LMFDB - Embedded newform 1521.4.a.x.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:	$4$
Coefficient field:	4.4.5054412.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 29x^{2} + 48 $ x^4 - 29*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.4
Root			$5.21898$ 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)

$f(q)$	$=$	$q+5.21898 q^{2} +19.2377 q^{4} -5.83936 q^{5} +31.3139 q^{7} +58.6495 q^{8} -30.4755 q^{10} -16.2773 q^{11} +163.426 q^{14} +152.189 q^{16} +54.0000 q^{17} -66.3500 q^{19} -112.336 q^{20} -84.9510 q^{22} +182.853 q^{23} -90.9019 q^{25} +602.408 q^{28} +164.853 q^{29} +58.9055 q^{31} +325.073 q^{32} +281.825 q^{34} -182.853 q^{35} +110.366 q^{37} -346.279 q^{38} -342.475 q^{40} +55.0357 q^{41} -113.147 q^{43} -313.139 q^{44} +954.305 q^{46} +514.089 q^{47} +637.559 q^{49} -474.415 q^{50} -242.559 q^{53} +95.0490 q^{55} +1836.54 q^{56} +860.364 q^{58} -265.036 q^{59} -468.098 q^{61} +307.426 q^{62} +479.042 q^{64} -852.919 q^{67} +1038.84 q^{68} -954.305 q^{70} +165.619 q^{71} -315.325 q^{73} +576.000 q^{74} -1276.42 q^{76} -509.706 q^{77} +479.608 q^{79} -888.684 q^{80} +287.230 q^{82} +574.235 q^{83} -315.325 q^{85} -590.512 q^{86} -954.657 q^{88} +66.7144 q^{89} +3517.68 q^{92} +2683.02 q^{94} +387.441 q^{95} +1438.25 q^{97} +3327.40 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 26 q^{4} - 20 q^{10} + 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} + 44 q^{25} + 48 q^{29} - 120 q^{35} - 468 q^{38} - 1268 q^{40} - 1064 q^{43} + 716 q^{49} + 864 q^{53} + 584 q^{55}+ \cdots + 3384 q^{95}+O(q^{100}) $ 4 * q + 26 * q^4 - 20 * q^10 + 348 * q^14 + 354 * q^16 + 216 * q^17 - 136 * q^22 + 120 * q^23 + 44 * q^25 + 48 * q^29 - 120 * q^35 - 468 * q^38 - 1268 * q^40 - 1064 * q^43 + 716 * q^49 + 864 * q^53 + 584 * q^55 + 3372 * q^56 - 2280 * q^61 + 924 * q^62 + 1050 * q^64 + 1404 * q^68 + 2304 * q^74 - 816 * q^77 + 288 * q^79 + 28 * q^82 - 2392 * q^88 + 8568 * q^92 + 6656 * q^94 + 3384 * q^95

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 $

$2$	5.21898	1.84519	0.922594	−	0.385773i	$-0.126065\pi$
$2$	5.21898	1.84519	0.922594	+	0.385773i	$0.126065\pi$
$3$	0	0
$3$	0	0
$4$	19.2377	2.40472
$5$	−5.83936	−0.522288	−0.261144	−	0.965300i	$-0.584100\pi$
$5$	−5.83936	−0.522288	−0.261144	+	0.965300i	$0.584100\pi$
$6$	0	0
$7$	31.3139	1.69079	0.845395	−	0.534141i	$-0.179365\pi$
$7$	31.3139	1.69079	0.845395	+	0.534141i	$0.179365\pi$
$8$	58.6495	2.59197
$9$	0	0
$10$	−30.4755	−0.963719
$11$	−16.2773	−0.446163	−0.223082	−	0.974800i	$-0.571612\pi$
$11$	−16.2773	−0.446163	−0.223082	+	0.974800i	$0.571612\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	163.426	3.11983
$15$	0	0
$16$	152.189	2.37795
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	−66.3500	−0.801144	−0.400572	−	0.916265i	$-0.631189\pi$
$19$	−66.3500	−0.801144	−0.400572	+	0.916265i	$0.631189\pi$
$20$	−112.336	−1.25595
$21$	0	0
$22$	−84.9510	−0.823255
$23$	182.853	1.65772	0.828858	−	0.559459i	$-0.188991\pi$
$23$	182.853	1.65772	0.828858	+	0.559459i	$0.188991\pi$
$24$	0	0
$25$	−90.9019	−0.727215
$26$	0	0
$27$	0	0
$28$	602.408	4.06587
$29$	164.853	1.05560	0.527800	−	0.849369i	$-0.323017\pi$
$29$	164.853	1.05560	0.527800	+	0.849369i	$0.323017\pi$
$30$	0	0
$31$	58.9055	0.341282	0.170641	−	0.985333i	$-0.445416\pi$
$31$	58.9055	0.341282	0.170641	+	0.985333i	$0.445416\pi$
$32$	325.073	1.79579
$33$	0	0
$34$	281.825	1.42155
$35$	−182.853	−0.883079
$36$	0	0
$37$	110.366	0.490382	0.245191	−	0.969475i	$-0.421149\pi$
$37$	110.366	0.490382	0.245191	+	0.969475i	$0.421149\pi$
$38$	−346.279	−1.47826
$39$	0	0
$40$	−342.475	−1.35375
$41$	55.0357	0.209637	0.104819	−	0.994491i	$-0.466574\pi$
$41$	55.0357	0.209637	0.104819	+	0.994491i	$0.466574\pi$
$42$	0	0
$43$	−113.147	−0.401274	−0.200637	−	0.979666i	$-0.564301\pi$
$43$	−113.147	−0.401274	−0.200637	+	0.979666i	$0.564301\pi$
$44$	−313.139	−1.07290
$45$	0	0
$46$	954.305	3.05880
$47$	514.089	1.59548	0.797740	−	0.603001i	$-0.206029\pi$
$47$	514.089	1.59548	0.797740	+	0.603001i	$0.206029\pi$
$48$	0	0
$49$	637.559	1.85877
$50$	−474.415	−1.34185
$51$	0	0
$52$	0	0
$53$	−242.559	−0.628641	−0.314321	−	0.949317i	$-0.601777\pi$
$53$	−242.559	−0.628641	−0.314321	+	0.949317i	$0.601777\pi$
$54$	0	0
$55$	95.0490	0.233026
$56$	1836.54	4.38247
$57$	0	0
$58$	860.364	1.94778
$59$	−265.036	−0.584825	−0.292413	−	0.956292i	$-0.594458\pi$
$59$	−265.036	−0.584825	−0.292413	+	0.956292i	$0.594458\pi$
$60$	0	0
$61$	−468.098	−0.982522	−0.491261	−	0.871013i	$-0.663464\pi$
$61$	−468.098	−0.982522	−0.491261	+	0.871013i	$0.663464\pi$
$62$	307.426	0.629729
$63$	0	0
$64$	479.042	0.935628
$65$	0	0
$66$	0	0
$67$	−852.919	−1.55523	−0.777617	−	0.628739i	$-0.783572\pi$
$67$	−852.919	−1.55523	−0.777617	+	0.628739i	$0.783572\pi$
$68$	1038.84	1.85261
$69$	0	0
$70$	−954.305	−1.62945
$71$	165.619	0.276836	0.138418	−	0.990374i	$-0.455798\pi$
$71$	165.619	0.276836	0.138418	+	0.990374i	$0.455798\pi$
$72$	0	0
$73$	−315.325	−0.505562	−0.252781	−	0.967524i	$-0.581345\pi$
$73$	−315.325	−0.505562	−0.252781	+	0.967524i	$0.581345\pi$
$74$	576.000	0.904846
$75$	0	0
$76$	−1276.42	−1.92653
$77$	−509.706	−0.754368
$78$	0	0
$79$	479.608	0.683039	0.341519	−	0.939875i	$-0.389058\pi$
$79$	479.608	0.683039	0.341519	+	0.939875i	$0.389058\pi$
$80$	−888.684	−1.24197
$81$	0	0
$82$	287.230	0.386820
$83$	574.235	0.759404	0.379702	−	0.925109i	$-0.376027\pi$
$83$	574.235	0.759404	0.379702	+	0.925109i	$0.376027\pi$
$84$	0	0
$85$	−315.325	−0.402374
$86$	−590.512	−0.740426
$87$	0	0
$88$	−954.657	−1.15644
$89$	66.7144	0.0794575	0.0397287	−	0.999211i	$-0.487351\pi$
$89$	66.7144	0.0794575	0.0397287	+	0.999211i	$0.487351\pi$
$90$	0	0
$91$	0	0
$92$	3517.68	3.98634
$93$	0	0
$94$	2683.02	2.94396
$95$	387.441	0.418428
$96$	0	0
$97$	1438.25	1.50549	0.752744	−	0.658313i	$-0.228730\pi$
$97$	1438.25	1.50549	0.752744	+	0.658313i	$0.228730\pi$
$98$	3327.40	3.42978
$99$	0	0
$100$	−1748.75	−1.74875
$101$	896.264	0.882986	0.441493	−	0.897265i	$-0.354449\pi$
$101$	896.264	0.882986	0.441493	+	0.897265i	$0.354449\pi$
$102$	0	0
$103$	−22.2644	−0.0212988	−0.0106494	−	0.999943i	$-0.503390\pi$
$103$	−22.2644	−0.0212988	−0.0106494	+	0.999943i	$0.503390\pi$
$104$	0	0
$105$	0	0
$106$	−1265.91	−1.15996
$107$	351.441	0.317524	0.158762	−	0.987317i	$-0.449250\pi$
$107$	351.441	0.317524	0.158762	+	0.987317i	$0.449250\pi$
$108$	0	0
$109$	−967.008	−0.849748	−0.424874	−	0.905252i	$-0.639682\pi$
$109$	−967.008	−0.849748	−0.424874	+	0.905252i	$0.639682\pi$
$110$	496.059	0.429976
$111$	0	0
$112$	4765.62	4.02061
$113$	−48.2943	−0.0402048	−0.0201024	−	0.999798i	$-0.506399\pi$
$113$	−48.2943	−0.0402048	−0.0201024	+	0.999798i	$0.506399\pi$
$114$	0	0
$115$	−1067.74	−0.865805
$116$	3171.40	2.53842
$117$	0	0
$118$	−1383.22	−1.07911
$119$	1690.95	1.30260
$120$	0	0
$121$	−1066.05	−0.800938
$122$	−2442.99	−1.81294
$123$	0	0
$124$	1133.21	0.820686
$125$	1260.73	0.902104
$126$	0	0
$127$	−1763.02	−1.23183	−0.615916	−	0.787812i	$-0.711214\pi$
$127$	−1763.02	−1.23183	−0.615916	+	0.787812i	$0.711214\pi$
$128$	−100.479	−0.0693844
$129$	0	0
$130$	0	0
$131$	−955.970	−0.637584	−0.318792	−	0.947825i	$-0.603277\pi$
$131$	−955.970	−0.637584	−0.318792	+	0.947825i	$0.603277\pi$
$132$	0	0
$133$	−2077.68	−1.35457
$134$	−4451.37	−2.86970
$135$	0	0
$136$	3167.07	1.99687
$137$	15.9215	0.00992894	0.00496447	−	0.999988i	$-0.498420\pi$
$137$	15.9215	0.00992894	0.00496447	+	0.999988i	$0.498420\pi$
$138$	0	0
$139$	2074.26	1.26573	0.632866	−	0.774261i	$-0.281878\pi$
$139$	2074.26	1.26573	0.632866	+	0.774261i	$0.281878\pi$
$140$	−3517.68	−2.12356
$141$	0	0
$142$	864.362	0.510815
$143$	0	0
$144$	0	0
$145$	−962.635	−0.551327
$146$	−1645.68	−0.932857
$147$	0	0
$148$	2123.20	1.17923
$149$	−2764.08	−1.51975	−0.759873	−	0.650071i	$-0.774739\pi$
$149$	−2764.08	−1.51975	−0.759873	+	0.650071i	$0.774739\pi$
$150$	0	0
$151$	1618.46	0.872239	0.436120	−	0.899889i	$-0.356352\pi$
$151$	1618.46	0.872239	0.436120	+	0.899889i	$0.356352\pi$
$152$	−3891.40	−2.07654
$153$	0	0
$154$	−2660.14	−1.39195
$155$	−343.970	−0.178247
$156$	0	0
$157$	−1109.97	−0.564237	−0.282119	−	0.959380i	$-0.591037\pi$
$157$	−1109.97	−0.564237	−0.282119	+	0.959380i	$0.591037\pi$
$158$	2503.06	1.26034
$159$	0	0
$160$	−1898.22	−0.937921
$161$	5725.83	2.80285
$162$	0	0
$163$	−233.201	−0.112060	−0.0560299	−	0.998429i	$-0.517844\pi$
$163$	−233.201	−0.112060	−0.0560299	+	0.998429i	$0.517844\pi$
$164$	1058.76	0.504119
$165$	0	0
$166$	2996.92	1.40124
$167$	215.405	0.0998118	0.0499059	−	0.998754i	$-0.484108\pi$
$167$	215.405	0.0998118	0.0499059	+	0.998754i	$0.484108\pi$
$168$	0	0
$169$	0	0
$170$	−1645.68	−0.742456
$171$	0	0
$172$	−2176.69	−0.964950
$173$	1383.15	0.607854	0.303927	−	0.952695i	$-0.401702\pi$
$173$	1383.15	0.607854	0.303927	+	0.952695i	$0.401702\pi$
$174$	0	0
$175$	−2846.49	−1.22957
$176$	−2477.22	−1.06095
$177$	0	0
$178$	348.181	0.146614
$179$	−3642.79	−1.52109	−0.760545	−	0.649285i	$-0.775068\pi$
$179$	−3642.79	−1.52109	−0.760545	+	0.649285i	$0.775068\pi$
$180$	0	0
$181$	2621.97	1.07674	0.538369	−	0.842709i	$-0.319041\pi$
$181$	2621.97	1.07674	0.538369	+	0.842709i	$0.319041\pi$
$182$	0	0
$183$	0	0
$184$	10724.2	4.29674
$185$	−644.469	−0.256121
$186$	0	0
$187$	−878.975	−0.343727
$188$	9889.91	3.83668
$189$	0	0
$190$	2022.05	0.772078
$191$	3419.32	1.29536	0.647679	−	0.761913i	$-0.275740\pi$
$191$	3419.32	1.29536	0.647679	+	0.761913i	$0.275740\pi$
$192$	0	0
$193$	1698.39	0.633435	0.316718	−	0.948520i	$-0.397419\pi$
$193$	1698.39	0.633435	0.316718	+	0.948520i	$0.397419\pi$
$194$	7506.20	2.77791
$195$	0	0
$196$	12265.2	4.46982
$197$	−2293.72	−0.829548	−0.414774	−	0.909925i	$-0.636139\pi$
$197$	−2293.72	−0.829548	−0.414774	+	0.909925i	$0.636139\pi$
$198$	0	0
$199$	900.981	0.320949	0.160474	−	0.987040i	$-0.448698\pi$
$199$	900.981	0.320949	0.160474	+	0.987040i	$0.448698\pi$
$200$	−5331.35	−1.88492
$201$	0	0
$202$	4677.58	1.62928
$203$	5162.18	1.78480
$204$	0	0
$205$	−321.373	−0.109491
$206$	−116.197	−0.0393002
$207$	0	0
$208$	0	0
$209$	1080.00	0.357441
$210$	0	0
$211$	431.019	0.140628	0.0703142	−	0.997525i	$-0.477600\pi$
$211$	431.019	0.140628	0.0703142	+	0.997525i	$0.477600\pi$
$212$	−4666.28	−1.51170
$213$	0	0
$214$	1834.17	0.585892
$215$	660.706	0.209580
$216$	0	0
$217$	1844.56	0.577036
$218$	−5046.79	−1.56794
$219$	0	0
$220$	1828.53	0.560361
$221$	0	0
$222$	0	0
$223$	−4104.30	−1.23249	−0.616243	−	0.787556i	$-0.711346\pi$
$223$	−4104.30	−1.23249	−0.616243	+	0.787556i	$0.711346\pi$
$224$	10179.3	3.03631
$225$	0	0
$226$	−252.047	−0.0741854
$227$	−1809.11	−0.528963	−0.264482	−	0.964391i	$-0.585201\pi$
$227$	−1809.11	−0.528963	−0.264482	+	0.964391i	$0.585201\pi$
$228$	0	0
$229$	−5249.33	−1.51478	−0.757392	−	0.652961i	$-0.773527\pi$
$229$	−5249.33	−1.51478	−0.757392	+	0.652961i	$0.773527\pi$
$230$	−5572.53	−1.59757
$231$	0	0
$232$	9668.54	2.73608
$233$	2808.88	0.789768	0.394884	−	0.918731i	$-0.370785\pi$
$233$	2808.88	0.789768	0.394884	+	0.918731i	$0.370785\pi$
$234$	0	0
$235$	−3001.95	−0.833300
$236$	−5098.69	−1.40634
$237$	0	0
$238$	8825.03	2.40354
$239$	−6712.01	−1.81659	−0.908293	−	0.418334i	$-0.862614\pi$
$239$	−6712.01	−1.81659	−0.908293	+	0.418334i	$0.862614\pi$
$240$	0	0
$241$	2519.11	0.673321	0.336661	−	0.941626i	$-0.390703\pi$
$241$	2519.11	0.673321	0.336661	+	0.941626i	$0.390703\pi$
$242$	−5563.69	−1.47788
$243$	0	0
$244$	−9005.15	−2.36269
$245$	−3722.93	−0.970814
$246$	0	0
$247$	0	0
$248$	3454.78	0.884591
$249$	0	0
$250$	6579.71	1.66455
$251$	−828.000	−0.208219	−0.104109	−	0.994566i	$-0.533199\pi$
$251$	−828.000	−0.208219	−0.104109	+	0.994566i	$0.533199\pi$
$252$	0	0
$253$	−2976.35	−0.739612
$254$	−9201.16	−2.27296
$255$	0	0
$256$	−4356.73	−1.06366
$257$	5840.76	1.41765	0.708826	−	0.705383i	$-0.249225\pi$
$257$	5840.76	1.41765	0.708826	+	0.705383i	$0.249225\pi$
$258$	0	0
$259$	3456.00	0.829133
$260$	0	0
$261$	0	0
$262$	−4989.19	−1.17646
$263$	4064.06	0.952854	0.476427	−	0.879214i	$-0.341932\pi$
$263$	4064.06	0.952854	0.476427	+	0.879214i	$0.341932\pi$
$264$	0	0
$265$	1416.39	0.328332
$266$	−10843.3	−2.49943
$267$	0	0
$268$	−16408.2	−3.73990
$269$	−1845.44	−0.418285	−0.209142	−	0.977885i	$-0.567067\pi$
$269$	−1845.44	−0.418285	−0.209142	+	0.977885i	$0.567067\pi$
$270$	0	0
$271$	−2106.78	−0.472242	−0.236121	−	0.971724i	$-0.575876\pi$
$271$	−2106.78	−0.472242	−0.236121	+	0.971724i	$0.575876\pi$
$272$	8218.19	1.83199
$273$	0	0
$274$	83.0939	0.0183208
$275$	1479.64	0.324457
$276$	0	0
$277$	−4781.94	−1.03725	−0.518626	−	0.855001i	$-0.673556\pi$
$277$	−4781.94	−1.03725	−0.518626	+	0.855001i	$0.673556\pi$
$278$	10825.5	2.33551
$279$	0	0
$280$	−10724.2	−2.28891
$281$	−5865.81	−1.24528	−0.622642	−	0.782507i	$-0.713941\pi$
$281$	−5865.81	−1.24528	−0.622642	+	0.782507i	$0.713941\pi$
$282$	0	0
$283$	−6407.02	−1.34579	−0.672894	−	0.739739i	$-0.734949\pi$
$283$	−6407.02	−1.34579	−0.672894	+	0.739739i	$0.734949\pi$
$284$	3186.14	0.665713
$285$	0	0
$286$	0	0
$287$	1723.38	0.354453
$288$	0	0
$289$	−1997.00	−0.406473
$290$	−5023.97	−1.01730
$291$	0	0
$292$	−6066.15	−1.21573
$293$	3010.24	0.600204	0.300102	−	0.953907i	$-0.402979\pi$
$293$	3010.24	0.600204	0.300102	+	0.953907i	$0.402979\pi$
$294$	0	0
$295$	1547.64	0.305447
$296$	6472.94	1.27105
$297$	0	0
$298$	−14425.7	−2.80422
$299$	0	0
$300$	0	0
$301$	−3543.07	−0.678470
$302$	8446.69	1.60944
$303$	0	0
$304$	−10097.7	−1.90508
$305$	2733.39	0.513159
$306$	0	0
$307$	−3341.84	−0.621266	−0.310633	−	0.950530i	$-0.600541\pi$
$307$	−3341.84	−0.621266	−0.310633	+	0.950530i	$0.600541\pi$
$308$	−9805.59	−1.81404
$309$	0	0
$310$	−1795.17	−0.328900
$311$	−8755.20	−1.59634	−0.798170	−	0.602432i	$-0.794199\pi$
$311$	−8755.20	−1.59634	−0.798170	+	0.602432i	$0.794199\pi$
$312$	0	0
$313$	1948.93	0.351949	0.175974	−	0.984395i	$-0.443692\pi$
$313$	1948.93	0.351949	0.175974	+	0.984395i	$0.443692\pi$
$314$	−5792.91	−1.04112
$315$	0	0
$316$	9226.57	1.64252
$317$	1940.43	0.343802	0.171901	−	0.985114i	$-0.445009\pi$
$317$	1940.43	0.343802	0.171901	+	0.985114i	$0.445009\pi$
$318$	0	0
$319$	−2683.36	−0.470970
$320$	−2797.29	−0.488667
$321$	0	0
$322$	29883.0	5.17178
$323$	−3582.90	−0.617207
$324$	0	0
$325$	0	0
$326$	−1217.07	−0.206771
$327$	0	0
$328$	3227.82	0.543373
$329$	16098.1	2.69762
$330$	0	0
$331$	−7402.13	−1.22918	−0.614589	−	0.788848i	$-0.710678\pi$
$331$	−7402.13	−1.22918	−0.614589	+	0.788848i	$0.710678\pi$
$332$	11047.0	1.82615
$333$	0	0
$334$	1124.20	0.184171
$335$	4980.50	0.812280
$336$	0	0
$337$	−5494.05	−0.888071	−0.444035	−	0.896009i	$-0.646454\pi$
$337$	−5494.05	−0.888071	−0.444035	+	0.896009i	$0.646454\pi$
$338$	0	0
$339$	0	0
$340$	−6066.15	−0.967597
$341$	−958.823	−0.152267
$342$	0	0
$343$	9223.77	1.45200
$344$	−6636.03	−1.04009
$345$	0	0
$346$	7218.62	1.12160
$347$	−3410.76	−0.527664	−0.263832	−	0.964569i	$-0.584986\pi$
$347$	−3410.76	−0.527664	−0.263832	+	0.964569i	$0.584986\pi$
$348$	0	0
$349$	−12629.8	−1.93712	−0.968562	−	0.248773i	$-0.919973\pi$
$349$	−12629.8	−1.93712	−0.968562	+	0.248773i	$0.919973\pi$
$350$	−14855.8	−2.26878
$351$	0	0
$352$	−5291.32	−0.801217
$353$	−2981.78	−0.449586	−0.224793	−	0.974407i	$-0.572171\pi$
$353$	−2981.78	−0.449586	−0.224793	+	0.974407i	$0.572171\pi$
$354$	0	0
$355$	−967.109	−0.144588
$356$	1283.43	0.191073
$357$	0	0
$358$	−19011.7	−2.80670
$359$	−8942.30	−1.31464	−0.657321	−	0.753611i	$-0.728310\pi$
$359$	−8942.30	−1.31464	−0.657321	+	0.753611i	$0.728310\pi$
$360$	0	0
$361$	−2456.68	−0.358168
$362$	13684.0	1.98678
$363$	0	0
$364$	0	0
$365$	1841.30	0.264049
$366$	0	0
$367$	−4735.26	−0.673511	−0.336756	−	0.941592i	$-0.609330\pi$
$367$	−4735.26	−0.673511	−0.336756	+	0.941592i	$0.609330\pi$
$368$	27828.1	3.94196
$369$	0	0
$370$	−3363.47	−0.472590
$371$	−7595.45	−1.06290
$372$	0	0
$373$	−8304.01	−1.15272	−0.576361	−	0.817195i	$-0.695528\pi$
$373$	−8304.01	−1.15272	−0.576361	+	0.817195i	$0.695528\pi$
$374$	−4587.35	−0.634241
$375$	0	0
$376$	30151.1	4.13543
$377$	0	0
$378$	0	0
$379$	4088.11	0.554069	0.277035	−	0.960860i	$-0.410648\pi$
$379$	4088.11	0.554069	0.277035	+	0.960860i	$0.410648\pi$
$380$	7453.50	1.00620
$381$	0	0
$382$	17845.4	2.39018
$383$	−13951.5	−1.86132	−0.930662	−	0.365879i	$-0.880768\pi$
$383$	−13951.5	−1.86132	−0.930662	+	0.365879i	$0.880768\pi$
$384$	0	0
$385$	2976.35	0.393997
$386$	8863.88	1.16881
$387$	0	0
$388$	27668.7	3.62027
$389$	2804.26	0.365506	0.182753	−	0.983159i	$-0.441499\pi$
$389$	2804.26	0.365506	0.182753	+	0.983159i	$0.441499\pi$
$390$	0	0
$391$	9874.06	1.27712
$392$	37392.5	4.81787
$393$	0	0
$394$	−11970.9	−1.53067
$395$	−2800.60	−0.356743
$396$	0	0
$397$	6556.18	0.828830	0.414415	−	0.910088i	$-0.363986\pi$
$397$	6556.18	0.828830	0.414415	+	0.910088i	$0.363986\pi$
$398$	4702.20	0.592211
$399$	0	0
$400$	−13834.2	−1.72928
$401$	4730.95	0.589157	0.294579	−	0.955627i	$-0.404821\pi$
$401$	4730.95	0.589157	0.294579	+	0.955627i	$0.404821\pi$
$402$	0	0
$403$	0	0
$404$	17242.1	2.12333
$405$	0	0
$406$	26941.3	3.29329
$407$	−1796.47	−0.218790
$408$	0	0
$409$	−12314.4	−1.48878	−0.744389	−	0.667746i	$-0.767259\pi$
$409$	−12314.4	−1.48878	−0.744389	+	0.667746i	$0.767259\pi$
$410$	−1677.24	−0.202032
$411$	0	0
$412$	−428.316	−0.0512175
$413$	−8299.29	−0.988817
$414$	0	0
$415$	−3353.16	−0.396627
$416$	0	0
$417$	0	0
$418$	5636.50	0.659546
$419$	5499.85	0.641254	0.320627	−	0.947206i	$-0.396106\pi$
$419$	5499.85	0.641254	0.320627	+	0.947206i	$0.396106\pi$
$420$	0	0
$421$	12629.7	1.46208	0.731039	−	0.682336i	$-0.239036\pi$
$421$	12629.7	1.46208	0.731039	+	0.682336i	$0.239036\pi$
$422$	2249.48	0.259486
$423$	0	0
$424$	−14225.9	−1.62942
$425$	−4908.70	−0.560252
$426$	0	0
$427$	−14658.0	−1.66124
$428$	6760.94	0.763557
$429$	0	0
$430$	3448.21	0.386715
$431$	−7191.15	−0.803679	−0.401840	−	0.915710i	$-0.631629\pi$
$431$	−7191.15	−0.803679	−0.401840	+	0.915710i	$0.631629\pi$
$432$	0	0
$433$	6062.68	0.672873	0.336436	−	0.941706i	$-0.390778\pi$
$433$	6062.68	0.672873	0.336436	+	0.941706i	$0.390778\pi$
$434$	9626.71	1.06474
$435$	0	0
$436$	−18603.0	−2.04340
$437$	−12132.3	−1.32807
$438$	0	0
$439$	11864.3	1.28986	0.644932	−	0.764240i	$-0.276886\pi$
$439$	11864.3	1.28986	0.644932	+	0.764240i	$0.276886\pi$
$440$	5574.58	0.603995
$441$	0	0
$442$	0	0
$443$	−10560.5	−1.13261	−0.566303	−	0.824197i	$-0.691627\pi$
$443$	−10560.5	−1.13261	−0.566303	+	0.824197i	$0.691627\pi$
$444$	0	0
$445$	−389.569	−0.0414997
$446$	−21420.3	−2.27417
$447$	0	0
$448$	15000.6	1.58195
$449$	12659.7	1.33062	0.665308	−	0.746569i	$-0.268300\pi$
$449$	12659.7	1.33062	0.665308	+	0.746569i	$0.268300\pi$
$450$	0	0
$451$	−895.834	−0.0935325
$452$	−929.072	−0.0966812
$453$	0	0
$454$	−9441.69	−0.976037
$455$	0	0
$456$	0	0
$457$	−1544.24	−0.158067	−0.0790336	−	0.996872i	$-0.525183\pi$
$457$	−1544.24	−0.158067	−0.0790336	+	0.996872i	$0.525183\pi$
$458$	−27396.1	−2.79506
$459$	0	0
$460$	−20541.0	−2.08202
$461$	13196.8	1.33327	0.666635	−	0.745384i	$-0.267734\pi$
$461$	13196.8	1.33327	0.666635	+	0.745384i	$0.267734\pi$
$462$	0	0
$463$	16309.2	1.63705	0.818524	−	0.574472i	$-0.194792\pi$
$463$	16309.2	1.63705	0.818524	+	0.574472i	$0.194792\pi$
$464$	25088.7	2.51016
$465$	0	0
$466$	14659.5	1.45727
$467$	14260.8	1.41308	0.706541	−	0.707672i	$-0.250255\pi$
$467$	14260.8	1.41308	0.706541	+	0.707672i	$0.250255\pi$
$468$	0	0
$469$	−26708.2	−2.62957
$470$	−15667.1	−1.53760
$471$	0	0
$472$	−15544.2	−1.51585
$473$	1841.73	0.179034
$474$	0	0
$475$	6031.34	0.582604
$476$	32530.0	3.13238
$477$	0	0
$478$	−35029.9	−3.35194
$479$	−18011.5	−1.71809	−0.859046	−	0.511899i	$-0.828942\pi$
$479$	−18011.5	−1.71809	−0.859046	+	0.511899i	$0.828942\pi$
$480$	0	0
$481$	0	0
$482$	13147.2	1.24240
$483$	0	0
$484$	−20508.4	−1.92603
$485$	−8398.46	−0.786298
$486$	0	0
$487$	−14043.3	−1.30670	−0.653351	−	0.757055i	$-0.726637\pi$
$487$	−14043.3	−1.30670	−0.653351	+	0.757055i	$0.726637\pi$
$488$	−27453.7	−2.54666
$489$	0	0
$490$	−19429.9	−1.79133
$491$	12966.1	1.19176	0.595878	−	0.803075i	$-0.296804\pi$
$491$	12966.1	1.19176	0.595878	+	0.803075i	$0.296804\pi$
$492$	0	0
$493$	8902.06	0.813242
$494$	0	0
$495$	0	0
$496$	8964.75	0.811551
$497$	5186.17	0.468072
$498$	0	0
$499$	−10215.5	−0.916453	−0.458227	−	0.888835i	$-0.651515\pi$
$499$	−10215.5	−0.916453	−0.458227	+	0.888835i	$0.651515\pi$
$500$	24253.6	2.16930
$501$	0	0
$502$	−4321.31	−0.384203
$503$	16632.0	1.47432	0.737161	−	0.675717i	$-0.236166\pi$
$503$	16632.0	1.47432	0.737161	+	0.675717i	$0.236166\pi$
$504$	0	0
$505$	−5233.61	−0.461173
$506$	−15533.5	−1.36472
$507$	0	0
$508$	−33916.5	−2.96221
$509$	−15235.3	−1.32671	−0.663354	−	0.748306i	$-0.730868\pi$
$509$	−15235.3	−1.32671	−0.663354	+	0.748306i	$0.730868\pi$
$510$	0	0
$511$	−9874.06	−0.854799
$512$	−21933.9	−1.89326
$513$	0	0
$514$	30482.8	2.61584
$515$	130.009	0.0111241
$516$	0	0
$517$	−8367.99	−0.711845
$518$	18036.8	1.52991
$519$	0	0
$520$	0	0
$521$	−2680.23	−0.225380	−0.112690	−	0.993630i	$-0.535947\pi$
$521$	−2680.23	−0.225380	−0.112690	+	0.993630i	$0.535947\pi$
$522$	0	0
$523$	−2410.38	−0.201527	−0.100764	−	0.994910i	$-0.532129\pi$
$523$	−2410.38	−0.201527	−0.100764	+	0.994910i	$0.532129\pi$
$524$	−18390.7	−1.53321
$525$	0	0
$526$	21210.2	1.75819
$527$	3180.90	0.262926
$528$	0	0
$529$	21268.2	1.74802
$530$	7392.09	0.605834
$531$	0	0
$532$	−39969.8	−3.25735
$533$	0	0
$534$	0	0
$535$	−2052.19	−0.165839
$536$	−50023.3	−4.03111
$537$	0	0
$538$	−9631.32	−0.771814
$539$	−10377.7	−0.829315
$540$	0	0
$541$	−9969.58	−0.792284	−0.396142	−	0.918189i	$-0.629651\pi$
$541$	−9969.58	−0.792284	−0.396142	+	0.918189i	$0.629651\pi$
$542$	−10995.2	−0.871375
$543$	0	0
$544$	17554.0	1.38349
$545$	5646.70	0.443813
$546$	0	0
$547$	16848.8	1.31701	0.658505	−	0.752576i	$-0.271189\pi$
$547$	16848.8	1.31701	0.658505	+	0.752576i	$0.271189\pi$
$548$	306.293	0.0238763
$549$	0	0
$550$	7722.20	0.598683
$551$	−10938.0	−0.845688
$552$	0	0
$553$	15018.4	1.15488
$554$	−24956.8	−1.91393
$555$	0	0
$556$	39904.2	3.04373
$557$	−3800.83	−0.289132	−0.144566	−	0.989495i	$-0.546179\pi$
$557$	−3800.83	−0.289132	−0.144566	+	0.989495i	$0.546179\pi$
$558$	0	0
$559$	0	0
$560$	−27828.1	−2.09992
$561$	0	0
$562$	−30613.5	−2.29778
$563$	15750.2	1.17903	0.589513	−	0.807759i	$-0.299320\pi$
$563$	15750.2	1.17903	0.589513	+	0.807759i	$0.299320\pi$
$564$	0	0
$565$	282.007	0.0209985
$566$	−33438.1	−2.48323
$567$	0	0
$568$	9713.48	0.717550
$569$	−17753.2	−1.30800	−0.654002	−	0.756493i	$-0.726911\pi$
$569$	−17753.2	−1.30800	−0.654002	+	0.756493i	$0.726911\pi$
$570$	0	0
$571$	25293.1	1.85374	0.926868	−	0.375388i	$-0.122491\pi$
$571$	25293.1	1.85374	0.926868	+	0.375388i	$0.122491\pi$
$572$	0	0
$573$	0	0
$574$	8994.29	0.654032
$575$	−16621.7	−1.20552
$576$	0	0
$577$	18488.8	1.33396	0.666982	−	0.745073i	$-0.267586\pi$
$577$	18488.8	1.33396	0.666982	+	0.745073i	$0.267586\pi$
$578$	−10422.3	−0.750018
$579$	0	0
$580$	−18518.9	−1.32579
$581$	17981.5	1.28399
$582$	0	0
$583$	3948.20	0.280477
$584$	−18493.7	−1.31040
$585$	0	0
$586$	15710.4	1.10749
$587$	−17376.7	−1.22183	−0.610914	−	0.791697i	$-0.709198\pi$
$587$	−17376.7	−1.22183	−0.610914	+	0.791697i	$0.709198\pi$
$588$	0	0
$589$	−3908.38	−0.273416
$590$	8077.09	0.563608
$591$	0	0
$592$	16796.5	1.16610
$593$	7991.09	0.553381	0.276690	−	0.960959i	$-0.410762\pi$
$593$	7991.09	0.553381	0.276690	+	0.960959i	$0.410762\pi$
$594$	0	0
$595$	−9874.06	−0.680331
$596$	−53174.7	−3.65456
$597$	0	0
$598$	0	0
$599$	−10386.5	−0.708480	−0.354240	−	0.935154i	$-0.615260\pi$
$599$	−10386.5	−0.708480	−0.354240	+	0.935154i	$0.615260\pi$
$600$	0	0
$601$	−9241.77	−0.627254	−0.313627	−	0.949546i	$-0.601544\pi$
$601$	−9241.77	−0.627254	−0.313627	+	0.949546i	$0.601544\pi$
$602$	−18491.2	−1.25190
$603$	0	0
$604$	31135.4	2.09749
$605$	6225.04	0.418320
$606$	0	0
$607$	18921.5	1.26524	0.632619	−	0.774463i	$-0.281980\pi$
$607$	18921.5	1.26524	0.632619	+	0.774463i	$0.281980\pi$
$608$	−21568.6	−1.43869
$609$	0	0
$610$	14265.5	0.946875
$611$	0	0
$612$	0	0
$613$	−17138.1	−1.12921	−0.564603	−	0.825363i	$-0.690971\pi$
$613$	−17138.1	−1.12921	−0.564603	+	0.825363i	$0.690971\pi$
$614$	−17441.0	−1.14635
$615$	0	0
$616$	−29894.0	−1.95530
$617$	18825.8	1.22836	0.614180	−	0.789166i	$-0.289487\pi$
$617$	18825.8	1.22836	0.614180	+	0.789166i	$0.289487\pi$
$618$	0	0
$619$	−1392.83	−0.0904404	−0.0452202	−	0.998977i	$-0.514399\pi$
$619$	−1392.83	−0.0904404	−0.0452202	+	0.998977i	$0.514399\pi$
$620$	−6617.21	−0.428635
$621$	0	0
$622$	−45693.2	−2.94555
$623$	2089.09	0.134346
$624$	0	0
$625$	4000.90	0.256057
$626$	10171.4	0.649412
$627$	0	0
$628$	−21353.3	−1.35683
$629$	5959.79	0.377794
$630$	0	0
$631$	−25488.4	−1.60804	−0.804022	−	0.594599i	$-0.797311\pi$
$631$	−25488.4	−1.60804	−0.804022	+	0.594599i	$0.797311\pi$
$632$	28128.8	1.77041
$633$	0	0
$634$	10127.1	0.634379
$635$	10294.9	0.643371
$636$	0	0
$637$	0	0
$638$	−14004.4	−0.869028
$639$	0	0
$640$	586.735	0.0362387
$641$	6066.41	0.373805	0.186902	−	0.982379i	$-0.440155\pi$
$641$	6066.41	0.373805	0.186902	+	0.982379i	$0.440155\pi$
$642$	0	0
$643$	1598.78	0.0980554	0.0490277	−	0.998797i	$-0.484388\pi$
$643$	1598.78	0.0980554	0.0490277	+	0.998797i	$0.484388\pi$
$644$	110152.	6.74006
$645$	0	0
$646$	−18699.1	−1.13886
$647$	−23067.2	−1.40164	−0.700822	−	0.713336i	$-0.747183\pi$
$647$	−23067.2	−1.40164	−0.700822	+	0.713336i	$0.747183\pi$
$648$	0	0
$649$	4314.07	0.260928
$650$	0	0
$651$	0	0
$652$	−4486.26	−0.269472
$653$	23743.9	1.42293	0.711463	−	0.702723i	$-0.248033\pi$
$653$	23743.9	1.42293	0.711463	+	0.702723i	$0.248033\pi$
$654$	0	0
$655$	5582.25	0.333002
$656$	8375.81	0.498507
$657$	0	0
$658$	84015.7	4.97762
$659$	7497.65	0.443197	0.221598	−	0.975138i	$-0.428873\pi$
$659$	7497.65	0.443197	0.221598	+	0.975138i	$0.428873\pi$
$660$	0	0
$661$	−1255.26	−0.0738638	−0.0369319	−	0.999318i	$-0.511758\pi$
$661$	−1255.26	−0.0738638	−0.0369319	+	0.999318i	$0.511758\pi$
$662$	−38631.5	−2.26806
$663$	0	0
$664$	33678.6	1.96835
$665$	12132.3	0.707474
$666$	0	0
$667$	30143.8	1.74989
$668$	4143.91	0.240019
$669$	0	0
$670$	25993.1	1.49881
$671$	7619.38	0.438365
$672$	0	0
$673$	1505.97	0.0862569	0.0431284	−	0.999070i	$-0.486268\pi$
$673$	1505.97	0.0862569	0.0431284	+	0.999070i	$0.486268\pi$
$674$	−28673.3	−1.63866
$675$	0	0
$676$	0	0
$677$	16201.4	0.919751	0.459876	−	0.887983i	$-0.347894\pi$
$677$	16201.4	0.919751	0.459876	+	0.887983i	$0.347894\pi$
$678$	0	0
$679$	45037.2	2.54546
$680$	−18493.7	−1.04294
$681$	0	0
$682$	−5004.08	−0.280962
$683$	−29090.4	−1.62974	−0.814870	−	0.579644i	$-0.803192\pi$
$683$	−29090.4	−1.62974	−0.814870	+	0.579644i	$0.803192\pi$
$684$	0	0
$685$	−92.9712	−0.00518576
$686$	48138.7	2.67922
$687$	0	0
$688$	−17219.7	−0.954208
$689$	0	0
$690$	0	0
$691$	940.952	0.0518025	0.0259012	−	0.999665i	$-0.491754\pi$
$691$	940.952	0.0518025	0.0259012	+	0.999665i	$0.491754\pi$
$692$	26608.6	1.46172
$693$	0	0
$694$	−17800.7	−0.973639
$695$	−12112.4	−0.661077
$696$	0	0
$697$	2971.93	0.161506
$698$	−65914.5	−3.57436
$699$	0	0
$700$	−54760.1	−2.95676
$701$	30713.9	1.65485	0.827424	−	0.561578i	$-0.189805\pi$
$701$	30713.9	1.65485	0.827424	+	0.561578i	$0.189805\pi$
$702$	0	0
$703$	−7322.81	−0.392866
$704$	−7797.51	−0.417443
$705$	0	0
$706$	−15561.8	−0.829571
$707$	28065.5	1.49294
$708$	0	0
$709$	25640.5	1.35818	0.679089	−	0.734056i	$-0.262375\pi$
$709$	25640.5	1.35818	0.679089	+	0.734056i	$0.262375\pi$
$710$	−5047.32	−0.266792
$711$	0	0
$712$	3912.77	0.205951
$713$	10771.0	0.565748
$714$	0	0
$715$	0	0
$716$	−70079.1	−3.65779
$717$	0	0
$718$	−46669.7	−2.42576
$719$	−20842.5	−1.08108	−0.540538	−	0.841319i	$-0.681779\pi$
$719$	−20842.5	−1.08108	−0.540538	+	0.841319i	$0.681779\pi$
$720$	0	0
$721$	−697.183	−0.0360117
$722$	−12821.3	−0.660888
$723$	0	0
$724$	50440.8	2.58925
$725$	−14985.4	−0.767649
$726$	0	0
$727$	−263.608	−0.0134480	−0.00672398	−	0.999977i	$-0.502140\pi$
$727$	−263.608	−0.0134480	−0.00672398	+	0.999977i	$0.502140\pi$
$728$	0	0
$729$	0	0
$730$	9609.69	0.487220
$731$	−6109.94	−0.309144
$732$	0	0
$733$	21835.5	1.10029	0.550146	−	0.835069i	$-0.314572\pi$
$733$	21835.5	1.10029	0.550146	+	0.835069i	$0.314572\pi$
$734$	−24713.2	−1.24275
$735$	0	0
$736$	59440.6	2.97692
$737$	13883.2	0.693888
$738$	0	0
$739$	−19536.4	−0.972476	−0.486238	−	0.873826i	$-0.661631\pi$
$739$	−19536.4	−0.972476	−0.486238	+	0.873826i	$0.661631\pi$
$740$	−12398.1	−0.615897
$741$	0	0
$742$	−39640.5	−1.96125
$743$	30353.1	1.49872	0.749360	−	0.662163i	$-0.230361\pi$
$743$	30353.1	1.49872	0.749360	+	0.662163i	$0.230361\pi$
$744$	0	0
$745$	16140.5	0.793746
$746$	−43338.4	−2.12699
$747$	0	0
$748$	−16909.5	−0.826567
$749$	11005.0	0.536867
$750$	0	0
$751$	−3904.20	−0.189702	−0.0948510	−	0.995491i	$-0.530237\pi$
$751$	−3904.20	−0.189702	−0.0948510	+	0.995491i	$0.530237\pi$
$752$	78238.5	3.79397
$753$	0	0
$754$	0	0
$755$	−9450.74	−0.455560
$756$	0	0
$757$	−2900.52	−0.139262	−0.0696308	−	0.997573i	$-0.522182\pi$
$757$	−2900.52	−0.139262	−0.0696308	+	0.997573i	$0.522182\pi$
$758$	21335.8	1.02236
$759$	0	0
$760$	22723.3	1.08455
$761$	33518.7	1.59665	0.798325	−	0.602227i	$-0.205720\pi$
$761$	33518.7	1.59665	0.798325	+	0.602227i	$0.205720\pi$
$762$	0	0
$763$	−30280.8	−1.43675
$764$	65780.0	3.11497
$765$	0	0
$766$	−72812.5	−3.43449
$767$	0	0
$768$	0	0
$769$	−552.769	−0.0259211	−0.0129606	−	0.999916i	$-0.504126\pi$
$769$	−552.769	−0.0259211	−0.0129606	+	0.999916i	$0.504126\pi$
$770$	15533.5	0.726999
$771$	0	0
$772$	32673.3	1.52323
$773$	−36498.8	−1.69828	−0.849141	−	0.528166i	$-0.822880\pi$
$773$	−36498.8	−1.69828	−0.849141	+	0.528166i	$0.822880\pi$
$774$	0	0
$775$	−5354.62	−0.248185
$776$	84352.8	3.90218
$777$	0	0
$778$	14635.4	0.674427
$779$	−3651.62	−0.167950
$780$	0	0
$781$	−2695.83	−0.123514
$782$	51532.5	2.35652
$783$	0	0
$784$	97029.2	4.42006
$785$	6481.51	0.294694
$786$	0	0
$787$	4284.79	0.194074	0.0970371	−	0.995281i	$-0.469063\pi$
$787$	4284.79	0.194074	0.0970371	+	0.995281i	$0.469063\pi$
$788$	−44126.0	−1.99483
$789$	0	0
$790$	−14616.3	−0.658258
$791$	−1512.28	−0.0679779
$792$	0	0
$793$	0	0
$794$	34216.6	1.52935
$795$	0	0
$796$	17332.8	0.771792
$797$	29538.5	1.31281	0.656405	−	0.754409i	$-0.272076\pi$
$797$	29538.5	1.31281	0.656405	+	0.754409i	$0.272076\pi$
$798$	0	0
$799$	27760.8	1.22917
$800$	−29549.8	−1.30593
$801$	0	0
$802$	24690.7	1.08711
$803$	5132.65	0.225563
$804$	0	0
$805$	−33435.2	−1.46389
$806$	0	0
$807$	0	0
$808$	52565.5	2.28867
$809$	895.586	0.0389211	0.0194605	−	0.999811i	$-0.493805\pi$
$809$	895.586	0.0389211	0.0194605	+	0.999811i	$0.493805\pi$
$810$	0	0
$811$	−20139.7	−0.872011	−0.436006	−	0.899944i	$-0.643607\pi$
$811$	−20139.7	−0.872011	−0.436006	+	0.899944i	$0.643607\pi$
$812$	99308.7	4.29194
$813$	0	0
$814$	−9375.73	−0.403709
$815$	1361.75	0.0585274
$816$	0	0
$817$	7507.31	0.321478
$818$	−64268.8	−2.74707
$819$	0	0
$820$	−6182.49	−0.263295
$821$	−17263.2	−0.733848	−0.366924	−	0.930251i	$-0.619589\pi$
$821$	−17263.2	−0.733848	−0.366924	+	0.930251i	$0.619589\pi$
$822$	0	0
$823$	12114.5	0.513104	0.256552	−	0.966530i	$-0.417413\pi$
$823$	12114.5	0.513104	0.256552	+	0.966530i	$0.417413\pi$
$824$	−1305.79	−0.0552057
$825$	0	0
$826$	−43313.8	−1.82455
$827$	31450.9	1.32243	0.661217	−	0.750194i	$-0.270040\pi$
$827$	31450.9	1.32243	0.661217	+	0.750194i	$0.270040\pi$
$828$	0	0
$829$	13760.4	0.576499	0.288250	−	0.957555i	$-0.406927\pi$
$829$	13760.4	0.576499	0.288250	+	0.957555i	$0.406927\pi$
$830$	−17500.1	−0.731852
$831$	0	0
$832$	0	0
$833$	34428.2	1.43201
$834$	0	0
$835$	−1257.83	−0.0521305
$836$	20776.8	0.859545
$837$	0	0
$838$	28703.6	1.18323
$839$	−9846.21	−0.405160	−0.202580	−	0.979266i	$-0.564933\pi$
$839$	−9846.21	−0.405160	−0.202580	+	0.979266i	$0.564933\pi$
$840$	0	0
$841$	2787.47	0.114292
$842$	65914.2	2.69781
$843$	0	0
$844$	8291.83	0.338171
$845$	0	0
$846$	0	0
$847$	−33382.1	−1.35422
$848$	−36914.7	−1.49488
$849$	0	0
$850$	−25618.4	−1.03377
$851$	20180.8	0.812914
$852$	0	0
$853$	27574.5	1.10684	0.553420	−	0.832903i	$-0.313323\pi$
$853$	27574.5	1.10684	0.553420	+	0.832903i	$0.313323\pi$
$854$	−76499.6	−3.06530
$855$	0	0
$856$	20611.9	0.823013
$857$	8046.95	0.320745	0.160373	−	0.987057i	$-0.448730\pi$
$857$	8046.95	0.320745	0.160373	+	0.987057i	$0.448730\pi$
$858$	0	0
$859$	2898.13	0.115114	0.0575570	−	0.998342i	$-0.481669\pi$
$859$	2898.13	0.115114	0.0575570	+	0.998342i	$0.481669\pi$
$860$	12710.5	0.503982
$861$	0	0
$862$	−37530.5	−1.48294
$863$	−4961.16	−0.195689	−0.0978447	−	0.995202i	$-0.531195\pi$
$863$	−4961.16	−0.195689	−0.0978447	+	0.995202i	$0.531195\pi$
$864$	0	0
$865$	−8076.69	−0.317475
$866$	31641.0	1.24158
$867$	0	0
$868$	35485.1	1.38761
$869$	−7806.72	−0.304747
$870$	0	0
$871$	0	0
$872$	−56714.5	−2.20252
$873$	0	0
$874$	−63318.2	−2.45054
$875$	39478.3	1.52527
$876$	0	0
$877$	1386.66	0.0533913	0.0266957	−	0.999644i	$-0.491502\pi$
$877$	1386.66	0.0533913	0.0266957	+	0.999644i	$0.491502\pi$
$878$	61919.3	2.38004
$879$	0	0
$880$	14465.4	0.554123
$881$	−9030.36	−0.345335	−0.172668	−	0.984980i	$-0.555239\pi$
$881$	−9030.36	−0.345335	−0.172668	+	0.984980i	$0.555239\pi$
$882$	0	0
$883$	15512.7	0.591216	0.295608	−	0.955309i	$-0.404478\pi$
$883$	15512.7	0.591216	0.295608	+	0.955309i	$0.404478\pi$
$884$	0	0
$885$	0	0
$886$	−55115.0	−2.08987
$887$	−7431.21	−0.281303	−0.140651	−	0.990059i	$-0.544920\pi$
$887$	−7431.21	−0.281303	−0.140651	+	0.990059i	$0.544920\pi$
$888$	0	0
$889$	−55207.0	−2.08277
$890$	−2033.15	−0.0765747
$891$	0	0
$892$	−78957.5	−2.96378
$893$	−34109.8	−1.27821
$894$	0	0
$895$	21271.6	0.794447
$896$	−3146.40	−0.117315
$897$	0	0
$898$	66070.5	2.45524
$899$	9710.74	0.360257
$900$	0	0
$901$	−13098.2	−0.484310
$902$	−4675.34	−0.172585
$903$	0	0
$904$	−2832.44	−0.104210
$905$	−15310.6	−0.562367
$906$	0	0
$907$	10550.2	0.386234	0.193117	−	0.981176i	$-0.438140\pi$
$907$	10550.2	0.386234	0.193117	+	0.981176i	$0.438140\pi$
$908$	−34803.1	−1.27201
$909$	0	0
$910$	0	0
$911$	−35703.3	−1.29847	−0.649234	−	0.760589i	$-0.724910\pi$
$911$	−35703.3	−1.29847	−0.649234	+	0.760589i	$0.724910\pi$
$912$	0	0
$913$	−9347.01	−0.338818
$914$	−8059.38	−0.291664
$915$	0	0
$916$	−100985.	−3.64263
$917$	−29935.1	−1.07802
$918$	0	0
$919$	−42896.6	−1.53975	−0.769873	−	0.638197i	$-0.779681\pi$
$919$	−42896.6	−1.53975	−0.769873	+	0.638197i	$0.779681\pi$
$920$	−62622.6	−2.24414
$921$	0	0
$922$	68874.0	2.46013
$923$	0	0
$924$	0	0
$925$	−10032.5	−0.356613
$926$	85117.5	3.02066
$927$	0	0
$928$	53589.3	1.89564
$929$	−10366.7	−0.366114	−0.183057	−	0.983102i	$-0.558599\pi$
$929$	−10366.7	−0.366114	−0.183057	+	0.983102i	$0.558599\pi$
$930$	0	0
$931$	−42302.0	−1.48914
$932$	54036.6	1.89917
$933$	0	0
$934$	74426.6	2.60740
$935$	5132.65	0.179525
$936$	0	0
$937$	20289.8	0.707405	0.353702	−	0.935358i	$-0.384923\pi$
$937$	20289.8	0.707405	0.353702	+	0.935358i	$0.384923\pi$
$938$	−139390.	−4.85206
$939$	0	0
$940$	−57750.7	−2.00385
$941$	37089.1	1.28488	0.642438	−	0.766337i	$-0.277923\pi$
$941$	37089.1	1.28488	0.642438	+	0.766337i	$0.277923\pi$
$942$	0	0
$943$	10063.4	0.347519
$944$	−40335.4	−1.39068
$945$	0	0
$946$	9611.96	0.330351
$947$	23458.2	0.804952	0.402476	−	0.915430i	$-0.368150\pi$
$947$	23458.2	0.804952	0.402476	+	0.915430i	$0.368150\pi$
$948$	0	0
$949$	0	0
$950$	31477.5	1.07501
$951$	0	0
$952$	99173.4	3.37629
$953$	−34695.5	−1.17933	−0.589663	−	0.807649i	$-0.700740\pi$
$953$	−34695.5	−1.17933	−0.589663	+	0.807649i	$0.700740\pi$
$954$	0	0
$955$	−19966.6	−0.676550
$956$	−129124.	−4.36838
$957$	0	0
$958$	−94001.6	−3.17020
$959$	498.563	0.0167878
$960$	0	0
$961$	−26321.1	−0.883527
$962$	0	0
$963$	0	0
$964$	48462.1	1.61915
$965$	−9917.53	−0.330836
$966$	0	0
$967$	−6289.66	−0.209164	−0.104582	−	0.994516i	$-0.533351\pi$
$967$	−6289.66	−0.209164	−0.104582	+	0.994516i	$0.533351\pi$
$968$	−62523.3	−2.07601
$969$	0	0
$970$	−43831.4	−1.45087
$971$	20185.9	0.667145	0.333573	−	0.942724i	$-0.391746\pi$
$971$	20185.9	0.667145	0.333573	+	0.942724i	$0.391746\pi$
$972$	0	0
$973$	64953.2	2.14009
$974$	−73291.8	−2.41111
$975$	0	0
$976$	−71239.2	−2.33639
$977$	44244.0	1.44881	0.724406	−	0.689373i	$-0.242114\pi$
$977$	44244.0	1.44881	0.724406	+	0.689373i	$0.242114\pi$
$978$	0	0
$979$	−1085.93	−0.0354510
$980$	−71620.8	−2.33453
$981$	0	0
$982$	67669.9	2.19901
$983$	8835.11	0.286670	0.143335	−	0.989674i	$-0.454217\pi$
$983$	8835.11	0.286670	0.143335	+	0.989674i	$0.454217\pi$
$984$	0	0
$985$	13393.9	0.433263
$986$	46459.6	1.50058
$987$	0	0
$988$	0	0
$989$	−20689.3	−0.665198
$990$	0	0
$991$	34915.1	1.11919	0.559594	−	0.828767i	$-0.310957\pi$
$991$	34915.1	1.11919	0.559594	+	0.828767i	$0.310957\pi$
$992$	19148.6	0.612872
$993$	0	0
$994$	27066.5	0.863680
$995$	−5261.15	−0.167628
$996$	0	0
$997$	37962.8	1.20591	0.602956	−	0.797774i	$-0.293989\pi$
$997$	37962.8	1.20591	0.602956	+	0.797774i	$0.293989\pi$
$998$	−53314.6	−1.69103
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.x.1.4		4
3.2	odd	2		507.4.a.j.1.1		4
13.5	odd	4		117.4.b.d.64.1		4
13.8	odd	4		117.4.b.d.64.4		4
13.12	even	2	inner	1521.4.a.x.1.1		4
39.5	even	4		39.4.b.a.25.4	yes	4
39.8	even	4		39.4.b.a.25.1	✓	4
39.38	odd	2		507.4.a.j.1.4		4
156.47	odd	4		624.4.c.e.337.3		4
156.83	odd	4		624.4.c.e.337.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.a.25.1	✓	4	39.8	even	4
39.4.b.a.25.4	yes	4	39.5	even	4
117.4.b.d.64.1		4	13.5	odd	4
117.4.b.d.64.4		4	13.8	odd	4
507.4.a.j.1.1		4	3.2	odd	2
507.4.a.j.1.4		4	39.38	odd	2
624.4.c.e.337.2		4	156.83	odd	4
624.4.c.e.337.3		4	156.47	odd	4
1521.4.a.x.1.1		4	13.12	even	2	inner
1521.4.a.x.1.4		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+5.21898 q^{2} +19.2377 q^{4} -5.83936 q^{5} +31.3139 q^{7} +58.6495 q^{8} -30.4755 q^{10} -16.2773 q^{11} +163.426 q^{14} +152.189 q^{16} +54.0000 q^{17} -66.3500 q^{19} -112.336 q^{20} -84.9510 q^{22} +182.853 q^{23} -90.9019 q^{25} +602.408 q^{28} +164.853 q^{29} +58.9055 q^{31} +325.073 q^{32} +281.825 q^{34} -182.853 q^{35} +110.366 q^{37} -346.279 q^{38} -342.475 q^{40} +55.0357 q^{41} -113.147 q^{43} -313.139 q^{44} +954.305 q^{46} +514.089 q^{47} +637.559 q^{49} -474.415 q^{50} -242.559 q^{53} +95.0490 q^{55} +1836.54 q^{56} +860.364 q^{58} -265.036 q^{59} -468.098 q^{61} +307.426 q^{62} +479.042 q^{64} -852.919 q^{67} +1038.84 q^{68} -954.305 q^{70} +165.619 q^{71} -315.325 q^{73} +576.000 q^{74} -1276.42 q^{76} -509.706 q^{77} +479.608 q^{79} -888.684 q^{80} +287.230 q^{82} +574.235 q^{83} -315.325 q^{85} -590.512 q^{86} -954.657 q^{88} +66.7144 q^{89} +3517.68 q^{92} +2683.02 q^{94} +387.441 q^{95} +1438.25 q^{97} +3327.40 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 26 q^{4} - 20 q^{10} + 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} + 44 q^{25} + 48 q^{29} - 120 q^{35} - 468 q^{38} - 1268 q^{40} - 1064 q^{43} + 716 q^{49} + 864 q^{53} + 584 q^{55}+ \cdots + 3384 q^{95}+O(q^{100}) \) 4 * q + 26 * q^4 - 20 * q^10 + 348 * q^14 + 354 * q^16 + 216 * q^17 - 136 * q^22 + 120 * q^23 + 44 * q^25 + 48 * q^29 - 120 * q^35 - 468 * q^38 - 1268 * q^40 - 1064 * q^43 + 716 * q^49 + 864 * q^53 + 584 * q^55 + 3372 * q^56 - 2280 * q^61 + 924 * q^62 + 1050 * q^64 + 1404 * q^68 + 2304 * q^74 - 816 * q^77 + 288 * q^79 + 28 * q^82 - 2392 * q^88 + 8568 * q^92 + 6656 * q^94 + 3384 * q^95

Introduction

L-functions

Modular forms

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Database

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