LMFDB - Embedded newform 1521.4.a.bn.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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} + \cdots - 6889792 $ x^18 - 100x^16 + 4066x^14 - 86197x^12 + 1016064x^10 - 6594119x^8 + 22251817x^6 - 37096332x^4 + 27824160x^2 - 6889792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7\cdot 13^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.66578$ 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)

$f(q)$	$=$	$q-3.66578 q^{2} +5.43796 q^{4} +9.45496 q^{5} -15.4112 q^{7} +9.39187 q^{8} +O(q^{10})$	$q-3.66578 q^{2} +5.43796 q^{4} +9.45496 q^{5} -15.4112 q^{7} +9.39187 q^{8} -34.6598 q^{10} -12.7273 q^{11} +56.4943 q^{14} -77.9323 q^{16} +18.9954 q^{17} -95.4839 q^{19} +51.4157 q^{20} +46.6556 q^{22} -104.290 q^{23} -35.6037 q^{25} -83.8058 q^{28} -23.3225 q^{29} +177.018 q^{31} +210.548 q^{32} -69.6329 q^{34} -145.713 q^{35} +350.533 q^{37} +350.023 q^{38} +88.7998 q^{40} -348.495 q^{41} +60.8686 q^{43} -69.2107 q^{44} +382.304 q^{46} -226.920 q^{47} -105.494 q^{49} +130.516 q^{50} +294.134 q^{53} -120.336 q^{55} -144.740 q^{56} +85.4954 q^{58} -596.423 q^{59} -487.247 q^{61} -648.908 q^{62} -148.364 q^{64} +943.554 q^{67} +103.296 q^{68} +534.151 q^{70} +1044.76 q^{71} +554.083 q^{73} -1284.98 q^{74} -519.238 q^{76} +196.144 q^{77} +126.990 q^{79} -736.846 q^{80} +1277.51 q^{82} +1448.23 q^{83} +179.601 q^{85} -223.131 q^{86} -119.533 q^{88} -247.561 q^{89} -567.125 q^{92} +831.840 q^{94} -902.797 q^{95} -136.799 q^{97} +386.716 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) $ 18 * q + 56 * q^4 + 94 * q^7	$ 18 q + 56 q^{4} + 94 q^{7} + 156 q^{10} + 88 q^{16} + 448 q^{19} - 256 q^{22} - 16 q^{25} + 1300 q^{28} + 818 q^{31} + 900 q^{34} + 524 q^{37} + 800 q^{40} + 752 q^{43} + 1650 q^{46} + 2948 q^{49} - 464 q^{55} + 1626 q^{58} - 1784 q^{61} - 4274 q^{64} + 2872 q^{67} + 5772 q^{70} + 3078 q^{73} + 5726 q^{76} + 3326 q^{79} - 1038 q^{82} + 2340 q^{85} + 780 q^{88} + 1220 q^{94} + 4630 q^{97}+O(q^{100}) $ 18 * q + 56 * q^4 + 94 * q^7 + 156 * q^10 + 88 * q^16 + 448 * q^19 - 256 * q^22 - 16 * q^25 + 1300 * q^28 + 818 * q^31 + 900 * q^34 + 524 * q^37 + 800 * q^40 + 752 * q^43 + 1650 * q^46 + 2948 * q^49 - 464 * q^55 + 1626 * q^58 - 1784 * q^61 - 4274 * q^64 + 2872 * q^67 + 5772 * q^70 + 3078 * q^73 + 5726 * q^76 + 3326 * q^79 - 1038 * q^82 + 2340 * q^85 + 780 * q^88 + 1220 * q^94 + 4630 * q^97

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 $

$2$	−3.66578	−1.29605	−0.648025	−	0.761619i	$-0.724405\pi$
$2$	−3.66578	−1.29605	−0.648025	+	0.761619i	$0.724405\pi$
$3$	0	0
$3$	0	0
$4$	5.43796	0.679745
$5$	9.45496	0.845677	0.422839	−	0.906205i	$-0.361034\pi$
$5$	9.45496	0.845677	0.422839	+	0.906205i	$0.361034\pi$
$6$	0	0
$7$	−15.4112	−0.832129	−0.416065	−	0.909335i	$-0.636591\pi$
$7$	−15.4112	−0.832129	−0.416065	+	0.909335i	$0.636591\pi$
$8$	9.39187	0.415066
$9$	0	0
$10$	−34.6598	−1.09604
$11$	−12.7273	−0.348858	−0.174429	−	0.984670i	$-0.555808\pi$
$11$	−12.7273	−0.348858	−0.174429	+	0.984670i	$0.555808\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	56.4943	1.07848
$15$	0	0
$16$	−77.9323	−1.21769
$17$	18.9954	0.271003	0.135502	−	0.990777i	$-0.456735\pi$
$17$	18.9954	0.271003	0.135502	+	0.990777i	$0.456735\pi$
$18$	0	0
$19$	−95.4839	−1.15292	−0.576461	−	0.817125i	$-0.695567\pi$
$19$	−95.4839	−1.15292	−0.576461	+	0.817125i	$0.695567\pi$
$20$	51.4157	0.574845
$21$	0	0
$22$	46.6556	0.452137
$23$	−104.290	−0.945476	−0.472738	−	0.881203i	$-0.656734\pi$
$23$	−104.290	−0.945476	−0.472738	+	0.881203i	$0.656734\pi$
$24$	0	0
$25$	−35.6037	−0.284830
$26$	0	0
$27$	0	0
$28$	−83.8058	−0.565636
$29$	−23.3225	−0.149341	−0.0746705	−	0.997208i	$-0.523790\pi$
$29$	−23.3225	−0.149341	−0.0746705	+	0.997208i	$0.523790\pi$
$30$	0	0
$31$	177.018	1.02559	0.512795	−	0.858511i	$-0.328610\pi$
$31$	177.018	1.02559	0.512795	+	0.858511i	$0.328610\pi$
$32$	210.548	1.16312
$33$	0	0
$34$	−69.6329	−0.351234
$35$	−145.713	−0.703713
$36$	0	0
$37$	350.533	1.55750	0.778748	−	0.627337i	$-0.215855\pi$
$37$	350.533	1.55750	0.778748	+	0.627337i	$0.215855\pi$
$38$	350.023	1.49424
$39$	0	0
$40$	88.7998	0.351012
$41$	−348.495	−1.32746	−0.663729	−	0.747973i	$-0.731027\pi$
$41$	−348.495	−1.32746	−0.663729	+	0.747973i	$0.731027\pi$
$42$	0	0
$43$	60.8686	0.215869	0.107935	−	0.994158i	$-0.465576\pi$
$43$	60.8686	0.215869	0.107935	+	0.994158i	$0.465576\pi$
$44$	−69.2107	−0.237134
$45$	0	0
$46$	382.304	1.22538
$47$	−226.920	−0.704249	−0.352124	−	0.935953i	$-0.614541\pi$
$47$	−226.920	−0.704249	−0.352124	+	0.935953i	$0.614541\pi$
$48$	0	0
$49$	−105.494	−0.307561
$50$	130.516	0.369154
$51$	0	0
$52$	0	0
$53$	294.134	0.762311	0.381155	−	0.924511i	$-0.375526\pi$
$53$	294.134	0.762311	0.381155	+	0.924511i	$0.375526\pi$
$54$	0	0
$55$	−120.336	−0.295021
$56$	−144.740	−0.345388
$57$	0	0
$58$	85.4954	0.193553
$59$	−596.423	−1.31606	−0.658031	−	0.752991i	$-0.728610\pi$
$59$	−596.423	−1.31606	−0.658031	+	0.752991i	$0.728610\pi$
$60$	0	0
$61$	−487.247	−1.02271	−0.511357	−	0.859368i	$-0.670857\pi$
$61$	−487.247	−1.02271	−0.511357	+	0.859368i	$0.670857\pi$
$62$	−648.908	−1.32922
$63$	0	0
$64$	−148.364	−0.289774
$65$	0	0
$66$	0	0
$67$	943.554	1.72050	0.860250	−	0.509873i	$-0.170307\pi$
$67$	943.554	1.72050	0.860250	+	0.509873i	$0.170307\pi$
$68$	103.296	0.184213
$69$	0	0
$70$	534.151	0.912047
$71$	1044.76	1.74633	0.873167	−	0.487422i	$-0.162063\pi$
$71$	1044.76	1.74633	0.873167	+	0.487422i	$0.162063\pi$
$72$	0	0
$73$	554.083	0.888364	0.444182	−	0.895937i	$-0.353494\pi$
$73$	554.083	0.888364	0.444182	+	0.895937i	$0.353494\pi$
$74$	−1284.98	−2.01859
$75$	0	0
$76$	−519.238	−0.783693
$77$	196.144	0.290295
$78$	0	0
$79$	126.990	0.180855	0.0904275	−	0.995903i	$-0.471177\pi$
$79$	126.990	0.180855	0.0904275	+	0.995903i	$0.471177\pi$
$80$	−736.846	−1.02977
$81$	0	0
$82$	1277.51	1.72045
$83$	1448.23	1.91523	0.957613	−	0.288058i	$-0.0930096\pi$
$83$	1448.23	1.91523	0.957613	+	0.288058i	$0.0930096\pi$
$84$	0	0
$85$	179.601	0.229181
$86$	−223.131	−0.279777
$87$	0	0
$88$	−119.533	−0.144799
$89$	−247.561	−0.294847	−0.147424	−	0.989073i	$-0.547098\pi$
$89$	−247.561	−0.294847	−0.147424	+	0.989073i	$0.547098\pi$
$90$	0	0
$91$	0	0
$92$	−567.125	−0.642683
$93$	0	0
$94$	831.840	0.912742
$95$	−902.797	−0.975000
$96$	0	0
$97$	−136.799	−0.143194	−0.0715970	−	0.997434i	$-0.522810\pi$
$97$	−136.799	−0.143194	−0.0715970	+	0.997434i	$0.522810\pi$
$98$	386.716	0.398615
$99$	0	0
$100$	−193.612	−0.193612
$101$	−1674.46	−1.64965	−0.824826	−	0.565386i	$-0.808727\pi$
$101$	−1674.46	−1.64965	−0.824826	+	0.565386i	$0.808727\pi$
$102$	0	0
$103$	−531.415	−0.508368	−0.254184	−	0.967156i	$-0.581807\pi$
$103$	−531.415	−0.508368	−0.254184	+	0.967156i	$0.581807\pi$
$104$	0	0
$105$	0	0
$106$	−1078.23	−0.987993
$107$	515.773	0.465997	0.232998	−	0.972477i	$-0.425146\pi$
$107$	515.773	0.465997	0.232998	+	0.972477i	$0.425146\pi$
$108$	0	0
$109$	1604.26	1.40973	0.704864	−	0.709343i	$-0.251008\pi$
$109$	1604.26	1.40973	0.704864	+	0.709343i	$0.251008\pi$
$110$	441.127	0.382362
$111$	0	0
$112$	1201.03	1.01328
$113$	−1907.87	−1.58830	−0.794149	−	0.607724i	$-0.792083\pi$
$113$	−1907.87	−1.58830	−0.794149	+	0.607724i	$0.792083\pi$
$114$	0	0
$115$	−986.057	−0.799568
$116$	−126.827	−0.101514
$117$	0	0
$118$	2186.36	1.70568
$119$	−292.742	−0.225510
$120$	0	0
$121$	−1169.02	−0.878298
$122$	1786.14	1.32549
$123$	0	0
$124$	962.615	0.697140
$125$	−1518.50	−1.08655
$126$	0	0
$127$	−771.937	−0.539357	−0.269678	−	0.962950i	$-0.586917\pi$
$127$	−771.937	−0.539357	−0.269678	+	0.962950i	$0.586917\pi$
$128$	−1140.51	−0.787562
$129$	0	0
$130$	0	0
$131$	−765.560	−0.510590	−0.255295	−	0.966863i	$-0.582173\pi$
$131$	−765.560	−0.510590	−0.255295	+	0.966863i	$0.582173\pi$
$132$	0	0
$133$	1471.53	0.959380
$134$	−3458.86	−2.22985
$135$	0	0
$136$	178.402	0.112484
$137$	−2654.87	−1.65562	−0.827812	−	0.561006i	$-0.810415\pi$
$137$	−2654.87	−1.65562	−0.827812	+	0.561006i	$0.810415\pi$
$138$	0	0
$139$	2879.66	1.75719	0.878596	−	0.477565i	$-0.158481\pi$
$139$	2879.66	1.75719	0.878596	+	0.477565i	$0.158481\pi$
$140$	−792.380	−0.478345
$141$	0	0
$142$	−3829.85	−2.26334
$143$	0	0
$144$	0	0
$145$	−220.514	−0.126294
$146$	−2031.15	−1.15136
$147$	0	0
$148$	1906.19	1.05870
$149$	2004.35	1.10203	0.551017	−	0.834494i	$-0.314240\pi$
$149$	2004.35	1.10203	0.551017	+	0.834494i	$0.314240\pi$
$150$	0	0
$151$	370.126	0.199473	0.0997365	−	0.995014i	$-0.468200\pi$
$151$	370.126	0.199473	0.0997365	+	0.995014i	$0.468200\pi$
$152$	−896.773	−0.478539
$153$	0	0
$154$	−719.021	−0.376236
$155$	1673.69	0.867319
$156$	0	0
$157$	−1148.00	−0.583568	−0.291784	−	0.956484i	$-0.594249\pi$
$157$	−1148.00	−0.583568	−0.291784	+	0.956484i	$0.594249\pi$
$158$	−465.519	−0.234397
$159$	0	0
$160$	1990.72	0.983627
$161$	1607.24	0.786758
$162$	0	0
$163$	2043.17	0.981798	0.490899	−	0.871217i	$-0.336668\pi$
$163$	2043.17	0.981798	0.490899	+	0.871217i	$0.336668\pi$
$164$	−1895.10	−0.902333
$165$	0	0
$166$	−5308.89	−2.48223
$167$	−813.252	−0.376834	−0.188417	−	0.982089i	$-0.560336\pi$
$167$	−813.252	−0.376834	−0.188417	+	0.982089i	$0.560336\pi$
$168$	0	0
$169$	0	0
$170$	−658.377	−0.297031
$171$	0	0
$172$	331.001	0.146736
$173$	303.878	0.133546	0.0667728	−	0.997768i	$-0.478730\pi$
$173$	303.878	0.133546	0.0667728	+	0.997768i	$0.478730\pi$
$174$	0	0
$175$	548.698	0.237015
$176$	991.869	0.424801
$177$	0	0
$178$	907.504	0.382137
$179$	1829.37	0.763874	0.381937	−	0.924188i	$-0.375257\pi$
$179$	1829.37	0.763874	0.381937	+	0.924188i	$0.375257\pi$
$180$	0	0
$181$	−1186.50	−0.487249	−0.243624	−	0.969870i	$-0.578336\pi$
$181$	−1186.50	−0.487249	−0.243624	+	0.969870i	$0.578336\pi$
$182$	0	0
$183$	0	0
$184$	−979.478	−0.392435
$185$	3314.28	1.31714
$186$	0	0
$187$	−241.760	−0.0945416
$188$	−1233.98	−0.478710
$189$	0	0
$190$	3309.46	1.26365
$191$	−1191.17	−0.451256	−0.225628	−	0.974214i	$-0.572443\pi$
$191$	−1191.17	−0.451256	−0.225628	+	0.974214i	$0.572443\pi$
$192$	0	0
$193$	−4000.67	−1.49210	−0.746048	−	0.665892i	$-0.768051\pi$
$193$	−4000.67	−1.49210	−0.746048	+	0.665892i	$0.768051\pi$
$194$	501.475	0.185586
$195$	0	0
$196$	−573.670	−0.209063
$197$	1676.21	0.606220	0.303110	−	0.952956i	$-0.401975\pi$
$197$	1676.21	0.606220	0.303110	+	0.952956i	$0.401975\pi$
$198$	0	0
$199$	3048.89	1.08608	0.543040	−	0.839707i	$-0.317273\pi$
$199$	3048.89	1.08608	0.543040	+	0.839707i	$0.317273\pi$
$200$	−334.386	−0.118223
$201$	0	0
$202$	6138.20	2.13803
$203$	359.429	0.124271
$204$	0	0
$205$	−3295.00	−1.12260
$206$	1948.05	0.658870
$207$	0	0
$208$	0	0
$209$	1215.26	0.402206
$210$	0	0
$211$	3454.40	1.12707	0.563533	−	0.826093i	$-0.309442\pi$
$211$	3454.40	1.12707	0.563533	+	0.826093i	$0.309442\pi$
$212$	1599.49	0.518177
$213$	0	0
$214$	−1890.71	−0.603955
$215$	575.510	0.182556
$216$	0	0
$217$	−2728.06	−0.853424
$218$	−5880.87	−1.82708
$219$	0	0
$220$	−654.385	−0.200539
$221$	0	0
$222$	0	0
$223$	467.158	0.140283	0.0701417	−	0.997537i	$-0.477655\pi$
$223$	467.158	0.140283	0.0701417	+	0.997537i	$0.477655\pi$
$224$	−3244.80	−0.967868
$225$	0	0
$226$	6993.85	2.05851
$227$	2265.23	0.662329	0.331165	−	0.943573i	$-0.392558\pi$
$227$	2265.23	0.662329	0.331165	+	0.943573i	$0.392558\pi$
$228$	0	0
$229$	102.372	0.0295410	0.0147705	−	0.999891i	$-0.495298\pi$
$229$	102.372	0.0295410	0.0147705	+	0.999891i	$0.495298\pi$
$230$	3614.67	1.03628
$231$	0	0
$232$	−219.042	−0.0619864
$233$	−2570.39	−0.722713	−0.361356	−	0.932428i	$-0.617686\pi$
$233$	−2570.39	−0.722713	−0.361356	+	0.932428i	$0.617686\pi$
$234$	0	0
$235$	−2145.52	−0.595567
$236$	−3243.32	−0.894586
$237$	0	0
$238$	1073.13	0.292272
$239$	−867.428	−0.234767	−0.117383	−	0.993087i	$-0.537451\pi$
$239$	−867.428	−0.234767	−0.117383	+	0.993087i	$0.537451\pi$
$240$	0	0
$241$	5242.46	1.40123	0.700615	−	0.713540i	$-0.252909\pi$
$241$	5242.46	1.40123	0.700615	+	0.713540i	$0.252909\pi$
$242$	4285.36	1.13832
$243$	0	0
$244$	−2649.63	−0.695185
$245$	−997.437	−0.260098
$246$	0	0
$247$	0	0
$248$	1662.53	0.425688
$249$	0	0
$250$	5566.50	1.40822
$251$	5721.44	1.43878	0.719390	−	0.694606i	$-0.244421\pi$
$251$	5721.44	1.43878	0.719390	+	0.694606i	$0.244421\pi$
$252$	0	0
$253$	1327.33	0.329837
$254$	2829.75	0.699034
$255$	0	0
$256$	5367.78	1.31049
$257$	7263.31	1.76293	0.881465	−	0.472250i	$-0.156558\pi$
$257$	7263.31	1.76293	0.881465	+	0.472250i	$0.156558\pi$
$258$	0	0
$259$	−5402.16	−1.29604
$260$	0	0
$261$	0	0
$262$	2806.38	0.661750
$263$	3168.26	0.742827	0.371413	−	0.928468i	$-0.378873\pi$
$263$	3168.26	0.742827	0.371413	+	0.928468i	$0.378873\pi$
$264$	0	0
$265$	2781.03	0.644669
$266$	−5394.30	−1.24340
$267$	0	0
$268$	5131.01	1.16950
$269$	3738.01	0.847251	0.423626	−	0.905837i	$-0.360757\pi$
$269$	3738.01	0.847251	0.423626	+	0.905837i	$0.360757\pi$
$270$	0	0
$271$	2397.74	0.537462	0.268731	−	0.963215i	$-0.413396\pi$
$271$	2397.74	0.537462	0.268731	+	0.963215i	$0.413396\pi$
$272$	−1480.35	−0.329998
$273$	0	0
$274$	9732.16	2.14577
$275$	453.140	0.0993651
$276$	0	0
$277$	2325.17	0.504353	0.252177	−	0.967681i	$-0.418854\pi$
$277$	2325.17	0.504353	0.252177	+	0.967681i	$0.418854\pi$
$278$	−10556.2	−2.27741
$279$	0	0
$280$	−1368.52	−0.292087
$281$	4700.23	0.997838	0.498919	−	0.866649i	$-0.333731\pi$
$281$	4700.23	0.997838	0.498919	+	0.866649i	$0.333731\pi$
$282$	0	0
$283$	3110.96	0.653454	0.326727	−	0.945119i	$-0.394054\pi$
$283$	3110.96	0.653454	0.326727	+	0.945119i	$0.394054\pi$
$284$	5681.34	1.18706
$285$	0	0
$286$	0	0
$287$	5370.74	1.10462
$288$	0	0
$289$	−4552.18	−0.926557
$290$	808.355	0.163684
$291$	0	0
$292$	3013.09	0.603861
$293$	−2367.81	−0.472113	−0.236057	−	0.971739i	$-0.575855\pi$
$293$	−2367.81	−0.472113	−0.236057	+	0.971739i	$0.575855\pi$
$294$	0	0
$295$	−5639.15	−1.11296
$296$	3292.17	0.646464
$297$	0	0
$298$	−7347.52	−1.42829
$299$	0	0
$300$	0	0
$301$	−938.061	−0.179631
$302$	−1356.80	−0.258527
$303$	0	0
$304$	7441.28	1.40390
$305$	−4606.90	−0.864886
$306$	0	0
$307$	3696.56	0.687210	0.343605	−	0.939114i	$-0.388352\pi$
$307$	3696.56	0.687210	0.343605	+	0.939114i	$0.388352\pi$
$308$	1066.62	0.197326
$309$	0	0
$310$	−6135.40	−1.12409
$311$	2806.06	0.511630	0.255815	−	0.966726i	$-0.417656\pi$
$311$	2806.06	0.511630	0.255815	+	0.966726i	$0.417656\pi$
$312$	0	0
$313$	8168.52	1.47512	0.737559	−	0.675283i	$-0.235978\pi$
$313$	8168.52	1.47512	0.737559	+	0.675283i	$0.235978\pi$
$314$	4208.31	0.756333
$315$	0	0
$316$	690.569	0.122935
$317$	−6882.13	−1.21937	−0.609683	−	0.792646i	$-0.708703\pi$
$317$	−6882.13	−1.21937	−0.609683	+	0.792646i	$0.708703\pi$
$318$	0	0
$319$	296.834	0.0520987
$320$	−1402.78	−0.245055
$321$	0	0
$322$	−5891.79	−1.01968
$323$	−1813.75	−0.312446
$324$	0	0
$325$	0	0
$326$	−7489.80	−1.27246
$327$	0	0
$328$	−3273.02	−0.550982
$329$	3497.12	0.586026
$330$	0	0
$331$	−2835.07	−0.470784	−0.235392	−	0.971900i	$-0.575637\pi$
$331$	−2835.07	−0.470784	−0.235392	+	0.971900i	$0.575637\pi$
$332$	7875.41	1.30187
$333$	0	0
$334$	2981.20	0.488396
$335$	8921.27	1.45499
$336$	0	0
$337$	1948.43	0.314948	0.157474	−	0.987523i	$-0.449665\pi$
$337$	1948.43	0.314948	0.157474	+	0.987523i	$0.449665\pi$
$338$	0	0
$339$	0	0
$340$	976.661	0.155785
$341$	−2252.96	−0.357785
$342$	0	0
$343$	6911.84	1.08806
$344$	571.670	0.0896000
$345$	0	0
$346$	−1113.95	−0.173082
$347$	8797.96	1.36109	0.680546	−	0.732705i	$-0.261743\pi$
$347$	8797.96	1.36109	0.680546	+	0.732705i	$0.261743\pi$
$348$	0	0
$349$	2479.39	0.380282	0.190141	−	0.981757i	$-0.439105\pi$
$349$	2479.39	0.380282	0.190141	+	0.981757i	$0.439105\pi$
$350$	−2011.41	−0.307184
$351$	0	0
$352$	−2679.71	−0.405764
$353$	8819.59	1.32980	0.664900	−	0.746933i	$-0.268474\pi$
$353$	8819.59	1.32980	0.664900	+	0.746933i	$0.268474\pi$
$354$	0	0
$355$	9878.12	1.47683
$356$	−1346.23	−0.200421
$357$	0	0
$358$	−6706.07	−0.990019
$359$	8233.51	1.21044	0.605220	−	0.796058i	$-0.293085\pi$
$359$	8233.51	1.21044	0.605220	+	0.796058i	$0.293085\pi$
$360$	0	0
$361$	2258.18	0.329229
$362$	4349.46	0.631499
$363$	0	0
$364$	0	0
$365$	5238.84	0.751269
$366$	0	0
$367$	6269.59	0.891744	0.445872	−	0.895097i	$-0.352894\pi$
$367$	6269.59	0.891744	0.445872	+	0.895097i	$0.352894\pi$
$368$	8127.55	1.15130
$369$	0	0
$370$	−12149.4	−1.70708
$371$	−4532.98	−0.634341
$372$	0	0
$373$	−4222.06	−0.586086	−0.293043	−	0.956099i	$-0.594668\pi$
$373$	−4222.06	−0.586086	−0.293043	+	0.956099i	$0.594668\pi$
$374$	886.241	0.122531
$375$	0	0
$376$	−2131.20	−0.292310
$377$	0	0
$378$	0	0
$379$	2627.37	0.356092	0.178046	−	0.984022i	$-0.443022\pi$
$379$	2627.37	0.356092	0.178046	+	0.984022i	$0.443022\pi$
$380$	−4909.38	−0.662752
$381$	0	0
$382$	4366.56	0.584850
$383$	−12709.7	−1.69566	−0.847829	−	0.530270i	$-0.822091\pi$
$383$	−12709.7	−1.69566	−0.847829	+	0.530270i	$0.822091\pi$
$384$	0	0
$385$	1854.53	0.245496
$386$	14665.6	1.93383
$387$	0	0
$388$	−743.907	−0.0973354
$389$	−13138.4	−1.71245	−0.856223	−	0.516607i	$-0.827195\pi$
$389$	−13138.4	−1.71245	−0.856223	+	0.516607i	$0.827195\pi$
$390$	0	0
$391$	−1981.03	−0.256227
$392$	−990.782	−0.127658
$393$	0	0
$394$	−6144.64	−0.785691
$395$	1200.69	0.152945
$396$	0	0
$397$	10395.1	1.31414	0.657069	−	0.753830i	$-0.271796\pi$
$397$	10395.1	1.31414	0.657069	+	0.753830i	$0.271796\pi$
$398$	−11176.6	−1.40761
$399$	0	0
$400$	2774.68	0.346835
$401$	−6784.67	−0.844913	−0.422457	−	0.906383i	$-0.638832\pi$
$401$	−6784.67	−0.844913	−0.422457	+	0.906383i	$0.638832\pi$
$402$	0	0
$403$	0	0
$404$	−9105.65	−1.12134
$405$	0	0
$406$	−1317.59	−0.161061
$407$	−4461.35	−0.543344
$408$	0	0
$409$	8002.13	0.967432	0.483716	−	0.875225i	$-0.339287\pi$
$409$	8002.13	0.967432	0.483716	+	0.875225i	$0.339287\pi$
$410$	12078.8	1.45495
$411$	0	0
$412$	−2889.82	−0.345561
$413$	9191.61	1.09513
$414$	0	0
$415$	13692.9	1.61966
$416$	0	0
$417$	0	0
$418$	−4454.86	−0.521279
$419$	12879.7	1.50170	0.750852	−	0.660470i	$-0.229643\pi$
$419$	12879.7	1.50170	0.750852	+	0.660470i	$0.229643\pi$
$420$	0	0
$421$	−12575.3	−1.45577	−0.727887	−	0.685697i	$-0.759498\pi$
$421$	−12575.3	−1.45577	−0.727887	+	0.685697i	$0.759498\pi$
$422$	−12663.1	−1.46073
$423$	0	0
$424$	2762.47	0.316409
$425$	−676.306	−0.0771898
$426$	0	0
$427$	7509.08	0.851030
$428$	2804.75	0.316759
$429$	0	0
$430$	−2109.70	−0.236601
$431$	7839.30	0.876116	0.438058	−	0.898947i	$-0.355666\pi$
$431$	7839.30	0.876116	0.438058	+	0.898947i	$0.355666\pi$
$432$	0	0
$433$	878.419	0.0974922	0.0487461	−	0.998811i	$-0.484477\pi$
$433$	878.419	0.0974922	0.0487461	+	0.998811i	$0.484477\pi$
$434$	10000.5	1.10608
$435$	0	0
$436$	8723.91	0.958256
$437$	9958.02	1.09006
$438$	0	0
$439$	−8427.48	−0.916222	−0.458111	−	0.888895i	$-0.651474\pi$
$439$	−8427.48	−0.916222	−0.458111	+	0.888895i	$0.651474\pi$
$440$	−1130.18	−0.122453
$441$	0	0
$442$	0	0
$443$	9616.41	1.03135	0.515677	−	0.856783i	$-0.327541\pi$
$443$	9616.41	1.03135	0.515677	+	0.856783i	$0.327541\pi$
$444$	0	0
$445$	−2340.68	−0.249346
$446$	−1712.50	−0.181814
$447$	0	0
$448$	2286.48	0.241129
$449$	−6147.42	−0.646136	−0.323068	−	0.946376i	$-0.604714\pi$
$449$	−6147.42	−0.646136	−0.323068	+	0.946376i	$0.604714\pi$
$450$	0	0
$451$	4435.41	0.463094
$452$	−10374.9	−1.07964
$453$	0	0
$454$	−8303.85	−0.858411
$455$	0	0
$456$	0	0
$457$	10966.2	1.12248	0.561242	−	0.827652i	$-0.310324\pi$
$457$	10966.2	1.12248	0.561242	+	0.827652i	$0.310324\pi$
$458$	−375.272	−0.0382867
$459$	0	0
$460$	−5362.14	−0.543503
$461$	19458.5	1.96588	0.982942	−	0.183916i	$-0.0588775\pi$
$461$	19458.5	1.96588	0.982942	+	0.183916i	$0.0588775\pi$
$462$	0	0
$463$	−10607.1	−1.06469	−0.532346	−	0.846527i	$-0.678690\pi$
$463$	−10607.1	−1.06469	−0.532346	+	0.846527i	$0.678690\pi$
$464$	1817.58	0.181851
$465$	0	0
$466$	9422.51	0.936672
$467$	−958.675	−0.0949940	−0.0474970	−	0.998871i	$-0.515124\pi$
$467$	−958.675	−0.0949940	−0.0474970	+	0.998871i	$0.515124\pi$
$468$	0	0
$469$	−14541.3	−1.43168
$470$	7865.01	0.771885
$471$	0	0
$472$	−5601.52	−0.546252
$473$	−774.695	−0.0753076
$474$	0	0
$475$	3399.58	0.328387
$476$	−1591.92	−0.153289
$477$	0	0
$478$	3179.80	0.304269
$479$	−10862.3	−1.03614	−0.518071	−	0.855338i	$-0.673350\pi$
$479$	−10862.3	−1.03614	−0.518071	+	0.855338i	$0.673350\pi$
$480$	0	0
$481$	0	0
$482$	−19217.7	−1.81606
$483$	0	0
$484$	−6357.06	−0.597019
$485$	−1293.43	−0.121096
$486$	0	0
$487$	17700.1	1.64695	0.823477	−	0.567350i	$-0.192031\pi$
$487$	17700.1	1.64695	0.823477	+	0.567350i	$0.192031\pi$
$488$	−4576.16	−0.424494
$489$	0	0
$490$	3656.39	0.337100
$491$	16743.5	1.53895	0.769476	−	0.638675i	$-0.220517\pi$
$491$	16743.5	1.53895	0.769476	+	0.638675i	$0.220517\pi$
$492$	0	0
$493$	−443.021	−0.0404719
$494$	0	0
$495$	0	0
$496$	−13795.4	−1.24885
$497$	−16101.0	−1.45317
$498$	0	0
$499$	−8847.23	−0.793700	−0.396850	−	0.917884i	$-0.629897\pi$
$499$	−8847.23	−0.793700	−0.396850	+	0.917884i	$0.629897\pi$
$500$	−8257.56	−0.738578
$501$	0	0
$502$	−20973.6	−1.86473
$503$	−13132.5	−1.16411	−0.582056	−	0.813149i	$-0.697751\pi$
$503$	−13132.5	−1.16411	−0.582056	+	0.813149i	$0.697751\pi$
$504$	0	0
$505$	−15831.9	−1.39507
$506$	−4865.71	−0.427485
$507$	0	0
$508$	−4197.76	−0.366625
$509$	−6678.39	−0.581561	−0.290780	−	0.956790i	$-0.593915\pi$
$509$	−6678.39	−0.581561	−0.290780	+	0.956790i	$0.593915\pi$
$510$	0	0
$511$	−8539.12	−0.739233
$512$	−10553.0	−0.910903
$513$	0	0
$514$	−26625.7	−2.28484
$515$	−5024.51	−0.429915
$516$	0	0
$517$	2888.09	0.245683
$518$	19803.1	1.67973
$519$	0	0
$520$	0	0
$521$	1463.98	0.123105	0.0615527	−	0.998104i	$-0.480395\pi$
$521$	1463.98	0.123105	0.0615527	+	0.998104i	$0.480395\pi$
$522$	0	0
$523$	8812.25	0.736774	0.368387	−	0.929673i	$-0.379910\pi$
$523$	8812.25	0.736774	0.368387	+	0.929673i	$0.379910\pi$
$524$	−4163.09	−0.347071
$525$	0	0
$526$	−11614.2	−0.962740
$527$	3362.52	0.277938
$528$	0	0
$529$	−1290.61	−0.106074
$530$	−10194.6	−0.835523
$531$	0	0
$532$	8002.11	0.652134
$533$	0	0
$534$	0	0
$535$	4876.61	0.394083
$536$	8861.74	0.714121
$537$	0	0
$538$	−13702.7	−1.09808
$539$	1342.65	0.107295
$540$	0	0
$541$	15404.2	1.22418	0.612088	−	0.790790i	$-0.290330\pi$
$541$	15404.2	1.22418	0.612088	+	0.790790i	$0.290330\pi$
$542$	−8789.59	−0.696578
$543$	0	0
$544$	3999.44	0.315210
$545$	15168.2	1.19217
$546$	0	0
$547$	−1332.47	−0.104154	−0.0520769	−	0.998643i	$-0.516584\pi$
$547$	−1332.47	−0.104154	−0.0520769	+	0.998643i	$0.516584\pi$
$548$	−14437.1	−1.12540
$549$	0	0
$550$	−1661.11	−0.128782
$551$	2226.93	0.172179
$552$	0	0
$553$	−1957.08	−0.150495
$554$	−8523.56	−0.653667
$555$	0	0
$556$	15659.5	1.19444
$557$	−8485.98	−0.645534	−0.322767	−	0.946478i	$-0.604613\pi$
$557$	−8485.98	−0.645534	−0.322767	+	0.946478i	$0.604613\pi$
$558$	0	0
$559$	0	0
$560$	11355.7	0.856905
$561$	0	0
$562$	−17230.0	−1.29325
$563$	−17985.2	−1.34633	−0.673166	−	0.739491i	$-0.735066\pi$
$563$	−17985.2	−1.34633	−0.673166	+	0.739491i	$0.735066\pi$
$564$	0	0
$565$	−18038.9	−1.34319
$566$	−11404.1	−0.846909
$567$	0	0
$568$	9812.21	0.724844
$569$	−16723.5	−1.23214	−0.616068	−	0.787693i	$-0.711275\pi$
$569$	−16723.5	−1.23214	−0.616068	+	0.787693i	$0.711275\pi$
$570$	0	0
$571$	16036.9	1.17535	0.587674	−	0.809097i	$-0.300044\pi$
$571$	16036.9	1.17535	0.587674	+	0.809097i	$0.300044\pi$
$572$	0	0
$573$	0	0
$574$	−19688.0	−1.43164
$575$	3713.11	0.269300
$576$	0	0
$577$	−20293.1	−1.46415	−0.732074	−	0.681225i	$-0.761447\pi$
$577$	−20293.1	−1.46415	−0.732074	+	0.681225i	$0.761447\pi$
$578$	16687.3	1.20086
$579$	0	0
$580$	−1199.15	−0.0858479
$581$	−22319.0	−1.59372
$582$	0	0
$583$	−3743.54	−0.265938
$584$	5203.88	0.368730
$585$	0	0
$586$	8679.89	0.611882
$587$	−9014.69	−0.633860	−0.316930	−	0.948449i	$-0.602652\pi$
$587$	−9014.69	−0.633860	−0.316930	+	0.948449i	$0.602652\pi$
$588$	0	0
$589$	−16902.3	−1.18243
$590$	20671.9	1.44246
$591$	0	0
$592$	−27317.9	−1.89655
$593$	2114.55	0.146432	0.0732161	−	0.997316i	$-0.476674\pi$
$593$	2114.55	0.146432	0.0732161	+	0.997316i	$0.476674\pi$
$594$	0	0
$595$	−2767.87	−0.190708
$596$	10899.6	0.749102
$597$	0	0
$598$	0	0
$599$	6336.81	0.432245	0.216123	−	0.976366i	$-0.430659\pi$
$599$	6336.81	0.432245	0.216123	+	0.976366i	$0.430659\pi$
$600$	0	0
$601$	14789.6	1.00379	0.501896	−	0.864928i	$-0.332636\pi$
$601$	14789.6	1.00379	0.501896	+	0.864928i	$0.332636\pi$
$602$	3438.73	0.232811
$603$	0	0
$604$	2012.73	0.135591
$605$	−11053.0	−0.742757
$606$	0	0
$607$	−4865.13	−0.325320	−0.162660	−	0.986682i	$-0.552007\pi$
$607$	−4865.13	−0.325320	−0.162660	+	0.986682i	$0.552007\pi$
$608$	−20103.9	−1.34099
$609$	0	0
$610$	16887.9	1.12094
$611$	0	0
$612$	0	0
$613$	−7699.32	−0.507296	−0.253648	−	0.967297i	$-0.581631\pi$
$613$	−7699.32	−0.507296	−0.253648	+	0.967297i	$0.581631\pi$
$614$	−13550.8	−0.890659
$615$	0	0
$616$	1842.16	0.120491
$617$	12024.2	0.784562	0.392281	−	0.919845i	$-0.371686\pi$
$617$	12024.2	0.784562	0.392281	+	0.919845i	$0.371686\pi$
$618$	0	0
$619$	−10305.6	−0.669170	−0.334585	−	0.942366i	$-0.608596\pi$
$619$	−10305.6	−0.669170	−0.334585	+	0.942366i	$0.608596\pi$
$620$	9101.49	0.589556
$621$	0	0
$622$	−10286.4	−0.663098
$623$	3815.22	0.245351
$624$	0	0
$625$	−9906.91	−0.634042
$626$	−29944.0	−1.91183
$627$	0	0
$628$	−6242.77	−0.396678
$629$	6658.52	0.422087
$630$	0	0
$631$	−15380.2	−0.970326	−0.485163	−	0.874424i	$-0.661240\pi$
$631$	−15380.2	−0.970326	−0.485163	+	0.874424i	$0.661240\pi$
$632$	1192.68	0.0750668
$633$	0	0
$634$	25228.4	1.58036
$635$	−7298.63	−0.456122
$636$	0	0
$637$	0	0
$638$	−1088.13	−0.0675226
$639$	0	0
$640$	−10783.5	−0.666023
$641$	11443.0	0.705101	0.352551	−	0.935793i	$-0.385314\pi$
$641$	11443.0	0.705101	0.352551	+	0.935793i	$0.385314\pi$
$642$	0	0
$643$	20154.3	1.23609	0.618046	−	0.786142i	$-0.287924\pi$
$643$	20154.3	1.23609	0.618046	+	0.786142i	$0.287924\pi$
$644$	8740.10	0.534795
$645$	0	0
$646$	6648.83	0.404945
$647$	−8876.33	−0.539358	−0.269679	−	0.962950i	$-0.586918\pi$
$647$	−8876.33	−0.539358	−0.269679	+	0.962950i	$0.586918\pi$
$648$	0	0
$649$	7590.87	0.459118
$650$	0	0
$651$	0	0
$652$	11110.7	0.667372
$653$	22064.9	1.32230	0.661152	−	0.750252i	$-0.270068\pi$
$653$	22064.9	1.32230	0.661152	+	0.750252i	$0.270068\pi$
$654$	0	0
$655$	−7238.34	−0.431794
$656$	27159.0	1.61643
$657$	0	0
$658$	−12819.7	−0.759519
$659$	−30027.8	−1.77499	−0.887495	−	0.460817i	$-0.847556\pi$
$659$	−30027.8	−1.77499	−0.887495	+	0.460817i	$0.847556\pi$
$660$	0	0
$661$	24272.9	1.42830	0.714149	−	0.699994i	$-0.246814\pi$
$661$	24272.9	1.42830	0.714149	+	0.699994i	$0.246814\pi$
$662$	10392.8	0.610160
$663$	0	0
$664$	13601.6	0.794945
$665$	13913.2	0.811326
$666$	0	0
$667$	2432.31	0.141198
$668$	−4422.43	−0.256151
$669$	0	0
$670$	−32703.4	−1.88574
$671$	6201.35	0.356782
$672$	0	0
$673$	30335.7	1.73752	0.868762	−	0.495230i	$-0.164916\pi$
$673$	30335.7	1.73752	0.868762	+	0.495230i	$0.164916\pi$
$674$	−7142.51	−0.408189
$675$	0	0
$676$	0	0
$677$	9984.71	0.566829	0.283415	−	0.958997i	$-0.408533\pi$
$677$	9984.71	0.566829	0.283415	+	0.958997i	$0.408533\pi$
$678$	0	0
$679$	2108.24	0.119156
$680$	1686.79	0.0951254
$681$	0	0
$682$	8258.87	0.463707
$683$	3099.24	0.173629	0.0868147	−	0.996224i	$-0.472331\pi$
$683$	3099.24	0.173629	0.0868147	+	0.996224i	$0.472331\pi$
$684$	0	0
$685$	−25101.6	−1.40012
$686$	−25337.3	−1.41018
$687$	0	0
$688$	−4743.63	−0.262862
$689$	0	0
$690$	0	0
$691$	33108.4	1.82273	0.911363	−	0.411604i	$-0.135031\pi$
$691$	33108.4	1.82273	0.911363	+	0.411604i	$0.135031\pi$
$692$	1652.48	0.0907770
$693$	0	0
$694$	−32251.4	−1.76404
$695$	27227.1	1.48602
$696$	0	0
$697$	−6619.79	−0.359745
$698$	−9088.89	−0.492865
$699$	0	0
$700$	2983.80	0.161110
$701$	−886.423	−0.0477600	−0.0238800	−	0.999715i	$-0.507602\pi$
$701$	−886.423	−0.0477600	−0.0238800	+	0.999715i	$0.507602\pi$
$702$	0	0
$703$	−33470.3	−1.79567
$704$	1888.28	0.101090
$705$	0	0
$706$	−32330.7	−1.72349
$707$	25805.5	1.37272
$708$	0	0
$709$	31368.8	1.66161	0.830803	−	0.556566i	$-0.187881\pi$
$709$	31368.8	1.66161	0.830803	+	0.556566i	$0.187881\pi$
$710$	−36211.0	−1.91405
$711$	0	0
$712$	−2325.06	−0.122381
$713$	−18461.2	−0.969672
$714$	0	0
$715$	0	0
$716$	9948.04	0.519240
$717$	0	0
$718$	−30182.2	−1.56879
$719$	−16101.2	−0.835152	−0.417576	−	0.908642i	$-0.637120\pi$
$719$	−16101.2	−0.835152	−0.417576	+	0.908642i	$0.637120\pi$
$720$	0	0
$721$	8189.77	0.423028
$722$	−8278.01	−0.426698
$723$	0	0
$724$	−6452.16	−0.331205
$725$	830.370	0.0425368
$726$	0	0
$727$	−2103.83	−0.107327	−0.0536635	−	0.998559i	$-0.517090\pi$
$727$	−2103.83	−0.107327	−0.0536635	+	0.998559i	$0.517090\pi$
$728$	0	0
$729$	0	0
$730$	−19204.4	−0.973682
$731$	1156.22	0.0585013
$732$	0	0
$733$	−8104.43	−0.408382	−0.204191	−	0.978931i	$-0.565456\pi$
$733$	−8104.43	−0.408382	−0.204191	+	0.978931i	$0.565456\pi$
$734$	−22983.0	−1.15574
$735$	0	0
$736$	−21958.0	−1.09971
$737$	−12008.9	−0.600209
$738$	0	0
$739$	6096.91	0.303489	0.151745	−	0.988420i	$-0.451511\pi$
$739$	6096.91	0.303489	0.151745	+	0.988420i	$0.451511\pi$
$740$	18022.9	0.895319
$741$	0	0
$742$	16616.9	0.822137
$743$	−37762.3	−1.86455	−0.932277	−	0.361745i	$-0.882181\pi$
$743$	−37762.3	−1.86455	−0.932277	+	0.361745i	$0.882181\pi$
$744$	0	0
$745$	18951.1	0.931964
$746$	15477.2	0.759596
$747$	0	0
$748$	−1314.68	−0.0642642
$749$	−7948.71	−0.387770
$750$	0	0
$751$	18803.3	0.913637	0.456818	−	0.889560i	$-0.348989\pi$
$751$	18803.3	0.913637	0.456818	+	0.889560i	$0.348989\pi$
$752$	17684.4	0.857558
$753$	0	0
$754$	0	0
$755$	3499.52	0.168690
$756$	0	0
$757$	−19451.0	−0.933893	−0.466947	−	0.884286i	$-0.654646\pi$
$757$	−19451.0	−0.933893	−0.466947	+	0.884286i	$0.654646\pi$
$758$	−9631.37	−0.461513
$759$	0	0
$760$	−8478.95	−0.404689
$761$	−7000.32	−0.333458	−0.166729	−	0.986003i	$-0.553320\pi$
$761$	−7000.32	−0.333458	−0.166729	+	0.986003i	$0.553320\pi$
$762$	0	0
$763$	−24723.7	−1.17308
$764$	−6477.53	−0.306739
$765$	0	0
$766$	46591.1	2.19766
$767$	0	0
$768$	0	0
$769$	21957.7	1.02967	0.514834	−	0.857290i	$-0.327854\pi$
$769$	21957.7	1.02967	0.514834	+	0.857290i	$0.327854\pi$
$770$	−6798.32	−0.318174
$771$	0	0
$772$	−21755.5	−1.01425
$773$	15048.7	0.700213	0.350107	−	0.936710i	$-0.386145\pi$
$773$	15048.7	0.700213	0.350107	+	0.936710i	$0.386145\pi$
$774$	0	0
$775$	−6302.49	−0.292119
$776$	−1284.80	−0.0594349
$777$	0	0
$778$	48162.4	2.21942
$779$	33275.7	1.53046
$780$	0	0
$781$	−13296.9	−0.609222
$782$	7262.02	0.332083
$783$	0	0
$784$	8221.35	0.374515
$785$	−10854.3	−0.493510
$786$	0	0
$787$	−132.925	−0.00602068	−0.00301034	−	0.999995i	$-0.500958\pi$
$787$	−132.925	−0.00602068	−0.00301034	+	0.999995i	$0.500958\pi$
$788$	9115.19	0.412075
$789$	0	0
$790$	−4401.47	−0.198224
$791$	29402.7	1.32167
$792$	0	0
$793$	0	0
$794$	−38106.0	−1.70319
$795$	0	0
$796$	16579.7	0.738258
$797$	−35040.6	−1.55734	−0.778672	−	0.627431i	$-0.784106\pi$
$797$	−35040.6	−1.55734	−0.778672	+	0.627431i	$0.784106\pi$
$798$	0	0
$799$	−4310.43	−0.190854
$800$	−7496.29	−0.331292
$801$	0	0
$802$	24871.1	1.09505
$803$	−7052.00	−0.309912
$804$	0	0
$805$	15196.4	0.665344
$806$	0	0
$807$	0	0
$808$	−15726.3	−0.684715
$809$	−25088.1	−1.09030	−0.545149	−	0.838339i	$-0.683527\pi$
$809$	−25088.1	−1.09030	−0.545149	+	0.838339i	$0.683527\pi$
$810$	0	0
$811$	16450.7	0.712286	0.356143	−	0.934432i	$-0.384092\pi$
$811$	16450.7	0.712286	0.356143	+	0.934432i	$0.384092\pi$
$812$	1954.56	0.0844726
$813$	0	0
$814$	16354.4	0.704201
$815$	19318.0	0.830284
$816$	0	0
$817$	−5811.97	−0.248880
$818$	−29334.1	−1.25384
$819$	0	0
$820$	−17918.1	−0.763083
$821$	27895.9	1.18584	0.592919	−	0.805262i	$-0.297976\pi$
$821$	27895.9	1.18584	0.592919	+	0.805262i	$0.297976\pi$
$822$	0	0
$823$	3885.73	0.164578	0.0822892	−	0.996608i	$-0.473777\pi$
$823$	3885.73	0.164578	0.0822892	+	0.996608i	$0.473777\pi$
$824$	−4990.98	−0.211006
$825$	0	0
$826$	−33694.5	−1.41935
$827$	−2853.98	−0.120003	−0.0600016	−	0.998198i	$-0.519111\pi$
$827$	−2853.98	−0.120003	−0.0600016	+	0.998198i	$0.519111\pi$
$828$	0	0
$829$	8704.35	0.364674	0.182337	−	0.983236i	$-0.441634\pi$
$829$	8704.35	0.364674	0.182337	+	0.983236i	$0.441634\pi$
$830$	−50195.4	−2.09916
$831$	0	0
$832$	0	0
$833$	−2003.89	−0.0833501
$834$	0	0
$835$	−7689.26	−0.318680
$836$	6608.51	0.273397
$837$	0	0
$838$	−47214.1	−1.94628
$839$	−21730.7	−0.894194	−0.447097	−	0.894485i	$-0.647542\pi$
$839$	−21730.7	−0.894194	−0.447097	+	0.894485i	$0.647542\pi$
$840$	0	0
$841$	−23845.1	−0.977697
$842$	46098.2	1.88676
$843$	0	0
$844$	18784.9	0.766118
$845$	0	0
$846$	0	0
$847$	18016.0	0.730858
$848$	−22922.6	−0.928259
$849$	0	0
$850$	2479.19	0.100042
$851$	−36557.1	−1.47258
$852$	0	0
$853$	−36279.1	−1.45624	−0.728120	−	0.685450i	$-0.759606\pi$
$853$	−36279.1	−1.45624	−0.728120	+	0.685450i	$0.759606\pi$
$854$	−27526.7	−1.10298
$855$	0	0
$856$	4844.07	0.193419
$857$	−45276.5	−1.80469	−0.902343	−	0.431018i	$-0.858155\pi$
$857$	−45276.5	−1.80469	−0.902343	+	0.431018i	$0.858155\pi$
$858$	0	0
$859$	−19162.8	−0.761149	−0.380575	−	0.924750i	$-0.624274\pi$
$859$	−19162.8	−0.761149	−0.380575	+	0.924750i	$0.624274\pi$
$860$	3129.60	0.124091
$861$	0	0
$862$	−28737.2	−1.13549
$863$	11531.7	0.454860	0.227430	−	0.973794i	$-0.426968\pi$
$863$	11531.7	0.454860	0.227430	+	0.973794i	$0.426968\pi$
$864$	0	0
$865$	2873.15	0.112937
$866$	−3220.09	−0.126355
$867$	0	0
$868$	−14835.1	−0.580111
$869$	−1616.25	−0.0630926
$870$	0	0
$871$	0	0
$872$	15067.0	0.585130
$873$	0	0
$874$	−36503.9	−1.41277
$875$	23402.0	0.904151
$876$	0	0
$877$	5495.83	0.211609	0.105804	−	0.994387i	$-0.466258\pi$
$877$	5495.83	0.211609	0.105804	+	0.994387i	$0.466258\pi$
$878$	30893.3	1.18747
$879$	0	0
$880$	9378.09	0.359245
$881$	−9701.52	−0.371002	−0.185501	−	0.982644i	$-0.559391\pi$
$881$	−9701.52	−0.371002	−0.185501	+	0.982644i	$0.559391\pi$
$882$	0	0
$883$	15919.0	0.606702	0.303351	−	0.952879i	$-0.401895\pi$
$883$	15919.0	0.606702	0.303351	+	0.952879i	$0.401895\pi$
$884$	0	0
$885$	0	0
$886$	−35251.7	−1.33669
$887$	42705.2	1.61657	0.808287	−	0.588789i	$-0.200395\pi$
$887$	42705.2	1.61657	0.808287	+	0.588789i	$0.200395\pi$
$888$	0	0
$889$	11896.5	0.448815
$890$	8580.42	0.323164
$891$	0	0
$892$	2540.39	0.0953570
$893$	21667.2	0.811944
$894$	0	0
$895$	17296.6	0.645991
$896$	17576.7	0.655353
$897$	0	0
$898$	22535.1	0.837424
$899$	−4128.50	−0.153163
$900$	0	0
$901$	5587.20	0.206589
$902$	−16259.2	−0.600192
$903$	0	0
$904$	−17918.5	−0.659248
$905$	−11218.3	−0.412055
$906$	0	0
$907$	−3188.34	−0.116722	−0.0583611	−	0.998296i	$-0.518587\pi$
$907$	−3188.34	−0.116722	−0.0583611	+	0.998296i	$0.518587\pi$
$908$	12318.2	0.450215
$909$	0	0
$910$	0	0
$911$	−29528.3	−1.07389	−0.536947	−	0.843616i	$-0.680423\pi$
$911$	−29528.3	−1.07389	−0.536947	+	0.843616i	$0.680423\pi$
$912$	0	0
$913$	−18432.1	−0.668141
$914$	−40199.5	−1.45479
$915$	0	0
$916$	556.692	0.0200804
$917$	11798.2	0.424877
$918$	0	0
$919$	−3132.67	−0.112445	−0.0562227	−	0.998418i	$-0.517906\pi$
$919$	−3132.67	−0.112445	−0.0562227	+	0.998418i	$0.517906\pi$
$920$	−9260.92	−0.331874
$921$	0	0
$922$	−71330.6	−2.54788
$923$	0	0
$924$	0	0
$925$	−12480.3	−0.443621
$926$	38883.2	1.37989
$927$	0	0
$928$	−4910.51	−0.173702
$929$	−21825.6	−0.770803	−0.385401	−	0.922749i	$-0.625937\pi$
$929$	−21825.6	−0.770803	−0.385401	+	0.922749i	$0.625937\pi$
$930$	0	0
$931$	10072.9	0.354594
$932$	−13977.7	−0.491261
$933$	0	0
$934$	3514.29	0.123117
$935$	−2285.84	−0.0799517
$936$	0	0
$937$	11985.0	0.417857	0.208928	−	0.977931i	$-0.433002\pi$
$937$	11985.0	0.417857	0.208928	+	0.977931i	$0.433002\pi$
$938$	53305.4	1.85553
$939$	0	0
$940$	−11667.3	−0.404834
$941$	18229.7	0.631531	0.315766	−	0.948837i	$-0.397739\pi$
$941$	18229.7	0.631531	0.315766	+	0.948837i	$0.397739\pi$
$942$	0	0
$943$	36344.5	1.25508
$944$	46480.6	1.60256
$945$	0	0
$946$	2839.86	0.0976024
$947$	−1762.07	−0.0604640	−0.0302320	−	0.999543i	$-0.509625\pi$
$947$	−1762.07	−0.0604640	−0.0302320	+	0.999543i	$0.509625\pi$
$948$	0	0
$949$	0	0
$950$	−12462.1	−0.425605
$951$	0	0
$952$	−2749.40	−0.0936014
$953$	30840.5	1.04829	0.524146	−	0.851629i	$-0.324385\pi$
$953$	30840.5	1.04829	0.524146	+	0.851629i	$0.324385\pi$
$954$	0	0
$955$	−11262.4	−0.381617
$956$	−4717.04	−0.159582
$957$	0	0
$958$	39818.9	1.34289
$959$	40914.8	1.37769
$960$	0	0
$961$	1544.24	0.0518359
$962$	0	0
$963$	0	0
$964$	28508.3	0.952479
$965$	−37826.2	−1.26183
$966$	0	0
$967$	−50612.3	−1.68312	−0.841562	−	0.540160i	$-0.818364\pi$
$967$	−50612.3	−1.68312	−0.841562	+	0.540160i	$0.818364\pi$
$968$	−10979.2	−0.364552
$969$	0	0
$970$	4741.42	0.156946
$971$	46156.7	1.52548	0.762739	−	0.646707i	$-0.223854\pi$
$971$	46156.7	1.52548	0.762739	+	0.646707i	$0.223854\pi$
$972$	0	0
$973$	−44379.2	−1.46221
$974$	−64884.6	−2.13453
$975$	0	0
$976$	37972.3	1.24535
$977$	−12445.5	−0.407540	−0.203770	−	0.979019i	$-0.565319\pi$
$977$	−12445.5	−0.407540	−0.203770	+	0.979019i	$0.565319\pi$
$978$	0	0
$979$	3150.79	0.102860
$980$	−5424.03	−0.176800
$981$	0	0
$982$	−61378.2	−1.99456
$983$	26127.4	0.847745	0.423873	−	0.905722i	$-0.360670\pi$
$983$	26127.4	0.847745	0.423873	+	0.905722i	$0.360670\pi$
$984$	0	0
$985$	15848.5	0.512667
$986$	1624.02	0.0524536
$987$	0	0
$988$	0	0
$989$	−6347.98	−0.204099
$990$	0	0
$991$	41054.6	1.31599	0.657993	−	0.753024i	$-0.271406\pi$
$991$	41054.6	1.31599	0.657993	+	0.753024i	$0.271406\pi$
$992$	37270.7	1.19289
$993$	0	0
$994$	59022.7	1.88339
$995$	28827.1	0.918474
$996$	0	0
$997$	12412.0	0.394276	0.197138	−	0.980376i	$-0.436835\pi$
$997$	12412.0	0.394276	0.197138	+	0.980376i	$0.436835\pi$
$998$	32432.0	1.02867
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bn.1.4	yes	18
3.2	odd	2	inner	1521.4.a.bn.1.15	yes	18
13.12	even	2		1521.4.a.bm.1.15	yes	18
39.38	odd	2		1521.4.a.bm.1.4	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.4.a.bm.1.4	✓	18	39.38	odd	2
1521.4.a.bm.1.15	yes	18	13.12	even	2
1521.4.a.bn.1.4	yes	18	1.1	even	1	trivial
1521.4.a.bn.1.15	yes	18	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-3.66578 q^{2} +5.43796 q^{4} +9.45496 q^{5} -15.4112 q^{7} +9.39187 q^{8} +O(q^{10})\)	\(q-3.66578 q^{2} +5.43796 q^{4} +9.45496 q^{5} -15.4112 q^{7} +9.39187 q^{8} -34.6598 q^{10} -12.7273 q^{11} +56.4943 q^{14} -77.9323 q^{16} +18.9954 q^{17} -95.4839 q^{19} +51.4157 q^{20} +46.6556 q^{22} -104.290 q^{23} -35.6037 q^{25} -83.8058 q^{28} -23.3225 q^{29} +177.018 q^{31} +210.548 q^{32} -69.6329 q^{34} -145.713 q^{35} +350.533 q^{37} +350.023 q^{38} +88.7998 q^{40} -348.495 q^{41} +60.8686 q^{43} -69.2107 q^{44} +382.304 q^{46} -226.920 q^{47} -105.494 q^{49} +130.516 q^{50} +294.134 q^{53} -120.336 q^{55} -144.740 q^{56} +85.4954 q^{58} -596.423 q^{59} -487.247 q^{61} -648.908 q^{62} -148.364 q^{64} +943.554 q^{67} +103.296 q^{68} +534.151 q^{70} +1044.76 q^{71} +554.083 q^{73} -1284.98 q^{74} -519.238 q^{76} +196.144 q^{77} +126.990 q^{79} -736.846 q^{80} +1277.51 q^{82} +1448.23 q^{83} +179.601 q^{85} -223.131 q^{86} -119.533 q^{88} -247.561 q^{89} -567.125 q^{92} +831.840 q^{94} -902.797 q^{95} -136.799 q^{97} +386.716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) \) 18 * q + 56 * q^4 + 94 * q^7	\( 18 q + 56 q^{4} + 94 q^{7} + 156 q^{10} + 88 q^{16} + 448 q^{19} - 256 q^{22} - 16 q^{25} + 1300 q^{28} + 818 q^{31} + 900 q^{34} + 524 q^{37} + 800 q^{40} + 752 q^{43} + 1650 q^{46} + 2948 q^{49} - 464 q^{55} + 1626 q^{58} - 1784 q^{61} - 4274 q^{64} + 2872 q^{67} + 5772 q^{70} + 3078 q^{73} + 5726 q^{76} + 3326 q^{79} - 1038 q^{82} + 2340 q^{85} + 780 q^{88} + 1220 q^{94} + 4630 q^{97}+O(q^{100}) \) 18 * q + 56 * q^4 + 94 * q^7 + 156 * q^10 + 88 * q^16 + 448 * q^19 - 256 * q^22 - 16 * q^25 + 1300 * q^28 + 818 * q^31 + 900 * q^34 + 524 * q^37 + 800 * q^40 + 752 * q^43 + 1650 * q^46 + 2948 * q^49 - 464 * q^55 + 1626 * q^58 - 1784 * q^61 - 4274 * q^64 + 2872 * q^67 + 5772 * q^70 + 3078 * q^73 + 5726 * q^76 + 3326 * q^79 - 1038 * q^82 + 2340 * q^85 + 780 * q^88 + 1220 * q^94 + 4630 * q^97

Introduction

L-functions

Modular forms

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