LMFDB - Embedded newform 1521.4.a.bk.1.4

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 $ x^10 - 70x^8 + 1645x^6 - 14700x^4 + 44100x^2 - 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.04224$ 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)

$f(q)$	$=$	$q-2.04224 q^{2} -3.82924 q^{4} -12.0825 q^{5} +29.7373 q^{7} +24.1582 q^{8} +O(q^{10})$	$q-2.04224 q^{2} -3.82924 q^{4} -12.0825 q^{5} +29.7373 q^{7} +24.1582 q^{8} +24.6753 q^{10} -28.0636 q^{11} -60.7308 q^{14} -18.7029 q^{16} +50.6556 q^{17} +105.148 q^{19} +46.2667 q^{20} +57.3126 q^{22} +160.592 q^{23} +20.9857 q^{25} -113.871 q^{28} -140.105 q^{29} +223.593 q^{31} -155.070 q^{32} -103.451 q^{34} -359.300 q^{35} -228.352 q^{37} -214.739 q^{38} -291.890 q^{40} +295.902 q^{41} +192.103 q^{43} +107.462 q^{44} -327.968 q^{46} -36.9300 q^{47} +541.307 q^{49} -42.8579 q^{50} -149.102 q^{53} +339.077 q^{55} +718.399 q^{56} +286.128 q^{58} -438.867 q^{59} +286.146 q^{61} -456.631 q^{62} +466.313 q^{64} -537.128 q^{67} -193.973 q^{68} +733.777 q^{70} -102.729 q^{71} -75.5209 q^{73} +466.350 q^{74} -402.639 q^{76} -834.535 q^{77} +17.5526 q^{79} +225.977 q^{80} -604.304 q^{82} -1463.08 q^{83} -612.044 q^{85} -392.322 q^{86} -677.965 q^{88} +334.905 q^{89} -614.946 q^{92} +75.4200 q^{94} -1270.45 q^{95} -748.756 q^{97} -1105.48 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 60 q^{4}+O(q^{10}) $ 10 * q + 60 * q^4	$ 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) $ 10 * q + 60 * q^4 + 80 * q^10 + 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 + 960 * q^25 - 990 * q^29 + 120 * q^35 - 1380 * q^38 + 2000 * q^40 - 740 * q^43 + 1550 * q^49 - 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 + 3140 * q^64 - 1200 * q^68 + 4380 * q^74 - 4320 * q^77 + 1100 * q^79 - 4780 * q^82 + 6340 * q^88 + 1740 * q^92 + 6460 * q^94 + 2760 * 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 $

$2$	−2.04224	−0.722042	−0.361021	−	0.932558i	$-0.617572\pi$
$2$	−2.04224	−0.722042	−0.361021	+	0.932558i	$0.617572\pi$
$3$	0	0
$3$	0	0
$4$	−3.82924	−0.478656
$5$	−12.0825	−1.08069	−0.540344	−	0.841444i	$-0.681706\pi$
$5$	−12.0825	−1.08069	−0.540344	+	0.841444i	$0.681706\pi$
$6$	0	0
$7$	29.7373	1.60566	0.802832	−	0.596206i	$-0.203326\pi$
$7$	29.7373	1.60566	0.802832	+	0.596206i	$0.203326\pi$
$8$	24.1582	1.06765
$9$	0	0
$10$	24.6753	0.780302
$11$	−28.0636	−0.769226	−0.384613	−	0.923078i	$-0.625665\pi$
$11$	−28.0636	−0.769226	−0.384613	+	0.923078i	$0.625665\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−60.7308	−1.15936
$15$	0	0
$16$	−18.7029	−0.292233
$17$	50.6556	0.722693	0.361347	−	0.932432i	$-0.382317\pi$
$17$	50.6556	0.722693	0.361347	+	0.932432i	$0.382317\pi$
$18$	0	0
$19$	105.148	1.26962	0.634808	−	0.772670i	$-0.281079\pi$
$19$	105.148	1.26962	0.634808	+	0.772670i	$0.281079\pi$
$20$	46.2667	0.517277
$21$	0	0
$22$	57.3126	0.555413
$23$	160.592	1.45590	0.727951	−	0.685629i	$-0.240473\pi$
$23$	160.592	1.45590	0.727951	+	0.685629i	$0.240473\pi$
$24$	0	0
$25$	20.9857	0.167886
$26$	0	0
$27$	0	0
$28$	−113.871	−0.768560
$29$	−140.105	−0.897132	−0.448566	−	0.893750i	$-0.648065\pi$
$29$	−140.105	−0.897132	−0.448566	+	0.893750i	$0.648065\pi$
$30$	0	0
$31$	223.593	1.29544	0.647718	−	0.761880i	$-0.275724\pi$
$31$	223.593	1.29544	0.647718	+	0.761880i	$0.275724\pi$
$32$	−155.070	−0.856647
$33$	0	0
$34$	−103.451	−0.521815
$35$	−359.300	−1.73522
$36$	0	0
$37$	−228.352	−1.01462	−0.507308	−	0.861765i	$-0.669359\pi$
$37$	−228.352	−1.01462	−0.507308	+	0.861765i	$0.669359\pi$
$38$	−214.739	−0.916716
$39$	0	0
$40$	−291.890	−1.15380
$41$	295.902	1.12713	0.563563	−	0.826073i	$-0.309430\pi$
$41$	295.902	1.12713	0.563563	+	0.826073i	$0.309430\pi$
$42$	0	0
$43$	192.103	0.681291	0.340645	−	0.940192i	$-0.389354\pi$
$43$	192.103	0.681291	0.340645	+	0.940192i	$0.389354\pi$
$44$	107.462	0.368194
$45$	0	0
$46$	−327.968	−1.05122
$47$	−36.9300	−0.114613	−0.0573063	−	0.998357i	$-0.518251\pi$
$47$	−36.9300	−0.114613	−0.0573063	+	0.998357i	$0.518251\pi$
$48$	0	0
$49$	541.307	1.57815
$50$	−42.8579	−0.121220
$51$	0	0
$52$	0	0
$53$	−149.102	−0.386429	−0.193214	−	0.981157i	$-0.561891\pi$
$53$	−149.102	−0.386429	−0.193214	+	0.981157i	$0.561891\pi$
$54$	0	0
$55$	339.077	0.831293
$56$	718.399	1.71429
$57$	0	0
$58$	286.128	0.647767
$59$	−438.867	−0.968400	−0.484200	−	0.874957i	$-0.660889\pi$
$59$	−438.867	−0.968400	−0.484200	+	0.874957i	$0.660889\pi$
$60$	0	0
$61$	286.146	0.600610	0.300305	−	0.953843i	$-0.402912\pi$
$61$	286.146	0.600610	0.300305	+	0.953843i	$0.402912\pi$
$62$	−456.631	−0.935358
$63$	0	0
$64$	466.313	0.910768
$65$	0	0
$66$	0	0
$67$	−537.128	−0.979412	−0.489706	−	0.871888i	$-0.662896\pi$
$67$	−537.128	−0.979412	−0.489706	+	0.871888i	$0.662896\pi$
$68$	−193.973	−0.345921
$69$	0	0
$70$	733.777	1.25290
$71$	−102.729	−0.171713	−0.0858567	−	0.996307i	$-0.527363\pi$
$71$	−102.729	−0.171713	−0.0858567	+	0.996307i	$0.527363\pi$
$72$	0	0
$73$	−75.5209	−0.121083	−0.0605414	−	0.998166i	$-0.519283\pi$
$73$	−75.5209	−0.121083	−0.0605414	+	0.998166i	$0.519283\pi$
$74$	466.350	0.732596
$75$	0	0
$76$	−402.639	−0.607709
$77$	−834.535	−1.23512
$78$	0	0
$79$	17.5526	0.0249978	0.0124989	−	0.999922i	$-0.496021\pi$
$79$	17.5526	0.0249978	0.0124989	+	0.999922i	$0.496021\pi$
$80$	225.977	0.315813
$81$	0	0
$82$	−604.304	−0.813831
$83$	−1463.08	−1.93487	−0.967434	−	0.253122i	$-0.918543\pi$
$83$	−1463.08	−1.93487	−0.967434	+	0.253122i	$0.918543\pi$
$84$	0	0
$85$	−612.044	−0.781005
$86$	−392.322	−0.491920
$87$	0	0
$88$	−677.965	−0.821265
$89$	334.905	0.398875	0.199438	−	0.979911i	$-0.436089\pi$
$89$	334.905	0.398875	0.199438	+	0.979911i	$0.436089\pi$
$90$	0	0
$91$	0	0
$92$	−614.946	−0.696876
$93$	0	0
$94$	75.4200	0.0827551
$95$	−1270.45	−1.37206
$96$	0	0
$97$	−748.756	−0.783760	−0.391880	−	0.920016i	$-0.628175\pi$
$97$	−748.756	−0.783760	−0.391880	+	0.920016i	$0.628175\pi$
$98$	−1105.48	−1.13949
$99$	0	0
$100$	−80.3594	−0.0803594
$101$	784.002	0.772387	0.386194	−	0.922418i	$-0.373790\pi$
$101$	784.002	0.772387	0.386194	+	0.922418i	$0.373790\pi$
$102$	0	0
$103$	396.040	0.378864	0.189432	−	0.981894i	$-0.439335\pi$
$103$	396.040	0.378864	0.189432	+	0.981894i	$0.439335\pi$
$104$	0	0
$105$	0	0
$106$	304.502	0.279018
$107$	1436.59	1.29795	0.648974	−	0.760810i	$-0.275198\pi$
$107$	1436.59	1.29795	0.648974	+	0.760810i	$0.275198\pi$
$108$	0	0
$109$	−1977.92	−1.73807	−0.869037	−	0.494746i	$-0.835261\pi$
$109$	−1977.92	−1.73807	−0.869037	+	0.494746i	$0.835261\pi$
$110$	−692.477	−0.600228
$111$	0	0
$112$	−556.175	−0.469228
$113$	−122.405	−0.101902	−0.0509509	−	0.998701i	$-0.516225\pi$
$113$	−122.405	−0.101902	−0.0509509	+	0.998701i	$0.516225\pi$
$114$	0	0
$115$	−1940.35	−1.57338
$116$	536.496	0.429417
$117$	0	0
$118$	896.273	0.699225
$119$	1506.36	1.16040
$120$	0	0
$121$	−543.436	−0.408291
$122$	−584.379	−0.433665
$123$	0	0
$124$	−856.192	−0.620067
$125$	1256.75	0.899256
$126$	0	0
$127$	2309.61	1.61374	0.806868	−	0.590731i	$-0.201161\pi$
$127$	2309.61	1.61374	0.806868	+	0.590731i	$0.201161\pi$
$128$	288.232	0.199034
$129$	0	0
$130$	0	0
$131$	1444.26	0.963250	0.481625	−	0.876377i	$-0.340047\pi$
$131$	1444.26	0.963250	0.481625	+	0.876377i	$0.340047\pi$
$132$	0	0
$133$	3126.83	2.03858
$134$	1096.94	0.707176
$135$	0	0
$136$	1223.75	0.771584
$137$	735.918	0.458932	0.229466	−	0.973317i	$-0.426302\pi$
$137$	735.918	0.458932	0.229466	+	0.973317i	$0.426302\pi$
$138$	0	0
$139$	−1505.14	−0.918449	−0.459225	−	0.888320i	$-0.651873\pi$
$139$	−1505.14	−0.918449	−0.459225	+	0.888320i	$0.651873\pi$
$140$	1375.85	0.830573
$141$	0	0
$142$	209.797	0.123984
$143$	0	0
$144$	0	0
$145$	1692.81	0.969520
$146$	154.232	0.0874269
$147$	0	0
$148$	874.415	0.485652
$149$	−427.843	−0.235237	−0.117618	−	0.993059i	$-0.537526\pi$
$149$	−427.843	−0.235237	−0.117618	+	0.993059i	$0.537526\pi$
$150$	0	0
$151$	−1601.83	−0.863278	−0.431639	−	0.902046i	$-0.642065\pi$
$151$	−1601.83	−0.863278	−0.431639	+	0.902046i	$0.642065\pi$
$152$	2540.20	1.35551
$153$	0	0
$154$	1704.32	0.891807
$155$	−2701.55	−1.39996
$156$	0	0
$157$	−730.346	−0.371261	−0.185631	−	0.982620i	$-0.559433\pi$
$157$	−730.346	−0.371261	−0.185631	+	0.982620i	$0.559433\pi$
$158$	−35.8467	−0.0180495
$159$	0	0
$160$	1873.62	0.925767
$161$	4775.58	2.33769
$162$	0	0
$163$	1898.36	0.912215	0.456107	−	0.889925i	$-0.349243\pi$
$163$	1898.36	0.912215	0.456107	+	0.889925i	$0.349243\pi$
$164$	−1133.08	−0.539505
$165$	0	0
$166$	2987.97	1.39706
$167$	−1427.50	−0.661457	−0.330729	−	0.943726i	$-0.607294\pi$
$167$	−1427.50	−0.661457	−0.330729	+	0.943726i	$0.607294\pi$
$168$	0	0
$169$	0	0
$170$	1249.94	0.563919
$171$	0	0
$172$	−735.611	−0.326104
$173$	2044.40	0.898454	0.449227	−	0.893418i	$-0.351699\pi$
$173$	2044.40	0.898454	0.449227	+	0.893418i	$0.351699\pi$
$174$	0	0
$175$	624.058	0.269568
$176$	524.871	0.224793
$177$	0	0
$178$	−683.958	−0.288004
$179$	3889.72	1.62420	0.812098	−	0.583521i	$-0.198325\pi$
$179$	3889.72	1.62420	0.812098	+	0.583521i	$0.198325\pi$
$180$	0	0
$181$	−2477.02	−1.01721	−0.508606	−	0.861000i	$-0.669839\pi$
$181$	−2477.02	−1.01721	−0.508606	+	0.861000i	$0.669839\pi$
$182$	0	0
$183$	0	0
$184$	3879.61	1.55440
$185$	2759.05	1.09648
$186$	0	0
$187$	−1421.58	−0.555914
$188$	141.414	0.0548600
$189$	0	0
$190$	2594.57	0.990683
$191$	−2276.81	−0.862535	−0.431267	−	0.902224i	$-0.641933\pi$
$191$	−2276.81	−0.862535	−0.431267	+	0.902224i	$0.641933\pi$
$192$	0	0
$193$	3922.42	1.46291	0.731456	−	0.681888i	$-0.238841\pi$
$193$	3922.42	1.46291	0.731456	+	0.681888i	$0.238841\pi$
$194$	1529.14	0.565907
$195$	0	0
$196$	−2072.80	−0.755392
$197$	5063.23	1.83117	0.915584	−	0.402128i	$-0.131729\pi$
$197$	5063.23	1.83117	0.915584	+	0.402128i	$0.131729\pi$
$198$	0	0
$199$	3270.06	1.16487	0.582433	−	0.812879i	$-0.302101\pi$
$199$	3270.06	1.16487	0.582433	+	0.812879i	$0.302101\pi$
$200$	506.977	0.179243
$201$	0	0
$202$	−1601.12	−0.557696
$203$	−4166.34	−1.44049
$204$	0	0
$205$	−3575.22	−1.21807
$206$	−808.810	−0.273555
$207$	0	0
$208$	0	0
$209$	−2950.84	−0.976622
$210$	0	0
$211$	−2812.18	−0.917527	−0.458764	−	0.888558i	$-0.651708\pi$
$211$	−2812.18	−0.917527	−0.458764	+	0.888558i	$0.651708\pi$
$212$	570.948	0.184966
$213$	0	0
$214$	−2933.87	−0.937173
$215$	−2321.08	−0.736262
$216$	0	0
$217$	6649.05	2.08003
$218$	4039.39	1.25496
$219$	0	0
$220$	−1298.41	−0.397903
$221$	0	0
$222$	0	0
$223$	−917.736	−0.275588	−0.137794	−	0.990461i	$-0.544001\pi$
$223$	−917.736	−0.275588	−0.137794	+	0.990461i	$0.544001\pi$
$224$	−4611.35	−1.37549
$225$	0	0
$226$	249.981	0.0735774
$227$	1336.39	0.390746	0.195373	−	0.980729i	$-0.437408\pi$
$227$	1336.39	0.390746	0.195373	+	0.980729i	$0.437408\pi$
$228$	0	0
$229$	164.820	0.0475617	0.0237808	−	0.999717i	$-0.492430\pi$
$229$	164.820	0.0475617	0.0237808	+	0.999717i	$0.492430\pi$
$230$	3962.66	1.13604
$231$	0	0
$232$	−3384.68	−0.957824
$233$	4243.42	1.19312	0.596558	−	0.802570i	$-0.296535\pi$
$233$	4243.42	1.19312	0.596558	+	0.802570i	$0.296535\pi$
$234$	0	0
$235$	446.205	0.123860
$236$	1680.53	0.463530
$237$	0	0
$238$	−3076.35	−0.837859
$239$	2491.07	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$239$	2491.07	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$240$	0	0
$241$	2917.40	0.779776	0.389888	−	0.920862i	$-0.372514\pi$
$241$	2917.40	0.779776	0.389888	+	0.920862i	$0.372514\pi$
$242$	1109.83	0.294803
$243$	0	0
$244$	−1095.72	−0.287485
$245$	−6540.32	−1.70549
$246$	0	0
$247$	0	0
$248$	5401.60	1.38307
$249$	0	0
$250$	−2566.58	−0.649300
$251$	1313.88	0.330403	0.165202	−	0.986260i	$-0.447173\pi$
$251$	1313.88	0.330403	0.165202	+	0.986260i	$0.447173\pi$
$252$	0	0
$253$	−4506.79	−1.11992
$254$	−4716.78	−1.16519
$255$	0	0
$256$	−4319.15	−1.05448
$257$	987.582	0.239703	0.119851	−	0.992792i	$-0.461758\pi$
$257$	987.582	0.239703	0.119851	+	0.992792i	$0.461758\pi$
$258$	0	0
$259$	−6790.57	−1.62913
$260$	0	0
$261$	0	0
$262$	−2949.53	−0.695507
$263$	6986.45	1.63803	0.819017	−	0.573769i	$-0.194519\pi$
$263$	6986.45	1.63803	0.819017	+	0.573769i	$0.194519\pi$
$264$	0	0
$265$	1801.52	0.417609
$266$	−6385.74	−1.47194
$267$	0	0
$268$	2056.79	0.468801
$269$	5904.34	1.33827	0.669134	−	0.743142i	$-0.266665\pi$
$269$	5904.34	1.33827	0.669134	+	0.743142i	$0.266665\pi$
$270$	0	0
$271$	−2131.54	−0.477793	−0.238897	−	0.971045i	$-0.576786\pi$
$271$	−2131.54	−0.477793	−0.238897	+	0.971045i	$0.576786\pi$
$272$	−947.408	−0.211195
$273$	0	0
$274$	−1502.92	−0.331368
$275$	−588.934	−0.129142
$276$	0	0
$277$	4032.41	0.874673	0.437336	−	0.899298i	$-0.355922\pi$
$277$	4032.41	0.874673	0.437336	+	0.899298i	$0.355922\pi$
$278$	3073.86	0.663159
$279$	0	0
$280$	−8680.03	−1.85261
$281$	2298.29	0.487916	0.243958	−	0.969786i	$-0.421554\pi$
$281$	2298.29	0.487916	0.243958	+	0.969786i	$0.421554\pi$
$282$	0	0
$283$	6656.80	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$283$	6656.80	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$284$	393.373	0.0821916
$285$	0	0
$286$	0	0
$287$	8799.33	1.80978
$288$	0	0
$289$	−2347.01	−0.477715
$290$	−3457.13	−0.700034
$291$	0	0
$292$	289.188	0.0579570
$293$	−7466.99	−1.48883	−0.744413	−	0.667719i	$-0.767271\pi$
$293$	−7466.99	−1.48883	−0.744413	+	0.667719i	$0.767271\pi$
$294$	0	0
$295$	5302.59	1.04654
$296$	−5516.57	−1.08326
$297$	0	0
$298$	873.759	0.169851
$299$	0	0
$300$	0	0
$301$	5712.64	1.09392
$302$	3271.32	0.623323
$303$	0	0
$304$	−1966.58	−0.371024
$305$	−3457.34	−0.649071
$306$	0	0
$307$	−3965.99	−0.737299	−0.368650	−	0.929568i	$-0.620180\pi$
$307$	−3965.99	−0.737299	−0.368650	+	0.929568i	$0.620180\pi$
$308$	3195.64	0.591196
$309$	0	0
$310$	5517.23	1.01083
$311$	−7372.29	−1.34419	−0.672097	−	0.740463i	$-0.734606\pi$
$311$	−7372.29	−1.34419	−0.672097	+	0.740463i	$0.734606\pi$
$312$	0	0
$313$	8249.55	1.48975	0.744875	−	0.667204i	$-0.232509\pi$
$313$	8249.55	1.48975	0.744875	+	0.667204i	$0.232509\pi$
$314$	1491.54	0.268066
$315$	0	0
$316$	−67.2133	−0.0119653
$317$	5575.26	0.987817	0.493909	−	0.869514i	$-0.335568\pi$
$317$	5575.26	0.987817	0.493909	+	0.869514i	$0.335568\pi$
$318$	0	0
$319$	3931.85	0.690098
$320$	−5634.21	−0.984256
$321$	0	0
$322$	−9752.88	−1.68791
$323$	5326.35	0.917543
$324$	0	0
$325$	0	0
$326$	−3876.91	−0.658657
$327$	0	0
$328$	7148.46	1.20338
$329$	−1098.20	−0.184029
$330$	0	0
$331$	−4157.36	−0.690361	−0.345180	−	0.938536i	$-0.612182\pi$
$331$	−4157.36	−0.690361	−0.345180	+	0.938536i	$0.612182\pi$
$332$	5602.50	0.926136
$333$	0	0
$334$	2915.30	0.477600
$335$	6489.82	1.05844
$336$	0	0
$337$	3225.18	0.521326	0.260663	−	0.965430i	$-0.416059\pi$
$337$	3225.18	0.521326	0.260663	+	0.965430i	$0.416059\pi$
$338$	0	0
$339$	0	0
$340$	2343.67	0.373833
$341$	−6274.82	−0.996483
$342$	0	0
$343$	5897.11	0.928321
$344$	4640.87	0.727381
$345$	0	0
$346$	−4175.15	−0.648721
$347$	−3290.49	−0.509057	−0.254529	−	0.967065i	$-0.581920\pi$
$347$	−3290.49	−0.509057	−0.254529	+	0.967065i	$0.581920\pi$
$348$	0	0
$349$	−4491.52	−0.688899	−0.344449	−	0.938805i	$-0.611934\pi$
$349$	−4491.52	−0.688899	−0.344449	+	0.938805i	$0.611934\pi$
$350$	−1274.48	−0.194639
$351$	0	0
$352$	4351.81	0.658955
$353$	−5897.88	−0.889270	−0.444635	−	0.895712i	$-0.646667\pi$
$353$	−5897.88	−0.889270	−0.444635	+	0.895712i	$0.646667\pi$
$354$	0	0
$355$	1241.21	0.185569
$356$	−1282.43	−0.190924
$357$	0	0
$358$	−7943.75	−1.17274
$359$	9277.20	1.36388	0.681938	−	0.731410i	$-0.261137\pi$
$359$	9277.20	1.36388	0.681938	+	0.731410i	$0.261137\pi$
$360$	0	0
$361$	4197.19	0.611924
$362$	5058.67	0.734469
$363$	0	0
$364$	0	0
$365$	912.477	0.130853
$366$	0	0
$367$	−6574.36	−0.935092	−0.467546	−	0.883969i	$-0.654862\pi$
$367$	−6574.36	−0.935092	−0.467546	+	0.883969i	$0.654862\pi$
$368$	−3003.54	−0.425463
$369$	0	0
$370$	−5634.65	−0.791707
$371$	−4433.89	−0.620474
$372$	0	0
$373$	−5345.55	−0.742043	−0.371021	−	0.928624i	$-0.620992\pi$
$373$	−5345.55	−0.742043	−0.371021	+	0.928624i	$0.620992\pi$
$374$	2903.20	0.401393
$375$	0	0
$376$	−892.162	−0.122366
$377$	0	0
$378$	0	0
$379$	−1038.51	−0.140751	−0.0703757	−	0.997521i	$-0.522420\pi$
$379$	−1038.51	−0.140751	−0.0703757	+	0.997521i	$0.522420\pi$
$380$	4864.87	0.656743
$381$	0	0
$382$	4649.80	0.622786
$383$	−6749.19	−0.900437	−0.450219	−	0.892918i	$-0.648654\pi$
$383$	−6749.19	−0.900437	−0.450219	+	0.892918i	$0.648654\pi$
$384$	0	0
$385$	10083.2	1.33478
$386$	−8010.54	−1.05628
$387$	0	0
$388$	2867.17	0.375151
$389$	1246.11	0.162417	0.0812083	−	0.996697i	$-0.474122\pi$
$389$	1246.11	0.162417	0.0812083	+	0.996697i	$0.474122\pi$
$390$	0	0
$391$	8134.89	1.05217
$392$	13077.0	1.68492
$393$	0	0
$394$	−10340.3	−1.32218
$395$	−212.079	−0.0270148
$396$	0	0
$397$	8355.69	1.05632	0.528161	−	0.849144i	$-0.322882\pi$
$397$	8355.69	1.05632	0.528161	+	0.849144i	$0.322882\pi$
$398$	−6678.25	−0.841082
$399$	0	0
$400$	−392.494	−0.0490618
$401$	−3283.66	−0.408923	−0.204461	−	0.978875i	$-0.565544\pi$
$401$	−3283.66	−0.408923	−0.204461	+	0.978875i	$0.565544\pi$
$402$	0	0
$403$	0	0
$404$	−3002.14	−0.369708
$405$	0	0
$406$	8508.68	1.04010
$407$	6408.37	0.780470
$408$	0	0
$409$	10928.9	1.32127	0.660636	−	0.750706i	$-0.270287\pi$
$409$	10928.9	1.32127	0.660636	+	0.750706i	$0.270287\pi$
$410$	7301.47	0.879497
$411$	0	0
$412$	−1516.53	−0.181345
$413$	−13050.7	−1.55492
$414$	0	0
$415$	17677.6	2.09099
$416$	0	0
$417$	0	0
$418$	6026.33	0.705162
$419$	−7302.94	−0.851485	−0.425742	−	0.904844i	$-0.639987\pi$
$419$	−7302.94	−0.851485	−0.425742	+	0.904844i	$0.639987\pi$
$420$	0	0
$421$	7580.99	0.877612	0.438806	−	0.898582i	$-0.355401\pi$
$421$	7580.99	0.877612	0.438806	+	0.898582i	$0.355401\pi$
$422$	5743.15	0.662493
$423$	0	0
$424$	−3602.03	−0.412571
$425$	1063.04	0.121330
$426$	0	0
$427$	8509.19	0.964377
$428$	−5501.06	−0.621270
$429$	0	0
$430$	4740.21	0.531612
$431$	10056.7	1.12394	0.561968	−	0.827159i	$-0.310044\pi$
$431$	10056.7	1.12394	0.561968	+	0.827159i	$0.310044\pi$
$432$	0	0
$433$	2733.38	0.303367	0.151683	−	0.988429i	$-0.451531\pi$
$433$	2733.38	0.303367	0.151683	+	0.988429i	$0.451531\pi$
$434$	−13579.0	−1.50187
$435$	0	0
$436$	7573.93	0.831939
$437$	16886.0	1.84844
$438$	0	0
$439$	6744.23	0.733223	0.366611	−	0.930374i	$-0.380518\pi$
$439$	6744.23	0.733223	0.366611	+	0.930374i	$0.380518\pi$
$440$	8191.48	0.887531
$441$	0	0
$442$	0	0
$443$	−8655.69	−0.928317	−0.464158	−	0.885752i	$-0.653643\pi$
$443$	−8655.69	−0.928317	−0.464158	+	0.885752i	$0.653643\pi$
$444$	0	0
$445$	−4046.48	−0.431059
$446$	1874.24	0.198986
$447$	0	0
$448$	13866.9	1.46239
$449$	6522.46	0.685555	0.342777	−	0.939417i	$-0.388632\pi$
$449$	6522.46	0.685555	0.342777	+	0.939417i	$0.388632\pi$
$450$	0	0
$451$	−8304.07	−0.867014
$452$	468.719	0.0487759
$453$	0	0
$454$	−2729.23	−0.282135
$455$	0	0
$456$	0	0
$457$	1551.23	0.158782	0.0793909	−	0.996844i	$-0.474702\pi$
$457$	1551.23	0.158782	0.0793909	+	0.996844i	$0.474702\pi$
$458$	−336.603	−0.0343415
$459$	0	0
$460$	7430.06	0.753105
$461$	7766.25	0.784621	0.392310	−	0.919833i	$-0.371676\pi$
$461$	7766.25	0.784621	0.392310	+	0.919833i	$0.371676\pi$
$462$	0	0
$463$	2004.52	0.201205	0.100603	−	0.994927i	$-0.467923\pi$
$463$	2004.52	0.201205	0.100603	+	0.994927i	$0.467923\pi$
$464$	2620.37	0.262172
$465$	0	0
$466$	−8666.10	−0.861479
$467$	−18674.3	−1.85042	−0.925209	−	0.379458i	$-0.876111\pi$
$467$	−18674.3	−1.85042	−0.925209	+	0.379458i	$0.876111\pi$
$468$	0	0
$469$	−15972.7	−1.57261
$470$	−911.259	−0.0894324
$471$	0	0
$472$	−10602.2	−1.03391
$473$	−5391.11	−0.524067
$474$	0	0
$475$	2206.61	0.213150
$476$	−5768.22	−0.555433
$477$	0	0
$478$	−5087.36	−0.486800
$479$	−9313.02	−0.888357	−0.444178	−	0.895938i	$-0.646504\pi$
$479$	−9313.02	−0.888357	−0.444178	+	0.895938i	$0.646504\pi$
$480$	0	0
$481$	0	0
$482$	−5958.03	−0.563031
$483$	0	0
$484$	2080.95	0.195431
$485$	9046.81	0.846999
$486$	0	0
$487$	3536.80	0.329092	0.164546	−	0.986369i	$-0.447384\pi$
$487$	3536.80	0.329092	0.164546	+	0.986369i	$0.447384\pi$
$488$	6912.76	0.641241
$489$	0	0
$490$	13356.9	1.23144
$491$	3361.78	0.308992	0.154496	−	0.987993i	$-0.450625\pi$
$491$	3361.78	0.308992	0.154496	+	0.987993i	$0.450625\pi$
$492$	0	0
$493$	−7097.10	−0.648351
$494$	0	0
$495$	0	0
$496$	−4181.84	−0.378569
$497$	−3054.87	−0.275714
$498$	0	0
$499$	−4027.43	−0.361308	−0.180654	−	0.983547i	$-0.557821\pi$
$499$	−4027.43	−0.361308	−0.180654	+	0.983547i	$0.557821\pi$
$500$	−4812.40	−0.430434
$501$	0	0
$502$	−2683.26	−0.238565
$503$	−1766.67	−0.156604	−0.0783022	−	0.996930i	$-0.524950\pi$
$503$	−1766.67	−0.156604	−0.0783022	+	0.996930i	$0.524950\pi$
$504$	0	0
$505$	−9472.67	−0.834709
$506$	9203.96	0.808628
$507$	0	0
$508$	−8844.05	−0.772424
$509$	−6816.27	−0.593567	−0.296784	−	0.954945i	$-0.595914\pi$
$509$	−6816.27	−0.593567	−0.296784	+	0.954945i	$0.595914\pi$
$510$	0	0
$511$	−2245.79	−0.194418
$512$	6514.89	0.562344
$513$	0	0
$514$	−2016.88	−0.173076
$515$	−4785.13	−0.409433
$516$	0	0
$517$	1036.39	0.0881630
$518$	13868.0	1.17630
$519$	0	0
$520$	0	0
$521$	5442.27	0.457640	0.228820	−	0.973469i	$-0.426513\pi$
$521$	5442.27	0.457640	0.228820	+	0.973469i	$0.426513\pi$
$522$	0	0
$523$	20728.5	1.73307	0.866535	−	0.499117i	$-0.166342\pi$
$523$	20728.5	1.73307	0.866535	+	0.499117i	$0.166342\pi$
$524$	−5530.43	−0.461065
$525$	0	0
$526$	−14268.0	−1.18273
$527$	11326.2	0.936202
$528$	0	0
$529$	13622.8	1.11965
$530$	−3679.13	−0.301531
$531$	0	0
$532$	−11973.4	−0.975775
$533$	0	0
$534$	0	0
$535$	−17357.5	−1.40268
$536$	−12976.0	−1.04567
$537$	0	0
$538$	−12058.1	−0.966285
$539$	−15191.0	−1.21396
$540$	0	0
$541$	8577.44	0.681651	0.340825	−	0.940127i	$-0.389294\pi$
$541$	8577.44	0.681651	0.340825	+	0.940127i	$0.389294\pi$
$542$	4353.13	0.344987
$543$	0	0
$544$	−7855.14	−0.619093
$545$	23898.1	1.87832
$546$	0	0
$547$	8723.99	0.681921	0.340961	−	0.940078i	$-0.389248\pi$
$547$	8723.99	0.681921	0.340961	+	0.940078i	$0.389248\pi$
$548$	−2818.01	−0.219671
$549$	0	0
$550$	1202.75	0.0932459
$551$	−14731.8	−1.13901
$552$	0	0
$553$	521.968	0.0401380
$554$	−8235.17	−0.631550
$555$	0	0
$556$	5763.56	0.439621
$557$	965.006	0.0734087	0.0367043	−	0.999326i	$-0.488314\pi$
$557$	965.006	0.0734087	0.0367043	+	0.999326i	$0.488314\pi$
$558$	0	0
$559$	0	0
$560$	6719.95	0.507089
$561$	0	0
$562$	−4693.66	−0.352296
$563$	14605.2	1.09331	0.546657	−	0.837357i	$-0.315900\pi$
$563$	14605.2	1.09331	0.546657	+	0.837357i	$0.315900\pi$
$564$	0	0
$565$	1478.95	0.110124
$566$	−13594.8	−1.00960
$567$	0	0
$568$	−2481.74	−0.183330
$569$	−7802.48	−0.574863	−0.287432	−	0.957801i	$-0.592801\pi$
$569$	−7802.48	−0.574863	−0.287432	+	0.957801i	$0.592801\pi$
$570$	0	0
$571$	11988.2	0.878618	0.439309	−	0.898336i	$-0.355223\pi$
$571$	11988.2	0.878618	0.439309	+	0.898336i	$0.355223\pi$
$572$	0	0
$573$	0	0
$574$	−17970.4	−1.30674
$575$	3370.14	0.244425
$576$	0	0
$577$	5576.90	0.402374	0.201187	−	0.979553i	$-0.435520\pi$
$577$	5576.90	0.402374	0.201187	+	0.979553i	$0.435520\pi$
$578$	4793.17	0.344930
$579$	0	0
$580$	−6482.19	−0.464066
$581$	−43508.1	−3.10675
$582$	0	0
$583$	4184.33	0.297251
$584$	−1824.45	−0.129274
$585$	0	0
$586$	15249.4	1.07499
$587$	−26754.0	−1.88119	−0.940593	−	0.339535i	$-0.889730\pi$
$587$	−26754.0	−1.88119	−0.940593	+	0.339535i	$0.889730\pi$
$588$	0	0
$589$	23510.5	1.64471
$590$	−10829.2	−0.755644
$591$	0	0
$592$	4270.85	0.296505
$593$	3589.40	0.248565	0.124283	−	0.992247i	$-0.460337\pi$
$593$	3589.40	0.248565	0.124283	+	0.992247i	$0.460337\pi$
$594$	0	0
$595$	−18200.5	−1.25403
$596$	1638.32	0.112597
$597$	0	0
$598$	0	0
$599$	7462.78	0.509050	0.254525	−	0.967066i	$-0.418081\pi$
$599$	7462.78	0.509050	0.254525	+	0.967066i	$0.418081\pi$
$600$	0	0
$601$	−16511.0	−1.12063	−0.560316	−	0.828279i	$-0.689320\pi$
$601$	−16511.0	−1.12063	−0.560316	+	0.828279i	$0.689320\pi$
$602$	−11666.6	−0.789858
$603$	0	0
$604$	6133.80	0.413213
$605$	6566.04	0.441235
$606$	0	0
$607$	−11953.6	−0.799309	−0.399654	−	0.916666i	$-0.630870\pi$
$607$	−11953.6	−0.799309	−0.399654	+	0.916666i	$0.630870\pi$
$608$	−16305.3	−1.08761
$609$	0	0
$610$	7060.73	0.468657
$611$	0	0
$612$	0	0
$613$	−4575.65	−0.301482	−0.150741	−	0.988573i	$-0.548166\pi$
$613$	−4575.65	−0.301482	−0.150741	+	0.988573i	$0.548166\pi$
$614$	8099.51	0.532361
$615$	0	0
$616$	−20160.9	−1.31868
$617$	19231.0	1.25480	0.627400	−	0.778697i	$-0.284119\pi$
$617$	19231.0	1.25480	0.627400	+	0.778697i	$0.284119\pi$
$618$	0	0
$619$	−11715.6	−0.760727	−0.380363	−	0.924837i	$-0.624201\pi$
$619$	−11715.6	−0.760727	−0.380363	+	0.924837i	$0.624201\pi$
$620$	10344.9	0.670099
$621$	0	0
$622$	15056.0	0.970564
$623$	9959.17	0.640459
$624$	0	0
$625$	−17807.8	−1.13970
$626$	−16847.6	−1.07566
$627$	0	0
$628$	2796.68	0.177706
$629$	−11567.3	−0.733256
$630$	0	0
$631$	−8780.09	−0.553930	−0.276965	−	0.960880i	$-0.589329\pi$
$631$	−8780.09	−0.553930	−0.276965	+	0.960880i	$0.589329\pi$
$632$	424.040	0.0266889
$633$	0	0
$634$	−11386.0	−0.713245
$635$	−27905.7	−1.74395
$636$	0	0
$637$	0	0
$638$	−8029.78	−0.498279
$639$	0	0
$640$	−3482.55	−0.215094
$641$	−24991.7	−1.53996	−0.769980	−	0.638068i	$-0.779734\pi$
$641$	−24991.7	−1.53996	−0.769980	+	0.638068i	$0.779734\pi$
$642$	0	0
$643$	−2353.86	−0.144365	−0.0721827	−	0.997391i	$-0.522996\pi$
$643$	−2353.86	−0.144365	−0.0721827	+	0.997391i	$0.522996\pi$
$644$	−18286.8	−1.11895
$645$	0	0
$646$	−10877.7	−0.662504
$647$	5910.80	0.359162	0.179581	−	0.983743i	$-0.442526\pi$
$647$	5910.80	0.359162	0.179581	+	0.983743i	$0.442526\pi$
$648$	0	0
$649$	12316.2	0.744918
$650$	0	0
$651$	0	0
$652$	−7269.28	−0.436637
$653$	−5924.34	−0.355034	−0.177517	−	0.984118i	$-0.556806\pi$
$653$	−5924.34	−0.355034	−0.177517	+	0.984118i	$0.556806\pi$
$654$	0	0
$655$	−17450.2	−1.04097
$656$	−5534.23	−0.329383
$657$	0	0
$658$	2242.79	0.132877
$659$	12839.5	0.758964	0.379482	−	0.925199i	$-0.376102\pi$
$659$	12839.5	0.758964	0.379482	+	0.925199i	$0.376102\pi$
$660$	0	0
$661$	10265.5	0.604057	0.302028	−	0.953299i	$-0.402336\pi$
$661$	10265.5	0.604057	0.302028	+	0.953299i	$0.402336\pi$
$662$	8490.35	0.498469
$663$	0	0
$664$	−35345.4	−2.06577
$665$	−37779.8	−2.20306
$666$	0	0
$667$	−22499.7	−1.30614
$668$	5466.25	0.316610
$669$	0	0
$670$	−13253.8	−0.764236
$671$	−8030.27	−0.462004
$672$	0	0
$673$	9862.82	0.564909	0.282454	−	0.959281i	$-0.408851\pi$
$673$	9862.82	0.564909	0.282454	+	0.959281i	$0.408851\pi$
$674$	−6586.61	−0.376419
$675$	0	0
$676$	0	0
$677$	−32615.5	−1.85158	−0.925788	−	0.378043i	$-0.876597\pi$
$677$	−32615.5	−1.85158	−0.925788	+	0.378043i	$0.876597\pi$
$678$	0	0
$679$	−22266.0	−1.25845
$680$	−14785.9	−0.833841
$681$	0	0
$682$	12814.7	0.719502
$683$	21627.4	1.21164	0.605820	−	0.795602i	$-0.292845\pi$
$683$	21627.4	1.21164	0.605820	+	0.795602i	$0.292845\pi$
$684$	0	0
$685$	−8891.70	−0.495962
$686$	−12043.3	−0.670286
$687$	0	0
$688$	−3592.90	−0.199096
$689$	0	0
$690$	0	0
$691$	14233.1	0.783580	0.391790	−	0.920055i	$-0.371856\pi$
$691$	14233.1	0.783580	0.391790	+	0.920055i	$0.371856\pi$
$692$	−7828.49	−0.430050
$693$	0	0
$694$	6719.98	0.367561
$695$	18185.8	0.992557
$696$	0	0
$697$	14989.1	0.814566
$698$	9172.78	0.497414
$699$	0	0
$700$	−2389.67	−0.129030
$701$	28747.0	1.54887	0.774437	−	0.632651i	$-0.218033\pi$
$701$	28747.0	1.54887	0.774437	+	0.632651i	$0.218033\pi$
$702$	0	0
$703$	−24010.8	−1.28817
$704$	−13086.4	−0.700586
$705$	0	0
$706$	12044.9	0.642090
$707$	23314.1	1.24019
$708$	0	0
$709$	−1818.65	−0.0963339	−0.0481670	−	0.998839i	$-0.515338\pi$
$709$	−1818.65	−0.0963339	−0.0481670	+	0.998839i	$0.515338\pi$
$710$	−2534.86	−0.133988
$711$	0	0
$712$	8090.70	0.425859
$713$	35907.3	1.88603
$714$	0	0
$715$	0	0
$716$	−14894.7	−0.777431
$717$	0	0
$718$	−18946.3	−0.984776
$719$	−26141.8	−1.35595	−0.677973	−	0.735087i	$-0.737141\pi$
$719$	−26141.8	−1.35595	−0.677973	+	0.735087i	$0.737141\pi$
$720$	0	0
$721$	11777.2	0.608328
$722$	−8571.68	−0.441835
$723$	0	0
$724$	9485.10	0.486894
$725$	−2940.20	−0.150616
$726$	0	0
$727$	−1340.10	−0.0683652	−0.0341826	−	0.999416i	$-0.510883\pi$
$727$	−1340.10	−0.0683652	−0.0341826	+	0.999416i	$0.510883\pi$
$728$	0	0
$729$	0	0
$730$	−1863.50	−0.0944811
$731$	9731.11	0.492364
$732$	0	0
$733$	−32517.1	−1.63854	−0.819269	−	0.573409i	$-0.805620\pi$
$733$	−32517.1	−1.63854	−0.819269	+	0.573409i	$0.805620\pi$
$734$	13426.4	0.675176
$735$	0	0
$736$	−24903.0	−1.24719
$737$	15073.7	0.753389
$738$	0	0
$739$	−22170.5	−1.10359	−0.551797	−	0.833978i	$-0.686058\pi$
$739$	−22170.5	−1.10359	−0.551797	+	0.833978i	$0.686058\pi$
$740$	−10565.1	−0.524838
$741$	0	0
$742$	9055.07	0.448008
$743$	30387.8	1.50043	0.750216	−	0.661193i	$-0.229949\pi$
$743$	30387.8	1.50043	0.750216	+	0.661193i	$0.229949\pi$
$744$	0	0
$745$	5169.39	0.254217
$746$	10916.9	0.535786
$747$	0	0
$748$	5443.57	0.266092
$749$	42720.3	2.08407
$750$	0	0
$751$	−16898.3	−0.821077	−0.410539	−	0.911843i	$-0.634659\pi$
$751$	−16898.3	−0.821077	−0.410539	+	0.911843i	$0.634659\pi$
$752$	690.699	0.0334936
$753$	0	0
$754$	0	0
$755$	19354.0	0.932934
$756$	0	0
$757$	32925.8	1.58086	0.790428	−	0.612555i	$-0.209858\pi$
$757$	32925.8	1.58086	0.790428	+	0.612555i	$0.209858\pi$
$758$	2120.89	0.101628
$759$	0	0
$760$	−30691.8	−1.46488
$761$	16792.0	0.799882	0.399941	−	0.916541i	$-0.369031\pi$
$761$	16792.0	0.799882	0.399941	+	0.916541i	$0.369031\pi$
$762$	0	0
$763$	−58817.9	−2.79076
$764$	8718.46	0.412857
$765$	0	0
$766$	13783.5	0.650153
$767$	0	0
$768$	0	0
$769$	−1541.57	−0.0722891	−0.0361445	−	0.999347i	$-0.511508\pi$
$769$	−1541.57	−0.0722891	−0.0361445	+	0.999347i	$0.511508\pi$
$770$	−20592.4	−0.963765
$771$	0	0
$772$	−15019.9	−0.700231
$773$	38071.2	1.77144	0.885721	−	0.464218i	$-0.153665\pi$
$773$	38071.2	1.77144	0.885721	+	0.464218i	$0.153665\pi$
$774$	0	0
$775$	4692.26	0.217485
$776$	−18088.6	−0.836782
$777$	0	0
$778$	−2544.85	−0.117272
$779$	31113.6	1.43102
$780$	0	0
$781$	2882.93	0.132086
$782$	−16613.4	−0.759712
$783$	0	0
$784$	−10124.0	−0.461189
$785$	8824.38	0.401217
$786$	0	0
$787$	20049.8	0.908129	0.454065	−	0.890969i	$-0.349973\pi$
$787$	20049.8	0.908129	0.454065	+	0.890969i	$0.349973\pi$
$788$	−19388.3	−0.876498
$789$	0	0
$790$	433.117	0.0195058
$791$	−3640.00	−0.163620
$792$	0	0
$793$	0	0
$794$	−17064.4	−0.762709
$795$	0	0
$796$	−12521.9	−0.557570
$797$	22401.9	0.995627	0.497813	−	0.867284i	$-0.334136\pi$
$797$	22401.9	0.995627	0.497813	+	0.867284i	$0.334136\pi$
$798$	0	0
$799$	−1870.71	−0.0828298
$800$	−3254.24	−0.143819
$801$	0	0
$802$	6706.02	0.295259
$803$	2119.39	0.0931401
$804$	0	0
$805$	−57700.7	−2.52631
$806$	0	0
$807$	0	0
$808$	18940.1	0.824640
$809$	41966.4	1.82381	0.911903	−	0.410406i	$-0.134613\pi$
$809$	41966.4	1.82381	0.911903	+	0.410406i	$0.134613\pi$
$810$	0	0
$811$	−13029.1	−0.564133	−0.282067	−	0.959395i	$-0.591020\pi$
$811$	−13029.1	−0.564133	−0.282067	+	0.959395i	$0.591020\pi$
$812$	15953.9	0.689500
$813$	0	0
$814$	−13087.4	−0.563532
$815$	−22936.8	−0.985819
$816$	0	0
$817$	20199.4	0.864977
$818$	−22319.5	−0.954014
$819$	0	0
$820$	13690.4	0.583036
$821$	−8898.14	−0.378255	−0.189127	−	0.981953i	$-0.560566\pi$
$821$	−8898.14	−0.378255	−0.189127	+	0.981953i	$0.560566\pi$
$822$	0	0
$823$	22723.6	0.962448	0.481224	−	0.876598i	$-0.340192\pi$
$823$	22723.6	0.962448	0.481224	+	0.876598i	$0.340192\pi$
$824$	9567.61	0.404494
$825$	0	0
$826$	26652.7	1.12272
$827$	−19073.3	−0.801989	−0.400994	−	0.916081i	$-0.631335\pi$
$827$	−19073.3	−0.801989	−0.400994	+	0.916081i	$0.631335\pi$
$828$	0	0
$829$	42503.8	1.78072	0.890361	−	0.455255i	$-0.150452\pi$
$829$	42503.8	1.78072	0.890361	+	0.455255i	$0.150452\pi$
$830$	−36102.0	−1.50978
$831$	0	0
$832$	0	0
$833$	27420.2	1.14052
$834$	0	0
$835$	17247.7	0.714829
$836$	11299.5	0.467465
$837$	0	0
$838$	14914.4	0.614808
$839$	19427.2	0.799408	0.399704	−	0.916644i	$-0.369113\pi$
$839$	19427.2	0.799408	0.399704	+	0.916644i	$0.369113\pi$
$840$	0	0
$841$	−4759.60	−0.195154
$842$	−15482.2	−0.633673
$843$	0	0
$844$	10768.5	0.439180
$845$	0	0
$846$	0	0
$847$	−16160.3	−0.655578
$848$	2788.64	0.112927
$849$	0	0
$850$	−2170.99	−0.0876052
$851$	−36671.5	−1.47718
$852$	0	0
$853$	26851.8	1.07783	0.538914	−	0.842361i	$-0.318835\pi$
$853$	26851.8	1.07783	0.538914	+	0.842361i	$0.318835\pi$
$854$	−17377.8	−0.696320
$855$	0	0
$856$	34705.4	1.38576
$857$	−41539.4	−1.65573	−0.827864	−	0.560929i	$-0.810444\pi$
$857$	−41539.4	−1.65573	−0.827864	+	0.560929i	$0.810444\pi$
$858$	0	0
$859$	−11936.2	−0.474107	−0.237054	−	0.971497i	$-0.576182\pi$
$859$	−11936.2	−0.474107	−0.237054	+	0.971497i	$0.576182\pi$
$860$	8887.99	0.352416
$861$	0	0
$862$	−20538.3	−0.811529
$863$	−41128.6	−1.62229	−0.811143	−	0.584848i	$-0.801154\pi$
$863$	−41128.6	−1.62229	−0.811143	+	0.584848i	$0.801154\pi$
$864$	0	0
$865$	−24701.3	−0.970948
$866$	−5582.22	−0.219043
$867$	0	0
$868$	−25460.8	−0.995619
$869$	−492.590	−0.0192290
$870$	0	0
$871$	0	0
$872$	−47782.9	−1.85566
$873$	0	0
$874$	−34485.3	−1.33465
$875$	37372.3	1.44390
$876$	0	0
$877$	−6406.86	−0.246687	−0.123343	−	0.992364i	$-0.539362\pi$
$877$	−6406.86	−0.246687	−0.123343	+	0.992364i	$0.539362\pi$
$878$	−13773.4	−0.529417
$879$	0	0
$880$	−6341.73	−0.242931
$881$	2938.07	0.112357	0.0561783	−	0.998421i	$-0.482108\pi$
$881$	2938.07	0.112357	0.0561783	+	0.998421i	$0.482108\pi$
$882$	0	0
$883$	3022.06	0.115176	0.0575881	−	0.998340i	$-0.481659\pi$
$883$	3022.06	0.115176	0.0575881	+	0.998340i	$0.481659\pi$
$884$	0	0
$885$	0	0
$886$	17677.0	0.670284
$887$	10060.4	0.380830	0.190415	−	0.981704i	$-0.439017\pi$
$887$	10060.4	0.380830	0.190415	+	0.981704i	$0.439017\pi$
$888$	0	0
$889$	68681.5	2.59112
$890$	8263.89	0.311243
$891$	0	0
$892$	3514.24	0.131912
$893$	−3883.13	−0.145514
$894$	0	0
$895$	−46997.3	−1.75525
$896$	8571.24	0.319582
$897$	0	0
$898$	−13320.5	−0.494999
$899$	−31326.5	−1.16218
$900$	0	0
$901$	−7552.84	−0.279269
$902$	16958.9	0.626020
$903$	0	0
$904$	−2957.09	−0.108796
$905$	29928.4	1.09929
$906$	0	0
$907$	−43158.6	−1.58000	−0.789999	−	0.613108i	$-0.789919\pi$
$907$	−43158.6	−1.58000	−0.789999	+	0.613108i	$0.789919\pi$
$908$	−5117.36	−0.187033
$909$	0	0
$910$	0	0
$911$	32665.9	1.18800	0.594001	−	0.804464i	$-0.297547\pi$
$911$	32665.9	1.18800	0.594001	+	0.804464i	$0.297547\pi$
$912$	0	0
$913$	41059.3	1.48835
$914$	−3167.98	−0.114647
$915$	0	0
$916$	−631.137	−0.0227657
$917$	42948.4	1.54665
$918$	0	0
$919$	18989.9	0.681633	0.340816	−	0.940130i	$-0.389297\pi$
$919$	18989.9	0.681633	0.340816	+	0.940130i	$0.389297\pi$
$920$	−46875.3	−1.67982
$921$	0	0
$922$	−15860.6	−0.566529
$923$	0	0
$924$	0	0
$925$	−4792.12	−0.170340
$926$	−4093.72	−0.145279
$927$	0	0
$928$	21726.0	0.768525
$929$	5596.81	0.197659	0.0988295	−	0.995104i	$-0.468490\pi$
$929$	5596.81	0.197659	0.0988295	+	0.995104i	$0.468490\pi$
$930$	0	0
$931$	56917.6	2.00365
$932$	−16249.1	−0.571091
$933$	0	0
$934$	38137.5	1.33608
$935$	17176.1	0.600770
$936$	0	0
$937$	40294.4	1.40487	0.702433	−	0.711750i	$-0.252097\pi$
$937$	40294.4	1.40487	0.702433	+	0.711750i	$0.252097\pi$
$938$	32620.2	1.13549
$939$	0	0
$940$	−1708.63	−0.0592865
$941$	−43648.8	−1.51213	−0.756063	−	0.654499i	$-0.772880\pi$
$941$	−43648.8	−1.51213	−0.756063	+	0.654499i	$0.772880\pi$
$942$	0	0
$943$	47519.5	1.64098
$944$	8208.10	0.282999
$945$	0	0
$946$	11010.0	0.378398
$947$	10486.6	0.359839	0.179919	−	0.983681i	$-0.442416\pi$
$947$	10486.6	0.359839	0.179919	+	0.983681i	$0.442416\pi$
$948$	0	0
$949$	0	0
$950$	−4506.44	−0.153903
$951$	0	0
$952$	36390.9	1.23890
$953$	−33058.8	−1.12369	−0.561846	−	0.827242i	$-0.689909\pi$
$953$	−33058.8	−1.12369	−0.561846	+	0.827242i	$0.689909\pi$
$954$	0	0
$955$	27509.5	0.932131
$956$	−9538.90	−0.322709
$957$	0	0
$958$	19019.5	0.641431
$959$	21884.2	0.736891
$960$	0	0
$961$	20202.8	0.678152
$962$	0	0
$963$	0	0
$964$	−11171.4	−0.373244
$965$	−47392.5	−1.58095
$966$	0	0
$967$	−53634.9	−1.78364	−0.891821	−	0.452389i	$-0.850572\pi$
$967$	−53634.9	−1.78364	−0.891821	+	0.452389i	$0.850572\pi$
$968$	−13128.4	−0.435913
$969$	0	0
$970$	−18475.8	−0.611569
$971$	4086.80	0.135069	0.0675344	−	0.997717i	$-0.478487\pi$
$971$	4086.80	0.135069	0.0675344	+	0.997717i	$0.478487\pi$
$972$	0	0
$973$	−44758.8	−1.47472
$974$	−7223.00	−0.237618
$975$	0	0
$976$	−5351.76	−0.175518
$977$	−14381.0	−0.470919	−0.235459	−	0.971884i	$-0.575659\pi$
$977$	−14381.0	−0.470919	−0.235459	+	0.971884i	$0.575659\pi$
$978$	0	0
$979$	−9398.64	−0.306825
$980$	25044.5	0.816343
$981$	0	0
$982$	−6865.57	−0.223105
$983$	12916.5	0.419099	0.209549	−	0.977798i	$-0.432800\pi$
$983$	12916.5	0.419099	0.209549	+	0.977798i	$0.432800\pi$
$984$	0	0
$985$	−61176.2	−1.97892
$986$	14494.0	0.468137
$987$	0	0
$988$	0	0
$989$	30850.3	0.991893
$990$	0	0
$991$	−5838.98	−0.187166	−0.0935829	−	0.995611i	$-0.529832\pi$
$991$	−5838.98	−0.187166	−0.0935829	+	0.995611i	$0.529832\pi$
$992$	−34672.5	−1.10973
$993$	0	0
$994$	6238.79	0.199077
$995$	−39510.3	−1.25886
$996$	0	0
$997$	44290.1	1.40690	0.703452	−	0.710743i	$-0.251641\pi$
$997$	44290.1	1.40690	0.703452	+	0.710743i	$0.251641\pi$
$998$	8224.99	0.260879
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bk.1.4		10
3.2	odd	2		507.4.a.r.1.7		10
13.6	odd	12		117.4.q.e.10.4		10
13.11	odd	12		117.4.q.e.82.4		10
13.12	even	2	inner	1521.4.a.bk.1.7		10
39.5	even	4		507.4.b.i.337.4		10
39.8	even	4		507.4.b.i.337.7		10
39.11	even	12		39.4.j.c.4.2	✓	10
39.32	even	12		39.4.j.c.10.2	yes	10
39.38	odd	2		507.4.a.r.1.4		10
156.11	odd	12		624.4.bv.h.433.4		10
156.71	odd	12		624.4.bv.h.49.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.2	✓	10	39.11	even	12
39.4.j.c.10.2	yes	10	39.32	even	12
117.4.q.e.10.4		10	13.6	odd	12
117.4.q.e.82.4		10	13.11	odd	12
507.4.a.r.1.4		10	39.38	odd	2
507.4.a.r.1.7		10	3.2	odd	2
507.4.b.i.337.4		10	39.5	even	4
507.4.b.i.337.7		10	39.8	even	4
624.4.bv.h.49.2		10	156.71	odd	12
624.4.bv.h.433.4		10	156.11	odd	12
1521.4.a.bk.1.4		10	1.1	even	1	trivial
1521.4.a.bk.1.7		10	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)

\(f(q)\)	\(=\)	\(q-2.04224 q^{2} -3.82924 q^{4} -12.0825 q^{5} +29.7373 q^{7} +24.1582 q^{8} +O(q^{10})\)	\(q-2.04224 q^{2} -3.82924 q^{4} -12.0825 q^{5} +29.7373 q^{7} +24.1582 q^{8} +24.6753 q^{10} -28.0636 q^{11} -60.7308 q^{14} -18.7029 q^{16} +50.6556 q^{17} +105.148 q^{19} +46.2667 q^{20} +57.3126 q^{22} +160.592 q^{23} +20.9857 q^{25} -113.871 q^{28} -140.105 q^{29} +223.593 q^{31} -155.070 q^{32} -103.451 q^{34} -359.300 q^{35} -228.352 q^{37} -214.739 q^{38} -291.890 q^{40} +295.902 q^{41} +192.103 q^{43} +107.462 q^{44} -327.968 q^{46} -36.9300 q^{47} +541.307 q^{49} -42.8579 q^{50} -149.102 q^{53} +339.077 q^{55} +718.399 q^{56} +286.128 q^{58} -438.867 q^{59} +286.146 q^{61} -456.631 q^{62} +466.313 q^{64} -537.128 q^{67} -193.973 q^{68} +733.777 q^{70} -102.729 q^{71} -75.5209 q^{73} +466.350 q^{74} -402.639 q^{76} -834.535 q^{77} +17.5526 q^{79} +225.977 q^{80} -604.304 q^{82} -1463.08 q^{83} -612.044 q^{85} -392.322 q^{86} -677.965 q^{88} +334.905 q^{89} -614.946 q^{92} +75.4200 q^{94} -1270.45 q^{95} -748.756 q^{97} -1105.48 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 60 q^{4}+O(q^{10}) \) 10 * q + 60 * q^4	\( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 10 * q + 60 * q^4 + 80 * q^10 + 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 + 960 * q^25 - 990 * q^29 + 120 * q^35 - 1380 * q^38 + 2000 * q^40 - 740 * q^43 + 1550 * q^49 - 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 + 3140 * q^64 - 1200 * q^68 + 4380 * q^74 - 4320 * q^77 + 1100 * q^79 - 4780 * q^82 + 6340 * q^88 + 1740 * q^92 + 6460 * q^94 + 2760 * q^95

Introduction

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