LMFDB - Embedded newform 1521.4.a.z.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 13x^{2} + 14x - 2 $ x^4 - 2x^3 - 13x^2 + 14*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.2
Root			$0.170498$ 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)

$f(q)$	$=$	$q-4.12311 q^{2} +9.00000 q^{4} +13.4424 q^{5} +31.4219 q^{7} -4.12311 q^{8} +O(q^{10})$	$q-4.12311 q^{2} +9.00000 q^{4} +13.4424 q^{5} +31.4219 q^{7} -4.12311 q^{8} -55.4243 q^{10} +40.4962 q^{11} -129.556 q^{14} -55.0000 q^{16} +43.1414 q^{17} +26.9779 q^{19} +120.981 q^{20} -166.970 q^{22} +19.0100 q^{23} +55.6971 q^{25} +282.797 q^{28} +154.111 q^{29} -308.270 q^{31} +259.756 q^{32} -177.877 q^{34} +422.384 q^{35} -43.5116 q^{37} -111.233 q^{38} -55.4243 q^{40} +47.8384 q^{41} +342.121 q^{43} +364.466 q^{44} -78.3802 q^{46} +133.468 q^{47} +644.334 q^{49} -229.645 q^{50} +438.454 q^{53} +544.364 q^{55} -129.556 q^{56} -635.418 q^{58} +590.553 q^{59} -541.304 q^{61} +1271.03 q^{62} -631.000 q^{64} +230.345 q^{67} +388.273 q^{68} -1741.54 q^{70} +449.412 q^{71} +389.711 q^{73} +179.403 q^{74} +242.801 q^{76} +1272.47 q^{77} -897.820 q^{79} -739.330 q^{80} -197.243 q^{82} -1300.24 q^{83} +579.923 q^{85} -1410.60 q^{86} -166.970 q^{88} -925.045 q^{89} +171.090 q^{92} -550.301 q^{94} +362.647 q^{95} +1560.49 q^{97} -2656.66 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{4}+O(q^{10}) $ 4 * q + 36 * q^4	$ 4 q + 36 q^{4} - 136 q^{10} - 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} - 120 q^{25} - 12 q^{29} + 804 q^{35} + 612 q^{38} - 136 q^{40} + 940 q^{43} + 692 q^{49} + 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} - 2524 q^{64} + 1296 q^{68} + 3060 q^{74} + 2976 q^{77} + 8 q^{79} + 68 q^{82} - 68 q^{88} + 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) $ 4 * q + 36 * q^4 - 136 * q^10 - 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 - 120 * q^25 - 12 * q^29 + 804 * q^35 + 612 * q^38 - 136 * q^40 + 940 * q^43 + 692 * q^49 + 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 - 2524 * q^64 + 1296 * q^68 + 3060 * q^74 + 2976 * q^77 + 8 * q^79 + 68 * q^82 - 68 * q^88 + 2484 * q^92 - 5372 * q^94 + 108 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.12311	−1.45774	−0.728869	−	0.684653i	$-0.759954\pi$
$2$	−4.12311	−1.45774	−0.728869	+	0.684653i	$0.759954\pi$
$3$	0	0
$3$	0	0
$4$	9.00000	1.12500
$5$	13.4424	1.20232	0.601161	−	0.799128i	$-0.294705\pi$
$5$	13.4424	1.20232	0.601161	+	0.799128i	$0.294705\pi$
$6$	0	0
$7$	31.4219	1.69662	0.848311	−	0.529498i	$-0.177620\pi$
$7$	31.4219	1.69662	0.848311	+	0.529498i	$0.177620\pi$
$8$	−4.12311	−0.182217
$9$	0	0
$10$	−55.4243	−1.75267
$11$	40.4962	1.11001	0.555003	−	0.831849i	$-0.312717\pi$
$11$	40.4962	1.11001	0.555003	+	0.831849i	$0.312717\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−129.556	−2.47323
$15$	0	0
$16$	−55.0000	−0.859375
$17$	43.1414	0.615490	0.307745	−	0.951469i	$-0.400426\pi$
$17$	43.1414	0.615490	0.307745	+	0.951469i	$0.400426\pi$
$18$	0	0
$19$	26.9779	0.325745	0.162873	−	0.986647i	$-0.447924\pi$
$19$	26.9779	0.325745	0.162873	+	0.986647i	$0.447924\pi$
$20$	120.981	1.35261
$21$	0	0
$22$	−166.970	−1.61810
$23$	19.0100	0.172342	0.0861709	−	0.996280i	$-0.472537\pi$
$23$	19.0100	0.172342	0.0861709	+	0.996280i	$0.472537\pi$
$24$	0	0
$25$	55.6971	0.445577
$26$	0	0
$27$	0	0
$28$	282.797	1.90870
$29$	154.111	0.986820	0.493410	−	0.869797i	$-0.335750\pi$
$29$	154.111	0.986820	0.493410	+	0.869797i	$0.335750\pi$
$30$	0	0
$31$	−308.270	−1.78603	−0.893016	−	0.450025i	$-0.851415\pi$
$31$	−308.270	−1.78603	−0.893016	+	0.450025i	$0.851415\pi$
$32$	259.756	1.43496
$33$	0	0
$34$	−177.877	−0.897223
$35$	422.384	2.03988
$36$	0	0
$37$	−43.5116	−0.193331	−0.0966657	−	0.995317i	$-0.530818\pi$
$37$	−43.5116	−0.193331	−0.0966657	+	0.995317i	$0.530818\pi$
$38$	−111.233	−0.474851
$39$	0	0
$40$	−55.4243	−0.219084
$41$	47.8384	0.182222	0.0911110	−	0.995841i	$-0.470958\pi$
$41$	47.8384	0.182222	0.0911110	+	0.995841i	$0.470958\pi$
$42$	0	0
$43$	342.121	1.21333	0.606663	−	0.794959i	$-0.292508\pi$
$43$	342.121	1.21333	0.606663	+	0.794959i	$0.292508\pi$
$44$	364.466	1.24876
$45$	0	0
$46$	−78.3802	−0.251229
$47$	133.468	0.414218	0.207109	−	0.978318i	$-0.433594\pi$
$47$	133.468	0.414218	0.207109	+	0.978318i	$0.433594\pi$
$48$	0	0
$49$	644.334	1.87853
$50$	−229.645	−0.649535
$51$	0	0
$52$	0	0
$53$	438.454	1.13635	0.568173	−	0.822909i	$-0.307650\pi$
$53$	438.454	1.13635	0.568173	+	0.822909i	$0.307650\pi$
$54$	0	0
$55$	544.364	1.33458
$56$	−129.556	−0.309154
$57$	0	0
$58$	−635.418	−1.43852
$59$	590.553	1.30311	0.651555	−	0.758601i	$-0.274117\pi$
$59$	590.553	1.30311	0.651555	+	0.758601i	$0.274117\pi$
$60$	0	0
$61$	−541.304	−1.13618	−0.568089	−	0.822967i	$-0.692317\pi$
$61$	−541.304	−1.13618	−0.568089	+	0.822967i	$0.692317\pi$
$62$	1271.03	2.60357
$63$	0	0
$64$	−631.000	−1.23242
$65$	0	0
$66$	0	0
$67$	230.345	0.420018	0.210009	−	0.977700i	$-0.432651\pi$
$67$	230.345	0.420018	0.210009	+	0.977700i	$0.432651\pi$
$68$	388.273	0.692426
$69$	0	0
$70$	−1741.54	−2.97362
$71$	449.412	0.751203	0.375601	−	0.926781i	$-0.377436\pi$
$71$	449.412	0.751203	0.375601	+	0.926781i	$0.377436\pi$
$72$	0	0
$73$	389.711	0.624826	0.312413	−	0.949946i	$-0.398863\pi$
$73$	389.711	0.624826	0.312413	+	0.949946i	$0.398863\pi$
$74$	179.403	0.281826
$75$	0	0
$76$	242.801	0.366464
$77$	1272.47	1.88326
$78$	0	0
$79$	−897.820	−1.27864	−0.639321	−	0.768940i	$-0.720784\pi$
$79$	−897.820	−1.27864	−0.639321	+	0.768940i	$0.720784\pi$
$80$	−739.330	−1.03325
$81$	0	0
$82$	−197.243	−0.265632
$83$	−1300.24	−1.71952	−0.859759	−	0.510700i	$-0.829386\pi$
$83$	−1300.24	−1.71952	−0.859759	+	0.510700i	$0.829386\pi$
$84$	0	0
$85$	579.923	0.740017
$86$	−1410.60	−1.76871
$87$	0	0
$88$	−166.970	−0.202262
$89$	−925.045	−1.10174	−0.550869	−	0.834592i	$-0.685703\pi$
$89$	−925.045	−1.10174	−0.550869	+	0.834592i	$0.685703\pi$
$90$	0	0
$91$	0	0
$92$	171.090	0.193884
$93$	0	0
$94$	−550.301	−0.603822
$95$	362.647	0.391651
$96$	0	0
$97$	1560.49	1.63344	0.816722	−	0.577031i	$-0.195789\pi$
$97$	1560.49	1.63344	0.816722	+	0.577031i	$0.195789\pi$
$98$	−2656.66	−2.73840
$99$	0	0
$100$	501.274	0.501274
$101$	−958.840	−0.944635	−0.472318	−	0.881428i	$-0.656582\pi$
$101$	−958.840	−0.944635	−0.472318	+	0.881428i	$0.656582\pi$
$102$	0	0
$103$	635.153	0.607606	0.303803	−	0.952735i	$-0.401743\pi$
$103$	635.153	0.607606	0.303803	+	0.952735i	$0.401743\pi$
$104$	0	0
$105$	0	0
$106$	−1807.79	−1.65649
$107$	1448.32	1.30855	0.654275	−	0.756257i	$-0.272974\pi$
$107$	1448.32	1.30855	0.654275	+	0.756257i	$0.272974\pi$
$108$	0	0
$109$	331.084	0.290937	0.145468	−	0.989363i	$-0.453531\pi$
$109$	331.084	0.290937	0.145468	+	0.989363i	$0.453531\pi$
$110$	−2244.47	−1.94547
$111$	0	0
$112$	−1728.20	−1.45803
$113$	−695.204	−0.578755	−0.289378	−	0.957215i	$-0.593448\pi$
$113$	−695.204	−0.578755	−0.289378	+	0.957215i	$0.593448\pi$
$114$	0	0
$115$	255.539	0.207210
$116$	1387.00	1.11017
$117$	0	0
$118$	−2434.91	−1.89959
$119$	1355.58	1.04425
$120$	0	0
$121$	308.940	0.232111
$122$	2231.85	1.65625
$123$	0	0
$124$	−2774.43	−2.00929
$125$	−931.594	−0.666595
$126$	0	0
$127$	247.154	0.172688	0.0863441	−	0.996265i	$-0.472482\pi$
$127$	247.154	0.172688	0.0863441	+	0.996265i	$0.472482\pi$
$128$	523.634	0.361587
$129$	0	0
$130$	0	0
$131$	−472.243	−0.314962	−0.157481	−	0.987522i	$-0.550337\pi$
$131$	−472.243	−0.314962	−0.157481	+	0.987522i	$0.550337\pi$
$132$	0	0
$133$	847.697	0.552667
$134$	−949.739	−0.612275
$135$	0	0
$136$	−177.877	−0.112153
$137$	−1830.70	−1.14166	−0.570829	−	0.821069i	$-0.693378\pi$
$137$	−1830.70	−1.14166	−0.570829	+	0.821069i	$0.693378\pi$
$138$	0	0
$139$	−100.000	−0.0610208	−0.0305104	−	0.999534i	$-0.509713\pi$
$139$	−100.000	−0.0610208	−0.0305104	+	0.999534i	$0.509713\pi$
$140$	3801.46	2.29487
$141$	0	0
$142$	−1852.97	−1.09506
$143$	0	0
$144$	0	0
$145$	2071.62	1.18647
$146$	−1606.82	−0.910832
$147$	0	0
$148$	−391.604	−0.217498
$149$	149.557	0.0822293	0.0411147	−	0.999154i	$-0.486909\pi$
$149$	149.557	0.0822293	0.0411147	+	0.999154i	$0.486909\pi$
$150$	0	0
$151$	800.032	0.431163	0.215582	−	0.976486i	$-0.430835\pi$
$151$	800.032	0.431163	0.215582	+	0.976486i	$0.430835\pi$
$152$	−111.233	−0.0593564
$153$	0	0
$154$	−5246.51	−2.74530
$155$	−4143.88	−2.14739
$156$	0	0
$157$	−2706.16	−1.37564	−0.687818	−	0.725884i	$-0.741431\pi$
$157$	−2706.16	−1.37564	−0.687818	+	0.725884i	$0.741431\pi$
$158$	3701.81	1.86392
$159$	0	0
$160$	3491.73	1.72528
$161$	597.330	0.292399
$162$	0	0
$163$	−3678.25	−1.76750	−0.883750	−	0.467959i	$-0.844990\pi$
$163$	−3678.25	−1.76750	−0.883750	+	0.467959i	$0.844990\pi$
$164$	430.546	0.205000
$165$	0	0
$166$	5361.03	2.50661
$167$	−3223.11	−1.49348	−0.746742	−	0.665114i	$-0.768383\pi$
$167$	−3223.11	−1.49348	−0.746742	+	0.665114i	$0.768383\pi$
$168$	0	0
$169$	0	0
$170$	−2391.08	−1.07875
$171$	0	0
$172$	3079.09	1.36499
$173$	2689.54	1.18198	0.590988	−	0.806680i	$-0.298738\pi$
$173$	2689.54	1.18198	0.590988	+	0.806680i	$0.298738\pi$
$174$	0	0
$175$	1750.11	0.755976
$176$	−2227.29	−0.953911
$177$	0	0
$178$	3814.06	1.60604
$179$	1524.04	0.636381	0.318191	−	0.948027i	$-0.396925\pi$
$179$	1524.04	0.636381	0.318191	+	0.948027i	$0.396925\pi$
$180$	0	0
$181$	476.881	0.195836	0.0979180	−	0.995194i	$-0.468782\pi$
$181$	476.881	0.195836	0.0979180	+	0.995194i	$0.468782\pi$
$182$	0	0
$183$	0	0
$184$	−78.3802	−0.0314036
$185$	−584.899	−0.232446
$186$	0	0
$187$	1747.06	0.683197
$188$	1201.21	0.465996
$189$	0	0
$190$	−1495.23	−0.570924
$191$	1369.74	0.518906	0.259453	−	0.965756i	$-0.416458\pi$
$191$	1369.74	0.518906	0.259453	+	0.965756i	$0.416458\pi$
$192$	0	0
$193$	2144.72	0.799898	0.399949	−	0.916537i	$-0.369028\pi$
$193$	2144.72	0.799898	0.399949	+	0.916537i	$0.369028\pi$
$194$	−6434.08	−2.38113
$195$	0	0
$196$	5799.01	2.11334
$197$	−239.739	−0.0867040	−0.0433520	−	0.999060i	$-0.513804\pi$
$197$	−239.739	−0.0867040	−0.0433520	+	0.999060i	$0.513804\pi$
$198$	0	0
$199$	−1589.94	−0.566371	−0.283185	−	0.959065i	$-0.591391\pi$
$199$	−1589.94	−0.566371	−0.283185	+	0.959065i	$0.591391\pi$
$200$	−229.645	−0.0811918
$201$	0	0
$202$	3953.40	1.37703
$203$	4842.47	1.67426
$204$	0	0
$205$	643.061	0.219090
$206$	−2618.80	−0.885731
$207$	0	0
$208$	0	0
$209$	1092.50	0.361579
$210$	0	0
$211$	−1872.85	−0.611055	−0.305527	−	0.952183i	$-0.598833\pi$
$211$	−1872.85	−0.611055	−0.305527	+	0.952183i	$0.598833\pi$
$212$	3946.09	1.27839
$213$	0	0
$214$	−5971.59	−1.90752
$215$	4598.92	1.45881
$216$	0	0
$217$	−9686.43	−3.03022
$218$	−1365.09	−0.424109
$219$	0	0
$220$	4899.28	1.50141
$221$	0	0
$222$	0	0
$223$	56.1283	0.0168548	0.00842742	−	0.999964i	$-0.497317\pi$
$223$	56.1283	0.0168548	0.00842742	+	0.999964i	$0.497317\pi$
$224$	8162.01	2.43459
$225$	0	0
$226$	2866.40	0.843673
$227$	−667.390	−0.195137	−0.0975687	−	0.995229i	$-0.531107\pi$
$227$	−667.390	−0.195137	−0.0975687	+	0.995229i	$0.531107\pi$
$228$	0	0
$229$	−723.299	−0.208720	−0.104360	−	0.994540i	$-0.533279\pi$
$229$	−723.299	−0.208720	−0.104360	+	0.994540i	$0.533279\pi$
$230$	−1053.62	−0.302058
$231$	0	0
$232$	−635.418	−0.179816
$233$	275.451	0.0774482	0.0387241	−	0.999250i	$-0.487671\pi$
$233$	275.451	0.0774482	0.0387241	+	0.999250i	$0.487671\pi$
$234$	0	0
$235$	1794.12	0.498024
$236$	5314.98	1.46600
$237$	0	0
$238$	−5589.22	−1.52225
$239$	−1529.39	−0.413925	−0.206963	−	0.978349i	$-0.566358\pi$
$239$	−1529.39	−0.413925	−0.206963	+	0.978349i	$0.566358\pi$
$240$	0	0
$241$	−975.526	−0.260743	−0.130372	−	0.991465i	$-0.541617\pi$
$241$	−975.526	−0.260743	−0.130372	+	0.991465i	$0.541617\pi$
$242$	−1273.79	−0.338357
$243$	0	0
$244$	−4871.74	−1.27820
$245$	8661.38	2.25859
$246$	0	0
$247$	0	0
$248$	1271.03	0.325446
$249$	0	0
$250$	3841.06	0.971720
$251$	1874.14	0.471294	0.235647	−	0.971839i	$-0.424279\pi$
$251$	1874.14	0.471294	0.235647	+	0.971839i	$0.424279\pi$
$252$	0	0
$253$	769.832	0.191300
$254$	−1019.04	−0.251734
$255$	0	0
$256$	2889.00	0.705322
$257$	1818.19	0.441305	0.220653	−	0.975352i	$-0.429181\pi$
$257$	1818.19	0.441305	0.220653	+	0.975352i	$0.429181\pi$
$258$	0	0
$259$	−1367.22	−0.328010
$260$	0	0
$261$	0	0
$262$	1947.11	0.459132
$263$	−673.799	−0.157978	−0.0789890	−	0.996875i	$-0.525169\pi$
$263$	−673.799	−0.157978	−0.0789890	+	0.996875i	$0.525169\pi$
$264$	0	0
$265$	5893.86	1.36625
$266$	−3495.14	−0.805643
$267$	0	0
$268$	2073.11	0.472520
$269$	3356.40	0.760756	0.380378	−	0.924831i	$-0.375794\pi$
$269$	3356.40	0.760756	0.380378	+	0.924831i	$0.375794\pi$
$270$	0	0
$271$	−8915.55	−1.99845	−0.999227	−	0.0393133i	$-0.987483\pi$
$271$	−8915.55	−1.99845	−0.999227	+	0.0393133i	$0.987483\pi$
$272$	−2372.78	−0.528937
$273$	0	0
$274$	7548.16	1.66424
$275$	2255.52	0.494593
$276$	0	0
$277$	4017.31	0.871396	0.435698	−	0.900093i	$-0.356502\pi$
$277$	4017.31	0.871396	0.435698	+	0.900093i	$0.356502\pi$
$278$	412.311	0.0889523
$279$	0	0
$280$	−1741.54	−0.371702
$281$	1841.12	0.390860	0.195430	−	0.980718i	$-0.437390\pi$
$281$	1841.12	0.390860	0.195430	+	0.980718i	$0.437390\pi$
$282$	0	0
$283$	4849.40	1.01861	0.509305	−	0.860586i	$-0.329902\pi$
$283$	4849.40	1.01861	0.509305	+	0.860586i	$0.329902\pi$
$284$	4044.71	0.845103
$285$	0	0
$286$	0	0
$287$	1503.17	0.309162
$288$	0	0
$289$	−3051.82	−0.621172
$290$	−8541.52	−1.72957
$291$	0	0
$292$	3507.40	0.702929
$293$	−1413.85	−0.281905	−0.140953	−	0.990016i	$-0.545017\pi$
$293$	−1413.85	−0.281905	−0.140953	+	0.990016i	$0.545017\pi$
$294$	0	0
$295$	7938.43	1.56676
$296$	179.403	0.0352283
$297$	0	0
$298$	−616.639	−0.119869
$299$	0	0
$300$	0	0
$301$	10750.1	2.05856
$302$	−3298.62	−0.628523
$303$	0	0
$304$	−1483.79	−0.279937
$305$	−7276.41	−1.36605
$306$	0	0
$307$	4625.64	0.859932	0.429966	−	0.902845i	$-0.358526\pi$
$307$	4625.64	0.859932	0.429966	+	0.902845i	$0.358526\pi$
$308$	11452.2	2.11867
$309$	0	0
$310$	17085.7	3.13032
$311$	6060.79	1.10507	0.552534	−	0.833490i	$-0.313661\pi$
$311$	6060.79	1.10507	0.552534	+	0.833490i	$0.313661\pi$
$312$	0	0
$313$	969.946	0.175158	0.0875792	−	0.996158i	$-0.472087\pi$
$313$	969.946	0.175158	0.0875792	+	0.996158i	$0.472087\pi$
$314$	11157.8	2.00532
$315$	0	0
$316$	−8080.38	−1.43847
$317$	8741.63	1.54883	0.774414	−	0.632679i	$-0.218045\pi$
$317$	8741.63	1.54883	0.774414	+	0.632679i	$0.218045\pi$
$318$	0	0
$319$	6240.92	1.09537
$320$	−8482.13	−1.48177
$321$	0	0
$322$	−2462.85	−0.426241
$323$	1163.87	0.200493
$324$	0	0
$325$	0	0
$326$	15165.8	2.57655
$327$	0	0
$328$	−197.243	−0.0332040
$329$	4193.81	0.702772
$330$	0	0
$331$	−6987.76	−1.16037	−0.580184	−	0.814485i	$-0.697019\pi$
$331$	−6987.76	−1.16037	−0.580184	+	0.814485i	$0.697019\pi$
$332$	−11702.2	−1.93446
$333$	0	0
$334$	13289.2	2.17711
$335$	3096.39	0.504996
$336$	0	0
$337$	−4156.59	−0.671881	−0.335940	−	0.941883i	$-0.609054\pi$
$337$	−4156.59	−0.671881	−0.335940	+	0.941883i	$0.609054\pi$
$338$	0	0
$339$	0	0
$340$	5219.30	0.832519
$341$	−12483.8	−1.98250
$342$	0	0
$343$	9468.49	1.49053
$344$	−1410.60	−0.221089
$345$	0	0
$346$	−11089.3	−1.72301
$347$	312.513	0.0483475	0.0241737	−	0.999708i	$-0.492305\pi$
$347$	312.513	0.0483475	0.0241737	+	0.999708i	$0.492305\pi$
$348$	0	0
$349$	4458.75	0.683872	0.341936	−	0.939723i	$-0.388917\pi$
$349$	4458.75	0.683872	0.341936	+	0.939723i	$0.388917\pi$
$350$	−7215.88	−1.10201
$351$	0	0
$352$	10519.1	1.59281
$353$	−2249.17	−0.339126	−0.169563	−	0.985519i	$-0.554236\pi$
$353$	−2249.17	−0.339126	−0.169563	+	0.985519i	$0.554236\pi$
$354$	0	0
$355$	6041.16	0.903187
$356$	−8325.41	−1.23945
$357$	0	0
$358$	−6283.78	−0.927677
$359$	−7842.79	−1.15300	−0.576499	−	0.817098i	$-0.695582\pi$
$359$	−7842.79	−1.15300	−0.576499	+	0.817098i	$0.695582\pi$
$360$	0	0
$361$	−6131.19	−0.893890
$362$	−1966.23	−0.285478
$363$	0	0
$364$	0	0
$365$	5238.64	0.751241
$366$	0	0
$367$	6660.24	0.947307	0.473653	−	0.880711i	$-0.342935\pi$
$367$	6660.24	0.947307	0.473653	+	0.880711i	$0.342935\pi$
$368$	−1045.55	−0.148106
$369$	0	0
$370$	2411.60	0.338846
$371$	13777.1	1.92795
$372$	0	0
$373$	−36.9873	−0.00513439	−0.00256720	−	0.999997i	$-0.500817\pi$
$373$	−36.9873	−0.00513439	−0.00256720	+	0.999997i	$0.500817\pi$
$374$	−7203.32	−0.995923
$375$	0	0
$376$	−550.301	−0.0754777
$377$	0	0
$378$	0	0
$379$	−12079.9	−1.63721	−0.818603	−	0.574360i	$-0.805251\pi$
$379$	−12079.9	−1.63721	−0.818603	+	0.574360i	$0.805251\pi$
$380$	3263.82	0.440607
$381$	0	0
$382$	−5647.59	−0.756429
$383$	−10567.4	−1.40984	−0.704919	−	0.709287i	$-0.749017\pi$
$383$	−10567.4	−1.40984	−0.704919	+	0.709287i	$0.749017\pi$
$384$	0	0
$385$	17104.9	2.26428
$386$	−8842.90	−1.16604
$387$	0	0
$388$	14044.4	1.83763
$389$	9757.49	1.27179	0.635893	−	0.771778i	$-0.280632\pi$
$389$	9757.49	1.27179	0.635893	+	0.771778i	$0.280632\pi$
$390$	0	0
$391$	820.119	0.106075
$392$	−2656.66	−0.342300
$393$	0	0
$394$	988.469	0.126392
$395$	−12068.8	−1.53734
$396$	0	0
$397$	14200.5	1.79522	0.897612	−	0.440786i	$-0.145300\pi$
$397$	14200.5	1.79522	0.897612	+	0.440786i	$0.145300\pi$
$398$	6555.48	0.825620
$399$	0	0
$400$	−3063.34	−0.382918
$401$	−12676.4	−1.57863	−0.789313	−	0.613992i	$-0.789563\pi$
$401$	−12676.4	−1.57863	−0.789313	+	0.613992i	$0.789563\pi$
$402$	0	0
$403$	0	0
$404$	−8629.56	−1.06271
$405$	0	0
$406$	−19966.0	−2.44063
$407$	−1762.05	−0.214599
$408$	0	0
$409$	−1533.68	−0.185417	−0.0927083	−	0.995693i	$-0.529552\pi$
$409$	−1533.68	−0.185417	−0.0927083	+	0.995693i	$0.529552\pi$
$410$	−2651.41	−0.319375
$411$	0	0
$412$	5716.38	0.683557
$413$	18556.3	2.21089
$414$	0	0
$415$	−17478.3	−2.06741
$416$	0	0
$417$	0	0
$418$	−4504.50	−0.527087
$419$	2165.89	0.252532	0.126266	−	0.991996i	$-0.459701\pi$
$419$	2165.89	0.252532	0.126266	+	0.991996i	$0.459701\pi$
$420$	0	0
$421$	−734.575	−0.0850380	−0.0425190	−	0.999096i	$-0.513538\pi$
$421$	−734.575	−0.0850380	−0.0425190	+	0.999096i	$0.513538\pi$
$422$	7721.98	0.890758
$423$	0	0
$424$	−1807.79	−0.207062
$425$	2402.85	0.274248
$426$	0	0
$427$	−17008.8	−1.92767
$428$	13034.9	1.47212
$429$	0	0
$430$	−18961.8	−2.12656
$431$	−13709.3	−1.53215	−0.766073	−	0.642754i	$-0.777792\pi$
$431$	−13709.3	−1.53215	−0.766073	+	0.642754i	$0.777792\pi$
$432$	0	0
$433$	10049.9	1.11540	0.557701	−	0.830042i	$-0.311684\pi$
$433$	10049.9	1.11540	0.557701	+	0.830042i	$0.311684\pi$
$434$	39938.2	4.41727
$435$	0	0
$436$	2979.76	0.327304
$437$	512.850	0.0561395
$438$	0	0
$439$	−8133.47	−0.884258	−0.442129	−	0.896951i	$-0.645777\pi$
$439$	−8133.47	−0.884258	−0.442129	+	0.896951i	$0.645777\pi$
$440$	−2244.47	−0.243184
$441$	0	0
$442$	0	0
$443$	2370.78	0.254264	0.127132	−	0.991886i	$-0.459423\pi$
$443$	2370.78	0.254264	0.127132	+	0.991886i	$0.459423\pi$
$444$	0	0
$445$	−12434.8	−1.32464
$446$	−231.423	−0.0245699
$447$	0	0
$448$	−19827.2	−2.09095
$449$	12923.2	1.35832	0.679158	−	0.733992i	$-0.262345\pi$
$449$	12923.2	1.35832	0.679158	+	0.733992i	$0.262345\pi$
$450$	0	0
$451$	1937.27	0.202267
$452$	−6256.84	−0.651099
$453$	0	0
$454$	2751.72	0.284459
$455$	0	0
$456$	0	0
$457$	−8401.26	−0.859944	−0.429972	−	0.902842i	$-0.641477\pi$
$457$	−8401.26	−0.859944	−0.429972	+	0.902842i	$0.641477\pi$
$458$	2982.24	0.304260
$459$	0	0
$460$	2299.85	0.233111
$461$	17627.3	1.78088	0.890441	−	0.455098i	$-0.150396\pi$
$461$	17627.3	1.78088	0.890441	+	0.455098i	$0.150396\pi$
$462$	0	0
$463$	5461.81	0.548233	0.274116	−	0.961697i	$-0.411615\pi$
$463$	5461.81	0.548233	0.274116	+	0.961697i	$0.411615\pi$
$464$	−8476.13	−0.848048
$465$	0	0
$466$	−1135.72	−0.112899
$467$	−8262.19	−0.818691	−0.409345	−	0.912379i	$-0.634243\pi$
$467$	−8262.19	−0.818691	−0.409345	+	0.912379i	$0.634243\pi$
$468$	0	0
$469$	7237.89	0.712611
$470$	−7397.35	−0.725988
$471$	0	0
$472$	−2434.91	−0.237449
$473$	13854.6	1.34680
$474$	0	0
$475$	1502.59	0.145145
$476$	12200.3	1.17479
$477$	0	0
$478$	6305.84	0.603395
$479$	−1575.87	−0.150320	−0.0751601	−	0.997171i	$-0.523947\pi$
$479$	−1575.87	−0.150320	−0.0751601	+	0.997171i	$0.523947\pi$
$480$	0	0
$481$	0	0
$482$	4022.20	0.380095
$483$	0	0
$484$	2780.46	0.261125
$485$	20976.7	1.96393
$486$	0	0
$487$	12595.7	1.17200	0.586001	−	0.810310i	$-0.300701\pi$
$487$	12595.7	1.17200	0.586001	+	0.810310i	$0.300701\pi$
$488$	2231.85	0.207031
$489$	0	0
$490$	−35711.8	−3.29244
$491$	1071.21	0.0984586	0.0492293	−	0.998788i	$-0.484323\pi$
$491$	1071.21	0.0984586	0.0492293	+	0.998788i	$0.484323\pi$
$492$	0	0
$493$	6648.59	0.607378
$494$	0	0
$495$	0	0
$496$	16954.9	1.53487
$497$	14121.4	1.27451
$498$	0	0
$499$	1422.30	0.127597	0.0637985	−	0.997963i	$-0.479678\pi$
$499$	1422.30	0.127597	0.0637985	+	0.997963i	$0.479678\pi$
$500$	−8384.35	−0.749919
$501$	0	0
$502$	−7727.28	−0.687022
$503$	9349.34	0.828760	0.414380	−	0.910104i	$-0.363998\pi$
$503$	9349.34	0.828760	0.414380	+	0.910104i	$0.363998\pi$
$504$	0	0
$505$	−12889.1	−1.13576
$506$	−3174.10	−0.278866
$507$	0	0
$508$	2224.39	0.194274
$509$	13736.3	1.19617	0.598086	−	0.801432i	$-0.295928\pi$
$509$	13736.3	1.19617	0.598086	+	0.801432i	$0.295928\pi$
$510$	0	0
$511$	12245.5	1.06009
$512$	−16100.7	−1.38976
$513$	0	0
$514$	−7496.58	−0.643307
$515$	8537.96	0.730538
$516$	0	0
$517$	5404.93	0.459785
$518$	5637.17	0.478153
$519$	0	0
$520$	0	0
$521$	−11052.3	−0.929386	−0.464693	−	0.885472i	$-0.653835\pi$
$521$	−11052.3	−0.929386	−0.464693	+	0.885472i	$0.653835\pi$
$522$	0	0
$523$	6477.04	0.541532	0.270766	−	0.962645i	$-0.412723\pi$
$523$	6477.04	0.541532	0.270766	+	0.962645i	$0.412723\pi$
$524$	−4250.19	−0.354332
$525$	0	0
$526$	2778.14	0.230290
$527$	−13299.2	−1.09929
$528$	0	0
$529$	−11805.6	−0.970298
$530$	−24301.0	−1.99164
$531$	0	0
$532$	7629.27	0.621750
$533$	0	0
$534$	0	0
$535$	19468.9	1.57330
$536$	−949.739	−0.0765344
$537$	0	0
$538$	−13838.8	−1.10898
$539$	26093.1	2.08517
$540$	0	0
$541$	−18341.5	−1.45761	−0.728803	−	0.684723i	$-0.759923\pi$
$541$	−18341.5	−1.45761	−0.728803	+	0.684723i	$0.759923\pi$
$542$	36759.7	2.91322
$543$	0	0
$544$	11206.2	0.883204
$545$	4450.55	0.349799
$546$	0	0
$547$	−18943.1	−1.48071	−0.740356	−	0.672215i	$-0.765343\pi$
$547$	−18943.1	−1.48071	−0.740356	+	0.672215i	$0.765343\pi$
$548$	−16476.3	−1.28436
$549$	0	0
$550$	−9299.75	−0.720987
$551$	4157.61	0.321452
$552$	0	0
$553$	−28211.2	−2.16937
$554$	−16563.8	−1.27027
$555$	0	0
$556$	−900.000	−0.0686484
$557$	415.532	0.0316098	0.0158049	−	0.999875i	$-0.494969\pi$
$557$	415.532	0.0316098	0.0158049	+	0.999875i	$0.494969\pi$
$558$	0	0
$559$	0	0
$560$	−23231.1	−1.75303
$561$	0	0
$562$	−7591.12	−0.569772
$563$	−18291.8	−1.36929	−0.684643	−	0.728879i	$-0.740042\pi$
$563$	−18291.8	−1.36929	−0.684643	+	0.728879i	$0.740042\pi$
$564$	0	0
$565$	−9345.19	−0.695850
$566$	−19994.6	−1.48487
$567$	0	0
$568$	−1852.97	−0.136882
$569$	−4347.47	−0.320308	−0.160154	−	0.987092i	$-0.551199\pi$
$569$	−4347.47	−0.320308	−0.160154	+	0.987092i	$0.551199\pi$
$570$	0	0
$571$	−16756.0	−1.22805	−0.614024	−	0.789288i	$-0.710450\pi$
$571$	−16756.0	−1.22805	−0.614024	+	0.789288i	$0.710450\pi$
$572$	0	0
$573$	0	0
$574$	−6197.74	−0.450677
$575$	1058.80	0.0767915
$576$	0	0
$577$	−19974.7	−1.44117	−0.720587	−	0.693364i	$-0.756128\pi$
$577$	−19974.7	−1.44117	−0.720587	+	0.693364i	$0.756128\pi$
$578$	12583.0	0.905506
$579$	0	0
$580$	18644.6	1.33478
$581$	−40856.0	−2.91737
$582$	0	0
$583$	17755.7	1.26135
$584$	−1606.82	−0.113854
$585$	0	0
$586$	5829.47	0.410944
$587$	−15748.7	−1.10735	−0.553677	−	0.832732i	$-0.686776\pi$
$587$	−15748.7	−1.10735	−0.553677	+	0.832732i	$0.686776\pi$
$588$	0	0
$589$	−8316.50	−0.581792
$590$	−32731.0	−2.28392
$591$	0	0
$592$	2393.14	0.166144
$593$	13318.4	0.922297	0.461148	−	0.887323i	$-0.347438\pi$
$593$	13318.4	0.922297	0.461148	+	0.887323i	$0.347438\pi$
$594$	0	0
$595$	18222.3	1.25553
$596$	1346.01	0.0925080
$597$	0	0
$598$	0	0
$599$	−2970.80	−0.202644	−0.101322	−	0.994854i	$-0.532307\pi$
$599$	−2970.80	−0.202644	−0.101322	+	0.994854i	$0.532307\pi$
$600$	0	0
$601$	10632.6	0.721654	0.360827	−	0.932633i	$-0.382495\pi$
$601$	10632.6	0.721654	0.360827	+	0.932633i	$0.382495\pi$
$602$	−44323.8	−3.00083
$603$	0	0
$604$	7200.29	0.485059
$605$	4152.88	0.279072
$606$	0	0
$607$	11587.9	0.774856	0.387428	−	0.921900i	$-0.373364\pi$
$607$	11587.9	0.774856	0.387428	+	0.921900i	$0.373364\pi$
$608$	7007.67	0.467432
$609$	0	0
$610$	30001.4	1.99135
$611$	0	0
$612$	0	0
$613$	−20792.3	−1.36998	−0.684988	−	0.728555i	$-0.740192\pi$
$613$	−20792.3	−1.36998	−0.684988	+	0.728555i	$0.740192\pi$
$614$	−19072.0	−1.25356
$615$	0	0
$616$	−5246.51	−0.343162
$617$	−1562.78	−0.101969	−0.0509846	−	0.998699i	$-0.516236\pi$
$617$	−1562.78	−0.101969	−0.0509846	+	0.998699i	$0.516236\pi$
$618$	0	0
$619$	−758.406	−0.0492454	−0.0246227	−	0.999697i	$-0.507838\pi$
$619$	−758.406	−0.0492454	−0.0246227	+	0.999697i	$0.507838\pi$
$620$	−37294.9	−2.41581
$621$	0	0
$622$	−24989.3	−1.61090
$623$	−29066.7	−1.86923
$624$	0	0
$625$	−19485.0	−1.24704
$626$	−3999.19	−0.255335
$627$	0	0
$628$	−24355.4	−1.54759
$629$	−1877.15	−0.118994
$630$	0	0
$631$	14265.2	0.899981	0.449990	−	0.893033i	$-0.351427\pi$
$631$	14265.2	0.899981	0.449990	+	0.893033i	$0.351427\pi$
$632$	3701.81	0.232990
$633$	0	0
$634$	−36042.7	−2.25779
$635$	3322.34	0.207627
$636$	0	0
$637$	0	0
$638$	−25732.0	−1.59677
$639$	0	0
$640$	7038.88	0.434744
$641$	−3985.64	−0.245590	−0.122795	−	0.992432i	$-0.539186\pi$
$641$	−3985.64	−0.245590	−0.122795	+	0.992432i	$0.539186\pi$
$642$	0	0
$643$	8156.55	0.500254	0.250127	−	0.968213i	$-0.419528\pi$
$643$	8156.55	0.500254	0.250127	+	0.968213i	$0.419528\pi$
$644$	5375.97	0.328949
$645$	0	0
$646$	−4798.74	−0.292266
$647$	−11279.2	−0.685368	−0.342684	−	0.939451i	$-0.611336\pi$
$647$	−11279.2	−0.685368	−0.342684	+	0.939451i	$0.611336\pi$
$648$	0	0
$649$	23915.2	1.44646
$650$	0	0
$651$	0	0
$652$	−33104.2	−1.98844
$653$	−6565.75	−0.393473	−0.196736	−	0.980456i	$-0.563034\pi$
$653$	−6565.75	−0.393473	−0.196736	+	0.980456i	$0.563034\pi$
$654$	0	0
$655$	−6348.06	−0.378686
$656$	−2631.11	−0.156597
$657$	0	0
$658$	−17291.5	−1.02446
$659$	−4799.35	−0.283696	−0.141848	−	0.989888i	$-0.545304\pi$
$659$	−4799.35	−0.283696	−0.141848	+	0.989888i	$0.545304\pi$
$660$	0	0
$661$	−15593.6	−0.917581	−0.458790	−	0.888545i	$-0.651717\pi$
$661$	−15593.6	−0.917581	−0.458790	+	0.888545i	$0.651717\pi$
$662$	28811.3	1.69151
$663$	0	0
$664$	5361.03	0.313326
$665$	11395.1	0.664483
$666$	0	0
$667$	2929.66	0.170070
$668$	−29008.0	−1.68017
$669$	0	0
$670$	−12766.7	−0.736152
$671$	−21920.8	−1.26116
$672$	0	0
$673$	−2205.54	−0.126326	−0.0631630	−	0.998003i	$-0.520119\pi$
$673$	−2205.54	−0.126326	−0.0631630	+	0.998003i	$0.520119\pi$
$674$	17138.1	0.979426
$675$	0	0
$676$	0	0
$677$	15046.4	0.854182	0.427091	−	0.904209i	$-0.359538\pi$
$677$	15046.4	0.854182	0.427091	+	0.904209i	$0.359538\pi$
$678$	0	0
$679$	49033.6	2.77134
$680$	−2391.08	−0.134844
$681$	0	0
$682$	51471.9	2.88997
$683$	30632.5	1.71614	0.858068	−	0.513537i	$-0.171665\pi$
$683$	30632.5	1.71614	0.858068	+	0.513537i	$0.171665\pi$
$684$	0	0
$685$	−24608.9	−1.37264
$686$	−39039.6	−2.17280
$687$	0	0
$688$	−18816.7	−1.04270
$689$	0	0
$690$	0	0
$691$	−2175.72	−0.119780	−0.0598901	−	0.998205i	$-0.519075\pi$
$691$	−2175.72	−0.119780	−0.0598901	+	0.998205i	$0.519075\pi$
$692$	24205.9	1.32972
$693$	0	0
$694$	−1288.52	−0.0704780
$695$	−1344.24	−0.0733666
$696$	0	0
$697$	2063.82	0.112156
$698$	−18383.9	−0.996906
$699$	0	0
$700$	15751.0	0.850473
$701$	32718.2	1.76284	0.881419	−	0.472335i	$-0.156589\pi$
$701$	32718.2	1.76284	0.881419	+	0.472335i	$0.156589\pi$
$702$	0	0
$703$	−1173.85	−0.0629768
$704$	−25553.1	−1.36799
$705$	0	0
$706$	9273.58	0.494356
$707$	−30128.6	−1.60269
$708$	0	0
$709$	25219.7	1.33589	0.667945	−	0.744211i	$-0.267174\pi$
$709$	25219.7	1.33589	0.667945	+	0.744211i	$0.267174\pi$
$710$	−24908.3	−1.31661
$711$	0	0
$712$	3814.06	0.200756
$713$	−5860.22	−0.307808
$714$	0	0
$715$	0	0
$716$	13716.4	0.715929
$717$	0	0
$718$	32336.6	1.68077
$719$	35466.2	1.83959	0.919796	−	0.392398i	$-0.128354\pi$
$719$	35466.2	1.83959	0.919796	+	0.392398i	$0.128354\pi$
$720$	0	0
$721$	19957.7	1.03088
$722$	25279.5	1.30306
$723$	0	0
$724$	4291.93	0.220315
$725$	8583.57	0.439704
$726$	0	0
$727$	−14262.2	−0.727588	−0.363794	−	0.931479i	$-0.618519\pi$
$727$	−14262.2	−0.727588	−0.363794	+	0.931479i	$0.618519\pi$
$728$	0	0
$729$	0	0
$730$	−21599.5	−1.09511
$731$	14759.6	0.746790
$732$	0	0
$733$	−16022.5	−0.807371	−0.403685	−	0.914898i	$-0.632271\pi$
$733$	−16022.5	−0.807371	−0.403685	+	0.914898i	$0.632271\pi$
$734$	−27460.9	−1.38093
$735$	0	0
$736$	4937.96	0.247304
$737$	9328.11	0.466222
$738$	0	0
$739$	−3796.21	−0.188966	−0.0944830	−	0.995526i	$-0.530120\pi$
$739$	−3796.21	−0.188966	−0.0944830	+	0.995526i	$0.530120\pi$
$740$	−5264.09	−0.261502
$741$	0	0
$742$	−56804.3	−2.81044
$743$	−30329.3	−1.49754	−0.748772	−	0.662827i	$-0.769356\pi$
$743$	−30329.3	−1.49754	−0.748772	+	0.662827i	$0.769356\pi$
$744$	0	0
$745$	2010.40	0.0988661
$746$	152.502	0.00748460
$747$	0	0
$748$	15723.6	0.768597
$749$	45509.1	2.22011
$750$	0	0
$751$	21551.9	1.04719	0.523595	−	0.851967i	$-0.324591\pi$
$751$	21551.9	1.04719	0.523595	+	0.851967i	$0.324591\pi$
$752$	−7340.72	−0.355969
$753$	0	0
$754$	0	0
$755$	10754.3	0.518397
$756$	0	0
$757$	−20417.6	−0.980306	−0.490153	−	0.871637i	$-0.663059\pi$
$757$	−20417.6	−0.980306	−0.490153	+	0.871637i	$0.663059\pi$
$758$	49806.5	2.38662
$759$	0	0
$760$	−1495.23	−0.0713655
$761$	−31375.4	−1.49456	−0.747278	−	0.664512i	$-0.768640\pi$
$761$	−31375.4	−1.49456	−0.747278	+	0.664512i	$0.768640\pi$
$762$	0	0
$763$	10403.3	0.493609
$764$	12327.7	0.583770
$765$	0	0
$766$	43570.5	2.05518
$767$	0	0
$768$	0	0
$769$	−12452.7	−0.583946	−0.291973	−	0.956427i	$-0.594312\pi$
$769$	−12452.7	−0.583946	−0.291973	+	0.956427i	$0.594312\pi$
$770$	−70525.5	−3.30073
$771$	0	0
$772$	19302.5	0.899885
$773$	−37449.8	−1.74253	−0.871264	−	0.490815i	$-0.836699\pi$
$773$	−37449.8	−1.74253	−0.871264	+	0.490815i	$0.836699\pi$
$774$	0	0
$775$	−17169.8	−0.795815
$776$	−6434.08	−0.297642
$777$	0	0
$778$	−40231.2	−1.85393
$779$	1290.58	0.0593580
$780$	0	0
$781$	18199.5	0.833839
$782$	−3381.44	−0.154629
$783$	0	0
$784$	−35438.4	−1.61436
$785$	−36377.1	−1.65396
$786$	0	0
$787$	26460.2	1.19848	0.599240	−	0.800570i	$-0.295470\pi$
$787$	26460.2	1.19848	0.599240	+	0.800570i	$0.295470\pi$
$788$	−2157.65	−0.0975420
$789$	0	0
$790$	49761.0	2.24104
$791$	−21844.6	−0.981928
$792$	0	0
$793$	0	0
$794$	−58550.3	−2.61697
$795$	0	0
$796$	−14309.4	−0.637167
$797$	−4749.47	−0.211085	−0.105543	−	0.994415i	$-0.533658\pi$
$797$	−4749.47	−0.211085	−0.105543	+	0.994415i	$0.533658\pi$
$798$	0	0
$799$	5757.99	0.254947
$800$	14467.6	0.639386
$801$	0	0
$802$	52266.1	2.30122
$803$	15781.8	0.693560
$804$	0	0
$805$	8029.53	0.351557
$806$	0	0
$807$	0	0
$808$	3953.40	0.172129
$809$	4464.04	0.194002	0.0970009	−	0.995284i	$-0.469075\pi$
$809$	4464.04	0.194002	0.0970009	+	0.995284i	$0.469075\pi$
$810$	0	0
$811$	20774.6	0.899499	0.449749	−	0.893155i	$-0.351513\pi$
$811$	20774.6	0.899499	0.449749	+	0.893155i	$0.351513\pi$
$812$	43582.2	1.88354
$813$	0	0
$814$	7265.13	0.312829
$815$	−49444.3	−2.12510
$816$	0	0
$817$	9229.73	0.395235
$818$	6323.51	0.270289
$819$	0	0
$820$	5787.55	0.246476
$821$	35769.2	1.52053	0.760264	−	0.649614i	$-0.225070\pi$
$821$	35769.2	1.52053	0.760264	+	0.649614i	$0.225070\pi$
$822$	0	0
$823$	2945.44	0.124753	0.0623764	−	0.998053i	$-0.480132\pi$
$823$	2945.44	0.124753	0.0623764	+	0.998053i	$0.480132\pi$
$824$	−2618.80	−0.110716
$825$	0	0
$826$	−76509.6	−3.22289
$827$	17878.6	0.751753	0.375877	−	0.926670i	$-0.377342\pi$
$827$	17878.6	0.751753	0.375877	+	0.926670i	$0.377342\pi$
$828$	0	0
$829$	13423.8	0.562398	0.281199	−	0.959649i	$-0.409268\pi$
$829$	13423.8	0.562398	0.281199	+	0.959649i	$0.409268\pi$
$830$	72065.0	3.01375
$831$	0	0
$832$	0	0
$833$	27797.5	1.15621
$834$	0	0
$835$	−43326.2	−1.79565
$836$	9832.53	0.406776
$837$	0	0
$838$	−8930.21	−0.368125
$839$	31542.7	1.29794	0.648971	−	0.760813i	$-0.275200\pi$
$839$	31542.7	1.29794	0.648971	+	0.760813i	$0.275200\pi$
$840$	0	0
$841$	−638.669	−0.0261867
$842$	3028.73	0.123963
$843$	0	0
$844$	−16855.7	−0.687437
$845$	0	0
$846$	0	0
$847$	9707.47	0.393805
$848$	−24115.0	−0.976547
$849$	0	0
$850$	−9907.22	−0.399782
$851$	−827.155	−0.0333191
$852$	0	0
$853$	21810.6	0.875476	0.437738	−	0.899103i	$-0.355780\pi$
$853$	21810.6	0.875476	0.437738	+	0.899103i	$0.355780\pi$
$854$	70129.1	2.81003
$855$	0	0
$856$	−5971.59	−0.238440
$857$	−33234.6	−1.32470	−0.662352	−	0.749193i	$-0.730442\pi$
$857$	−33234.6	−1.32470	−0.662352	+	0.749193i	$0.730442\pi$
$858$	0	0
$859$	42697.7	1.69596	0.847978	−	0.530032i	$-0.177820\pi$
$859$	42697.7	1.69596	0.847978	+	0.530032i	$0.177820\pi$
$860$	41390.3	1.64116
$861$	0	0
$862$	56525.0	2.23347
$863$	27419.2	1.08153	0.540765	−	0.841174i	$-0.318135\pi$
$863$	27419.2	1.08153	0.540765	+	0.841174i	$0.318135\pi$
$864$	0	0
$865$	36153.8	1.42112
$866$	−41437.0	−1.62596
$867$	0	0
$868$	−87177.9	−3.40900
$869$	−36358.3	−1.41930
$870$	0	0
$871$	0	0
$872$	−1365.09	−0.0530137
$873$	0	0
$874$	−2114.54	−0.0818367
$875$	−29272.4	−1.13096
$876$	0	0
$877$	−42604.2	−1.64041	−0.820206	−	0.572068i	$-0.806141\pi$
$877$	−42604.2	−1.64041	−0.820206	+	0.572068i	$0.806141\pi$
$878$	33535.2	1.28902
$879$	0	0
$880$	−29940.0	−1.14691
$881$	4699.40	0.179712	0.0898562	−	0.995955i	$-0.471359\pi$
$881$	4699.40	0.179712	0.0898562	+	0.995955i	$0.471359\pi$
$882$	0	0
$883$	22233.5	0.847358	0.423679	−	0.905812i	$-0.360738\pi$
$883$	22233.5	0.847358	0.423679	+	0.905812i	$0.360738\pi$
$884$	0	0
$885$	0	0
$886$	−9774.98	−0.370651
$887$	8852.46	0.335103	0.167552	−	0.985863i	$-0.446414\pi$
$887$	8852.46	0.335103	0.167552	+	0.985863i	$0.446414\pi$
$888$	0	0
$889$	7766.05	0.292986
$890$	51270.0	1.93098
$891$	0	0
$892$	505.155	0.0189617
$893$	3600.68	0.134930
$894$	0	0
$895$	20486.7	0.765135
$896$	16453.6	0.613477
$897$	0	0
$898$	−53283.7	−1.98007
$899$	−47508.0	−1.76249
$900$	0	0
$901$	18915.5	0.699410
$902$	−7987.58	−0.294853
$903$	0	0
$904$	2866.40	0.105459
$905$	6410.41	0.235458
$906$	0	0
$907$	37172.4	1.36085	0.680424	−	0.732818i	$-0.261795\pi$
$907$	37172.4	1.36085	0.680424	+	0.732818i	$0.261795\pi$
$908$	−6006.51	−0.219530
$909$	0	0
$910$	0	0
$911$	−38035.1	−1.38327	−0.691635	−	0.722247i	$-0.743109\pi$
$911$	−38035.1	−1.38327	−0.691635	+	0.722247i	$0.743109\pi$
$912$	0	0
$913$	−52654.8	−1.90867
$914$	34639.3	1.25357
$915$	0	0
$916$	−6509.70	−0.234810
$917$	−14838.8	−0.534372
$918$	0	0
$919$	−8352.27	−0.299800	−0.149900	−	0.988701i	$-0.547895\pi$
$919$	−8352.27	−0.299800	−0.149900	+	0.988701i	$0.547895\pi$
$920$	−1053.62	−0.0377573
$921$	0	0
$922$	−72679.4	−2.59606
$923$	0	0
$924$	0	0
$925$	−2423.47	−0.0861440
$926$	−22519.6	−0.799179
$927$	0	0
$928$	40031.3	1.41605
$929$	20232.1	0.714524	0.357262	−	0.934004i	$-0.383710\pi$
$929$	20232.1	0.714524	0.357262	+	0.934004i	$0.383710\pi$
$930$	0	0
$931$	17382.8	0.611921
$932$	2479.06	0.0871292
$933$	0	0
$934$	34065.9	1.19344
$935$	23484.7	0.821423
$936$	0	0
$937$	−27766.9	−0.968095	−0.484048	−	0.875042i	$-0.660834\pi$
$937$	−27766.9	−0.968095	−0.484048	+	0.875042i	$0.660834\pi$
$938$	−29842.6	−1.03880
$939$	0	0
$940$	16147.1	0.560277
$941$	−400.765	−0.0138837	−0.00694185	−	0.999976i	$-0.502210\pi$
$941$	−400.765	−0.0138837	−0.00694185	+	0.999976i	$0.502210\pi$
$942$	0	0
$943$	909.408	0.0314045
$944$	−32480.4	−1.11986
$945$	0	0
$946$	−57124.0	−1.96328
$947$	21804.4	0.748201	0.374101	−	0.927388i	$-0.377951\pi$
$947$	21804.4	0.748201	0.374101	+	0.927388i	$0.377951\pi$
$948$	0	0
$949$	0	0
$950$	−6195.35	−0.211583
$951$	0	0
$952$	−5589.22	−0.190281
$953$	−30480.4	−1.03605	−0.518026	−	0.855365i	$-0.673333\pi$
$953$	−30480.4	−1.03605	−0.518026	+	0.855365i	$0.673333\pi$
$954$	0	0
$955$	18412.6	0.623892
$956$	−13764.5	−0.465666
$957$	0	0
$958$	6497.48	0.219127
$959$	−57524.0	−1.93696
$960$	0	0
$961$	65239.6	2.18991
$962$	0	0
$963$	0	0
$964$	−8779.73	−0.293336
$965$	28830.1	0.961734
$966$	0	0
$967$	−23864.1	−0.793608	−0.396804	−	0.917903i	$-0.629881\pi$
$967$	−23864.1	−0.793608	−0.396804	+	0.917903i	$0.629881\pi$
$968$	−1273.79	−0.0422947
$969$	0	0
$970$	−86489.2	−2.86289
$971$	14010.3	0.463040	0.231520	−	0.972830i	$-0.425630\pi$
$971$	14010.3	0.463040	0.231520	+	0.972830i	$0.425630\pi$
$972$	0	0
$973$	−3142.19	−0.103529
$974$	−51933.3	−1.70847
$975$	0	0
$976$	29771.7	0.976404
$977$	−25448.2	−0.833326	−0.416663	−	0.909061i	$-0.636800\pi$
$977$	−25448.2	−0.833326	−0.416663	+	0.909061i	$0.636800\pi$
$978$	0	0
$979$	−37460.8	−1.22293
$980$	77952.4	2.54092
$981$	0	0
$982$	−4416.72	−0.143527
$983$	52479.9	1.70280	0.851399	−	0.524519i	$-0.175755\pi$
$983$	52479.9	1.70280	0.851399	+	0.524519i	$0.175755\pi$
$984$	0	0
$985$	−3222.66	−0.104246
$986$	−27412.8	−0.885398
$987$	0	0
$988$	0	0
$989$	6503.73	0.209107
$990$	0	0
$991$	38048.5	1.21963	0.609814	−	0.792545i	$-0.291244\pi$
$991$	38048.5	1.21963	0.609814	+	0.792545i	$0.291244\pi$
$992$	−80075.0	−2.56289
$993$	0	0
$994$	−58223.9	−1.85790
$995$	−21372.5	−0.680960
$996$	0	0
$997$	−31236.6	−0.992251	−0.496125	−	0.868251i	$-0.665244\pi$
$997$	−31236.6	−0.992251	−0.496125	+	0.868251i	$0.665244\pi$
$998$	−5864.30	−0.186003
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.z.1.2		4
3.2	odd	2		507.4.a.k.1.3		4
13.6	odd	12		117.4.q.d.10.2		4
13.11	odd	12		117.4.q.d.82.2		4
13.12	even	2	inner	1521.4.a.z.1.3		4
39.5	even	4		507.4.b.e.337.2		4
39.8	even	4		507.4.b.e.337.3		4
39.11	even	12		39.4.j.b.4.1	✓	4
39.32	even	12		39.4.j.b.10.1	yes	4
39.38	odd	2		507.4.a.k.1.2		4
156.11	odd	12		624.4.bv.c.433.1		4
156.71	odd	12		624.4.bv.c.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.b.4.1	✓	4	39.11	even	12
39.4.j.b.10.1	yes	4	39.32	even	12
117.4.q.d.10.2		4	13.6	odd	12
117.4.q.d.82.2		4	13.11	odd	12
507.4.a.k.1.2		4	39.38	odd	2
507.4.a.k.1.3		4	3.2	odd	2
507.4.b.e.337.2		4	39.5	even	4
507.4.b.e.337.3		4	39.8	even	4
624.4.bv.c.49.2		4	156.71	odd	12
624.4.bv.c.433.1		4	156.11	odd	12
1521.4.a.z.1.2		4	1.1	even	1	trivial
1521.4.a.z.1.3		4	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-4.12311 q^{2} +9.00000 q^{4} +13.4424 q^{5} +31.4219 q^{7} -4.12311 q^{8} +O(q^{10})\)	\(q-4.12311 q^{2} +9.00000 q^{4} +13.4424 q^{5} +31.4219 q^{7} -4.12311 q^{8} -55.4243 q^{10} +40.4962 q^{11} -129.556 q^{14} -55.0000 q^{16} +43.1414 q^{17} +26.9779 q^{19} +120.981 q^{20} -166.970 q^{22} +19.0100 q^{23} +55.6971 q^{25} +282.797 q^{28} +154.111 q^{29} -308.270 q^{31} +259.756 q^{32} -177.877 q^{34} +422.384 q^{35} -43.5116 q^{37} -111.233 q^{38} -55.4243 q^{40} +47.8384 q^{41} +342.121 q^{43} +364.466 q^{44} -78.3802 q^{46} +133.468 q^{47} +644.334 q^{49} -229.645 q^{50} +438.454 q^{53} +544.364 q^{55} -129.556 q^{56} -635.418 q^{58} +590.553 q^{59} -541.304 q^{61} +1271.03 q^{62} -631.000 q^{64} +230.345 q^{67} +388.273 q^{68} -1741.54 q^{70} +449.412 q^{71} +389.711 q^{73} +179.403 q^{74} +242.801 q^{76} +1272.47 q^{77} -897.820 q^{79} -739.330 q^{80} -197.243 q^{82} -1300.24 q^{83} +579.923 q^{85} -1410.60 q^{86} -166.970 q^{88} -925.045 q^{89} +171.090 q^{92} -550.301 q^{94} +362.647 q^{95} +1560.49 q^{97} -2656.66 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{4}+O(q^{10}) \) 4 * q + 36 * q^4	\( 4 q + 36 q^{4} - 136 q^{10} - 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} - 120 q^{25} - 12 q^{29} + 804 q^{35} + 612 q^{38} - 136 q^{40} + 940 q^{43} + 692 q^{49} + 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} - 2524 q^{64} + 1296 q^{68} + 3060 q^{74} + 2976 q^{77} + 8 q^{79} + 68 q^{82} - 68 q^{88} + 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100}) \) 4 * q + 36 * q^4 - 136 * q^10 - 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 - 120 * q^25 - 12 * q^29 + 804 * q^35 + 612 * q^38 - 136 * q^40 + 940 * q^43 + 692 * q^49 + 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 - 2524 * q^64 + 1296 * q^68 + 3060 * q^74 + 2976 * q^77 + 8 * q^79 + 68 * q^82 - 68 * q^88 + 2484 * q^92 - 5372 * q^94 + 108 * q^95

Introduction

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