LMFDB - Embedded newform 1521.4.a.bk.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:	$10$
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			1.2
Root			$-5.04537$ 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)

$f(q)$	$=$	$q-5.04537 q^{2} +17.4557 q^{4} -20.1174 q^{5} -15.4279 q^{7} -47.7076 q^{8} +O(q^{10})$	$q-5.04537 q^{2} +17.4557 q^{4} -20.1174 q^{5} -15.4279 q^{7} -47.7076 q^{8} +101.500 q^{10} +26.9372 q^{11} +77.8394 q^{14} +101.057 q^{16} -23.2334 q^{17} +45.0794 q^{19} -351.164 q^{20} -135.908 q^{22} +142.010 q^{23} +279.710 q^{25} -269.305 q^{28} -2.29068 q^{29} +37.7740 q^{31} -128.207 q^{32} +117.221 q^{34} +310.369 q^{35} -313.840 q^{37} -227.442 q^{38} +959.753 q^{40} -5.86820 q^{41} -360.898 q^{43} +470.209 q^{44} -716.493 q^{46} -209.748 q^{47} -104.980 q^{49} -1411.24 q^{50} -276.886 q^{53} -541.906 q^{55} +736.028 q^{56} +11.5573 q^{58} -543.189 q^{59} +205.788 q^{61} -190.583 q^{62} -161.602 q^{64} -492.578 q^{67} -405.557 q^{68} -1565.93 q^{70} +826.859 q^{71} -66.1205 q^{73} +1583.44 q^{74} +786.894 q^{76} -415.584 q^{77} +317.642 q^{79} -2033.00 q^{80} +29.6072 q^{82} +141.450 q^{83} +467.396 q^{85} +1820.86 q^{86} -1285.11 q^{88} +641.320 q^{89} +2478.89 q^{92} +1058.26 q^{94} -906.880 q^{95} +1114.92 q^{97} +529.663 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.04537	−1.78381	−0.891903	−	0.452226i	$-0.850630\pi$
$2$	−5.04537	−1.78381	−0.891903	+	0.452226i	$0.850630\pi$
$3$	0	0
$3$	0	0
$4$	17.4557	2.18197
$5$	−20.1174	−1.79935	−0.899677	−	0.436556i	$-0.856198\pi$
$5$	−20.1174	−1.79935	−0.899677	+	0.436556i	$0.856198\pi$
$6$	0	0
$7$	−15.4279	−0.833028	−0.416514	−	0.909129i	$-0.636748\pi$
$7$	−15.4279	−0.833028	−0.416514	+	0.909129i	$0.636748\pi$
$8$	−47.7076	−2.10840
$9$	0	0
$10$	101.500	3.20970
$11$	26.9372	0.738352	0.369176	−	0.929359i	$-0.379640\pi$
$11$	26.9372	0.738352	0.369176	+	0.929359i	$0.379640\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	77.8394	1.48596
$15$	0	0
$16$	101.057	1.57901
$17$	−23.2334	−0.331467	−0.165733	−	0.986171i	$-0.552999\pi$
$17$	−23.2334	−0.331467	−0.165733	+	0.986171i	$0.552999\pi$
$18$	0	0
$19$	45.0794	0.544312	0.272156	−	0.962253i	$-0.412263\pi$
$19$	45.0794	0.544312	0.272156	+	0.962253i	$0.412263\pi$
$20$	−351.164	−3.92613
$21$	0	0
$22$	−135.908	−1.31708
$23$	142.010	1.28744	0.643720	−	0.765261i	$-0.277390\pi$
$23$	142.010	1.28744	0.643720	+	0.765261i	$0.277390\pi$
$24$	0	0
$25$	279.710	2.23768
$26$	0	0
$27$	0	0
$28$	−269.305	−1.81764
$29$	−2.29068	−0.0146679	−0.00733394	−	0.999973i	$-0.502334\pi$
$29$	−2.29068	−0.0146679	−0.00733394	+	0.999973i	$0.502334\pi$
$30$	0	0
$31$	37.7740	0.218852	0.109426	−	0.993995i	$-0.465099\pi$
$31$	37.7740	0.218852	0.109426	+	0.993995i	$0.465099\pi$
$32$	−128.207	−0.708251
$33$	0	0
$34$	117.221	0.591273
$35$	310.369	1.49891
$36$	0	0
$37$	−313.840	−1.39446	−0.697228	−	0.716849i	$-0.745584\pi$
$37$	−313.840	−1.39446	−0.697228	+	0.716849i	$0.745584\pi$
$38$	−227.442	−0.970947
$39$	0	0
$40$	959.753	3.79376
$41$	−5.86820	−0.0223527	−0.0111763	−	0.999938i	$-0.503558\pi$
$41$	−5.86820	−0.0223527	−0.0111763	+	0.999938i	$0.503558\pi$
$42$	0	0
$43$	−360.898	−1.27992	−0.639958	−	0.768410i	$-0.721048\pi$
$43$	−360.898	−1.27992	−0.639958	+	0.768410i	$0.721048\pi$
$44$	470.209	1.61106
$45$	0	0
$46$	−716.493	−2.29655
$47$	−209.748	−0.650956	−0.325478	−	0.945550i	$-0.605525\pi$
$47$	−209.748	−0.650956	−0.325478	+	0.945550i	$0.605525\pi$
$48$	0	0
$49$	−104.980	−0.306064
$50$	−1411.24	−3.99158
$51$	0	0
$52$	0	0
$53$	−276.886	−0.717609	−0.358804	−	0.933413i	$-0.616815\pi$
$53$	−276.886	−0.717609	−0.358804	+	0.933413i	$0.616815\pi$
$54$	0	0
$55$	−541.906	−1.32856
$56$	736.028	1.75636
$57$	0	0
$58$	11.5573	0.0261647
$59$	−543.189	−1.19860	−0.599298	−	0.800526i	$-0.704553\pi$
$59$	−543.189	−1.19860	−0.599298	+	0.800526i	$0.704553\pi$
$60$	0	0
$61$	205.788	0.431942	0.215971	−	0.976400i	$-0.430708\pi$
$61$	205.788	0.431942	0.215971	+	0.976400i	$0.430708\pi$
$62$	−190.583	−0.390389
$63$	0	0
$64$	−161.602	−0.315629
$65$	0	0
$66$	0	0
$67$	−492.578	−0.898179	−0.449090	−	0.893487i	$-0.648252\pi$
$67$	−492.578	−0.898179	−0.449090	+	0.893487i	$0.648252\pi$
$68$	−405.557	−0.723249
$69$	0	0
$70$	−1565.93	−2.67377
$71$	826.859	1.38211	0.691057	−	0.722800i	$-0.257145\pi$
$71$	826.859	1.38211	0.691057	+	0.722800i	$0.257145\pi$
$72$	0	0
$73$	−66.1205	−0.106011	−0.0530056	−	0.998594i	$-0.516880\pi$
$73$	−66.1205	−0.106011	−0.0530056	+	0.998594i	$0.516880\pi$
$74$	1583.44	2.48744
$75$	0	0
$76$	786.894	1.18767
$77$	−415.584	−0.615068
$78$	0	0
$79$	317.642	0.452374	0.226187	−	0.974084i	$-0.427374\pi$
$79$	317.642	0.452374	0.226187	+	0.974084i	$0.427374\pi$
$80$	−2033.00	−2.84120
$81$	0	0
$82$	29.6072	0.0398728
$83$	141.450	0.187063	0.0935313	−	0.995616i	$-0.470184\pi$
$83$	141.450	0.187063	0.0935313	+	0.995616i	$0.470184\pi$
$84$	0	0
$85$	467.396	0.596426
$86$	1820.86	2.28312
$87$	0	0
$88$	−1285.11	−1.55674
$89$	641.320	0.763818	0.381909	−	0.924200i	$-0.375267\pi$
$89$	641.320	0.763818	0.381909	+	0.924200i	$0.375267\pi$
$90$	0	0
$91$	0	0
$92$	2478.89	2.80915
$93$	0	0
$94$	1058.26	1.16118
$95$	−906.880	−0.979410
$96$	0	0
$97$	1114.92	1.16704	0.583522	−	0.812097i	$-0.301674\pi$
$97$	1114.92	1.16704	0.583522	+	0.812097i	$0.301674\pi$
$98$	529.663	0.545960
$99$	0	0
$100$	4882.53	4.88253
$101$	−1589.91	−1.56635	−0.783177	−	0.621799i	$-0.786402\pi$
$101$	−1589.91	−1.56635	−0.783177	+	0.621799i	$0.786402\pi$
$102$	0	0
$103$	527.502	0.504625	0.252312	−	0.967646i	$-0.418809\pi$
$103$	527.502	0.504625	0.252312	+	0.967646i	$0.418809\pi$
$104$	0	0
$105$	0	0
$106$	1396.99	1.28008
$107$	−1751.96	−1.58289	−0.791443	−	0.611243i	$-0.790670\pi$
$107$	−1751.96	−1.58289	−0.791443	+	0.611243i	$0.790670\pi$
$108$	0	0
$109$	−967.122	−0.849848	−0.424924	−	0.905229i	$-0.639699\pi$
$109$	−967.122	−0.849848	−0.424924	+	0.905229i	$0.639699\pi$
$110$	2734.12	2.36989
$111$	0	0
$112$	−1559.09	−1.31536
$113$	−1910.97	−1.59088	−0.795439	−	0.606033i	$-0.792760\pi$
$113$	−1910.97	−1.59088	−0.795439	+	0.606033i	$0.792760\pi$
$114$	0	0
$115$	−2856.87	−2.31656
$116$	−39.9855	−0.0320048
$117$	0	0
$118$	2740.59	2.13806
$119$	358.443	0.276121
$120$	0	0
$121$	−605.387	−0.454836
$122$	−1038.28	−0.770501
$123$	0	0
$124$	659.372	0.477527
$125$	−3112.35	−2.22702
$126$	0	0
$127$	1233.11	0.861584	0.430792	−	0.902451i	$-0.358234\pi$
$127$	1233.11	0.861584	0.430792	+	0.902451i	$0.358234\pi$
$128$	1841.00	1.27127
$129$	0	0
$130$	0	0
$131$	1274.90	0.850292	0.425146	−	0.905125i	$-0.360223\pi$
$131$	1274.90	0.850292	0.425146	+	0.905125i	$0.360223\pi$
$132$	0	0
$133$	−695.480	−0.453427
$134$	2485.24	1.60218
$135$	0	0
$136$	1108.41	0.698864
$137$	2031.32	1.26677	0.633385	−	0.773837i	$-0.281665\pi$
$137$	2031.32	1.26677	0.633385	+	0.773837i	$0.281665\pi$
$138$	0	0
$139$	−1445.66	−0.882156	−0.441078	−	0.897469i	$-0.645404\pi$
$139$	−1445.66	−0.882156	−0.441078	+	0.897469i	$0.645404\pi$
$140$	5417.72	3.27058
$141$	0	0
$142$	−4171.81	−2.46542
$143$	0	0
$144$	0	0
$145$	46.0825	0.0263927
$146$	333.602	0.189103
$147$	0	0
$148$	−5478.30	−3.04266
$149$	966.318	0.531301	0.265650	−	0.964069i	$-0.414413\pi$
$149$	966.318	0.531301	0.265650	+	0.964069i	$0.414413\pi$
$150$	0	0
$151$	−1463.09	−0.788505	−0.394252	−	0.919002i	$-0.628996\pi$
$151$	−1463.09	−0.788505	−0.394252	+	0.919002i	$0.628996\pi$
$152$	−2150.63	−1.14763
$153$	0	0
$154$	2096.78	1.09716
$155$	−759.914	−0.393792
$156$	0	0
$157$	−66.0424	−0.0335717	−0.0167859	−	0.999859i	$-0.505343\pi$
$157$	−66.0424	−0.0335717	−0.0167859	+	0.999859i	$0.505343\pi$
$158$	−1602.62	−0.806948
$159$	0	0
$160$	2579.19	1.27439
$161$	−2190.92	−1.07247
$162$	0	0
$163$	−3525.24	−1.69398	−0.846988	−	0.531612i	$-0.821587\pi$
$163$	−3525.24	−1.69398	−0.846988	+	0.531612i	$0.821587\pi$
$164$	−102.434	−0.0487728
$165$	0	0
$166$	−713.669	−0.333683
$167$	260.652	0.120777	0.0603887	−	0.998175i	$-0.480766\pi$
$167$	260.652	0.120777	0.0603887	+	0.998175i	$0.480766\pi$
$168$	0	0
$169$	0	0
$170$	−2358.19	−1.06391
$171$	0	0
$172$	−6299.74	−2.79273
$173$	−911.753	−0.400689	−0.200345	−	0.979725i	$-0.564206\pi$
$173$	−911.753	−0.400689	−0.200345	+	0.979725i	$0.564206\pi$
$174$	0	0
$175$	−4315.33	−1.86405
$176$	2722.18	1.16587
$177$	0	0
$178$	−3235.70	−1.36250
$179$	−2690.48	−1.12344	−0.561721	−	0.827327i	$-0.689861\pi$
$179$	−2690.48	−1.12344	−0.561721	+	0.827327i	$0.689861\pi$
$180$	0	0
$181$	4773.85	1.96043	0.980213	−	0.197944i	$-0.0634265\pi$
$181$	4773.85	1.96043	0.980213	+	0.197944i	$0.0634265\pi$
$182$	0	0
$183$	0	0
$184$	−6774.96	−2.71444
$185$	6313.63	2.50912
$186$	0	0
$187$	−625.844	−0.244739
$188$	−3661.31	−1.42036
$189$	0	0
$190$	4575.54	1.74708
$191$	2057.47	0.779443	0.389721	−	0.920933i	$-0.372571\pi$
$191$	2057.47	0.779443	0.389721	+	0.920933i	$0.372571\pi$
$192$	0	0
$193$	−729.873	−0.272215	−0.136107	−	0.990694i	$-0.543459\pi$
$193$	−729.873	−0.272215	−0.136107	+	0.990694i	$0.543459\pi$
$194$	−5625.20	−2.08178
$195$	0	0
$196$	−1832.50	−0.667822
$197$	−1701.11	−0.615225	−0.307613	−	0.951512i	$-0.599530\pi$
$197$	−1701.11	−0.615225	−0.307613	+	0.951512i	$0.599530\pi$
$198$	0	0
$199$	−1840.88	−0.655761	−0.327881	−	0.944719i	$-0.606334\pi$
$199$	−1840.88	−0.655761	−0.327881	+	0.944719i	$0.606334\pi$
$200$	−13344.3	−4.71792
$201$	0	0
$202$	8021.66	2.79407
$203$	35.3404	0.0122188
$204$	0	0
$205$	118.053	0.0402204
$206$	−2661.44	−0.900153
$207$	0	0
$208$	0	0
$209$	1214.31	0.401894
$210$	0	0
$211$	142.970	0.0466467	0.0233234	−	0.999728i	$-0.492575\pi$
$211$	142.970	0.0466467	0.0233234	+	0.999728i	$0.492575\pi$
$212$	−4833.25	−1.56580
$213$	0	0
$214$	8839.31	2.82356
$215$	7260.32	2.30302
$216$	0	0
$217$	−582.773	−0.182310
$218$	4879.48	1.51597
$219$	0	0
$220$	−9459.37	−2.89887
$221$	0	0
$222$	0	0
$223$	−2338.71	−0.702295	−0.351148	−	0.936320i	$-0.614208\pi$
$223$	−2338.71	−0.702295	−0.351148	+	0.936320i	$0.614208\pi$
$224$	1977.96	0.589993
$225$	0	0
$226$	9641.57	2.83782
$227$	−3279.36	−0.958850	−0.479425	−	0.877583i	$-0.659155\pi$
$227$	−3279.36	−0.958850	−0.479425	+	0.877583i	$0.659155\pi$
$228$	0	0
$229$	1143.72	0.330041	0.165021	−	0.986290i	$-0.447231\pi$
$229$	1143.72	0.330041	0.165021	+	0.986290i	$0.447231\pi$
$230$	14414.0	4.13230
$231$	0	0
$232$	109.283	0.0309258
$233$	4238.17	1.19164	0.595819	−	0.803118i	$-0.296827\pi$
$233$	4238.17	1.19164	0.595819	+	0.803118i	$0.296827\pi$
$234$	0	0
$235$	4219.59	1.17130
$236$	−9481.75	−2.61529
$237$	0	0
$238$	−1808.48	−0.492547
$239$	3310.03	0.895849	0.447924	−	0.894072i	$-0.352163\pi$
$239$	3310.03	0.895849	0.447924	+	0.894072i	$0.352163\pi$
$240$	0	0
$241$	5702.86	1.52429	0.762145	−	0.647407i	$-0.224146\pi$
$241$	5702.86	1.52429	0.762145	+	0.647407i	$0.224146\pi$
$242$	3054.40	0.811340
$243$	0	0
$244$	3592.18	0.942483
$245$	2111.93	0.550718
$246$	0	0
$247$	0	0
$248$	−1802.11	−0.461427
$249$	0	0
$250$	15703.0	3.97257
$251$	3910.23	0.983313	0.491657	−	0.870789i	$-0.336392\pi$
$251$	3910.23	0.983313	0.491657	+	0.870789i	$0.336392\pi$
$252$	0	0
$253$	3825.35	0.950585
$254$	−6221.51	−1.53690
$255$	0	0
$256$	−7995.69	−1.95207
$257$	−6972.80	−1.69242	−0.846209	−	0.532851i	$-0.821121\pi$
$257$	−6972.80	−1.69242	−0.846209	+	0.532851i	$0.821121\pi$
$258$	0	0
$259$	4841.88	1.16162
$260$	0	0
$261$	0	0
$262$	−6432.32	−1.51676
$263$	−281.691	−0.0660449	−0.0330224	−	0.999455i	$-0.510513\pi$
$263$	−281.691	−0.0660449	−0.0330224	+	0.999455i	$0.510513\pi$
$264$	0	0
$265$	5570.23	1.29123
$266$	3508.95	0.808826
$267$	0	0
$268$	−8598.31	−1.95980
$269$	−4333.13	−0.982139	−0.491070	−	0.871120i	$-0.663394\pi$
$269$	−4333.13	−0.982139	−0.491070	+	0.871120i	$0.663394\pi$
$270$	0	0
$271$	−428.596	−0.0960715	−0.0480357	−	0.998846i	$-0.515296\pi$
$271$	−428.596	−0.0960715	−0.0480357	+	0.998846i	$0.515296\pi$
$272$	−2347.89	−0.523390
$273$	0	0
$274$	−10248.8	−2.25967
$275$	7534.59	1.65219
$276$	0	0
$277$	8938.75	1.93891	0.969454	−	0.245274i	$-0.0788779\pi$
$277$	8938.75	1.93891	0.969454	+	0.245274i	$0.0788779\pi$
$278$	7293.91	1.57360
$279$	0	0
$280$	−14807.0	−3.16031
$281$	775.819	0.164703	0.0823514	−	0.996603i	$-0.473757\pi$
$281$	775.819	0.164703	0.0823514	+	0.996603i	$0.473757\pi$
$282$	0	0
$283$	−4014.53	−0.843248	−0.421624	−	0.906771i	$-0.638540\pi$
$283$	−4014.53	−0.843248	−0.421624	+	0.906771i	$0.638540\pi$
$284$	14433.4	3.01573
$285$	0	0
$286$	0	0
$287$	90.5340	0.0186204
$288$	0	0
$289$	−4373.21	−0.890130
$290$	−232.503	−0.0470795
$291$	0	0
$292$	−1154.18	−0.231313
$293$	−4956.21	−0.988208	−0.494104	−	0.869403i	$-0.664504\pi$
$293$	−4956.21	−0.988208	−0.494104	+	0.869403i	$0.664504\pi$
$294$	0	0
$295$	10927.5	2.15670
$296$	14972.5	2.94007
$297$	0	0
$298$	−4875.43	−0.947738
$299$	0	0
$300$	0	0
$301$	5567.89	1.06621
$302$	7381.80	1.40654
$303$	0	0
$304$	4555.58	0.859474
$305$	−4139.92	−0.777217
$306$	0	0
$307$	3894.90	0.724084	0.362042	−	0.932162i	$-0.382080\pi$
$307$	3894.90	0.724084	0.362042	+	0.932162i	$0.382080\pi$
$308$	−7254.33	−1.34206
$309$	0	0
$310$	3834.04	0.702448
$311$	−3097.44	−0.564758	−0.282379	−	0.959303i	$-0.591124\pi$
$311$	−3097.44	−0.564758	−0.282379	+	0.959303i	$0.591124\pi$
$312$	0	0
$313$	4487.36	0.810353	0.405177	−	0.914238i	$-0.367210\pi$
$313$	4487.36	0.810353	0.405177	+	0.914238i	$0.367210\pi$
$314$	333.208	0.0598854
$315$	0	0
$316$	5544.68	0.987065
$317$	6820.62	1.20847	0.604233	−	0.796807i	$-0.293479\pi$
$317$	6820.62	1.20847	0.604233	+	0.796807i	$0.293479\pi$
$318$	0	0
$319$	−61.7045	−0.0108301
$320$	3251.01	0.567928
$321$	0	0
$322$	11054.0	1.91309
$323$	−1047.35	−0.180421
$324$	0	0
$325$	0	0
$326$	17786.1	3.02173
$327$	0	0
$328$	279.958	0.0471283
$329$	3235.98	0.542265
$330$	0	0
$331$	−5012.96	−0.832438	−0.416219	−	0.909264i	$-0.636645\pi$
$331$	−5012.96	−0.832438	−0.416219	+	0.909264i	$0.636645\pi$
$332$	2469.12	0.408164
$333$	0	0
$334$	−1315.08	−0.215444
$335$	9909.39	1.61614
$336$	0	0
$337$	3220.79	0.520616	0.260308	−	0.965526i	$-0.416176\pi$
$337$	3220.79	0.520616	0.260308	+	0.965526i	$0.416176\pi$
$338$	0	0
$339$	0	0
$340$	8158.74	1.30138
$341$	1017.52	0.161590
$342$	0	0
$343$	6911.39	1.08799
$344$	17217.6	2.69858
$345$	0	0
$346$	4600.13	0.714753
$347$	3360.71	0.519921	0.259960	−	0.965619i	$-0.416290\pi$
$347$	3360.71	0.519921	0.259960	+	0.965619i	$0.416290\pi$
$348$	0	0
$349$	−4591.32	−0.704205	−0.352102	−	0.935961i	$-0.614533\pi$
$349$	−4591.32	−0.704205	−0.352102	+	0.935961i	$0.614533\pi$
$350$	21772.4	3.32510
$351$	0	0
$352$	−3453.54	−0.522938
$353$	−1740.09	−0.262367	−0.131183	−	0.991358i	$-0.541878\pi$
$353$	−1740.09	−0.262367	−0.131183	+	0.991358i	$0.541878\pi$
$354$	0	0
$355$	−16634.2	−2.48691
$356$	11194.7	1.66662
$357$	0	0
$358$	13574.5	2.00400
$359$	−1425.49	−0.209567	−0.104784	−	0.994495i	$-0.533415\pi$
$359$	−1425.49	−0.209567	−0.104784	+	0.994495i	$0.533415\pi$
$360$	0	0
$361$	−4826.85	−0.703725
$362$	−24085.8	−3.49702
$363$	0	0
$364$	0	0
$365$	1330.17	0.190752
$366$	0	0
$367$	10849.2	1.54312	0.771559	−	0.636158i	$-0.219477\pi$
$367$	10849.2	1.54312	0.771559	+	0.636158i	$0.219477\pi$
$368$	14351.1	2.03288
$369$	0	0
$370$	−31854.6	−4.47579
$371$	4271.77	0.597788
$372$	0	0
$373$	−494.749	−0.0686786	−0.0343393	−	0.999410i	$-0.510933\pi$
$373$	−494.749	−0.0686786	−0.0343393	+	0.999410i	$0.510933\pi$
$374$	3157.61	0.436567
$375$	0	0
$376$	10006.6	1.37248
$377$	0	0
$378$	0	0
$379$	12643.3	1.71357	0.856786	−	0.515671i	$-0.172457\pi$
$379$	12643.3	1.71357	0.856786	+	0.515671i	$0.172457\pi$
$380$	−15830.3	−2.13704
$381$	0	0
$382$	−10380.7	−1.39037
$383$	2321.51	0.309722	0.154861	−	0.987936i	$-0.450507\pi$
$383$	2321.51	0.309722	0.154861	+	0.987936i	$0.450507\pi$
$384$	0	0
$385$	8360.47	1.10673
$386$	3682.48	0.485578
$387$	0	0
$388$	19461.8	2.54645
$389$	10477.6	1.36564	0.682821	−	0.730586i	$-0.260753\pi$
$389$	10477.6	1.36564	0.682821	+	0.730586i	$0.260753\pi$
$390$	0	0
$391$	−3299.38	−0.426744
$392$	5008.35	0.645306
$393$	0	0
$394$	8582.75	1.09744
$395$	−6390.14	−0.813982
$396$	0	0
$397$	1766.85	0.223364	0.111682	−	0.993744i	$-0.464376\pi$
$397$	1766.85	0.223364	0.111682	+	0.993744i	$0.464376\pi$
$398$	9287.91	1.16975
$399$	0	0
$400$	28266.5	3.53332
$401$	−5001.65	−0.622869	−0.311434	−	0.950268i	$-0.600809\pi$
$401$	−5001.65	−0.622869	−0.311434	+	0.950268i	$0.600809\pi$
$402$	0	0
$403$	0	0
$404$	−27753.0	−3.41773
$405$	0	0
$406$	−178.305	−0.0217959
$407$	−8453.96	−1.02960
$408$	0	0
$409$	11208.6	1.35508	0.677541	−	0.735485i	$-0.263046\pi$
$409$	11208.6	1.35508	0.677541	+	0.735485i	$0.263046\pi$
$410$	−595.620	−0.0717453
$411$	0	0
$412$	9207.94	1.10107
$413$	8380.26	0.998464
$414$	0	0
$415$	−2845.61	−0.336592
$416$	0	0
$417$	0	0
$418$	−6126.66	−0.716901
$419$	−3285.19	−0.383036	−0.191518	−	0.981489i	$-0.561341\pi$
$419$	−3285.19	−0.383036	−0.191518	+	0.981489i	$0.561341\pi$
$420$	0	0
$421$	13289.9	1.53850	0.769250	−	0.638948i	$-0.220630\pi$
$421$	13289.9	1.53850	0.769250	+	0.638948i	$0.220630\pi$
$422$	−721.336	−0.0832087
$423$	0	0
$424$	13209.6	1.51301
$425$	−6498.61	−0.741715
$426$	0	0
$427$	−3174.88	−0.359820
$428$	−30581.8	−3.45380
$429$	0	0
$430$	−36631.0	−4.10815
$431$	−13692.3	−1.53024	−0.765119	−	0.643889i	$-0.777320\pi$
$431$	−13692.3	−1.53024	−0.765119	+	0.643889i	$0.777320\pi$
$432$	0	0
$433$	−10002.5	−1.11014	−0.555070	−	0.831804i	$-0.687308\pi$
$433$	−10002.5	−1.11014	−0.555070	+	0.831804i	$0.687308\pi$
$434$	2940.30	0.325205
$435$	0	0
$436$	−16881.8	−1.85434
$437$	6401.73	0.700769
$438$	0	0
$439$	4487.21	0.487842	0.243921	−	0.969795i	$-0.421566\pi$
$439$	4487.21	0.487842	0.243921	+	0.969795i	$0.421566\pi$
$440$	25853.1	2.80113
$441$	0	0
$442$	0	0
$443$	−2035.35	−0.218290	−0.109145	−	0.994026i	$-0.534811\pi$
$443$	−2035.35	−0.218290	−0.109145	+	0.994026i	$0.534811\pi$
$444$	0	0
$445$	−12901.7	−1.37438
$446$	11799.7	1.25276
$447$	0	0
$448$	2493.18	0.262928
$449$	3575.58	0.375818	0.187909	−	0.982186i	$-0.439829\pi$
$449$	3575.58	0.375818	0.187909	+	0.982186i	$0.439829\pi$
$450$	0	0
$451$	−158.073	−0.0165041
$452$	−33357.4	−3.47124
$453$	0	0
$454$	16545.6	1.71040
$455$	0	0
$456$	0	0
$457$	−8058.60	−0.824869	−0.412435	−	0.910987i	$-0.635321\pi$
$457$	−8058.60	−0.824869	−0.412435	+	0.910987i	$0.635321\pi$
$458$	−5770.51	−0.588730
$459$	0	0
$460$	−49868.8	−5.05466
$461$	11692.1	1.18125	0.590626	−	0.806945i	$-0.298881\pi$
$461$	11692.1	1.18125	0.590626	+	0.806945i	$0.298881\pi$
$462$	0	0
$463$	−1732.40	−0.173891	−0.0869455	−	0.996213i	$-0.527711\pi$
$463$	−1732.40	−0.173891	−0.0869455	+	0.996213i	$0.527711\pi$
$464$	−231.489	−0.0231607
$465$	0	0
$466$	−21383.1	−2.12565
$467$	−10769.9	−1.06718	−0.533588	−	0.845745i	$-0.679157\pi$
$467$	−10769.9	−1.06718	−0.533588	+	0.845745i	$0.679157\pi$
$468$	0	0
$469$	7599.44	0.748208
$470$	−21289.4	−2.08937
$471$	0	0
$472$	25914.2	2.52712
$473$	−9721.58	−0.945029
$474$	0	0
$475$	12609.1	1.21799
$476$	6256.88	0.602487
$477$	0	0
$478$	−16700.3	−1.59802
$479$	7911.64	0.754680	0.377340	−	0.926075i	$-0.376839\pi$
$479$	7911.64	0.754680	0.377340	+	0.926075i	$0.376839\pi$
$480$	0	0
$481$	0	0
$482$	−28773.0	−2.71904
$483$	0	0
$484$	−10567.5	−0.992438
$485$	−22429.3	−2.09993
$486$	0	0
$487$	10964.8	1.02025	0.510125	−	0.860100i	$-0.329599\pi$
$487$	10964.8	1.02025	0.510125	+	0.860100i	$0.329599\pi$
$488$	−9817.66	−0.910707
$489$	0	0
$490$	−10655.4	−0.982375
$491$	9139.37	0.840029	0.420014	−	0.907518i	$-0.362025\pi$
$491$	9139.37	0.840029	0.420014	+	0.907518i	$0.362025\pi$
$492$	0	0
$493$	53.2204	0.00486192
$494$	0	0
$495$	0	0
$496$	3817.31	0.345569
$497$	−12756.7	−1.15134
$498$	0	0
$499$	12577.5	1.12835	0.564175	−	0.825655i	$-0.309194\pi$
$499$	12577.5	1.12835	0.564175	+	0.825655i	$0.309194\pi$
$500$	−54328.4	−4.85928
$501$	0	0
$502$	−19728.6	−1.75404
$503$	−13214.6	−1.17139	−0.585696	−	0.810531i	$-0.699179\pi$
$503$	−13214.6	−1.17139	−0.585696	+	0.810531i	$0.699179\pi$
$504$	0	0
$505$	31984.8	2.81842
$506$	−19300.3	−1.69566
$507$	0	0
$508$	21524.9	1.87995
$509$	21719.8	1.89138	0.945689	−	0.325073i	$-0.105389\pi$
$509$	21719.8	1.89138	0.945689	+	0.325073i	$0.105389\pi$
$510$	0	0
$511$	1020.10	0.0883103
$512$	25613.2	2.21085
$513$	0	0
$514$	35180.4	3.01895
$515$	−10612.0	−0.907999
$516$	0	0
$517$	−5650.03	−0.480635
$518$	−24429.1	−2.07211
$519$	0	0
$520$	0	0
$521$	4627.05	0.389088	0.194544	−	0.980894i	$-0.437677\pi$
$521$	4627.05	0.389088	0.194544	+	0.980894i	$0.437677\pi$
$522$	0	0
$523$	13784.0	1.15245	0.576224	−	0.817292i	$-0.304526\pi$
$523$	13784.0	1.15245	0.576224	+	0.817292i	$0.304526\pi$
$524$	22254.3	1.85531
$525$	0	0
$526$	1421.23	0.117811
$527$	−877.619	−0.0725421
$528$	0	0
$529$	7999.85	0.657504
$530$	−28103.9	−2.30331
$531$	0	0
$532$	−12140.1	−0.989362
$533$	0	0
$534$	0	0
$535$	35245.0	2.84817
$536$	23499.7	1.89372
$537$	0	0
$538$	21862.2	1.75195
$539$	−2827.87	−0.225983
$540$	0	0
$541$	−454.638	−0.0361302	−0.0180651	−	0.999837i	$-0.505751\pi$
$541$	−454.638	−0.0361302	−0.0180651	+	0.999837i	$0.505751\pi$
$542$	2162.43	0.171373
$543$	0	0
$544$	2978.69	0.234762
$545$	19456.0	1.52918
$546$	0	0
$547$	11611.4	0.907621	0.453810	−	0.891098i	$-0.350064\pi$
$547$	11611.4	0.907621	0.453810	+	0.891098i	$0.350064\pi$
$548$	35458.2	2.76405
$549$	0	0
$550$	−38014.8	−2.94719
$551$	−103.263	−0.00798390
$552$	0	0
$553$	−4900.55	−0.376840
$554$	−45099.3	−3.45864
$555$	0	0
$556$	−25235.1	−1.92483
$557$	4883.93	0.371524	0.185762	−	0.982595i	$-0.440525\pi$
$557$	4883.93	0.371524	0.185762	+	0.982595i	$0.440525\pi$
$558$	0	0
$559$	0	0
$560$	31364.9	2.36680
$561$	0	0
$562$	−3914.29	−0.293798
$563$	−19617.3	−1.46851	−0.734256	−	0.678873i	$-0.762469\pi$
$563$	−19617.3	−1.46851	−0.734256	+	0.678873i	$0.762469\pi$
$564$	0	0
$565$	38443.8	2.86256
$566$	20254.8	1.50419
$567$	0	0
$568$	−39447.5	−2.91405
$569$	−7475.19	−0.550749	−0.275374	−	0.961337i	$-0.588802\pi$
$569$	−7475.19	−0.550749	−0.275374	+	0.961337i	$0.588802\pi$
$570$	0	0
$571$	−7799.56	−0.571631	−0.285816	−	0.958285i	$-0.592265\pi$
$571$	−7799.56	−0.571631	−0.285816	+	0.958285i	$0.592265\pi$
$572$	0	0
$573$	0	0
$574$	−456.777	−0.0332152
$575$	39721.6	2.88088
$576$	0	0
$577$	−13136.1	−0.947771	−0.473885	−	0.880587i	$-0.657149\pi$
$577$	−13136.1	−0.947771	−0.473885	+	0.880587i	$0.657149\pi$
$578$	22064.4	1.58782
$579$	0	0
$580$	804.404	0.0575880
$581$	−2182.28	−0.155828
$582$	0	0
$583$	−7458.54	−0.529848
$584$	3154.45	0.223514
$585$	0	0
$586$	25005.9	1.76277
$587$	22150.7	1.55751	0.778754	−	0.627330i	$-0.215852\pi$
$587$	22150.7	1.55751	0.778754	+	0.627330i	$0.215852\pi$
$588$	0	0
$589$	1702.83	0.119124
$590$	−55133.4	−3.84713
$591$	0	0
$592$	−31715.6	−2.20186
$593$	−22770.3	−1.57684	−0.788419	−	0.615138i	$-0.789100\pi$
$593$	−22770.3	−1.57684	−0.788419	+	0.615138i	$0.789100\pi$
$594$	0	0
$595$	−7210.94	−0.496840
$596$	16867.8	1.15928
$597$	0	0
$598$	0	0
$599$	7214.11	0.492088	0.246044	−	0.969259i	$-0.420869\pi$
$599$	7214.11	0.492088	0.246044	+	0.969259i	$0.420869\pi$
$600$	0	0
$601$	27276.7	1.85132	0.925658	−	0.378360i	$-0.123512\pi$
$601$	27276.7	1.85132	0.925658	+	0.378360i	$0.123512\pi$
$602$	−28092.1	−1.90191
$603$	0	0
$604$	−25539.2	−1.72049
$605$	12178.8	0.818412
$606$	0	0
$607$	11566.2	0.773403	0.386701	−	0.922205i	$-0.373614\pi$
$607$	11566.2	0.773403	0.386701	+	0.922205i	$0.373614\pi$
$608$	−5779.50	−0.385509
$609$	0	0
$610$	20887.4	1.38640
$611$	0	0
$612$	0	0
$613$	−24476.6	−1.61272	−0.806362	−	0.591422i	$-0.798567\pi$
$613$	−24476.6	−1.61272	−0.806362	+	0.591422i	$0.798567\pi$
$614$	−19651.2	−1.29163
$615$	0	0
$616$	19826.5	1.29681
$617$	2423.28	0.158116	0.0790581	−	0.996870i	$-0.474809\pi$
$617$	2423.28	0.158116	0.0790581	+	0.996870i	$0.474809\pi$
$618$	0	0
$619$	−17223.4	−1.11836	−0.559182	−	0.829045i	$-0.688885\pi$
$619$	−17223.4	−1.11836	−0.559182	+	0.829045i	$0.688885\pi$
$620$	−13264.8	−0.859240
$621$	0	0
$622$	15627.7	1.00742
$623$	−9894.22	−0.636282
$624$	0	0
$625$	27648.7	1.76952
$626$	−22640.4	−1.44551
$627$	0	0
$628$	−1152.82	−0.0732523
$629$	7291.57	0.462216
$630$	0	0
$631$	−9242.58	−0.583108	−0.291554	−	0.956554i	$-0.594172\pi$
$631$	−9242.58	−0.583108	−0.291554	+	0.956554i	$0.594172\pi$
$632$	−15154.0	−0.953786
$633$	0	0
$634$	−34412.5	−2.15567
$635$	−24807.0	−1.55029
$636$	0	0
$637$	0	0
$638$	311.322	0.0193187
$639$	0	0
$640$	−37036.1	−2.28747
$641$	11598.8	0.714706	0.357353	−	0.933969i	$-0.383679\pi$
$641$	11598.8	0.714706	0.357353	+	0.933969i	$0.383679\pi$
$642$	0	0
$643$	−25363.9	−1.55561	−0.777804	−	0.628507i	$-0.783666\pi$
$643$	−25363.9	−1.55561	−0.777804	+	0.628507i	$0.783666\pi$
$644$	−38244.0	−2.34010
$645$	0	0
$646$	5284.26	0.321837
$647$	6590.09	0.400438	0.200219	−	0.979751i	$-0.435835\pi$
$647$	6590.09	0.400438	0.200219	+	0.979751i	$0.435835\pi$
$648$	0	0
$649$	−14632.0	−0.884985
$650$	0	0
$651$	0	0
$652$	−61535.6	−3.69620
$653$	−15698.2	−0.940765	−0.470382	−	0.882463i	$-0.655884\pi$
$653$	−15698.2	−0.940765	−0.470382	+	0.882463i	$0.655884\pi$
$654$	0	0
$655$	−25647.6	−1.52998
$656$	−593.021	−0.0352951
$657$	0	0
$658$	−16326.7	−0.967295
$659$	−2840.81	−0.167925	−0.0839624	−	0.996469i	$-0.526758\pi$
$659$	−2840.81	−0.167925	−0.0839624	+	0.996469i	$0.526758\pi$
$660$	0	0
$661$	−20819.1	−1.22507	−0.612533	−	0.790445i	$-0.709849\pi$
$661$	−20819.1	−1.22507	−0.612533	+	0.790445i	$0.709849\pi$
$662$	25292.2	1.48491
$663$	0	0
$664$	−6748.26	−0.394403
$665$	13991.3	0.815876
$666$	0	0
$667$	−325.300	−0.0188840
$668$	4549.87	0.263532
$669$	0	0
$670$	−49996.5	−2.88289
$671$	5543.36	0.318925
$672$	0	0
$673$	31264.5	1.79073	0.895364	−	0.445335i	$-0.146916\pi$
$673$	31264.5	1.79073	0.895364	+	0.445335i	$0.146916\pi$
$674$	−16250.1	−0.928679
$675$	0	0
$676$	0	0
$677$	27953.2	1.58689	0.793447	−	0.608639i	$-0.208284\pi$
$677$	27953.2	1.58689	0.793447	+	0.608639i	$0.208284\pi$
$678$	0	0
$679$	−17200.9	−0.972180
$680$	−22298.4	−1.25750
$681$	0	0
$682$	−5133.79	−0.288245
$683$	−34709.9	−1.94456	−0.972282	−	0.233811i	$-0.924880\pi$
$683$	−34709.9	−1.94456	−0.972282	+	0.233811i	$0.924880\pi$
$684$	0	0
$685$	−40864.9	−2.27937
$686$	−34870.5	−1.94076
$687$	0	0
$688$	−36471.1	−2.02100
$689$	0	0
$690$	0	0
$691$	11860.8	0.652974	0.326487	−	0.945202i	$-0.394135\pi$
$691$	11860.8	0.652974	0.326487	+	0.945202i	$0.394135\pi$
$692$	−15915.3	−0.874291
$693$	0	0
$694$	−16956.0	−0.927438
$695$	29083.0	1.58731
$696$	0	0
$697$	136.338	0.00740916
$698$	23164.9	1.25617
$699$	0	0
$700$	−75327.2	−4.06729
$701$	−16100.5	−0.867486	−0.433743	−	0.901037i	$-0.642807\pi$
$701$	−16100.5	−0.867486	−0.433743	+	0.901037i	$0.642807\pi$
$702$	0	0
$703$	−14147.7	−0.759019
$704$	−4353.10	−0.233045
$705$	0	0
$706$	8779.38	0.468012
$707$	24528.9	1.30482
$708$	0	0
$709$	28465.5	1.50782	0.753910	−	0.656978i	$-0.228166\pi$
$709$	28465.5	1.50782	0.753910	+	0.656978i	$0.228166\pi$
$710$	83925.9	4.43617
$711$	0	0
$712$	−30595.9	−1.61043
$713$	5364.28	0.281759
$714$	0	0
$715$	0	0
$716$	−46964.4	−2.45131
$717$	0	0
$718$	7192.14	0.373828
$719$	27822.1	1.44310	0.721550	−	0.692362i	$-0.243430\pi$
$719$	27822.1	1.44310	0.721550	+	0.692362i	$0.243430\pi$
$720$	0	0
$721$	−8138.25	−0.420367
$722$	24353.2	1.25531
$723$	0	0
$724$	83331.0	4.27758
$725$	−640.725	−0.0328220
$726$	0	0
$727$	866.153	0.0441869	0.0220934	−	0.999756i	$-0.492967\pi$
$727$	866.153	0.0441869	0.0220934	+	0.999756i	$0.492967\pi$
$728$	0	0
$729$	0	0
$730$	−6711.20	−0.340264
$731$	8384.90	0.424250
$732$	0	0
$733$	23120.1	1.16502	0.582511	−	0.812823i	$-0.302070\pi$
$733$	23120.1	1.16502	0.582511	+	0.812823i	$0.302070\pi$
$734$	−54738.3	−2.75262
$735$	0	0
$736$	−18206.7	−0.911831
$737$	−13268.7	−0.663172
$738$	0	0
$739$	−20765.1	−1.03364	−0.516819	−	0.856095i	$-0.672884\pi$
$739$	−20765.1	−1.03364	−0.516819	+	0.856095i	$0.672884\pi$
$740$	110209.	5.47482
$741$	0	0
$742$	−21552.7	−1.06634
$743$	−36995.8	−1.82671	−0.913354	−	0.407166i	$-0.866517\pi$
$743$	−36995.8	−1.82671	−0.913354	+	0.407166i	$0.866517\pi$
$744$	0	0
$745$	−19439.8	−0.955999
$746$	2496.19	0.122509
$747$	0	0
$748$	−10924.6	−0.534013
$749$	27029.1	1.31859
$750$	0	0
$751$	−34429.1	−1.67289	−0.836443	−	0.548055i	$-0.815369\pi$
$751$	−34429.1	−1.67289	−0.836443	+	0.548055i	$0.815369\pi$
$752$	−21196.5	−1.02787
$753$	0	0
$754$	0	0
$755$	29433.5	1.41880
$756$	0	0
$757$	4268.88	0.204960	0.102480	−	0.994735i	$-0.467322\pi$
$757$	4268.88	0.204960	0.102480	+	0.994735i	$0.467322\pi$
$758$	−63790.2	−3.05668
$759$	0	0
$760$	43265.1	2.06499
$761$	−7342.11	−0.349739	−0.174869	−	0.984592i	$-0.555950\pi$
$761$	−7342.11	−0.349739	−0.174869	+	0.984592i	$0.555950\pi$
$762$	0	0
$763$	14920.7	0.707947
$764$	35914.7	1.70072
$765$	0	0
$766$	−11712.9	−0.552484
$767$	0	0
$768$	0	0
$769$	14728.5	0.690666	0.345333	−	0.938480i	$-0.387766\pi$
$769$	14728.5	0.690666	0.345333	+	0.938480i	$0.387766\pi$
$770$	−42181.7	−1.97418
$771$	0	0
$772$	−12740.5	−0.593963
$773$	9937.41	0.462385	0.231193	−	0.972908i	$-0.425737\pi$
$773$	9937.41	0.462385	0.231193	+	0.972908i	$0.425737\pi$
$774$	0	0
$775$	10565.7	0.489719
$776$	−53190.3	−2.46059
$777$	0	0
$778$	−52863.3	−2.43604
$779$	−264.535	−0.0121668
$780$	0	0
$781$	22273.3	1.02049
$782$	16646.6	0.761229
$783$	0	0
$784$	−10608.9	−0.483279
$785$	1328.60	0.0604074
$786$	0	0
$787$	−36305.0	−1.64439	−0.822194	−	0.569207i	$-0.807250\pi$
$787$	−36305.0	−1.64439	−0.822194	+	0.569207i	$0.807250\pi$
$788$	−29694.2	−1.34240
$789$	0	0
$790$	32240.6	1.45199
$791$	29482.3	1.32525
$792$	0	0
$793$	0	0
$794$	−8914.40	−0.398439
$795$	0	0
$796$	−32133.9	−1.43085
$797$	9150.73	0.406694	0.203347	−	0.979107i	$-0.434818\pi$
$797$	9150.73	0.406694	0.203347	+	0.979107i	$0.434818\pi$
$798$	0	0
$799$	4873.17	0.215770
$800$	−35860.7	−1.58484
$801$	0	0
$802$	25235.1	1.11108
$803$	−1781.10	−0.0782736
$804$	0	0
$805$	44075.5	1.92976
$806$	0	0
$807$	0	0
$808$	75850.7	3.30250
$809$	−30349.9	−1.31897	−0.659484	−	0.751719i	$-0.729225\pi$
$809$	−30349.9	−1.31897	−0.659484	+	0.751719i	$0.729225\pi$
$810$	0	0
$811$	4238.72	0.183529	0.0917643	−	0.995781i	$-0.470749\pi$
$811$	4238.72	0.183529	0.0917643	+	0.995781i	$0.470749\pi$
$812$	616.892	0.0266609
$813$	0	0
$814$	42653.3	1.83661
$815$	70918.6	3.04806
$816$	0	0
$817$	−16269.1	−0.696674
$818$	−56551.4	−2.41721
$819$	0	0
$820$	2060.70	0.0877595
$821$	5934.48	0.252271	0.126136	−	0.992013i	$-0.459743\pi$
$821$	5934.48	0.252271	0.126136	+	0.992013i	$0.459743\pi$
$822$	0	0
$823$	−22459.6	−0.951265	−0.475633	−	0.879644i	$-0.657781\pi$
$823$	−22459.6	−0.951265	−0.475633	+	0.879644i	$0.657781\pi$
$824$	−25165.9	−1.06395
$825$	0	0
$826$	−42281.5	−1.78107
$827$	20138.4	0.846772	0.423386	−	0.905949i	$-0.360841\pi$
$827$	20138.4	0.846772	0.423386	+	0.905949i	$0.360841\pi$
$828$	0	0
$829$	5845.70	0.244909	0.122455	−	0.992474i	$-0.460923\pi$
$829$	5845.70	0.244909	0.122455	+	0.992474i	$0.460923\pi$
$830$	14357.2	0.600415
$831$	0	0
$832$	0	0
$833$	2439.05	0.101450
$834$	0	0
$835$	−5243.63	−0.217322
$836$	21196.7	0.876919
$837$	0	0
$838$	16575.0	0.683261
$839$	−27779.9	−1.14311	−0.571554	−	0.820564i	$-0.693659\pi$
$839$	−27779.9	−1.14311	−0.571554	+	0.820564i	$0.693659\pi$
$840$	0	0
$841$	−24383.8	−0.999785
$842$	−67052.3	−2.74439
$843$	0	0
$844$	2495.65	0.101782
$845$	0	0
$846$	0	0
$847$	9339.85	0.378891
$848$	−27981.2	−1.13311
$849$	0	0
$850$	32787.9	1.32308
$851$	−44568.4	−1.79528
$852$	0	0
$853$	−16480.5	−0.661526	−0.330763	−	0.943714i	$-0.607306\pi$
$853$	−16480.5	−0.661526	−0.330763	+	0.943714i	$0.607306\pi$
$854$	16018.4	0.641849
$855$	0	0
$856$	83582.1	3.33736
$857$	45445.2	1.81141	0.905704	−	0.423910i	$-0.139343\pi$
$857$	45445.2	1.81141	0.905704	+	0.423910i	$0.139343\pi$
$858$	0	0
$859$	−28243.1	−1.12182	−0.560909	−	0.827877i	$-0.689548\pi$
$859$	−28243.1	−1.12182	−0.560909	+	0.827877i	$0.689548\pi$
$860$	126734.	5.02512
$861$	0	0
$862$	69082.5	2.72965
$863$	−328.319	−0.0129503	−0.00647514	−	0.999979i	$-0.502061\pi$
$863$	−328.319	−0.0129503	−0.00647514	+	0.999979i	$0.502061\pi$
$864$	0	0
$865$	18342.1	0.720982
$866$	50466.4	1.98027
$867$	0	0
$868$	−10172.7	−0.397793
$869$	8556.40	0.334011
$870$	0	0
$871$	0	0
$872$	46139.1	1.79182
$873$	0	0
$874$	−32299.1	−1.25004
$875$	48017.0	1.85517
$876$	0	0
$877$	−42480.0	−1.63563	−0.817815	−	0.575482i	$-0.804815\pi$
$877$	−42480.0	−1.63563	−0.817815	+	0.575482i	$0.804815\pi$
$878$	−22639.6	−0.870216
$879$	0	0
$880$	−54763.3	−2.09781
$881$	5220.95	0.199657	0.0998287	−	0.995005i	$-0.468171\pi$
$881$	5220.95	0.199657	0.0998287	+	0.995005i	$0.468171\pi$
$882$	0	0
$883$	11790.3	0.449349	0.224674	−	0.974434i	$-0.427868\pi$
$883$	11790.3	0.449349	0.224674	+	0.974434i	$0.427868\pi$
$884$	0	0
$885$	0	0
$886$	10269.1	0.389387
$887$	49352.3	1.86819	0.934097	−	0.357019i	$-0.116207\pi$
$887$	49352.3	1.86819	0.934097	+	0.357019i	$0.116207\pi$
$888$	0	0
$889$	−19024.4	−0.717724
$890$	65093.8	2.45163
$891$	0	0
$892$	−40823.9	−1.53238
$893$	−9455.33	−0.354323
$894$	0	0
$895$	54125.5	2.02147
$896$	−28402.7	−1.05900
$897$	0	0
$898$	−18040.1	−0.670386
$899$	−86.5281	−0.00321009
$900$	0	0
$901$	6433.02	0.237863
$902$	797.536	0.0294402
$903$	0	0
$904$	91168.1	3.35421
$905$	−96037.3	−3.52750
$906$	0	0
$907$	6088.94	0.222910	0.111455	−	0.993769i	$-0.464449\pi$
$907$	6088.94	0.222910	0.111455	+	0.993769i	$0.464449\pi$
$908$	−57243.7	−2.09218
$909$	0	0
$910$	0	0
$911$	30301.7	1.10202	0.551010	−	0.834499i	$-0.314243\pi$
$911$	30301.7	1.10202	0.551010	+	0.834499i	$0.314243\pi$
$912$	0	0
$913$	3810.28	0.138118
$914$	40658.6	1.47141
$915$	0	0
$916$	19964.5	0.720139
$917$	−19669.0	−0.708317
$918$	0	0
$919$	34695.8	1.24538	0.622692	−	0.782467i	$-0.286039\pi$
$919$	34695.8	1.24538	0.622692	+	0.782467i	$0.286039\pi$
$920$	136295.	4.88424
$921$	0	0
$922$	−58991.2	−2.10713
$923$	0	0
$924$	0	0
$925$	−87783.9	−3.12034
$926$	8740.61	0.310188
$927$	0	0
$928$	293.681	0.0103885
$929$	−16990.8	−0.600054	−0.300027	−	0.953931i	$-0.596996\pi$
$929$	−16990.8	−0.600054	−0.300027	+	0.953931i	$0.596996\pi$
$930$	0	0
$931$	−4732.44	−0.166594
$932$	73980.4	2.60012
$933$	0	0
$934$	54338.0	1.90363
$935$	12590.3	0.440372
$936$	0	0
$937$	9307.86	0.324519	0.162260	−	0.986748i	$-0.448122\pi$
$937$	9307.86	0.324519	0.162260	+	0.986748i	$0.448122\pi$
$938$	−38342.0	−1.33466
$939$	0	0
$940$	73656.0	2.55574
$941$	52285.3	1.81132	0.905659	−	0.424006i	$-0.139377\pi$
$941$	52285.3	1.81132	0.905659	+	0.424006i	$0.139377\pi$
$942$	0	0
$943$	−833.344	−0.0287777
$944$	−54892.8	−1.89260
$945$	0	0
$946$	49048.9	1.68575
$947$	12348.5	0.423731	0.211865	−	0.977299i	$-0.432046\pi$
$947$	12348.5	0.423731	0.211865	+	0.977299i	$0.432046\pi$
$948$	0	0
$949$	0	0
$950$	−63617.7	−2.17267
$951$	0	0
$952$	−17100.5	−0.582174
$953$	25631.0	0.871218	0.435609	−	0.900136i	$-0.356533\pi$
$953$	25631.0	0.871218	0.435609	+	0.900136i	$0.356533\pi$
$954$	0	0
$955$	−41391.0	−1.40249
$956$	57778.9	1.95471
$957$	0	0
$958$	−39917.1	−1.34620
$959$	−31339.0	−1.05525
$960$	0	0
$961$	−28364.1	−0.952104
$962$	0	0
$963$	0	0
$964$	99547.7	3.32595
$965$	14683.1	0.489811
$966$	0	0
$967$	11185.6	0.371981	0.185991	−	0.982552i	$-0.440451\pi$
$967$	11185.6	0.371981	0.185991	+	0.982552i	$0.440451\pi$
$968$	28881.6	0.958977
$969$	0	0
$970$	113164.	3.74586
$971$	23541.3	0.778039	0.389019	−	0.921230i	$-0.372814\pi$
$971$	23541.3	0.778039	0.389019	+	0.921230i	$0.372814\pi$
$972$	0	0
$973$	22303.6	0.734860
$974$	−55321.3	−1.81993
$975$	0	0
$976$	20796.3	0.682041
$977$	24198.8	0.792415	0.396207	−	0.918161i	$-0.370326\pi$
$977$	24198.8	0.792415	0.396207	+	0.918161i	$0.370326\pi$
$978$	0	0
$979$	17275.4	0.563966
$980$	36865.2	1.20165
$981$	0	0
$982$	−46111.5	−1.49845
$983$	−33757.4	−1.09532	−0.547658	−	0.836702i	$-0.684480\pi$
$983$	−33757.4	−1.09532	−0.547658	+	0.836702i	$0.684480\pi$
$984$	0	0
$985$	34222.0	1.10701
$986$	−268.516	−0.00867272
$987$	0	0
$988$	0	0
$989$	−51251.1	−1.64782
$990$	0	0
$991$	25298.5	0.810933	0.405466	−	0.914110i	$-0.367109\pi$
$991$	25298.5	0.810933	0.405466	+	0.914110i	$0.367109\pi$
$992$	−4842.89	−0.155002
$993$	0	0
$994$	64362.2	2.05377
$995$	37033.7	1.17995
$996$	0	0
$997$	−28766.1	−0.913773	−0.456886	−	0.889525i	$-0.651035\pi$
$997$	−28766.1	−0.913773	−0.456886	+	0.889525i	$0.651035\pi$
$998$	−63458.2	−2.01276
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bk.1.2		10
3.2	odd	2		507.4.a.r.1.9		10
13.2	odd	12		117.4.q.e.82.1		10
13.7	odd	12		117.4.q.e.10.1		10
13.12	even	2	inner	1521.4.a.bk.1.9		10
39.2	even	12		39.4.j.c.4.5	✓	10
39.5	even	4		507.4.b.i.337.2		10
39.8	even	4		507.4.b.i.337.9		10
39.20	even	12		39.4.j.c.10.5	yes	10
39.38	odd	2		507.4.a.r.1.2		10
156.59	odd	12		624.4.bv.h.49.5		10
156.119	odd	12		624.4.bv.h.433.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.5	✓	10	39.2	even	12
39.4.j.c.10.5	yes	10	39.20	even	12
117.4.q.e.10.1		10	13.7	odd	12
117.4.q.e.82.1		10	13.2	odd	12
507.4.a.r.1.2		10	39.38	odd	2
507.4.a.r.1.9		10	3.2	odd	2
507.4.b.i.337.2		10	39.5	even	4
507.4.b.i.337.9		10	39.8	even	4
624.4.bv.h.49.5		10	156.59	odd	12
624.4.bv.h.433.1		10	156.119	odd	12
1521.4.a.bk.1.2		10	1.1	even	1	trivial
1521.4.a.bk.1.9		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)

\(f(q)\)	\(=\)	\(q-5.04537 q^{2} +17.4557 q^{4} -20.1174 q^{5} -15.4279 q^{7} -47.7076 q^{8} +O(q^{10})\)	\(q-5.04537 q^{2} +17.4557 q^{4} -20.1174 q^{5} -15.4279 q^{7} -47.7076 q^{8} +101.500 q^{10} +26.9372 q^{11} +77.8394 q^{14} +101.057 q^{16} -23.2334 q^{17} +45.0794 q^{19} -351.164 q^{20} -135.908 q^{22} +142.010 q^{23} +279.710 q^{25} -269.305 q^{28} -2.29068 q^{29} +37.7740 q^{31} -128.207 q^{32} +117.221 q^{34} +310.369 q^{35} -313.840 q^{37} -227.442 q^{38} +959.753 q^{40} -5.86820 q^{41} -360.898 q^{43} +470.209 q^{44} -716.493 q^{46} -209.748 q^{47} -104.980 q^{49} -1411.24 q^{50} -276.886 q^{53} -541.906 q^{55} +736.028 q^{56} +11.5573 q^{58} -543.189 q^{59} +205.788 q^{61} -190.583 q^{62} -161.602 q^{64} -492.578 q^{67} -405.557 q^{68} -1565.93 q^{70} +826.859 q^{71} -66.1205 q^{73} +1583.44 q^{74} +786.894 q^{76} -415.584 q^{77} +317.642 q^{79} -2033.00 q^{80} +29.6072 q^{82} +141.450 q^{83} +467.396 q^{85} +1820.86 q^{86} -1285.11 q^{88} +641.320 q^{89} +2478.89 q^{92} +1058.26 q^{94} -906.880 q^{95} +1114.92 q^{97} +529.663 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 60 q^{4}+O(q^{10}) \) 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

L-functions

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