LMFDB - Embedded newform 1521.4.a.bf.1.5

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.107680$ 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)

$f(q)$	$=$	$q-0.447278 q^{2} -7.79994 q^{4} -1.93073 q^{5} -8.14537 q^{7} +7.06697 q^{8} +O(q^{10})$	$q-0.447278 q^{2} -7.79994 q^{4} -1.93073 q^{5} -8.14537 q^{7} +7.06697 q^{8} +0.863573 q^{10} -8.40842 q^{11} +3.64325 q^{14} +59.2386 q^{16} +52.1271 q^{17} +48.8304 q^{19} +15.0596 q^{20} +3.76090 q^{22} -88.9229 q^{23} -121.272 q^{25} +63.5334 q^{28} -191.979 q^{29} +115.257 q^{31} -83.0319 q^{32} -23.3153 q^{34} +15.7265 q^{35} -136.716 q^{37} -21.8408 q^{38} -13.6444 q^{40} -436.077 q^{41} +202.048 q^{43} +65.5852 q^{44} +39.7733 q^{46} -618.160 q^{47} -276.653 q^{49} +54.2425 q^{50} +453.170 q^{53} +16.2344 q^{55} -57.5631 q^{56} +85.8679 q^{58} -500.044 q^{59} +480.502 q^{61} -51.5522 q^{62} -436.771 q^{64} +886.769 q^{67} -406.589 q^{68} -7.03412 q^{70} -123.732 q^{71} +673.168 q^{73} +61.1501 q^{74} -380.875 q^{76} +68.4897 q^{77} -681.298 q^{79} -114.374 q^{80} +195.048 q^{82} +939.418 q^{83} -100.643 q^{85} -90.3716 q^{86} -59.4220 q^{88} +754.979 q^{89} +693.594 q^{92} +276.489 q^{94} -94.2783 q^{95} +1051.10 q^{97} +123.741 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) $ 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8	$ 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) $ 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 - 54 * q^10 - 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 + 352 * q^19 - 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 + 940 * q^28 + 97 * q^29 + 717 * q^31 - 707 * q^32 + 632 * q^34 + 418 * q^35 + 1108 * q^37 + 660 * q^38 - 1506 * q^40 - 334 * q^41 + 242 * q^43 + 307 * q^44 + 979 * q^46 + 184 * q^47 - 38 * q^49 + 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 + 1161 * q^58 - 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 + 2308 * q^67 - 2785 * q^68 - 1420 * q^70 - 96 * q^71 + 2505 * q^73 + 1191 * q^74 + 2409 * q^76 + 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 1517 * q^82 - 1539 * q^83 + 4296 * q^85 + 3763 * q^86 - 3716 * q^88 + 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 + 1445 * q^97 - 1457 * q^98

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 $

$2$	−0.447278	−0.158137	−0.0790684	−	0.996869i	$-0.525195\pi$
$2$	−0.447278	−0.158137	−0.0790684	+	0.996869i	$0.525195\pi$
$3$	0	0
$3$	0	0
$4$	−7.79994	−0.974993
$5$	−1.93073	−0.172690	−0.0863448	−	0.996265i	$-0.527519\pi$
$5$	−1.93073	−0.172690	−0.0863448	+	0.996265i	$0.527519\pi$
$6$	0	0
$7$	−8.14537	−0.439809	−0.219904	−	0.975521i	$-0.570575\pi$
$7$	−8.14537	−0.439809	−0.219904	+	0.975521i	$0.570575\pi$
$8$	7.06697	0.312319
$9$	0	0
$10$	0.863573	0.0273086
$11$	−8.40842	−0.230476	−0.115238	−	0.993338i	$-0.536763\pi$
$11$	−8.40842	−0.230476	−0.115238	+	0.993338i	$0.536763\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	3.64325	0.0695499
$15$	0	0
$16$	59.2386	0.925604
$17$	52.1271	0.743688	0.371844	−	0.928295i	$-0.378726\pi$
$17$	52.1271	0.743688	0.371844	+	0.928295i	$0.378726\pi$
$18$	0	0
$19$	48.8304	0.589604	0.294802	−	0.955558i	$-0.404746\pi$
$19$	48.8304	0.589604	0.294802	+	0.955558i	$0.404746\pi$
$20$	15.0596	0.168371
$21$	0	0
$22$	3.76090	0.0364467
$23$	−88.9229	−0.806162	−0.403081	−	0.915164i	$-0.632061\pi$
$23$	−88.9229	−0.806162	−0.403081	+	0.915164i	$0.632061\pi$
$24$	0	0
$25$	−121.272	−0.970178
$26$	0	0
$27$	0	0
$28$	63.5334	0.428810
$29$	−191.979	−1.22930	−0.614648	−	0.788802i	$-0.710702\pi$
$29$	−191.979	−1.22930	−0.614648	+	0.788802i	$0.710702\pi$
$30$	0	0
$31$	115.257	0.667769	0.333885	−	0.942614i	$-0.391640\pi$
$31$	115.257	0.667769	0.333885	+	0.942614i	$0.391640\pi$
$32$	−83.0319	−0.458691
$33$	0	0
$34$	−23.3153	−0.117604
$35$	15.7265	0.0759504
$36$	0	0
$37$	−136.716	−0.607459	−0.303729	−	0.952758i	$-0.598232\pi$
$37$	−136.716	−0.607459	−0.303729	+	0.952758i	$0.598232\pi$
$38$	−21.8408	−0.0932380
$39$	0	0
$40$	−13.6444	−0.0539342
$41$	−436.077	−1.66107	−0.830535	−	0.556967i	$-0.811965\pi$
$41$	−436.077	−1.66107	−0.830535	+	0.556967i	$0.811965\pi$
$42$	0	0
$43$	202.048	0.716559	0.358279	−	0.933614i	$-0.383364\pi$
$43$	202.048	0.716559	0.358279	+	0.933614i	$0.383364\pi$
$44$	65.5852	0.224712
$45$	0	0
$46$	39.7733	0.127484
$47$	−618.160	−1.91847	−0.959233	−	0.282617i	$-0.908798\pi$
$47$	−618.160	−1.91847	−0.959233	+	0.282617i	$0.908798\pi$
$48$	0	0
$49$	−276.653	−0.806568
$50$	54.2425	0.153421
$51$	0	0
$52$	0	0
$53$	453.170	1.17448	0.587242	−	0.809411i	$-0.300214\pi$
$53$	453.170	1.17448	0.587242	+	0.809411i	$0.300214\pi$
$54$	0	0
$55$	16.2344	0.0398008
$56$	−57.5631	−0.137361
$57$	0	0
$58$	85.8679	0.194397
$59$	−500.044	−1.10339	−0.551697	−	0.834045i	$-0.686019\pi$
$59$	−500.044	−1.10339	−0.551697	+	0.834045i	$0.686019\pi$
$60$	0	0
$61$	480.502	1.00856	0.504279	−	0.863541i	$-0.331758\pi$
$61$	480.502	1.00856	0.504279	+	0.863541i	$0.331758\pi$
$62$	−51.5522	−0.105599
$63$	0	0
$64$	−436.771	−0.853068
$65$	0	0
$66$	0	0
$67$	886.769	1.61696	0.808478	−	0.588526i	$-0.200292\pi$
$67$	886.769	1.61696	0.808478	+	0.588526i	$0.200292\pi$
$68$	−406.589	−0.725090
$69$	0	0
$70$	−7.03412	−0.0120105
$71$	−123.732	−0.206821	−0.103410	−	0.994639i	$-0.532975\pi$
$71$	−123.732	−0.206821	−0.103410	+	0.994639i	$0.532975\pi$
$72$	0	0
$73$	673.168	1.07929	0.539646	−	0.841892i	$-0.318558\pi$
$73$	673.168	1.07929	0.539646	+	0.841892i	$0.318558\pi$
$74$	61.1501	0.0960616
$75$	0	0
$76$	−380.875	−0.574859
$77$	68.4897	0.101365
$78$	0	0
$79$	−681.298	−0.970279	−0.485139	−	0.874437i	$-0.661231\pi$
$79$	−681.298	−0.970279	−0.485139	+	0.874437i	$0.661231\pi$
$80$	−114.374	−0.159842
$81$	0	0
$82$	195.048	0.262676
$83$	939.418	1.24234	0.621172	−	0.783675i	$-0.286657\pi$
$83$	939.418	1.24234	0.621172	+	0.783675i	$0.286657\pi$
$84$	0	0
$85$	−100.643	−0.128427
$86$	−90.3716	−0.113314
$87$	0	0
$88$	−59.4220	−0.0719819
$89$	754.979	0.899187	0.449593	−	0.893233i	$-0.351569\pi$
$89$	754.979	0.899187	0.449593	+	0.893233i	$0.351569\pi$
$90$	0	0
$91$	0	0
$92$	693.594	0.786002
$93$	0	0
$94$	276.489	0.303380
$95$	−94.2783	−0.101818
$96$	0	0
$97$	1051.10	1.10024	0.550118	−	0.835087i	$-0.314583\pi$
$97$	1051.10	1.10024	0.550118	+	0.835087i	$0.314583\pi$
$98$	123.741	0.127548
$99$	0	0
$100$	945.917	0.945917
$101$	−599.873	−0.590986	−0.295493	−	0.955345i	$-0.595484\pi$
$101$	−599.873	−0.590986	−0.295493	+	0.955345i	$0.595484\pi$
$102$	0	0
$103$	−293.312	−0.280591	−0.140295	−	0.990110i	$-0.544805\pi$
$103$	−293.312	−0.280591	−0.140295	+	0.990110i	$0.544805\pi$
$104$	0	0
$105$	0	0
$106$	−202.693	−0.185729
$107$	−1533.69	−1.38568	−0.692839	−	0.721092i	$-0.743640\pi$
$107$	−1533.69	−1.38568	−0.692839	+	0.721092i	$0.743640\pi$
$108$	0	0
$109$	590.718	0.519088	0.259544	−	0.965731i	$-0.416428\pi$
$109$	590.718	0.519088	0.259544	+	0.965731i	$0.416428\pi$
$110$	−7.26128	−0.00629396
$111$	0	0
$112$	−482.521	−0.407089
$113$	653.785	0.544274	0.272137	−	0.962259i	$-0.412270\pi$
$113$	653.785	0.544274	0.272137	+	0.962259i	$0.412270\pi$
$114$	0	0
$115$	171.686	0.139216
$116$	1497.42	1.19855
$117$	0	0
$118$	223.659	0.174487
$119$	−424.595	−0.327080
$120$	0	0
$121$	−1260.30	−0.946881
$122$	−214.918	−0.159490
$123$	0	0
$124$	−899.002	−0.651070
$125$	475.485	0.340229
$126$	0	0
$127$	−141.203	−0.0986595	−0.0493297	−	0.998783i	$-0.515709\pi$
$127$	−141.203	−0.0986595	−0.0493297	+	0.998783i	$0.515709\pi$
$128$	859.613	0.593592
$129$	0	0
$130$	0	0
$131$	731.910	0.488147	0.244074	−	0.969757i	$-0.421516\pi$
$131$	731.910	0.488147	0.244074	+	0.969757i	$0.421516\pi$
$132$	0	0
$133$	−397.742	−0.259313
$134$	−396.633	−0.255700
$135$	0	0
$136$	368.381	0.232268
$137$	−1762.27	−1.09898	−0.549492	−	0.835499i	$-0.685179\pi$
$137$	−1762.27	−1.09898	−0.549492	+	0.835499i	$0.685179\pi$
$138$	0	0
$139$	−664.776	−0.405652	−0.202826	−	0.979215i	$-0.565013\pi$
$139$	−664.776	−0.405652	−0.202826	+	0.979215i	$0.565013\pi$
$140$	−122.666	−0.0740511
$141$	0	0
$142$	55.3425	0.0327059
$143$	0	0
$144$	0	0
$145$	370.659	0.212286
$146$	−301.094	−0.170676
$147$	0	0
$148$	1066.38	0.592268
$149$	3300.71	1.81480	0.907399	−	0.420270i	$-0.138064\pi$
$149$	3300.71	1.81480	0.907399	+	0.420270i	$0.138064\pi$
$150$	0	0
$151$	1464.15	0.789079	0.394540	−	0.918879i	$-0.370904\pi$
$151$	1464.15	0.789079	0.394540	+	0.918879i	$0.370904\pi$
$152$	345.083	0.184144
$153$	0	0
$154$	−30.6339	−0.0160296
$155$	−222.531	−0.115317
$156$	0	0
$157$	1535.32	0.780456	0.390228	−	0.920718i	$-0.372396\pi$
$157$	1535.32	0.780456	0.390228	+	0.920718i	$0.372396\pi$
$158$	304.730	0.153437
$159$	0	0
$160$	160.312	0.0792111
$161$	724.310	0.354557
$162$	0	0
$163$	793.554	0.381325	0.190663	−	0.981656i	$-0.438936\pi$
$163$	793.554	0.381325	0.190663	+	0.981656i	$0.438936\pi$
$164$	3401.38	1.61953
$165$	0	0
$166$	−420.181	−0.196460
$167$	1336.87	0.619460	0.309730	−	0.950825i	$-0.399761\pi$
$167$	1336.87	0.619460	0.309730	+	0.950825i	$0.399761\pi$
$168$	0	0
$169$	0	0
$170$	45.0156	0.0203090
$171$	0	0
$172$	−1575.96	−0.698640
$173$	−568.460	−0.249822	−0.124911	−	0.992168i	$-0.539865\pi$
$173$	−568.460	−0.249822	−0.124911	+	0.992168i	$0.539865\pi$
$174$	0	0
$175$	987.808	0.426693
$176$	−498.103	−0.213329
$177$	0	0
$178$	−337.686	−0.142194
$179$	−1546.00	−0.645552	−0.322776	−	0.946475i	$-0.604616\pi$
$179$	−1546.00	−0.645552	−0.322776	+	0.946475i	$0.604616\pi$
$180$	0	0
$181$	3408.96	1.39992	0.699960	−	0.714182i	$-0.253201\pi$
$181$	3408.96	1.39992	0.699960	+	0.714182i	$0.253201\pi$
$182$	0	0
$183$	0	0
$184$	−628.416	−0.251780
$185$	263.962	0.104902
$186$	0	0
$187$	−438.307	−0.171402
$188$	4821.61	1.87049
$189$	0	0
$190$	42.1686	0.0161012
$191$	3464.71	1.31255	0.656277	−	0.754520i	$-0.272130\pi$
$191$	3464.71	1.31255	0.656277	+	0.754520i	$0.272130\pi$
$192$	0	0
$193$	4652.24	1.73510	0.867552	−	0.497346i	$-0.165692\pi$
$193$	4652.24	1.73510	0.867552	+	0.497346i	$0.165692\pi$
$194$	−470.134	−0.173988
$195$	0	0
$196$	2157.88	0.786398
$197$	2870.98	1.03832	0.519160	−	0.854677i	$-0.326245\pi$
$197$	2870.98	1.03832	0.519160	+	0.854677i	$0.326245\pi$
$198$	0	0
$199$	25.0330	0.00891730	0.00445865	−	0.999990i	$-0.498581\pi$
$199$	25.0330	0.00891730	0.00445865	+	0.999990i	$0.498581\pi$
$200$	−857.028	−0.303005
$201$	0	0
$202$	268.310	0.0934565
$203$	1563.74	0.540655
$204$	0	0
$205$	841.947	0.286849
$206$	131.192	0.0443717
$207$	0	0
$208$	0	0
$209$	−410.587	−0.135889
$210$	0	0
$211$	−3605.91	−1.17650	−0.588249	−	0.808679i	$-0.700183\pi$
$211$	−3605.91	−1.17650	−0.588249	+	0.808679i	$0.700183\pi$
$212$	−3534.70	−1.14511
$213$	0	0
$214$	685.987	0.219127
$215$	−390.100	−0.123742
$216$	0	0
$217$	−938.815	−0.293691
$218$	−264.215	−0.0820868
$219$	0	0
$220$	−126.627	−0.0388054
$221$	0	0
$222$	0	0
$223$	−4304.37	−1.29256	−0.646282	−	0.763098i	$-0.723677\pi$
$223$	−4304.37	−1.29256	−0.646282	+	0.763098i	$0.723677\pi$
$224$	676.326	0.201736
$225$	0	0
$226$	−292.424	−0.0860697
$227$	5475.19	1.60089	0.800443	−	0.599409i	$-0.204598\pi$
$227$	5475.19	1.60089	0.800443	+	0.599409i	$0.204598\pi$
$228$	0	0
$229$	378.108	0.109110	0.0545548	−	0.998511i	$-0.482626\pi$
$229$	378.108	0.109110	0.0545548	+	0.998511i	$0.482626\pi$
$230$	−76.7914	−0.0220151
$231$	0	0
$232$	−1356.71	−0.383932
$233$	−2547.96	−0.716405	−0.358202	−	0.933644i	$-0.616610\pi$
$233$	−2547.96	−0.716405	−0.358202	+	0.933644i	$0.616610\pi$
$234$	0	0
$235$	1193.50	0.331299
$236$	3900.32	1.07580
$237$	0	0
$238$	189.912	0.0517234
$239$	6313.91	1.70884	0.854420	−	0.519583i	$-0.173913\pi$
$239$	6313.91	1.70884	0.854420	+	0.519583i	$0.173913\pi$
$240$	0	0
$241$	−6763.73	−1.80784	−0.903921	−	0.427699i	$-0.859324\pi$
$241$	−6763.73	−1.80784	−0.903921	+	0.427699i	$0.859324\pi$
$242$	563.704	0.149737
$243$	0	0
$244$	−3747.89	−0.983336
$245$	534.142	0.139286
$246$	0	0
$247$	0	0
$248$	814.521	0.208557
$249$	0	0
$250$	−212.674	−0.0538027
$251$	3416.00	0.859028	0.429514	−	0.903060i	$-0.358685\pi$
$251$	3416.00	0.859028	0.429514	+	0.903060i	$0.358685\pi$
$252$	0	0
$253$	747.701	0.185801
$254$	63.1571	0.0156017
$255$	0	0
$256$	3109.68	0.759199
$257$	−7002.36	−1.69959	−0.849796	−	0.527112i	$-0.823275\pi$
$257$	−7002.36	−1.69959	−0.849796	+	0.527112i	$0.823275\pi$
$258$	0	0
$259$	1113.60	0.267166
$260$	0	0
$261$	0	0
$262$	−327.367	−0.0771940
$263$	−3369.76	−0.790071	−0.395035	−	0.918666i	$-0.629268\pi$
$263$	−3369.76	−0.790071	−0.395035	+	0.918666i	$0.629268\pi$
$264$	0	0
$265$	−874.948	−0.202821
$266$	177.901	0.0410069
$267$	0	0
$268$	−6916.75	−1.57652
$269$	7801.70	1.76832	0.884160	−	0.467185i	$-0.154732\pi$
$269$	7801.70	1.76832	0.884160	+	0.467185i	$0.154732\pi$
$270$	0	0
$271$	4410.05	0.988528	0.494264	−	0.869312i	$-0.335438\pi$
$271$	4410.05	0.988528	0.494264	+	0.869312i	$0.335438\pi$
$272$	3087.94	0.688360
$273$	0	0
$274$	788.225	0.173790
$275$	1019.71	0.223603
$276$	0	0
$277$	−4636.44	−1.00569	−0.502846	−	0.864376i	$-0.667714\pi$
$277$	−4636.44	−1.00569	−0.502846	+	0.864376i	$0.667714\pi$
$278$	297.340	0.0641485
$279$	0	0
$280$	111.139	0.0237207
$281$	−1090.50	−0.231508	−0.115754	−	0.993278i	$-0.536928\pi$
$281$	−1090.50	−0.231508	−0.115754	+	0.993278i	$0.536928\pi$
$282$	0	0
$283$	1140.52	0.239565	0.119782	−	0.992800i	$-0.461780\pi$
$283$	1140.52	0.239565	0.119782	+	0.992800i	$0.461780\pi$
$284$	965.101	0.201649
$285$	0	0
$286$	0	0
$287$	3552.01	0.730553
$288$	0	0
$289$	−2195.76	−0.446929
$290$	−165.788	−0.0335703
$291$	0	0
$292$	−5250.67	−1.05230
$293$	335.079	0.0668106	0.0334053	−	0.999442i	$-0.489365\pi$
$293$	335.079	0.0668106	0.0334053	+	0.999442i	$0.489365\pi$
$294$	0	0
$295$	965.449	0.190545
$296$	−966.168	−0.189721
$297$	0	0
$298$	−1476.34	−0.286986
$299$	0	0
$300$	0	0
$301$	−1645.76	−0.315149
$302$	−654.883	−0.124782
$303$	0	0
$304$	2892.65	0.545739
$305$	−927.719	−0.174167
$306$	0	0
$307$	2540.04	0.472207	0.236104	−	0.971728i	$-0.424130\pi$
$307$	2540.04	0.472207	0.236104	+	0.971728i	$0.424130\pi$
$308$	−534.215	−0.0988303
$309$	0	0
$310$	99.5332	0.0182358
$311$	−2376.36	−0.433283	−0.216642	−	0.976251i	$-0.569510\pi$
$311$	−2376.36	−0.433283	−0.216642	+	0.976251i	$0.569510\pi$
$312$	0	0
$313$	1315.78	0.237612	0.118806	−	0.992917i	$-0.462093\pi$
$313$	1315.78	0.237612	0.118806	+	0.992917i	$0.462093\pi$
$314$	−686.713	−0.123419
$315$	0	0
$316$	5314.09	0.946015
$317$	6043.13	1.07071	0.535357	−	0.844626i	$-0.320177\pi$
$317$	6043.13	1.07071	0.535357	+	0.844626i	$0.320177\pi$
$318$	0	0
$319$	1614.24	0.283323
$320$	843.286	0.147316
$321$	0	0
$322$	−323.968	−0.0560685
$323$	2545.39	0.438481
$324$	0	0
$325$	0	0
$326$	−354.940	−0.0603015
$327$	0	0
$328$	−3081.75	−0.518784
$329$	5035.14	0.843758
$330$	0	0
$331$	8168.18	1.35639	0.678193	−	0.734884i	$-0.262763\pi$
$331$	8168.18	1.35639	0.678193	+	0.734884i	$0.262763\pi$
$332$	−7327.40	−1.21128
$333$	0	0
$334$	−597.951	−0.0979594
$335$	−1712.11	−0.279232
$336$	0	0
$337$	3076.32	0.497263	0.248631	−	0.968598i	$-0.420019\pi$
$337$	3076.32	0.497263	0.248631	+	0.968598i	$0.420019\pi$
$338$	0	0
$339$	0	0
$340$	785.012	0.125215
$341$	−969.133	−0.153905
$342$	0	0
$343$	5047.30	0.794544
$344$	1427.87	0.223795
$345$	0	0
$346$	254.260	0.0395061
$347$	8754.07	1.35430	0.677152	−	0.735844i	$-0.263214\pi$
$347$	8754.07	1.35430	0.677152	+	0.735844i	$0.263214\pi$
$348$	0	0
$349$	1064.97	0.163343	0.0816715	−	0.996659i	$-0.473974\pi$
$349$	1064.97	0.163343	0.0816715	+	0.996659i	$0.473974\pi$
$350$	−441.825	−0.0674758
$351$	0	0
$352$	698.167	0.105717
$353$	−5047.62	−0.761070	−0.380535	−	0.924767i	$-0.624260\pi$
$353$	−5047.62	−0.761070	−0.380535	+	0.924767i	$0.624260\pi$
$354$	0	0
$355$	238.893	0.0357158
$356$	−5888.79	−0.876700
$357$	0	0
$358$	691.494	0.102085
$359$	−8152.39	−1.19851	−0.599257	−	0.800557i	$-0.704537\pi$
$359$	−8152.39	−1.19851	−0.599257	+	0.800557i	$0.704537\pi$
$360$	0	0
$361$	−4474.59	−0.652367
$362$	−1524.75	−0.221379
$363$	0	0
$364$	0	0
$365$	−1299.71	−0.186383
$366$	0	0
$367$	11636.3	1.65507	0.827537	−	0.561412i	$-0.189742\pi$
$367$	11636.3	1.65507	0.827537	+	0.561412i	$0.189742\pi$
$368$	−5267.67	−0.746186
$369$	0	0
$370$	−118.064	−0.0165888
$371$	−3691.24	−0.516548
$372$	0	0
$373$	4720.59	0.655289	0.327644	−	0.944801i	$-0.393745\pi$
$373$	4720.59	0.655289	0.327644	+	0.944801i	$0.393745\pi$
$374$	196.045	0.0271049
$375$	0	0
$376$	−4368.52	−0.599173
$377$	0	0
$378$	0	0
$379$	12933.7	1.75293	0.876464	−	0.481467i	$-0.159896\pi$
$379$	12933.7	1.75293	0.876464	+	0.481467i	$0.159896\pi$
$380$	735.365	0.0992722
$381$	0	0
$382$	−1549.69	−0.207563
$383$	−554.945	−0.0740376	−0.0370188	−	0.999315i	$-0.511786\pi$
$383$	−554.945	−0.0740376	−0.0370188	+	0.999315i	$0.511786\pi$
$384$	0	0
$385$	−132.235	−0.0175047
$386$	−2080.84	−0.274384
$387$	0	0
$388$	−8198.51	−1.07272
$389$	−5091.35	−0.663604	−0.331802	−	0.943349i	$-0.607657\pi$
$389$	−5091.35	−0.663604	−0.331802	+	0.943349i	$0.607657\pi$
$390$	0	0
$391$	−4635.30	−0.599532
$392$	−1955.10	−0.251907
$393$	0	0
$394$	−1284.13	−0.164196
$395$	1315.40	0.167557
$396$	0	0
$397$	10254.6	1.29638	0.648188	−	0.761480i	$-0.275527\pi$
$397$	10254.6	1.29638	0.648188	+	0.761480i	$0.275527\pi$
$398$	−11.1967	−0.00141015
$399$	0	0
$400$	−7184.00	−0.898001
$401$	−505.816	−0.0629906	−0.0314953	−	0.999504i	$-0.510027\pi$
$401$	−505.816	−0.0629906	−0.0314953	+	0.999504i	$0.510027\pi$
$402$	0	0
$403$	0	0
$404$	4678.97	0.576207
$405$	0	0
$406$	−699.426	−0.0854974
$407$	1149.57	0.140005
$408$	0	0
$409$	2219.21	0.268295	0.134148	−	0.990961i	$-0.457170\pi$
$409$	2219.21	0.268295	0.134148	+	0.990961i	$0.457170\pi$
$410$	−376.585	−0.0453614
$411$	0	0
$412$	2287.82	0.273574
$413$	4073.04	0.485282
$414$	0	0
$415$	−1813.76	−0.214540
$416$	0	0
$417$	0	0
$418$	183.646	0.0214891
$419$	−10249.3	−1.19501	−0.597506	−	0.801864i	$-0.703842\pi$
$419$	−10249.3	−1.19501	−0.597506	+	0.801864i	$0.703842\pi$
$420$	0	0
$421$	97.9194	0.0113356	0.00566781	−	0.999984i	$-0.498196\pi$
$421$	97.9194	0.0113356	0.00566781	+	0.999984i	$0.498196\pi$
$422$	1612.85	0.186048
$423$	0	0
$424$	3202.54	0.366814
$425$	−6321.58	−0.721510
$426$	0	0
$427$	−3913.87	−0.443572
$428$	11962.7	1.35103
$429$	0	0
$430$	174.483	0.0195682
$431$	−11727.2	−1.31062	−0.655310	−	0.755360i	$-0.727462\pi$
$431$	−11727.2	−1.31062	−0.655310	+	0.755360i	$0.727462\pi$
$432$	0	0
$433$	6945.13	0.770812	0.385406	−	0.922747i	$-0.374061\pi$
$433$	6945.13	0.770812	0.385406	+	0.922747i	$0.374061\pi$
$434$	419.911	0.0464433
$435$	0	0
$436$	−4607.57	−0.506107
$437$	−4342.15	−0.475316
$438$	0	0
$439$	11446.3	1.24442	0.622210	−	0.782851i	$-0.286235\pi$
$439$	11446.3	1.24442	0.622210	+	0.782851i	$0.286235\pi$
$440$	114.728	0.0124305
$441$	0	0
$442$	0	0
$443$	10513.6	1.12758	0.563789	−	0.825919i	$-0.309343\pi$
$443$	10513.6	1.12758	0.563789	+	0.825919i	$0.309343\pi$
$444$	0	0
$445$	−1457.66	−0.155280
$446$	1925.25	0.204402
$447$	0	0
$448$	3557.66	0.375187
$449$	9862.37	1.03660	0.518300	−	0.855199i	$-0.326565\pi$
$449$	9862.37	1.03660	0.518300	+	0.855199i	$0.326565\pi$
$450$	0	0
$451$	3666.72	0.382836
$452$	−5099.49	−0.530663
$453$	0	0
$454$	−2448.93	−0.253159
$455$	0	0
$456$	0	0
$457$	−5990.48	−0.613179	−0.306590	−	0.951842i	$-0.599188\pi$
$457$	−5990.48	−0.613179	−0.306590	+	0.951842i	$0.599188\pi$
$458$	−169.120	−0.0172542
$459$	0	0
$460$	−1339.14	−0.135734
$461$	−16579.5	−1.67502	−0.837510	−	0.546423i	$-0.815989\pi$
$461$	−16579.5	−1.67502	−0.837510	+	0.546423i	$0.815989\pi$
$462$	0	0
$463$	3606.65	0.362020	0.181010	−	0.983481i	$-0.442063\pi$
$463$	3606.65	0.362020	0.181010	+	0.983481i	$0.442063\pi$
$464$	−11372.6	−1.13784
$465$	0	0
$466$	1139.65	0.113290
$467$	6789.61	0.672775	0.336387	−	0.941724i	$-0.390795\pi$
$467$	6789.61	0.672775	0.336387	+	0.941724i	$0.390795\pi$
$468$	0	0
$469$	−7223.06	−0.711152
$470$	−533.826	−0.0523906
$471$	0	0
$472$	−3533.80	−0.344611
$473$	−1698.90	−0.165149
$474$	0	0
$475$	−5921.78	−0.572021
$476$	3311.82	0.318901
$477$	0	0
$478$	−2824.07	−0.270230
$479$	12688.3	1.21032	0.605161	−	0.796103i	$-0.293109\pi$
$479$	12688.3	1.21032	0.605161	+	0.796103i	$0.293109\pi$
$480$	0	0
$481$	0	0
$482$	3025.27	0.285886
$483$	0	0
$484$	9830.26	0.923202
$485$	−2029.39	−0.189999
$486$	0	0
$487$	−1217.28	−0.113265	−0.0566325	−	0.998395i	$-0.518036\pi$
$487$	−1217.28	−0.113265	−0.0566325	+	0.998395i	$0.518036\pi$
$488$	3395.69	0.314991
$489$	0	0
$490$	−238.910	−0.0220262
$491$	10496.3	0.964747	0.482374	−	0.875966i	$-0.339775\pi$
$491$	10496.3	0.964747	0.482374	+	0.875966i	$0.339775\pi$
$492$	0	0
$493$	−10007.3	−0.914211
$494$	0	0
$495$	0	0
$496$	6827.69	0.618090
$497$	1007.84	0.0909615
$498$	0	0
$499$	−10516.5	−0.943450	−0.471725	−	0.881746i	$-0.656368\pi$
$499$	−10516.5	−0.943450	−0.471725	+	0.881746i	$0.656368\pi$
$500$	−3708.75	−0.331721
$501$	0	0
$502$	−1527.90	−0.135844
$503$	−13400.9	−1.18790	−0.593952	−	0.804500i	$-0.702433\pi$
$503$	−13400.9	−1.18790	−0.593952	+	0.804500i	$0.702433\pi$
$504$	0	0
$505$	1158.19	0.102057
$506$	−334.430	−0.0293819
$507$	0	0
$508$	1101.38	0.0961923
$509$	−4683.90	−0.407878	−0.203939	−	0.978984i	$-0.565374\pi$
$509$	−4683.90	−0.407878	−0.203939	+	0.978984i	$0.565374\pi$
$510$	0	0
$511$	−5483.20	−0.474682
$512$	−8267.80	−0.713649
$513$	0	0
$514$	3132.00	0.268768
$515$	566.305	0.0484551
$516$	0	0
$517$	5197.75	0.442160
$518$	−498.090	−0.0422487
$519$	0	0
$520$	0	0
$521$	−15034.6	−1.26426	−0.632128	−	0.774864i	$-0.717818\pi$
$521$	−15034.6	−1.26426	−0.632128	+	0.774864i	$0.717818\pi$
$522$	0	0
$523$	8009.33	0.669643	0.334822	−	0.942282i	$-0.391324\pi$
$523$	8009.33	0.669643	0.334822	+	0.942282i	$0.391324\pi$
$524$	−5708.86	−0.475940
$525$	0	0
$526$	1507.22	0.124939
$527$	6008.04	0.496612
$528$	0	0
$529$	−4259.71	−0.350103
$530$	391.345	0.0320735
$531$	0	0
$532$	3102.36	0.252828
$533$	0	0
$534$	0	0
$535$	2961.14	0.239292
$536$	6266.77	0.505006
$537$	0	0
$538$	−3489.53	−0.279636
$539$	2326.21	0.185894
$540$	0	0
$541$	20833.2	1.65561	0.827807	−	0.561013i	$-0.189588\pi$
$541$	20833.2	1.65561	0.827807	+	0.561013i	$0.189588\pi$
$542$	−1972.52	−0.156323
$543$	0	0
$544$	−4328.22	−0.341123
$545$	−1140.52	−0.0896411
$546$	0	0
$547$	−14247.2	−1.11365	−0.556825	−	0.830630i	$-0.687981\pi$
$547$	−14247.2	−1.11365	−0.556825	+	0.830630i	$0.687981\pi$
$548$	13745.6	1.07150
$549$	0	0
$550$	−456.093	−0.0353598
$551$	−9374.41	−0.724797
$552$	0	0
$553$	5549.43	0.426737
$554$	2073.78	0.159037
$555$	0	0
$556$	5185.22	0.395508
$557$	−2007.94	−0.152745	−0.0763725	−	0.997079i	$-0.524334\pi$
$557$	−2007.94	−0.152745	−0.0763725	+	0.997079i	$0.524334\pi$
$558$	0	0
$559$	0	0
$560$	931.616	0.0703000
$561$	0	0
$562$	487.758	0.0366100
$563$	−10562.1	−0.790659	−0.395330	−	0.918539i	$-0.629370\pi$
$563$	−10562.1	−0.790659	−0.395330	+	0.918539i	$0.629370\pi$
$564$	0	0
$565$	−1262.28	−0.0939905
$566$	−510.129	−0.0378840
$567$	0	0
$568$	−874.409	−0.0645940
$569$	23207.2	1.70983	0.854916	−	0.518766i	$-0.173608\pi$
$569$	23207.2	1.70983	0.854916	+	0.518766i	$0.173608\pi$
$570$	0	0
$571$	7987.78	0.585426	0.292713	−	0.956200i	$-0.405442\pi$
$571$	7987.78	0.585426	0.292713	+	0.956200i	$0.405442\pi$
$572$	0	0
$573$	0	0
$574$	−1588.74	−0.115527
$575$	10783.9	0.782120
$576$	0	0
$577$	3224.41	0.232641	0.116321	−	0.993212i	$-0.462890\pi$
$577$	3224.41	0.232641	0.116321	+	0.993212i	$0.462890\pi$
$578$	982.116	0.0706759
$579$	0	0
$580$	−2891.12	−0.206978
$581$	−7651.90	−0.546393
$582$	0	0
$583$	−3810.44	−0.270690
$584$	4757.26	0.337084
$585$	0	0
$586$	−149.873	−0.0105652
$587$	661.034	0.0464800	0.0232400	−	0.999730i	$-0.492602\pi$
$587$	661.034	0.0464800	0.0232400	+	0.999730i	$0.492602\pi$
$588$	0	0
$589$	5628.07	0.393719
$590$	−431.824	−0.0301321
$591$	0	0
$592$	−8098.87	−0.562266
$593$	10172.5	0.704443	0.352221	−	0.935917i	$-0.385426\pi$
$593$	10172.5	0.704443	0.352221	+	0.935917i	$0.385426\pi$
$594$	0	0
$595$	819.777	0.0564834
$596$	−25745.4	−1.76942
$597$	0	0
$598$	0	0
$599$	−23462.2	−1.60040	−0.800200	−	0.599733i	$-0.795274\pi$
$599$	−23462.2	−1.60040	−0.800200	+	0.599733i	$0.795274\pi$
$600$	0	0
$601$	−10482.6	−0.711474	−0.355737	−	0.934586i	$-0.615770\pi$
$601$	−10482.6	−0.711474	−0.355737	+	0.934586i	$0.615770\pi$
$602$	736.111	0.0498366
$603$	0	0
$604$	−11420.3	−0.769347
$605$	2433.29	0.163516
$606$	0	0
$607$	−24949.8	−1.66834	−0.834169	−	0.551508i	$-0.814053\pi$
$607$	−24949.8	−1.66834	−0.834169	+	0.551508i	$0.814053\pi$
$608$	−4054.48	−0.270446
$609$	0	0
$610$	414.949	0.0275423
$611$	0	0
$612$	0	0
$613$	8238.86	0.542846	0.271423	−	0.962460i	$-0.412506\pi$
$613$	8238.86	0.542846	0.271423	+	0.962460i	$0.412506\pi$
$614$	−1136.10	−0.0746733
$615$	0	0
$616$	484.014	0.0316583
$617$	−5248.85	−0.342481	−0.171241	−	0.985229i	$-0.554778\pi$
$617$	−5248.85	−0.342481	−0.171241	+	0.985229i	$0.554778\pi$
$618$	0	0
$619$	−17820.9	−1.15716	−0.578580	−	0.815626i	$-0.696393\pi$
$619$	−17820.9	−1.15716	−0.578580	+	0.815626i	$0.696393\pi$
$620$	1735.73	0.112433
$621$	0	0
$622$	1062.89	0.0685180
$623$	−6149.58	−0.395470
$624$	0	0
$625$	14241.0	0.911424
$626$	−588.522	−0.0375752
$627$	0	0
$628$	−11975.4	−0.760938
$629$	−7126.62	−0.451760
$630$	0	0
$631$	25445.0	1.60531	0.802653	−	0.596446i	$-0.203421\pi$
$631$	25445.0	1.60531	0.802653	+	0.596446i	$0.203421\pi$
$632$	−4814.71	−0.303036
$633$	0	0
$634$	−2702.96	−0.169319
$635$	272.625	0.0170375
$636$	0	0
$637$	0	0
$638$	−722.013	−0.0448037
$639$	0	0
$640$	−1659.68	−0.102507
$641$	18701.0	1.15233	0.576167	−	0.817332i	$-0.304548\pi$
$641$	18701.0	1.15233	0.576167	+	0.817332i	$0.304548\pi$
$642$	0	0
$643$	−28465.9	−1.74586	−0.872928	−	0.487849i	$-0.837782\pi$
$643$	−28465.9	−1.74586	−0.872928	+	0.487849i	$0.837782\pi$
$644$	−5649.58	−0.345690
$645$	0	0
$646$	−1138.50	−0.0693399
$647$	10924.6	0.663818	0.331909	−	0.943311i	$-0.392307\pi$
$647$	10924.6	0.663818	0.331909	+	0.943311i	$0.392307\pi$
$648$	0	0
$649$	4204.58	0.254305
$650$	0	0
$651$	0	0
$652$	−6189.68	−0.371789
$653$	27110.9	1.62470	0.812351	−	0.583169i	$-0.198187\pi$
$653$	27110.9	1.62470	0.812351	+	0.583169i	$0.198187\pi$
$654$	0	0
$655$	−1413.12	−0.0842980
$656$	−25832.6	−1.53749
$657$	0	0
$658$	−2252.11	−0.133429
$659$	−1727.08	−0.102090	−0.0510452	−	0.998696i	$-0.516255\pi$
$659$	−1727.08	−0.102090	−0.0510452	+	0.998696i	$0.516255\pi$
$660$	0	0
$661$	20651.1	1.21518	0.607592	−	0.794249i	$-0.292136\pi$
$661$	20651.1	1.21518	0.607592	+	0.794249i	$0.292136\pi$
$662$	−3653.45	−0.214494
$663$	0	0
$664$	6638.84	0.388007
$665$	767.932	0.0447806
$666$	0	0
$667$	17071.3	0.991010
$668$	−10427.5	−0.603969
$669$	0	0
$670$	765.790	0.0441568
$671$	−4040.26	−0.232448
$672$	0	0
$673$	10858.0	0.621907	0.310954	−	0.950425i	$-0.399352\pi$
$673$	10858.0	0.621907	0.310954	+	0.950425i	$0.399352\pi$
$674$	−1375.97	−0.0786355
$675$	0	0
$676$	0	0
$677$	−29132.9	−1.65387	−0.826933	−	0.562301i	$-0.809916\pi$
$677$	−29132.9	−1.65387	−0.826933	+	0.562301i	$0.809916\pi$
$678$	0	0
$679$	−8561.59	−0.483894
$680$	−711.244	−0.0401102
$681$	0	0
$682$	433.472	0.0243380
$683$	9443.65	0.529065	0.264532	−	0.964377i	$-0.414782\pi$
$683$	9443.65	0.529065	0.264532	+	0.964377i	$0.414782\pi$
$684$	0	0
$685$	3402.46	0.189783
$686$	−2257.55	−0.125647
$687$	0	0
$688$	11969.0	0.663249
$689$	0	0
$690$	0	0
$691$	−23700.1	−1.30477	−0.652384	−	0.757889i	$-0.726231\pi$
$691$	−23700.1	−1.30477	−0.652384	+	0.757889i	$0.726231\pi$
$692$	4433.96	0.243575
$693$	0	0
$694$	−3915.51	−0.214165
$695$	1283.50	0.0700519
$696$	0	0
$697$	−22731.5	−1.23532
$698$	−476.340	−0.0258305
$699$	0	0
$700$	−7704.84	−0.416022
$701$	−21643.1	−1.16612	−0.583059	−	0.812430i	$-0.698144\pi$
$701$	−21643.1	−1.16612	−0.583059	+	0.812430i	$0.698144\pi$
$702$	0	0
$703$	−6675.91	−0.358160
$704$	3672.55	0.196611
$705$	0	0
$706$	2257.69	0.120353
$707$	4886.18	0.259921
$708$	0	0
$709$	−28965.3	−1.53429	−0.767146	−	0.641472i	$-0.778324\pi$
$709$	−28965.3	−1.53429	−0.767146	+	0.641472i	$0.778324\pi$
$710$	−106.851	−0.00564798
$711$	0	0
$712$	5335.41	0.280833
$713$	−10249.0	−0.538330
$714$	0	0
$715$	0	0
$716$	12058.7	0.629408
$717$	0	0
$718$	3646.39	0.189529
$719$	21256.5	1.10255	0.551276	−	0.834323i	$-0.314141\pi$
$719$	21256.5	1.10255	0.551276	+	0.834323i	$0.314141\pi$
$720$	0	0
$721$	2389.13	0.123406
$722$	2001.39	0.103163
$723$	0	0
$724$	−26589.7	−1.36491
$725$	23281.7	1.19264
$726$	0	0
$727$	−11868.9	−0.605494	−0.302747	−	0.953071i	$-0.597904\pi$
$727$	−11868.9	−0.605494	−0.302747	+	0.953071i	$0.597904\pi$
$728$	0	0
$729$	0	0
$730$	581.330	0.0294739
$731$	10532.2	0.532896
$732$	0	0
$733$	27296.4	1.37546	0.687732	−	0.725965i	$-0.258607\pi$
$733$	27296.4	1.37546	0.687732	+	0.725965i	$0.258607\pi$
$734$	−5204.68	−0.261728
$735$	0	0
$736$	7383.44	0.369779
$737$	−7456.32	−0.372669
$738$	0	0
$739$	−26424.6	−1.31535	−0.657674	−	0.753302i	$-0.728460\pi$
$739$	−26424.6	−1.31535	−0.657674	+	0.753302i	$0.728460\pi$
$740$	−2058.89	−0.102279
$741$	0	0
$742$	1651.01	0.0816853
$743$	25974.8	1.28253	0.641267	−	0.767318i	$-0.278409\pi$
$743$	25974.8	1.28253	0.641267	+	0.767318i	$0.278409\pi$
$744$	0	0
$745$	−6372.78	−0.313397
$746$	−2111.42	−0.103625
$747$	0	0
$748$	3418.77	0.167116
$749$	12492.5	0.609433
$750$	0	0
$751$	20234.8	0.983195	0.491598	−	0.870823i	$-0.336413\pi$
$751$	20234.8	0.983195	0.491598	+	0.870823i	$0.336413\pi$
$752$	−36619.0	−1.77574
$753$	0	0
$754$	0	0
$755$	−2826.88	−0.136266
$756$	0	0
$757$	−3363.48	−0.161490	−0.0807449	−	0.996735i	$-0.525730\pi$
$757$	−3363.48	−0.161490	−0.0807449	+	0.996735i	$0.525730\pi$
$758$	−5784.97	−0.277202
$759$	0	0
$760$	−666.262	−0.0317998
$761$	21311.3	1.01516	0.507579	−	0.861605i	$-0.330541\pi$
$761$	21311.3	1.01516	0.507579	+	0.861605i	$0.330541\pi$
$762$	0	0
$763$	−4811.62	−0.228299
$764$	−27024.5	−1.27973
$765$	0	0
$766$	248.215	0.0117081
$767$	0	0
$768$	0	0
$769$	−34989.6	−1.64078	−0.820388	−	0.571807i	$-0.806242\pi$
$769$	−34989.6	−1.64078	−0.820388	+	0.571807i	$0.806242\pi$
$770$	59.1458	0.00276814
$771$	0	0
$772$	−36287.2	−1.69171
$773$	−20034.1	−0.932181	−0.466091	−	0.884737i	$-0.654338\pi$
$773$	−20034.1	−0.932181	−0.466091	+	0.884737i	$0.654338\pi$
$774$	0	0
$775$	−13977.5	−0.647855
$776$	7428.09	0.343625
$777$	0	0
$778$	2277.25	0.104940
$779$	−21293.9	−0.979373
$780$	0	0
$781$	1040.39	0.0476671
$782$	2073.27	0.0948081
$783$	0	0
$784$	−16388.5	−0.746563
$785$	−2964.28	−0.134777
$786$	0	0
$787$	5591.98	0.253282	0.126641	−	0.991949i	$-0.459580\pi$
$787$	5591.98	0.253282	0.126641	+	0.991949i	$0.459580\pi$
$788$	−22393.5	−1.01235
$789$	0	0
$790$	−588.351	−0.0264969
$791$	−5325.32	−0.239376
$792$	0	0
$793$	0	0
$794$	−4586.64	−0.205005
$795$	0	0
$796$	−195.256	−0.00869430
$797$	12738.6	0.566152	0.283076	−	0.959098i	$-0.408645\pi$
$797$	12738.6	0.566152	0.283076	+	0.959098i	$0.408645\pi$
$798$	0	0
$799$	−32222.9	−1.42674
$800$	10069.5	0.445012
$801$	0	0
$802$	226.240	0.00996113
$803$	−5660.28	−0.248751
$804$	0	0
$805$	−1398.45	−0.0612283
$806$	0	0
$807$	0	0
$808$	−4239.28	−0.184576
$809$	8806.48	0.382719	0.191359	−	0.981520i	$-0.438710\pi$
$809$	8806.48	0.382719	0.191359	+	0.981520i	$0.438710\pi$
$810$	0	0
$811$	−10565.8	−0.457478	−0.228739	−	0.973488i	$-0.573460\pi$
$811$	−10565.8	−0.457478	−0.228739	+	0.973488i	$0.573460\pi$
$812$	−12197.1	−0.527134
$813$	0	0
$814$	−514.176	−0.0221399
$815$	−1532.14	−0.0658509
$816$	0	0
$817$	9866.09	0.422486
$818$	−992.604	−0.0424273
$819$	0	0
$820$	−6567.14	−0.279676
$821$	−422.966	−0.0179800	−0.00899002	−	0.999960i	$-0.502862\pi$
$821$	−422.966	−0.0179800	−0.00899002	+	0.999960i	$0.502862\pi$
$822$	0	0
$823$	11148.8	0.472202	0.236101	−	0.971728i	$-0.424130\pi$
$823$	11148.8	0.472202	0.236101	+	0.971728i	$0.424130\pi$
$824$	−2072.83	−0.0876339
$825$	0	0
$826$	−1821.78	−0.0767409
$827$	813.158	0.0341914	0.0170957	−	0.999854i	$-0.494558\pi$
$827$	813.158	0.0341914	0.0170957	+	0.999854i	$0.494558\pi$
$828$	0	0
$829$	−28320.8	−1.18652	−0.593258	−	0.805012i	$-0.702159\pi$
$829$	−28320.8	−1.18652	−0.593258	+	0.805012i	$0.702159\pi$
$830$	811.255	0.0339266
$831$	0	0
$832$	0	0
$833$	−14421.1	−0.599835
$834$	0	0
$835$	−2581.13	−0.106974
$836$	3202.55	0.132491
$837$	0	0
$838$	4584.28	0.188975
$839$	22139.3	0.911006	0.455503	−	0.890234i	$-0.349459\pi$
$839$	22139.3	0.911006	0.455503	+	0.890234i	$0.349459\pi$
$840$	0	0
$841$	12466.8	0.511166
$842$	−43.7972	−0.00179258
$843$	0	0
$844$	28125.9	1.14708
$845$	0	0
$846$	0	0
$847$	10265.6	0.416446
$848$	26845.2	1.08711
$849$	0	0
$850$	2827.50	0.114097
$851$	12157.2	0.489710
$852$	0	0
$853$	−14080.5	−0.565191	−0.282595	−	0.959239i	$-0.591195\pi$
$853$	−14080.5	−0.565191	−0.282595	+	0.959239i	$0.591195\pi$
$854$	1750.59	0.0701451
$855$	0	0
$856$	−10838.6	−0.432774
$857$	−10907.7	−0.434772	−0.217386	−	0.976086i	$-0.569753\pi$
$857$	−10907.7	−0.434772	−0.217386	+	0.976086i	$0.569753\pi$
$858$	0	0
$859$	7739.24	0.307403	0.153702	−	0.988117i	$-0.450881\pi$
$859$	7739.24	0.307403	0.153702	+	0.988117i	$0.450881\pi$
$860$	3042.76	0.120648
$861$	0	0
$862$	5245.30	0.207257
$863$	29072.5	1.14674	0.573372	−	0.819295i	$-0.305635\pi$
$863$	29072.5	1.14674	0.573372	+	0.819295i	$0.305635\pi$
$864$	0	0
$865$	1097.54	0.0431417
$866$	−3106.40	−0.121894
$867$	0	0
$868$	7322.70	0.286346
$869$	5728.64	0.223626
$870$	0	0
$871$	0	0
$872$	4174.59	0.162121
$873$	0	0
$874$	1942.15	0.0751649
$875$	−3873.00	−0.149636
$876$	0	0
$877$	2391.96	0.0920988	0.0460494	−	0.998939i	$-0.485337\pi$
$877$	2391.96	0.0920988	0.0460494	+	0.998939i	$0.485337\pi$
$878$	−5119.66	−0.196788
$879$	0	0
$880$	961.702	0.0368397
$881$	11975.8	0.457974	0.228987	−	0.973430i	$-0.426459\pi$
$881$	11975.8	0.457974	0.228987	+	0.973430i	$0.426459\pi$
$882$	0	0
$883$	−44712.4	−1.70407	−0.852035	−	0.523485i	$-0.824632\pi$
$883$	−44712.4	−1.70407	−0.852035	+	0.523485i	$0.824632\pi$
$884$	0	0
$885$	0	0
$886$	−4702.51	−0.178311
$887$	−17023.3	−0.644403	−0.322202	−	0.946671i	$-0.604423\pi$
$887$	−17023.3	−0.644403	−0.322202	+	0.946671i	$0.604423\pi$
$888$	0	0
$889$	1150.15	0.0433913
$890$	651.979	0.0245555
$891$	0	0
$892$	33573.8	1.26024
$893$	−30185.0	−1.13113
$894$	0	0
$895$	2984.91	0.111480
$896$	−7001.87	−0.261067
$897$	0	0
$898$	−4411.22	−0.163925
$899$	−22127.0	−0.820886
$900$	0	0
$901$	23622.5	0.873449
$902$	−1640.04	−0.0605405
$903$	0	0
$904$	4620.28	0.169987
$905$	−6581.77	−0.241752
$906$	0	0
$907$	2303.59	0.0843324	0.0421662	−	0.999111i	$-0.486574\pi$
$907$	2303.59	0.0843324	0.0421662	+	0.999111i	$0.486574\pi$
$908$	−42706.2	−1.56085
$909$	0	0
$910$	0	0
$911$	12897.0	0.469042	0.234521	−	0.972111i	$-0.424648\pi$
$911$	12897.0	0.469042	0.234521	+	0.972111i	$0.424648\pi$
$912$	0	0
$913$	−7899.01	−0.286330
$914$	2679.41	0.0969662
$915$	0	0
$916$	−2949.22	−0.106381
$917$	−5961.68	−0.214691
$918$	0	0
$919$	16785.2	0.602496	0.301248	−	0.953546i	$-0.402597\pi$
$919$	16785.2	0.602496	0.301248	+	0.953546i	$0.402597\pi$
$920$	1213.30	0.0434797
$921$	0	0
$922$	7415.64	0.264882
$923$	0	0
$924$	0	0
$925$	16579.9	0.589344
$926$	−1613.18	−0.0572487
$927$	0	0
$928$	15940.4	0.563866
$929$	−16348.0	−0.577352	−0.288676	−	0.957427i	$-0.593215\pi$
$929$	−16348.0	−0.577352	−0.288676	+	0.957427i	$0.593215\pi$
$930$	0	0
$931$	−13509.1	−0.475556
$932$	19873.9	0.698489
$933$	0	0
$934$	−3036.85	−0.106390
$935$	846.251	0.0295993
$936$	0	0
$937$	−26247.5	−0.915121	−0.457560	−	0.889179i	$-0.651277\pi$
$937$	−26247.5	−0.915121	−0.457560	+	0.889179i	$0.651277\pi$
$938$	3230.72	0.112459
$939$	0	0
$940$	−9309.22	−0.323014
$941$	−43838.4	−1.51869	−0.759347	−	0.650686i	$-0.774481\pi$
$941$	−43838.4	−1.51869	−0.759347	+	0.650686i	$0.774481\pi$
$942$	0	0
$943$	38777.3	1.33909
$944$	−29621.9	−1.02130
$945$	0	0
$946$	759.882	0.0261162
$947$	45077.2	1.54679	0.773395	−	0.633924i	$-0.218557\pi$
$947$	45077.2	1.54679	0.773395	+	0.633924i	$0.218557\pi$
$948$	0	0
$949$	0	0
$950$	2648.68	0.0904575
$951$	0	0
$952$	−3000.60	−0.102153
$953$	−26479.1	−0.900046	−0.450023	−	0.893017i	$-0.648584\pi$
$953$	−26479.1	−0.900046	−0.450023	+	0.893017i	$0.648584\pi$
$954$	0	0
$955$	−6689.41	−0.226664
$956$	−49248.1	−1.66611
$957$	0	0
$958$	−5675.21	−0.191396
$959$	14354.3	0.483343
$960$	0	0
$961$	−16506.7	−0.554084
$962$	0	0
$963$	0	0
$964$	52756.7	1.76263
$965$	−8982.20	−0.299635
$966$	0	0
$967$	−602.475	−0.0200355	−0.0100177	−	0.999950i	$-0.503189\pi$
$967$	−602.475	−0.0200355	−0.0100177	+	0.999950i	$0.503189\pi$
$968$	−8906.49	−0.295729
$969$	0	0
$970$	907.701	0.0300459
$971$	11330.2	0.374464	0.187232	−	0.982316i	$-0.440048\pi$
$971$	11330.2	0.374464	0.187232	+	0.982316i	$0.440048\pi$
$972$	0	0
$973$	5414.85	0.178409
$974$	544.461	0.0179114
$975$	0	0
$976$	28464.3	0.933524
$977$	−55633.6	−1.82178	−0.910888	−	0.412653i	$-0.864602\pi$
$977$	−55633.6	−1.82178	−0.910888	+	0.412653i	$0.864602\pi$
$978$	0	0
$979$	−6348.18	−0.207241
$980$	−4166.27	−0.135803
$981$	0	0
$982$	−4694.76	−0.152562
$983$	6884.90	0.223392	0.111696	−	0.993742i	$-0.464372\pi$
$983$	6884.90	0.223392	0.111696	+	0.993742i	$0.464372\pi$
$984$	0	0
$985$	−5543.08	−0.179307
$986$	4476.05	0.144570
$987$	0	0
$988$	0	0
$989$	−17966.7	−0.577662
$990$	0	0
$991$	52646.1	1.68755	0.843773	−	0.536700i	$-0.180329\pi$
$991$	52646.1	1.68755	0.843773	+	0.536700i	$0.180329\pi$
$992$	−9570.05	−0.306300
$993$	0	0
$994$	−450.786	−0.0143844
$995$	−48.3319	−0.00153992
$996$	0	0
$997$	26100.7	0.829106	0.414553	−	0.910025i	$-0.363938\pi$
$997$	26100.7	0.829106	0.414553	+	0.910025i	$0.363938\pi$
$998$	4703.78	0.149194
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bf.1.5		9
3.2	odd	2		507.4.a.p.1.5	yes	9
13.12	even	2		1521.4.a.bi.1.5		9
39.5	even	4		507.4.b.k.337.8		18
39.8	even	4		507.4.b.k.337.11		18
39.38	odd	2		507.4.a.o.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.5	✓	9	39.38	odd	2
507.4.a.p.1.5	yes	9	3.2	odd	2
507.4.b.k.337.8		18	39.5	even	4
507.4.b.k.337.11		18	39.8	even	4
1521.4.a.bf.1.5		9	1.1	even	1	trivial
1521.4.a.bi.1.5		9	13.12	even	2

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)

\(f(q)\)	\(=\)	\(q-0.447278 q^{2} -7.79994 q^{4} -1.93073 q^{5} -8.14537 q^{7} +7.06697 q^{8} +O(q^{10})\)	\(q-0.447278 q^{2} -7.79994 q^{4} -1.93073 q^{5} -8.14537 q^{7} +7.06697 q^{8} +0.863573 q^{10} -8.40842 q^{11} +3.64325 q^{14} +59.2386 q^{16} +52.1271 q^{17} +48.8304 q^{19} +15.0596 q^{20} +3.76090 q^{22} -88.9229 q^{23} -121.272 q^{25} +63.5334 q^{28} -191.979 q^{29} +115.257 q^{31} -83.0319 q^{32} -23.3153 q^{34} +15.7265 q^{35} -136.716 q^{37} -21.8408 q^{38} -13.6444 q^{40} -436.077 q^{41} +202.048 q^{43} +65.5852 q^{44} +39.7733 q^{46} -618.160 q^{47} -276.653 q^{49} +54.2425 q^{50} +453.170 q^{53} +16.2344 q^{55} -57.5631 q^{56} +85.8679 q^{58} -500.044 q^{59} +480.502 q^{61} -51.5522 q^{62} -436.771 q^{64} +886.769 q^{67} -406.589 q^{68} -7.03412 q^{70} -123.732 q^{71} +673.168 q^{73} +61.1501 q^{74} -380.875 q^{76} +68.4897 q^{77} -681.298 q^{79} -114.374 q^{80} +195.048 q^{82} +939.418 q^{83} -100.643 q^{85} -90.3716 q^{86} -59.4220 q^{88} +754.979 q^{89} +693.594 q^{92} +276.489 q^{94} -94.2783 q^{95} +1051.10 q^{97} +123.741 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8	\( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) \) 9 * q - 6 * q^2 + 44 * q^4 - 33 * q^5 + 83 * q^7 - 87 * q^8 - 54 * q^10 - 85 * q^11 - 158 * q^14 + 216 * q^16 - 178 * q^17 + 352 * q^19 - 402 * q^20 - 630 * q^22 - 150 * q^23 - 20 * q^25 + 940 * q^28 + 97 * q^29 + 717 * q^31 - 707 * q^32 + 632 * q^34 + 418 * q^35 + 1108 * q^37 + 660 * q^38 - 1506 * q^40 - 334 * q^41 + 242 * q^43 + 307 * q^44 + 979 * q^46 + 184 * q^47 - 38 * q^49 + 2031 * q^50 + 151 * q^53 + 2064 * q^55 - 2276 * q^56 + 1161 * q^58 - 537 * q^59 - 1340 * q^61 - 347 * q^62 + 893 * q^64 + 2308 * q^67 - 2785 * q^68 - 1420 * q^70 - 96 * q^71 + 2505 * q^73 + 1191 * q^74 + 2409 * q^76 + 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 1517 * q^82 - 1539 * q^83 + 4296 * q^85 + 3763 * q^86 - 3716 * q^88 + 592 * q^89 - 515 * q^92 - 692 * q^94 - 4158 * q^95 + 1445 * q^97 - 1457 * q^98

Introduction

L-functions

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