.387522\pi$
$812$	−34147.3	−1.47578
$813$	0	0
$814$	2230.77	0.0960546
$815$	−6010.68	−0.258337
$816$	0	0
$817$	11598.0	0.496649
$818$	9312.03	0.398029
$819$	0	0
$820$	13936.6	0.593523
$821$	26861.9	1.14188	0.570942	−	0.820990i	$-0.306578\pi$
$821$	26861.9	1.14188	0.570942	+	0.820990i	$0.306578\pi$
$822$	0	0
$823$	−5205.94	−0.220495	−0.110248	−	0.993904i	$-0.535164\pi$
$823$	−5205.94	−0.220495	−0.110248	+	0.993904i	$0.535164\pi$
$824$	31886.9	1.34810
$825$	0	0
$826$	−14388.3	−0.606092
$827$	−46621.5	−1.96032	−0.980162	−	0.198200i	$-0.936491\pi$
$827$	−46621.5	−1.96032	−0.980162	+	0.198200i	$0.936491\pi$
$828$	0	0
$829$	−41829.7	−1.75248	−0.876241	−	0.481874i	$-0.839956\pi$
$829$	−41829.7	−1.75248	−0.876241	+	0.481874i	$0.839956\pi$
$830$	4880.87	0.204118
$831$	0	0
$832$	0	0
$833$	26707.0	1.11086
$834$	0	0
$835$	29384.3	1.21783
$836$	33027.4	1.36636
$837$	0	0
$838$	−8236.37	−0.339524
$839$	−12685.1	−0.521975	−0.260988	−	0.965342i	$-0.584048\pi$
$839$	−12685.1	−0.521975	−0.260988	+	0.965342i	$0.584048\pi$
$840$	0	0
$841$	13614.9	0.558238
$842$	−2225.97	−0.0911069
$843$	0	0
$844$	26490.1	1.08036
$845$	0	0
$846$	0	0
$847$	26078.2	1.05792
$848$	2593.77	0.105036
$849$	0	0
$850$	−1978.83	−0.0798509
$851$	4237.85	0.170707
$852$	0	0
$853$	−37493.3	−1.50498	−0.752488	−	0.658606i	$-0.771146\pi$
$853$	−37493.3	−1.50498	−0.752488	+	0.658606i	$0.771146\pi$
$854$	−4320.01	−0.173100
$855$	0	0
$856$	−136.723	−0.00545921
$857$	11826.3	0.471386	0.235693	−	0.971828i	$-0.424264\pi$
$857$	11826.3	0.471386	0.235693	+	0.971828i	$0.424264\pi$
$858$	0	0
$859$	−36498.7	−1.44973	−0.724866	−	0.688890i	$-0.758099\pi$
$859$	−36498.7	−1.44973	−0.724866	+	0.688890i	$0.758099\pi$
$860$	5556.14	0.220306
$861$	0	0
$862$	7286.72	0.287920
$863$	−2292.79	−0.0904372	−0.0452186	−	0.998977i	$-0.514398\pi$
$863$	−2292.79	−0.0904372	−0.0452186	+	0.998977i	$0.514398\pi$
$864$	0	0
$865$	9635.92	0.378764
$866$	−4836.24	−0.189772
$867$	0	0
$868$	5607.37	0.219270
$869$	7054.45	0.275381
$870$	0	0
$871$	0	0
$872$	11014.6	0.427753
$873$	0	0
$874$	−23051.3	−0.892132
$875$	45113.2	1.74298
$876$	0	0
$877$	−20800.3	−0.800885	−0.400442	−	0.916322i	$-0.631144\pi$
$877$	−20800.3	−0.800885	−0.400442	+	0.916322i	$0.631144\pi$
$878$	−8867.68	−0.340854
$879$	0	0
$880$	7873.52	0.301610
$881$	19255.9	0.736377	0.368188	−	0.929751i	$-0.379978\pi$
$881$	19255.9	0.736377	0.368188	+	0.929751i	$0.379978\pi$
$882$	0	0
$883$	1744.49	0.0664857	0.0332429	−	0.999447i	$-0.489417\pi$
$883$	1744.49	0.0664857	0.0332429	+	0.999447i	$0.489417\pi$
$884$	0	0
$885$	0	0
$886$	−10002.7	−0.379284
$887$	−2970.70	−0.112454	−0.0562268	−	0.998418i	$-0.517907\pi$
$887$	−2970.70	−0.112454	−0.0562268	+	0.998418i	$0.517907\pi$
$888$	0	0
$889$	−74634.6	−2.81571
$890$	2453.87	0.0924202
$891$	0	0
$892$	−26217.3	−0.984104
$893$	−64843.6	−2.42991
$894$	0	0
$895$	18252.5	0.681691
$896$	−38810.9	−1.44708
$897$	0	0
$898$	−601.703	−0.0223598
$899$	−6240.66	−0.231521
$900$	0	0
$901$	7349.12	0.271736
$902$	−16637.1	−0.614140
$903$	0	0
$904$	−29298.4	−1.07793
$905$	−8400.32	−0.308548
$906$	0	0
$907$	33003.3	1.20822	0.604111	−	0.796900i	$-0.293528\pi$
$907$	33003.3	1.20822	0.604111	+	0.796900i	$0.293528\pi$
$908$	−33697.7	−1.23160
$909$	0	0
$910$	0	0
$911$	−14977.0	−0.544686	−0.272343	−	0.962200i	$-0.587799\pi$
$911$	−14977.0	−0.544686	−0.272343	+	0.962200i	$0.587799\pi$
$912$	0	0
$913$	15859.3	0.574881
$914$	−24400.1	−0.883022
$915$	0	0
$916$	27117.7	0.978161
$917$	−22291.9	−0.802773
$918$	0	0
$919$	11425.4	0.410107	0.205054	−	0.978751i	$-0.434263\pi$
$919$	11425.4	0.410107	0.205054	+	0.978751i	$0.434263\pi$
$920$	−26143.1	−0.936862
$921$	0	0
$922$	17196.9	0.614263
$923$	0	0
$924$	0	0
$925$	906.820	0.0322336
$926$	−5226.55	−0.185481
$927$	0	0
$928$	36537.3	1.29245
$929$	−11954.2	−0.422179	−0.211090	−	0.977467i	$-0.567701\pi$
$929$	−11954.2	−0.422179	−0.211090	+	0.977467i	$0.567701\pi$
$930$	0	0
$931$	66571.7	2.34350
$932$	34438.2	1.21037
$933$	0	0
$934$	−1663.74	−0.0582861
$935$	22308.7	0.780290
$936$	0	0
$937$	42546.4	1.48338	0.741692	−	0.670740i	$-0.234024\pi$
$937$	42546.4	1.48338	0.741692	+	0.670740i	$0.234024\pi$
$938$	−19364.1	−0.674051
$939$	0	0
$940$	−31064.0	−1.07787
$941$	−20665.1	−0.715903	−0.357951	−	0.933740i	$-0.616525\pi$
$941$	−20665.1	−0.715903	−0.357951	+	0.933740i	$0.616525\pi$
$942$	0	0
$943$	−31605.9	−1.09144
$944$	−5583.37	−0.192503
$945$	0	0
$946$	−6632.72	−0.227958
$947$	9493.73	0.325771	0.162885	−	0.986645i	$-0.447920\pi$
$947$	9493.73	0.325771	0.162885	+	0.986645i	$0.447920\pi$
$948$	0	0
$949$	0	0
$950$	−4932.56	−0.168456
$951$	0	0
$952$	−29341.3	−0.998904
$953$	−53334.1	−1.81287	−0.906433	−	0.422349i	$-0.861206\pi$
$953$	−53334.1	−1.81287	−0.906433	+	0.422349i	$0.861206\pi$
$954$	0	0
$955$	43749.2	1.48240
$956$	12495.9	0.422746
$957$	0	0
$958$	28402.2	0.957864
$959$	6651.58	0.223974
$960$	0	0
$961$	−28766.2	−0.965601
$962$	0	0
$963$	0	0
$964$	−27456.1	−0.917326
$965$	24453.3	0.815730
$966$	0	0
$967$	−42110.1	−1.40038	−0.700191	−	0.713956i	$-0.746902\pi$
$967$	−42110.1	−1.40038	−0.700191	+	0.713956i	$0.746902\pi$
$968$	−17687.3	−0.587285
$969$	0	0
$970$	−3093.47	−0.102397
$971$	13827.7	0.457006	0.228503	−	0.973543i	$-0.426617\pi$
$971$	13827.7	0.457006	0.228503	+	0.973543i	$0.426617\pi$
$972$	0	0
$973$	−23267.4	−0.766618
$974$	−13262.3	−0.436295
$975$	0	0
$976$	−1676.38	−0.0549791
$977$	−1133.89	−0.0371302	−0.0185651	−	0.999828i	$-0.505910\pi$
$977$	−1133.89	−0.0371302	−0.0185651	+	0.999828i	$0.505910\pi$
$978$	0	0
$979$	7973.31	0.260294
$980$	31891.9	1.03954
$981$	0	0
$982$	−21116.7	−0.686214
$983$	−26250.2	−0.851729	−0.425865	−	0.904787i	$-0.640030\pi$
$983$	−26250.2	−0.851729	−0.425865	+	0.904787i	$0.640030\pi$
$984$	0	0
$985$	12414.0	0.401565
$986$	13793.7	0.445517
$987$	0	0
$988$	0	0
$989$	−12600.3	−0.405124
$990$	0	0
$991$	−29360.4	−0.941133	−0.470566	−	0.882365i	$-0.655950\pi$
$991$	−29360.4	−0.941133	−0.470566	+	0.882365i	$0.655950\pi$
$992$	−5999.84	−0.192031
$993$	0	0
$994$	−15145.7	−0.483293
$995$	54384.5	1.73277
$996$	0	0
$997$	16835.9	0.534803	0.267401	−	0.963585i	$-0.413835\pi$
$997$	16835.9	0.534803	0.267401	+	0.963585i	$0.413835\pi$
$998$	−13663.7	−0.433384
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.v.1.3		4
3.2	odd	2		507.4.a.m.1.2		4
13.3	even	3		117.4.g.e.100.2		8
13.9	even	3		117.4.g.e.55.2		8
13.12	even	2		1521.4.a.bb.1.2		4
39.5	even	4		507.4.b.h.337.5		8
39.8	even	4		507.4.b.h.337.4		8
39.29	odd	6		39.4.e.c.22.3	yes	8
39.35	odd	6		39.4.e.c.16.3	✓	8
39.38	odd	2		507.4.a.i.1.3		4
156.35	even	6		624.4.q.i.289.4		8
156.107	even	6		624.4.q.i.529.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.3	✓	8	39.35	odd	6
39.4.e.c.22.3	yes	8	39.29	odd	6
117.4.g.e.55.2		8	13.9	even	3
117.4.g.e.100.2		8	13.3	even	3
507.4.a.i.1.3		4	39.38	odd	2
507.4.a.m.1.2		4	3.2	odd	2
507.4.b.h.337.4		8	39.8	even	4
507.4.b.h.337.5		8	39.5	even	4
624.4.q.i.289.4		8	156.35	even	6
624.4.q.i.529.4		8	156.107	even	6
1521.4.a.v.1.3		4	1.1	even	1	trivial
1521.4.a.bb.1.2		4	13.12	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46610 q^{2} -5.85055 q^{4} -9.85055 q^{5} +29.9396 q^{7} -20.3063 q^{8} +O(q^{10})\)	\(q+1.46610 q^{2} -5.85055 q^{4} -9.85055 q^{5} +29.9396 q^{7} -20.3063 q^{8} -14.4419 q^{10} -46.9257 q^{11} +43.8945 q^{14} +17.0333 q^{16} +48.2616 q^{17} +120.300 q^{19} +57.6311 q^{20} -68.7979 q^{22} -130.697 q^{23} -27.9667 q^{25} -175.163 q^{28} +194.946 q^{29} -32.0123 q^{31} +187.423 q^{32} +70.7565 q^{34} -294.921 q^{35} -32.4250 q^{37} +176.372 q^{38} +200.028 q^{40} +241.825 q^{41} +96.4087 q^{43} +274.541 q^{44} -191.615 q^{46} -539.015 q^{47} +553.380 q^{49} -41.0021 q^{50} +152.277 q^{53} +462.244 q^{55} -607.963 q^{56} +285.810 q^{58} -327.792 q^{59} -98.4180 q^{61} -46.9332 q^{62} +138.515 q^{64} -441.151 q^{67} -282.357 q^{68} -432.385 q^{70} -345.049 q^{71} +773.839 q^{73} -47.5383 q^{74} -703.822 q^{76} -1404.94 q^{77} -150.332 q^{79} -167.787 q^{80} +354.540 q^{82} -337.966 q^{83} -475.403 q^{85} +141.345 q^{86} +952.889 q^{88} -169.913 q^{89} +764.649 q^{92} -790.250 q^{94} -1185.02 q^{95} +214.201 q^{97} +811.311 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8	\( 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29} + 26 q^{31} - 654 q^{32} + 1062 q^{34} - 88 q^{35} + 102 q^{37} - 332 q^{38} - 998 q^{40} + 1054 q^{41} + 450 q^{43} + 44 q^{44} - 172 q^{46} + 96 q^{47} + 1070 q^{49} - 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} + 722 q^{58} - 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} - 1134 q^{67} - 1786 q^{68} - 2324 q^{70} - 1064 q^{71} + 952 q^{73} - 1158 q^{74} - 1708 q^{76} - 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 1734 q^{82} + 404 q^{83} - 1394 q^{85} - 3168 q^{86} + 3060 q^{88} - 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} + 2166 q^{97} + 1906 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 - 62 * q^10 - 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 + 124 * q^19 + 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 - 144 * q^28 - 194 * q^29 + 26 * q^31 - 654 * q^32 + 1062 * q^34 - 88 * q^35 + 102 * q^37 - 332 * q^38 - 998 * q^40 + 1054 * q^41 + 450 * q^43 + 44 * q^44 - 172 * q^46 + 96 * q^47 + 1070 * q^49 - 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 + 722 * q^58 - 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 - 1134 * q^67 - 1786 * q^68 - 2324 * q^70 - 1064 * q^71 + 952 * q^73 - 1158 * q^74 - 1708 * q^76 - 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 1734 * q^82 + 404 * q^83 - 1394 * q^85 - 3168 * q^86 + 3060 * q^88 - 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 + 2166 * q^97 + 1906 * q^98

Introduction

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