LMFDB - Embedded newform 1521.4.a.be.1.8

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:	$1$
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} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
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			1.8
Root			$4.23649$ 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)

$f(q)$	$=$	$q+3.23649 q^{2} +2.47490 q^{4} +13.5815 q^{5} +1.42933 q^{7} -17.8820 q^{8} +O(q^{10})$	$q+3.23649 q^{2} +2.47490 q^{4} +13.5815 q^{5} +1.42933 q^{7} -17.8820 q^{8} +43.9566 q^{10} -54.5673 q^{11} +4.62601 q^{14} -77.6741 q^{16} +114.413 q^{17} -104.933 q^{19} +33.6129 q^{20} -176.607 q^{22} +64.5438 q^{23} +59.4583 q^{25} +3.53743 q^{28} +60.8037 q^{29} -148.902 q^{31} -108.336 q^{32} +370.296 q^{34} +19.4125 q^{35} -20.9326 q^{37} -339.615 q^{38} -242.865 q^{40} -371.761 q^{41} +40.2951 q^{43} -135.048 q^{44} +208.896 q^{46} -639.802 q^{47} -340.957 q^{49} +192.436 q^{50} -102.124 q^{53} -741.108 q^{55} -25.5592 q^{56} +196.791 q^{58} +704.586 q^{59} -819.087 q^{61} -481.919 q^{62} +270.764 q^{64} +574.918 q^{67} +283.159 q^{68} +62.8283 q^{70} -365.790 q^{71} -965.233 q^{73} -67.7482 q^{74} -259.698 q^{76} -77.9945 q^{77} +580.173 q^{79} -1054.93 q^{80} -1203.20 q^{82} -175.372 q^{83} +1553.90 q^{85} +130.415 q^{86} +975.770 q^{88} +20.0351 q^{89} +159.739 q^{92} -2070.72 q^{94} -1425.15 q^{95} +1226.72 q^{97} -1103.51 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8}+O(q^{10}) $ 9 * q - 8 * q^2 + 32 * q^4 - 41 * q^5 + q^7 - 111 * q^8	$ 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8} + 198 q^{10} - 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} - 72 q^{19} - 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} + 132 q^{28} + 547 q^{29} - 521 q^{31} - 721 q^{32} - 100 q^{34} - 138 q^{35} + 584 q^{37} + 416 q^{38} + 1342 q^{40} - 482 q^{41} + 158 q^{43} - 1453 q^{44} + 1537 q^{46} - 1500 q^{47} + 642 q^{49} - 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} + 1455 q^{58} - 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} + 252 q^{67} + 1579 q^{68} + 2492 q^{70} - 2352 q^{71} + 903 q^{73} - 1037 q^{74} - 485 q^{76} + 1686 q^{77} - 115 q^{79} - 5701 q^{80} - 5147 q^{82} - 1207 q^{83} + 4308 q^{85} - 5691 q^{86} - 484 q^{88} - 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} + 2155 q^{97} - 5593 q^{98}+O(q^{100}) $ 9 * q - 8 * q^2 + 32 * q^4 - 41 * q^5 + q^7 - 111 * q^8 + 198 * q^10 - 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 - 72 * q^19 - 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 + 132 * q^28 + 547 * q^29 - 521 * q^31 - 721 * q^32 - 100 * q^34 - 138 * q^35 + 584 * q^37 + 416 * q^38 + 1342 * q^40 - 482 * q^41 + 158 * q^43 - 1453 * q^44 + 1537 * q^46 - 1500 * q^47 + 642 * q^49 - 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 + 1455 * q^58 - 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 + 252 * q^67 + 1579 * q^68 + 2492 * q^70 - 2352 * q^71 + 903 * q^73 - 1037 * q^74 - 485 * q^76 + 1686 * q^77 - 115 * q^79 - 5701 * q^80 - 5147 * q^82 - 1207 * q^83 + 4308 * q^85 - 5691 * q^86 - 484 * q^88 - 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 + 2155 * q^97 - 5593 * 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 $

$2$	3.23649	1.14427	0.572137	−	0.820158i	$-0.306115\pi$
$2$	3.23649	1.14427	0.572137	+	0.820158i	$0.306115\pi$
$3$	0	0
$3$	0	0
$4$	2.47490	0.309362
$5$	13.5815	1.21477	0.607385	−	0.794408i	$-0.292219\pi$
$5$	13.5815	1.21477	0.607385	+	0.794408i	$0.292219\pi$
$6$	0	0
$7$	1.42933	0.0771763	0.0385882	−	0.999255i	$-0.487714\pi$
$7$	1.42933	0.0771763	0.0385882	+	0.999255i	$0.487714\pi$
$8$	−17.8820	−0.790279
$9$	0	0
$10$	43.9566	1.39003
$11$	−54.5673	−1.49570	−0.747848	−	0.663870i	$-0.768913\pi$
$11$	−54.5673	−1.49570	−0.747848	+	0.663870i	$0.768913\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	4.62601	0.0883109
$15$	0	0
$16$	−77.6741	−1.21366
$17$	114.413	1.63230	0.816151	−	0.577839i	$-0.196104\pi$
$17$	114.413	1.63230	0.816151	+	0.577839i	$0.196104\pi$
$18$	0	0
$19$	−104.933	−1.26701	−0.633507	−	0.773737i	$-0.718385\pi$
$19$	−104.933	−1.26701	−0.633507	+	0.773737i	$0.718385\pi$
$20$	33.6129	0.375804
$21$	0	0
$22$	−176.607	−1.71149
$23$	64.5438	0.585144	0.292572	−	0.956244i	$-0.405489\pi$
$23$	64.5438	0.585144	0.292572	+	0.956244i	$0.405489\pi$
$24$	0	0
$25$	59.4583	0.475666
$26$	0	0
$27$	0	0
$28$	3.53743	0.0238754
$29$	60.8037	0.389343	0.194672	−	0.980868i	$-0.437636\pi$
$29$	60.8037	0.389343	0.194672	+	0.980868i	$0.437636\pi$
$30$	0	0
$31$	−148.902	−0.862694	−0.431347	−	0.902186i	$-0.641961\pi$
$31$	−148.902	−0.862694	−0.431347	+	0.902186i	$0.641961\pi$
$32$	−108.336	−0.598477
$33$	0	0
$34$	370.296	1.86780
$35$	19.4125	0.0937515
$36$	0	0
$37$	−20.9326	−0.0930079	−0.0465040	−	0.998918i	$-0.514808\pi$
$37$	−20.9326	−0.0930079	−0.0465040	+	0.998918i	$0.514808\pi$
$38$	−339.615	−1.44981
$39$	0	0
$40$	−242.865	−0.960007
$41$	−371.761	−1.41608	−0.708040	−	0.706173i	$-0.750420\pi$
$41$	−371.761	−1.41608	−0.708040	+	0.706173i	$0.750420\pi$
$42$	0	0
$43$	40.2951	0.142906	0.0714528	−	0.997444i	$-0.477236\pi$
$43$	40.2951	0.142906	0.0714528	+	0.997444i	$0.477236\pi$
$44$	−135.048	−0.462712
$45$	0	0
$46$	208.896	0.669565
$47$	−639.802	−1.98563	−0.992816	−	0.119652i	$-0.961822\pi$
$47$	−639.802	−1.98563	−0.992816	+	0.119652i	$0.961822\pi$
$48$	0	0
$49$	−340.957	−0.994044
$50$	192.436	0.544292
$51$	0	0
$52$	0	0
$53$	−102.124	−0.264676	−0.132338	−	0.991205i	$-0.542248\pi$
$53$	−102.124	−0.264676	−0.132338	+	0.991205i	$0.542248\pi$
$54$	0	0
$55$	−741.108	−1.81693
$56$	−25.5592	−0.0609908
$57$	0	0
$58$	196.791	0.445515
$59$	704.586	1.55473	0.777367	−	0.629048i	$-0.216555\pi$
$59$	704.586	1.55473	0.777367	+	0.629048i	$0.216555\pi$
$60$	0	0
$61$	−819.087	−1.71924	−0.859618	−	0.510938i	$-0.829298\pi$
$61$	−819.087	−1.71924	−0.859618	+	0.510938i	$0.829298\pi$
$62$	−481.919	−0.987158
$63$	0	0
$64$	270.764	0.528835
$65$	0	0
$66$	0	0
$67$	574.918	1.04832	0.524159	−	0.851620i	$-0.324380\pi$
$67$	574.918	1.04832	0.524159	+	0.851620i	$0.324380\pi$
$68$	283.159	0.504972
$69$	0	0
$70$	62.8283	0.107277
$71$	−365.790	−0.611427	−0.305714	−	0.952124i	$-0.598895\pi$
$71$	−365.790	−0.611427	−0.305714	+	0.952124i	$0.598895\pi$
$72$	0	0
$73$	−965.233	−1.54756	−0.773780	−	0.633454i	$-0.781637\pi$
$73$	−965.233	−1.54756	−0.773780	+	0.633454i	$0.781637\pi$
$74$	−67.7482	−0.106427
$75$	0	0
$76$	−259.698	−0.391966
$77$	−77.9945	−0.115432
$78$	0	0
$79$	580.173	0.826261	0.413130	−	0.910672i	$-0.364435\pi$
$79$	580.173	0.826261	0.413130	+	0.910672i	$0.364435\pi$
$80$	−1054.93	−1.47431
$81$	0	0
$82$	−1203.20	−1.62038
$83$	−175.372	−0.231923	−0.115962	−	0.993254i	$-0.536995\pi$
$83$	−175.372	−0.231923	−0.115962	+	0.993254i	$0.536995\pi$
$84$	0	0
$85$	1553.90	1.98287
$86$	130.415	0.163523
$87$	0	0
$88$	975.770	1.18202
$89$	20.0351	0.0238620	0.0119310	−	0.999929i	$-0.496202\pi$
$89$	20.0351	0.0238620	0.0119310	+	0.999929i	$0.496202\pi$
$90$	0	0
$91$	0	0
$92$	159.739	0.181021
$93$	0	0
$94$	−2070.72	−2.27211
$95$	−1425.15	−1.53913
$96$	0	0
$97$	1226.72	1.28406	0.642031	−	0.766678i	$-0.278092\pi$
$97$	1226.72	1.28406	0.642031	+	0.766678i	$0.278092\pi$
$98$	−1103.51	−1.13746
$99$	0	0
$100$	147.153	0.147153
$101$	57.9799	0.0571209	0.0285605	−	0.999592i	$-0.490908\pi$
$101$	57.9799	0.0571209	0.0285605	+	0.999592i	$0.490908\pi$
$102$	0	0
$103$	−954.989	−0.913571	−0.456786	−	0.889577i	$-0.650999\pi$
$103$	−954.989	−0.913571	−0.456786	+	0.889577i	$0.650999\pi$
$104$	0	0
$105$	0	0
$106$	−330.524	−0.302861
$107$	−1604.68	−1.44982	−0.724908	−	0.688846i	$-0.758118\pi$
$107$	−1604.68	−1.44982	−0.724908	+	0.688846i	$0.758118\pi$
$108$	0	0
$109$	−1534.93	−1.34880	−0.674401	−	0.738365i	$-0.735598\pi$
$109$	−1534.93	−1.34880	−0.674401	+	0.738365i	$0.735598\pi$
$110$	−2398.59	−2.07906
$111$	0	0
$112$	−111.022	−0.0936656
$113$	1789.20	1.48951	0.744753	−	0.667341i	$-0.232567\pi$
$113$	1789.20	1.48951	0.744753	+	0.667341i	$0.232567\pi$
$114$	0	0
$115$	876.604	0.710815
$116$	150.483	0.120448
$117$	0	0
$118$	2280.39	1.77904
$119$	163.533	0.125975
$120$	0	0
$121$	1646.59	1.23711
$122$	−2650.97	−1.96728
$123$	0	0
$124$	−368.516	−0.266885
$125$	−890.158	−0.636945
$126$	0	0
$127$	1648.30	1.15168	0.575838	−	0.817564i	$-0.304676\pi$
$127$	1648.30	1.15168	0.575838	+	0.817564i	$0.304676\pi$
$128$	1743.01	1.20361
$129$	0	0
$130$	0	0
$131$	−2278.88	−1.51990	−0.759948	−	0.649983i	$-0.774776\pi$
$131$	−2278.88	−1.51990	−0.759948	+	0.649983i	$0.774776\pi$
$132$	0	0
$133$	−149.983	−0.0977835
$134$	1860.72	1.19956
$135$	0	0
$136$	−2045.92	−1.28997
$137$	719.324	0.448584	0.224292	−	0.974522i	$-0.427993\pi$
$137$	719.324	0.448584	0.224292	+	0.974522i	$0.427993\pi$
$138$	0	0
$139$	1777.65	1.08474	0.542369	−	0.840140i	$-0.317527\pi$
$139$	1777.65	1.08474	0.542369	+	0.840140i	$0.317527\pi$
$140$	48.0438	0.0290032
$141$	0	0
$142$	−1183.88	−0.699640
$143$	0	0
$144$	0	0
$145$	825.807	0.472963
$146$	−3123.97	−1.77083
$147$	0	0
$148$	−51.8060	−0.0287731
$149$	−266.172	−0.146347	−0.0731734	−	0.997319i	$-0.523313\pi$
$149$	−266.172	−0.146347	−0.0731734	+	0.997319i	$0.523313\pi$
$150$	0	0
$151$	−1417.27	−0.763812	−0.381906	−	0.924201i	$-0.624732\pi$
$151$	−1417.27	−0.763812	−0.381906	+	0.924201i	$0.624732\pi$
$152$	1876.41	1.00129
$153$	0	0
$154$	−252.429	−0.132086
$155$	−2022.31	−1.04797
$156$	0	0
$157$	35.6073	0.0181005	0.00905023	−	0.999959i	$-0.497119\pi$
$157$	35.6073	0.0181005	0.00905023	+	0.999959i	$0.497119\pi$
$158$	1877.73	0.945468
$159$	0	0
$160$	−1471.37	−0.727012
$161$	92.2541	0.0451593
$162$	0	0
$163$	97.7228	0.0469585	0.0234793	−	0.999724i	$-0.492526\pi$
$163$	97.7228	0.0469585	0.0234793	+	0.999724i	$0.492526\pi$
$164$	−920.069	−0.438081
$165$	0	0
$166$	−567.592	−0.265383
$167$	−2715.91	−1.25846	−0.629231	−	0.777218i	$-0.716630\pi$
$167$	−2715.91	−1.25846	−0.629231	+	0.777218i	$0.716630\pi$
$168$	0	0
$169$	0	0
$170$	5029.19	2.26895
$171$	0	0
$172$	99.7262	0.0442096
$173$	−2074.88	−0.911852	−0.455926	−	0.890018i	$-0.650692\pi$
$173$	−2074.88	−0.911852	−0.455926	+	0.890018i	$0.650692\pi$
$174$	0	0
$175$	84.9853	0.0367102
$176$	4238.46	1.81526
$177$	0	0
$178$	64.8435	0.0273046
$179$	−1023.50	−0.427372	−0.213686	−	0.976902i	$-0.568547\pi$
$179$	−1023.50	−0.427372	−0.213686	+	0.976902i	$0.568547\pi$
$180$	0	0
$181$	1773.72	0.728395	0.364197	−	0.931322i	$-0.381343\pi$
$181$	1773.72	0.728395	0.364197	+	0.931322i	$0.381343\pi$
$182$	0	0
$183$	0	0
$184$	−1154.17	−0.462427
$185$	−284.297	−0.112983
$186$	0	0
$187$	−6243.18	−2.44143
$188$	−1583.44	−0.614279
$189$	0	0
$190$	−4612.49	−1.76119
$191$	−2251.37	−0.852896	−0.426448	−	0.904512i	$-0.640235\pi$
$191$	−2251.37	−0.852896	−0.426448	+	0.904512i	$0.640235\pi$
$192$	0	0
$193$	3876.48	1.44578	0.722889	−	0.690964i	$-0.242814\pi$
$193$	3876.48	1.44578	0.722889	+	0.690964i	$0.242814\pi$
$194$	3970.26	1.46932
$195$	0	0
$196$	−843.834	−0.307520
$197$	756.708	0.273671	0.136835	−	0.990594i	$-0.456307\pi$
$197$	756.708	0.273671	0.136835	+	0.990594i	$0.456307\pi$
$198$	0	0
$199$	−1986.32	−0.707572	−0.353786	−	0.935326i	$-0.615106\pi$
$199$	−1986.32	−0.707572	−0.353786	+	0.935326i	$0.615106\pi$
$200$	−1063.23	−0.375909
$201$	0	0
$202$	187.652	0.0653620
$203$	86.9082	0.0300481
$204$	0	0
$205$	−5049.08	−1.72021
$206$	−3090.82	−1.04538
$207$	0	0
$208$	0	0
$209$	5725.91	1.89507
$210$	0	0
$211$	−390.462	−0.127396	−0.0636980	−	0.997969i	$-0.520289\pi$
$211$	−390.462	−0.127396	−0.0636980	+	0.997969i	$0.520289\pi$
$212$	−252.746	−0.0818806
$213$	0	0
$214$	−5193.54	−1.65899
$215$	547.269	0.173597
$216$	0	0
$217$	−212.829	−0.0665795
$218$	−4967.79	−1.54340
$219$	0	0
$220$	−1834.17	−0.562088
$221$	0	0
$222$	0	0
$223$	−1814.65	−0.544922	−0.272461	−	0.962167i	$-0.587838\pi$
$223$	−1814.65	−0.544922	−0.272461	+	0.962167i	$0.587838\pi$
$224$	−154.847	−0.0461883
$225$	0	0
$226$	5790.75	1.70440
$227$	1046.66	0.306032	0.153016	−	0.988224i	$-0.451101\pi$
$227$	1046.66	0.306032	0.153016	+	0.988224i	$0.451101\pi$
$228$	0	0
$229$	6018.10	1.73662	0.868312	−	0.496018i	$-0.165205\pi$
$229$	6018.10	1.73662	0.868312	+	0.496018i	$0.165205\pi$
$230$	2837.12	0.813367
$231$	0	0
$232$	−1087.29	−0.307690
$233$	4312.29	1.21248	0.606239	−	0.795282i	$-0.292677\pi$
$233$	4312.29	1.21248	0.606239	+	0.795282i	$0.292677\pi$
$234$	0	0
$235$	−8689.50	−2.41209
$236$	1743.78	0.480976
$237$	0	0
$238$	529.273	0.144150
$239$	−3341.91	−0.904478	−0.452239	−	0.891897i	$-0.649375\pi$
$239$	−3341.91	−0.904478	−0.452239	+	0.891897i	$0.649375\pi$
$240$	0	0
$241$	1981.50	0.529625	0.264812	−	0.964300i	$-0.414690\pi$
$241$	1981.50	0.529625	0.264812	+	0.964300i	$0.414690\pi$
$242$	5329.18	1.41559
$243$	0	0
$244$	−2027.16	−0.531866
$245$	−4630.72	−1.20753
$246$	0	0
$247$	0	0
$248$	2662.65	0.681768
$249$	0	0
$250$	−2880.99	−0.728839
$251$	−484.149	−0.121750	−0.0608749	−	0.998145i	$-0.519389\pi$
$251$	−484.149	−0.121750	−0.0608749	+	0.998145i	$0.519389\pi$
$252$	0	0
$253$	−3521.98	−0.875197
$254$	5334.71	1.31783
$255$	0	0
$256$	3475.14	0.848423
$257$	−1964.83	−0.476897	−0.238448	−	0.971155i	$-0.576639\pi$
$257$	−1964.83	−0.476897	−0.238448	+	0.971155i	$0.576639\pi$
$258$	0	0
$259$	−29.9195	−0.00717801
$260$	0	0
$261$	0	0
$262$	−7375.58	−1.73918
$263$	−4886.27	−1.14563	−0.572814	−	0.819685i	$-0.694148\pi$
$263$	−4886.27	−1.14563	−0.572814	+	0.819685i	$0.694148\pi$
$264$	0	0
$265$	−1387.00	−0.321520
$266$	−485.420	−0.111891
$267$	0	0
$268$	1422.86	0.324310
$269$	2379.98	0.539443	0.269722	−	0.962938i	$-0.413068\pi$
$269$	2379.98	0.539443	0.269722	+	0.962938i	$0.413068\pi$
$270$	0	0
$271$	−5468.73	−1.22584	−0.612919	−	0.790146i	$-0.710005\pi$
$271$	−5468.73	−1.22584	−0.612919	+	0.790146i	$0.710005\pi$
$272$	−8886.89	−1.98105
$273$	0	0
$274$	2328.09	0.513303
$275$	−3244.48	−0.711452
$276$	0	0
$277$	5298.94	1.14940	0.574698	−	0.818366i	$-0.305120\pi$
$277$	5298.94	1.14940	0.574698	+	0.818366i	$0.305120\pi$
$278$	5753.37	1.24124
$279$	0	0
$280$	−347.133	−0.0740898
$281$	−6632.52	−1.40805	−0.704027	−	0.710173i	$-0.748617\pi$
$281$	−6632.52	−1.40805	−0.704027	+	0.710173i	$0.748617\pi$
$282$	0	0
$283$	−3693.82	−0.775882	−0.387941	−	0.921684i	$-0.626814\pi$
$283$	−3693.82	−0.775882	−0.387941	+	0.921684i	$0.626814\pi$
$284$	−905.294	−0.189152
$285$	0	0
$286$	0	0
$287$	−531.367	−0.109288
$288$	0	0
$289$	8177.24	1.66441
$290$	2672.72	0.541199
$291$	0	0
$292$	−2388.85	−0.478757
$293$	−3051.86	−0.608503	−0.304252	−	0.952592i	$-0.598406\pi$
$293$	−3051.86	−0.608503	−0.304252	+	0.952592i	$0.598406\pi$
$294$	0	0
$295$	9569.36	1.88864
$296$	374.316	0.0735022
$297$	0	0
$298$	−861.465	−0.167461
$299$	0	0
$300$	0	0
$301$	57.5948	0.0110289
$302$	−4586.98	−0.874010
$303$	0	0
$304$	8150.57	1.53772
$305$	−11124.5	−2.08848
$306$	0	0
$307$	4737.05	0.880643	0.440322	−	0.897840i	$-0.354864\pi$
$307$	4737.05	0.880643	0.440322	+	0.897840i	$0.354864\pi$
$308$	−193.028	−0.0357104
$309$	0	0
$310$	−6545.20	−1.19917
$311$	10095.5	1.84072	0.920359	−	0.391076i	$-0.127897\pi$
$311$	10095.5	1.84072	0.920359	+	0.391076i	$0.127897\pi$
$312$	0	0
$313$	5235.36	0.945432	0.472716	−	0.881215i	$-0.343274\pi$
$313$	5235.36	0.945432	0.472716	+	0.881215i	$0.343274\pi$
$314$	115.243	0.0207119
$315$	0	0
$316$	1435.87	0.255614
$317$	−6701.12	−1.18730	−0.593648	−	0.804725i	$-0.702313\pi$
$317$	−6701.12	−1.18730	−0.593648	+	0.804725i	$0.702313\pi$
$318$	0	0
$319$	−3317.89	−0.582339
$320$	3677.39	0.642413
$321$	0	0
$322$	298.580	0.0516746
$323$	−12005.7	−2.06815
$324$	0	0
$325$	0	0
$326$	316.279	0.0537334
$327$	0	0
$328$	6647.81	1.11910
$329$	−914.485	−0.153244
$330$	0	0
$331$	764.044	0.126875	0.0634376	−	0.997986i	$-0.479794\pi$
$331$	764.044	0.126875	0.0634376	+	0.997986i	$0.479794\pi$
$332$	−434.029	−0.0717482
$333$	0	0
$334$	−8790.02	−1.44002
$335$	7808.27	1.27347
$336$	0	0
$337$	3310.33	0.535089	0.267545	−	0.963545i	$-0.413788\pi$
$337$	3310.33	0.535089	0.267545	+	0.963545i	$0.413788\pi$
$338$	0	0
$339$	0	0
$340$	3845.74	0.613425
$341$	8125.15	1.29033
$342$	0	0
$343$	−977.598	−0.153893
$344$	−720.555	−0.112935
$345$	0	0
$346$	−6715.35	−1.04341
$347$	−869.809	−0.134564	−0.0672821	−	0.997734i	$-0.521433\pi$
$347$	−869.809	−0.134564	−0.0672821	+	0.997734i	$0.521433\pi$
$348$	0	0
$349$	−10694.9	−1.64036	−0.820181	−	0.572104i	$-0.806127\pi$
$349$	−10694.9	−1.64036	−0.820181	+	0.572104i	$0.806127\pi$
$350$	275.054	0.0420065
$351$	0	0
$352$	5911.60	0.895140
$353$	1514.92	0.228417	0.114208	−	0.993457i	$-0.463567\pi$
$353$	1514.92	0.228417	0.114208	+	0.993457i	$0.463567\pi$
$354$	0	0
$355$	−4968.00	−0.742744
$356$	49.5848	0.00738200
$357$	0	0
$358$	−3312.54	−0.489031
$359$	7006.81	1.03010	0.515049	−	0.857160i	$-0.327774\pi$
$359$	7006.81	1.03010	0.515049	+	0.857160i	$0.327774\pi$
$360$	0	0
$361$	4151.92	0.605325
$362$	5740.63	0.833483
$363$	0	0
$364$	0	0
$365$	−13109.3	−1.87993
$366$	0	0
$367$	−572.331	−0.0814045	−0.0407022	−	0.999171i	$-0.512960\pi$
$367$	−572.331	−0.0814045	−0.0407022	+	0.999171i	$0.512960\pi$
$368$	−5013.38	−0.710164
$369$	0	0
$370$	−920.125	−0.129284
$371$	−145.968	−0.0204267
$372$	0	0
$373$	2405.41	0.333907	0.166954	−	0.985965i	$-0.446607\pi$
$373$	2405.41	0.333907	0.166954	+	0.985965i	$0.446607\pi$
$374$	−20206.0	−2.79366
$375$	0	0
$376$	11440.9	1.56920
$377$	0	0
$378$	0	0
$379$	3161.53	0.428488	0.214244	−	0.976780i	$-0.431271\pi$
$379$	3161.53	0.428488	0.214244	+	0.976780i	$0.431271\pi$
$380$	−3527.10	−0.476149
$381$	0	0
$382$	−7286.53	−0.975946
$383$	−7568.71	−1.00977	−0.504887	−	0.863186i	$-0.668466\pi$
$383$	−7568.71	−1.00977	−0.504887	+	0.863186i	$0.668466\pi$
$384$	0	0
$385$	−1059.28	−0.140224
$386$	12546.2	1.65437
$387$	0	0
$388$	3036.00	0.397240
$389$	6757.17	0.880726	0.440363	−	0.897820i	$-0.354850\pi$
$389$	6757.17	0.880726	0.440363	+	0.897820i	$0.354850\pi$
$390$	0	0
$391$	7384.62	0.955131
$392$	6096.98	0.785572
$393$	0	0
$394$	2449.08	0.313154
$395$	7879.65	1.00372
$396$	0	0
$397$	7313.92	0.924623	0.462312	−	0.886718i	$-0.347020\pi$
$397$	7313.92	0.924623	0.462312	+	0.886718i	$0.347020\pi$
$398$	−6428.73	−0.809656
$399$	0	0
$400$	−4618.37	−0.577296
$401$	5349.32	0.666165	0.333082	−	0.942898i	$-0.391911\pi$
$401$	5349.32	0.666165	0.333082	+	0.942898i	$0.391911\pi$
$402$	0	0
$403$	0	0
$404$	143.494	0.0176711
$405$	0	0
$406$	281.278	0.0343832
$407$	1142.23	0.139112
$408$	0	0
$409$	13444.2	1.62537	0.812683	−	0.582706i	$-0.198006\pi$
$409$	13444.2	1.62537	0.812683	+	0.582706i	$0.198006\pi$
$410$	−16341.3	−1.96839
$411$	0	0
$412$	−2363.50	−0.282624
$413$	1007.08	0.119989
$414$	0	0
$415$	−2381.83	−0.281733
$416$	0	0
$417$	0	0
$418$	18531.9	2.16848
$419$	8313.65	0.969328	0.484664	−	0.874700i	$-0.338942\pi$
$419$	8313.65	0.969328	0.484664	+	0.874700i	$0.338942\pi$
$420$	0	0
$421$	3491.02	0.404138	0.202069	−	0.979371i	$-0.435234\pi$
$421$	3491.02	0.404138	0.202069	+	0.979371i	$0.435234\pi$
$422$	−1263.73	−0.145776
$423$	0	0
$424$	1826.18	0.209167
$425$	6802.77	0.776431
$426$	0	0
$427$	−1170.74	−0.132684
$428$	−3971.42	−0.448518
$429$	0	0
$430$	1771.23	0.198643
$431$	−1269.80	−0.141912	−0.0709560	−	0.997479i	$-0.522605\pi$
$431$	−1269.80	−0.141912	−0.0709560	+	0.997479i	$0.522605\pi$
$432$	0	0
$433$	6157.71	0.683419	0.341710	−	0.939806i	$-0.388994\pi$
$433$	6157.71	0.683419	0.341710	+	0.939806i	$0.388994\pi$
$434$	−688.819	−0.0761852
$435$	0	0
$436$	−3798.79	−0.417268
$437$	−6772.77	−0.741386
$438$	0	0
$439$	−2466.57	−0.268162	−0.134081	−	0.990970i	$-0.542808\pi$
$439$	−2466.57	−0.268162	−0.134081	+	0.990970i	$0.542808\pi$
$440$	13252.5	1.43588
$441$	0	0
$442$	0	0
$443$	1858.98	0.199374	0.0996871	−	0.995019i	$-0.468216\pi$
$443$	1858.98	0.199374	0.0996871	+	0.995019i	$0.468216\pi$
$444$	0	0
$445$	272.108	0.0289868
$446$	−5873.09	−0.623540
$447$	0	0
$448$	387.010	0.0408136
$449$	5176.22	0.544055	0.272028	−	0.962289i	$-0.412306\pi$
$449$	5176.22	0.544055	0.272028	+	0.962289i	$0.412306\pi$
$450$	0	0
$451$	20286.0	2.11802
$452$	4428.09	0.460797
$453$	0	0
$454$	3387.51	0.350184
$455$	0	0
$456$	0	0
$457$	3327.37	0.340586	0.170293	−	0.985393i	$-0.445529\pi$
$457$	3327.37	0.340586	0.170293	+	0.985393i	$0.445529\pi$
$458$	19477.5	1.98717
$459$	0	0
$460$	2169.50	0.219899
$461$	8992.19	0.908477	0.454238	−	0.890880i	$-0.349911\pi$
$461$	8992.19	0.908477	0.454238	+	0.890880i	$0.349911\pi$
$462$	0	0
$463$	13143.4	1.31928	0.659640	−	0.751582i	$-0.270709\pi$
$463$	13143.4	1.31928	0.659640	+	0.751582i	$0.270709\pi$
$464$	−4722.87	−0.472529
$465$	0	0
$466$	13956.7	1.38741
$467$	−7798.23	−0.772718	−0.386359	−	0.922349i	$-0.626267\pi$
$467$	−7798.23	−0.772718	−0.386359	+	0.922349i	$0.626267\pi$
$468$	0	0
$469$	821.745	0.0809054
$470$	−28123.5	−2.76009
$471$	0	0
$472$	−12599.4	−1.22867
$473$	−2198.79	−0.213743
$474$	0	0
$475$	−6239.13	−0.602676
$476$	404.727	0.0389719
$477$	0	0
$478$	−10816.1	−1.03497
$479$	7341.94	0.700338	0.350169	−	0.936687i	$-0.386124\pi$
$479$	7341.94	0.700338	0.350169	+	0.936687i	$0.386124\pi$
$480$	0	0
$481$	0	0
$482$	6413.11	0.606036
$483$	0	0
$484$	4075.14	0.382714
$485$	16660.7	1.55984
$486$	0	0
$487$	2684.77	0.249812	0.124906	−	0.992169i	$-0.460137\pi$
$487$	2684.77	0.249812	0.124906	+	0.992169i	$0.460137\pi$
$488$	14646.9	1.35867
$489$	0	0
$490$	−14987.3	−1.38175
$491$	9212.74	0.846772	0.423386	−	0.905949i	$-0.360841\pi$
$491$	9212.74	0.846772	0.423386	+	0.905949i	$0.360841\pi$
$492$	0	0
$493$	6956.70	0.635526
$494$	0	0
$495$	0	0
$496$	11565.8	1.04701
$497$	−522.834	−0.0471877
$498$	0	0
$499$	11401.9	1.02289	0.511443	−	0.859317i	$-0.329111\pi$
$499$	11401.9	1.02289	0.511443	+	0.859317i	$0.329111\pi$
$500$	−2203.05	−0.197047
$501$	0	0
$502$	−1566.94	−0.139315
$503$	−1603.20	−0.142114	−0.0710570	−	0.997472i	$-0.522637\pi$
$503$	−1603.20	−0.142114	−0.0710570	+	0.997472i	$0.522637\pi$
$504$	0	0
$505$	787.456	0.0693888
$506$	−11398.9	−1.00147
$507$	0	0
$508$	4079.37	0.356285
$509$	−3427.55	−0.298475	−0.149237	−	0.988801i	$-0.547682\pi$
$509$	−3427.55	−0.298475	−0.149237	+	0.988801i	$0.547682\pi$
$510$	0	0
$511$	−1379.63	−0.119435
$512$	−2696.83	−0.232781
$513$	0	0
$514$	−6359.15	−0.545701
$515$	−12970.2	−1.10978
$516$	0	0
$517$	34912.3	2.96990
$518$	−96.8342	−0.00821361
$519$	0	0
$520$	0	0
$521$	5244.77	0.441032	0.220516	−	0.975383i	$-0.429226\pi$
$521$	5244.77	0.441032	0.220516	+	0.975383i	$0.429226\pi$
$522$	0	0
$523$	−15817.1	−1.32243	−0.661216	−	0.750196i	$-0.729959\pi$
$523$	−15817.1	−1.32243	−0.661216	+	0.750196i	$0.729959\pi$
$524$	−5639.99	−0.470198
$525$	0	0
$526$	−15814.4	−1.31091
$527$	−17036.2	−1.40818
$528$	0	0
$529$	−8001.10	−0.657607
$530$	−4489.02	−0.367907
$531$	0	0
$532$	−371.193	−0.0302505
$533$	0	0
$534$	0	0
$535$	−21794.0	−1.76119
$536$	−10280.7	−0.828464
$537$	0	0
$538$	7702.81	0.617271
$539$	18605.1	1.48679
$540$	0	0
$541$	5694.69	0.452558	0.226279	−	0.974063i	$-0.427344\pi$
$541$	5694.69	0.452558	0.226279	+	0.974063i	$0.427344\pi$
$542$	−17699.5	−1.40269
$543$	0	0
$544$	−12395.0	−0.976895
$545$	−20846.7	−1.63848
$546$	0	0
$547$	−20598.9	−1.61014	−0.805070	−	0.593180i	$-0.797872\pi$
$547$	−20598.9	−1.61014	−0.805070	+	0.593180i	$0.797872\pi$
$548$	1780.25	0.138775
$549$	0	0
$550$	−10500.7	−0.814096
$551$	−6380.31	−0.493304
$552$	0	0
$553$	829.257	0.0637678
$554$	17150.0	1.31522
$555$	0	0
$556$	4399.51	0.335577
$557$	−18421.7	−1.40135	−0.700674	−	0.713482i	$-0.747117\pi$
$557$	−18421.7	−1.40135	−0.700674	+	0.713482i	$0.747117\pi$
$558$	0	0
$559$	0	0
$560$	−1507.84	−0.113782
$561$	0	0
$562$	−21466.1	−1.61120
$563$	6516.79	0.487833	0.243916	−	0.969796i	$-0.421568\pi$
$563$	6516.79	0.487833	0.243916	+	0.969796i	$0.421568\pi$
$564$	0	0
$565$	24300.1	1.80941
$566$	−11955.0	−0.887822
$567$	0	0
$568$	6541.05	0.483198
$569$	24233.0	1.78541	0.892706	−	0.450640i	$-0.148804\pi$
$569$	24233.0	1.78541	0.892706	+	0.450640i	$0.148804\pi$
$570$	0	0
$571$	11060.7	0.810640	0.405320	−	0.914175i	$-0.367160\pi$
$571$	11060.7	0.810640	0.405320	+	0.914175i	$0.367160\pi$
$572$	0	0
$573$	0	0
$574$	−1719.77	−0.125055
$575$	3837.66	0.278333
$576$	0	0
$577$	−19118.6	−1.37941	−0.689703	−	0.724092i	$-0.742259\pi$
$577$	−19118.6	−1.37941	−0.689703	+	0.724092i	$0.742259\pi$
$578$	26465.6	1.90454
$579$	0	0
$580$	2043.79	0.146317
$581$	−250.664	−0.0178990
$582$	0	0
$583$	5572.63	0.395874
$584$	17260.3	1.22300
$585$	0	0
$586$	−9877.32	−0.696294
$587$	21636.2	1.52133	0.760667	−	0.649142i	$-0.224872\pi$
$587$	21636.2	1.52133	0.760667	+	0.649142i	$0.224872\pi$
$588$	0	0
$589$	15624.7	1.09305
$590$	30971.2	2.16112
$591$	0	0
$592$	1625.92	0.112880
$593$	−2148.58	−0.148789	−0.0743943	−	0.997229i	$-0.523702\pi$
$593$	−2148.58	−0.148789	−0.0743943	+	0.997229i	$0.523702\pi$
$594$	0	0
$595$	2221.03	0.153031
$596$	−658.749	−0.0452742
$597$	0	0
$598$	0	0
$599$	−27957.0	−1.90700	−0.953499	−	0.301397i	$-0.902547\pi$
$599$	−27957.0	−1.90700	−0.953499	+	0.301397i	$0.902547\pi$
$600$	0	0
$601$	−13217.8	−0.897111	−0.448555	−	0.893755i	$-0.648061\pi$
$601$	−13217.8	−0.897111	−0.448555	+	0.893755i	$0.648061\pi$
$602$	186.405	0.0126201
$603$	0	0
$604$	−3507.59	−0.236295
$605$	22363.2	1.50280
$606$	0	0
$607$	−23642.4	−1.58091	−0.790457	−	0.612518i	$-0.790157\pi$
$607$	−23642.4	−1.58091	−0.790457	+	0.612518i	$0.790157\pi$
$608$	11368.0	0.758279
$609$	0	0
$610$	−36004.3	−2.38979
$611$	0	0
$612$	0	0
$613$	−8117.53	−0.534851	−0.267426	−	0.963578i	$-0.586173\pi$
$613$	−8117.53	−0.534851	−0.267426	+	0.963578i	$0.586173\pi$
$614$	15331.4	1.00770
$615$	0	0
$616$	1394.69	0.0912237
$617$	−20928.0	−1.36553	−0.682764	−	0.730639i	$-0.739222\pi$
$617$	−20928.0	−1.36553	−0.682764	+	0.730639i	$0.739222\pi$
$618$	0	0
$619$	−3529.43	−0.229175	−0.114588	−	0.993413i	$-0.536555\pi$
$619$	−3529.43	−0.229175	−0.114588	+	0.993413i	$0.536555\pi$
$620$	−5005.01	−0.324204
$621$	0	0
$622$	32674.0	2.10628
$623$	28.6367	0.00184158
$624$	0	0
$625$	−19522.0	−1.24941
$626$	16944.2	1.08183
$627$	0	0
$628$	88.1244	0.00559959
$629$	−2394.95	−0.151817
$630$	0	0
$631$	10466.2	0.660307	0.330154	−	0.943927i	$-0.392899\pi$
$631$	10466.2	0.660307	0.330154	+	0.943927i	$0.392899\pi$
$632$	−10374.6	−0.652976
$633$	0	0
$634$	−21688.1	−1.35859
$635$	22386.4	1.39902
$636$	0	0
$637$	0	0
$638$	−10738.3	−0.666356
$639$	0	0
$640$	23672.8	1.46211
$641$	26594.1	1.63870	0.819349	−	0.573295i	$-0.194335\pi$
$641$	26594.1	1.63870	0.819349	+	0.573295i	$0.194335\pi$
$642$	0	0
$643$	−6965.66	−0.427214	−0.213607	−	0.976920i	$-0.568521\pi$
$643$	−6965.66	−0.427214	−0.213607	+	0.976920i	$0.568521\pi$
$644$	228.319	0.0139706
$645$	0	0
$646$	−38856.2	−2.36653
$647$	−3956.40	−0.240405	−0.120203	−	0.992749i	$-0.538354\pi$
$647$	−3956.40	−0.240405	−0.120203	+	0.992749i	$0.538354\pi$
$648$	0	0
$649$	−38447.3	−2.32541
$650$	0	0
$651$	0	0
$652$	241.854	0.0145272
$653$	−24311.1	−1.45691	−0.728457	−	0.685091i	$-0.759762\pi$
$653$	−24311.1	−1.45691	−0.728457	+	0.685091i	$0.759762\pi$
$654$	0	0
$655$	−30950.7	−1.84632
$656$	28876.2	1.71864
$657$	0	0
$658$	−2959.73	−0.175353
$659$	−8891.33	−0.525580	−0.262790	−	0.964853i	$-0.584643\pi$
$659$	−8891.33	−0.525580	−0.262790	+	0.964853i	$0.584643\pi$
$660$	0	0
$661$	994.333	0.0585099	0.0292550	−	0.999572i	$-0.490687\pi$
$661$	994.333	0.0585099	0.0292550	+	0.999572i	$0.490687\pi$
$662$	2472.82	0.145180
$663$	0	0
$664$	3136.00	0.183284
$665$	−2037.01	−0.118785
$666$	0	0
$667$	3924.50	0.227822
$668$	−6721.59	−0.389321
$669$	0	0
$670$	25271.4	1.45719
$671$	44695.4	2.57145
$672$	0	0
$673$	5228.19	0.299453	0.149726	−	0.988727i	$-0.452161\pi$
$673$	5228.19	0.299453	0.149726	+	0.988727i	$0.452161\pi$
$674$	10713.9	0.612288
$675$	0	0
$676$	0	0
$677$	10208.1	0.579512	0.289756	−	0.957101i	$-0.406426\pi$
$677$	10208.1	0.579512	0.289756	+	0.957101i	$0.406426\pi$
$678$	0	0
$679$	1753.38	0.0990993
$680$	−27786.8	−1.56702
$681$	0	0
$682$	26297.0	1.47649
$683$	−15514.5	−0.869172	−0.434586	−	0.900630i	$-0.643105\pi$
$683$	−15514.5	−0.869172	−0.434586	+	0.900630i	$0.643105\pi$
$684$	0	0
$685$	9769.53	0.544927
$686$	−3163.99	−0.176096
$687$	0	0
$688$	−3129.88	−0.173438
$689$	0	0
$690$	0	0
$691$	−17826.4	−0.981403	−0.490701	−	0.871328i	$-0.663259\pi$
$691$	−17826.4	−0.981403	−0.490701	+	0.871328i	$0.663259\pi$
$692$	−5135.12	−0.282092
$693$	0	0
$694$	−2815.13	−0.153978
$695$	24143.3	1.31771
$696$	0	0
$697$	−42534.1	−2.31147
$698$	−34614.1	−1.87702
$699$	0	0
$700$	210.330	0.0113567
$701$	−9266.46	−0.499271	−0.249636	−	0.968340i	$-0.580311\pi$
$701$	−9266.46	−0.499271	−0.249636	+	0.968340i	$0.580311\pi$
$702$	0	0
$703$	2196.52	0.117842
$704$	−14774.8	−0.790977
$705$	0	0
$706$	4903.04	0.261371
$707$	82.8722	0.00440839
$708$	0	0
$709$	−5844.49	−0.309583	−0.154792	−	0.987947i	$-0.549471\pi$
$709$	−5844.49	−0.309583	−0.154792	+	0.987947i	$0.549471\pi$
$710$	−16078.9	−0.849902
$711$	0	0
$712$	−358.267	−0.0188576
$713$	−9610.67	−0.504800
$714$	0	0
$715$	0	0
$716$	−2533.04	−0.132213
$717$	0	0
$718$	22677.5	1.17872
$719$	−11776.3	−0.610825	−0.305412	−	0.952220i	$-0.598794\pi$
$719$	−11776.3	−0.610825	−0.305412	+	0.952220i	$0.598794\pi$
$720$	0	0
$721$	−1364.99	−0.0705061
$722$	13437.7	0.692658
$723$	0	0
$724$	4389.77	0.225338
$725$	3615.28	0.185197
$726$	0	0
$727$	26770.6	1.36570	0.682851	−	0.730558i	$-0.260740\pi$
$727$	26770.6	1.36570	0.682851	+	0.730558i	$0.260740\pi$
$728$	0	0
$729$	0	0
$730$	−42428.3	−2.15115
$731$	4610.26	0.233265
$732$	0	0
$733$	21653.8	1.09113	0.545567	−	0.838067i	$-0.316314\pi$
$733$	21653.8	1.09113	0.545567	+	0.838067i	$0.316314\pi$
$734$	−1852.35	−0.0931490
$735$	0	0
$736$	−6992.41	−0.350195
$737$	−31371.7	−1.56797
$738$	0	0
$739$	9986.74	0.497115	0.248558	−	0.968617i	$-0.420043\pi$
$739$	9986.74	0.497115	0.248558	+	0.968617i	$0.420043\pi$
$740$	−703.605	−0.0349527
$741$	0	0
$742$	−472.426	−0.0233737
$743$	7505.34	0.370584	0.185292	−	0.982683i	$-0.440677\pi$
$743$	7505.34	0.370584	0.185292	+	0.982683i	$0.440677\pi$
$744$	0	0
$745$	−3615.03	−0.177778
$746$	7785.10	0.382081
$747$	0	0
$748$	−15451.2	−0.755285
$749$	−2293.61	−0.111891
$750$	0	0
$751$	13441.9	0.653134	0.326567	−	0.945174i	$-0.394108\pi$
$751$	13441.9	0.653134	0.326567	+	0.945174i	$0.394108\pi$
$752$	49696.0	2.40988
$753$	0	0
$754$	0	0
$755$	−19248.7	−0.927856
$756$	0	0
$757$	32951.1	1.58207	0.791035	−	0.611771i	$-0.209543\pi$
$757$	32951.1	1.58207	0.791035	+	0.611771i	$0.209543\pi$
$758$	10232.3	0.490308
$759$	0	0
$760$	25484.5	1.21634
$761$	−9866.20	−0.469973	−0.234986	−	0.971999i	$-0.575505\pi$
$761$	−9866.20	−0.469973	−0.234986	+	0.971999i	$0.575505\pi$
$762$	0	0
$763$	−2193.91	−0.104096
$764$	−5571.90	−0.263854
$765$	0	0
$766$	−24496.1	−1.15546
$767$	0	0
$768$	0	0
$769$	31948.6	1.49818	0.749088	−	0.662471i	$-0.230492\pi$
$769$	31948.6	1.49818	0.749088	+	0.662471i	$0.230492\pi$
$770$	−3428.37	−0.160454
$771$	0	0
$772$	9593.89	0.447269
$773$	10227.7	0.475893	0.237946	−	0.971278i	$-0.423526\pi$
$773$	10227.7	0.475893	0.237946	+	0.971278i	$0.423526\pi$
$774$	0	0
$775$	−8853.43	−0.410354
$776$	−21936.1	−1.01477
$777$	0	0
$778$	21869.6	1.00779
$779$	39009.9	1.79419
$780$	0	0
$781$	19960.2	0.914510
$782$	23900.3	1.09293
$783$	0	0
$784$	26483.5	1.20643
$785$	483.602	0.0219879
$786$	0	0
$787$	18651.6	0.844802	0.422401	−	0.906409i	$-0.361187\pi$
$787$	18651.6	0.844802	0.422401	+	0.906409i	$0.361187\pi$
$788$	1872.77	0.0846634
$789$	0	0
$790$	25502.4	1.14853
$791$	2557.35	0.114955
$792$	0	0
$793$	0	0
$794$	23671.5	1.05802
$795$	0	0
$796$	−4915.95	−0.218896
$797$	−32566.8	−1.44740	−0.723698	−	0.690117i	$-0.757559\pi$
$797$	−32566.8	−1.44740	−0.723698	+	0.690117i	$0.757559\pi$
$798$	0	0
$799$	−73201.4	−3.24115
$800$	−6441.47	−0.284675
$801$	0	0
$802$	17313.0	0.762275
$803$	52670.1	2.31468
$804$	0	0
$805$	1252.95	0.0548581
$806$	0	0
$807$	0	0
$808$	−1036.79	−0.0451415
$809$	−23895.3	−1.03846	−0.519231	−	0.854634i	$-0.673781\pi$
$809$	−23895.3	−1.03846	−0.519231	+	0.854634i	$0.673781\pi$
$810$	0	0
$811$	−219.216	−0.00949165	−0.00474582	−	0.999989i	$-0.501511\pi$
$811$	−219.216	−0.00949165	−0.00474582	+	0.999989i	$0.501511\pi$
$812$	215.089	0.00929574
$813$	0	0
$814$	3696.83	0.159182
$815$	1327.23	0.0570438
$816$	0	0
$817$	−4228.28	−0.181063
$818$	43512.2	1.85986
$819$	0	0
$820$	−12496.0	−0.532168
$821$	7550.84	0.320982	0.160491	−	0.987037i	$-0.448692\pi$
$821$	7550.84	0.320982	0.160491	+	0.987037i	$0.448692\pi$
$822$	0	0
$823$	−14082.8	−0.596471	−0.298235	−	0.954492i	$-0.596398\pi$
$823$	−14082.8	−0.596471	−0.298235	+	0.954492i	$0.596398\pi$
$824$	17077.1	0.721976
$825$	0	0
$826$	3259.42	0.137300
$827$	5424.18	0.228074	0.114037	−	0.993477i	$-0.463622\pi$
$827$	5424.18	0.228074	0.114037	+	0.993477i	$0.463622\pi$
$828$	0	0
$829$	26888.3	1.12650	0.563251	−	0.826286i	$-0.309550\pi$
$829$	26888.3	1.12650	0.563251	+	0.826286i	$0.309550\pi$
$830$	−7708.77	−0.322380
$831$	0	0
$832$	0	0
$833$	−39009.8	−1.62258
$834$	0	0
$835$	−36886.2	−1.52874
$836$	14171.0	0.586262
$837$	0	0
$838$	26907.1	1.10918
$839$	−23680.9	−0.974441	−0.487220	−	0.873279i	$-0.661989\pi$
$839$	−23680.9	−0.974441	−0.487220	+	0.873279i	$0.661989\pi$
$840$	0	0
$841$	−20691.9	−0.848412
$842$	11298.7	0.462444
$843$	0	0
$844$	−966.354	−0.0394115
$845$	0	0
$846$	0	0
$847$	2353.51	0.0954754
$848$	7932.38	0.321225
$849$	0	0
$850$	22017.1	0.888449
$851$	−1351.07	−0.0544230
$852$	0	0
$853$	−9653.82	−0.387504	−0.193752	−	0.981051i	$-0.562066\pi$
$853$	−9653.82	−0.387504	−0.193752	+	0.981051i	$0.562066\pi$
$854$	−3789.10	−0.151827
$855$	0	0
$856$	28694.8	1.14576
$857$	−41438.8	−1.65172	−0.825859	−	0.563877i	$-0.809309\pi$
$857$	−41438.8	−1.65172	−0.825859	+	0.563877i	$0.809309\pi$
$858$	0	0
$859$	−20696.9	−0.822085	−0.411042	−	0.911616i	$-0.634835\pi$
$859$	−20696.9	−0.822085	−0.411042	+	0.911616i	$0.634835\pi$
$860$	1354.43	0.0537045
$861$	0	0
$862$	−4109.70	−0.162386
$863$	−19513.8	−0.769707	−0.384854	−	0.922978i	$-0.625748\pi$
$863$	−19513.8	−0.769707	−0.384854	+	0.922978i	$0.625748\pi$
$864$	0	0
$865$	−28180.1	−1.10769
$866$	19929.4	0.782018
$867$	0	0
$868$	−526.729	−0.0205972
$869$	−31658.5	−1.23583
$870$	0	0
$871$	0	0
$872$	27447.5	1.06593
$873$	0	0
$874$	−21920.0	−0.848348
$875$	−1272.33	−0.0491571
$876$	0	0
$877$	35595.5	1.37055	0.685277	−	0.728282i	$-0.259681\pi$
$877$	35595.5	1.37055	0.685277	+	0.728282i	$0.259681\pi$
$878$	−7983.05	−0.306851
$879$	0	0
$880$	57564.9	2.20513
$881$	−4277.22	−0.163568	−0.0817838	−	0.996650i	$-0.526062\pi$
$881$	−4277.22	−0.163568	−0.0817838	+	0.996650i	$0.526062\pi$
$882$	0	0
$883$	355.428	0.0135460	0.00677300	−	0.999977i	$-0.497844\pi$
$883$	355.428	0.0135460	0.00677300	+	0.999977i	$0.497844\pi$
$884$	0	0
$885$	0	0
$886$	6016.58	0.228139
$887$	7133.87	0.270047	0.135024	−	0.990842i	$-0.456889\pi$
$887$	7133.87	0.270047	0.135024	+	0.990842i	$0.456889\pi$
$888$	0	0
$889$	2355.96	0.0888821
$890$	880.675	0.0331689
$891$	0	0
$892$	−4491.06	−0.168578
$893$	67136.3	2.51582
$894$	0	0
$895$	−13900.6	−0.519159
$896$	2491.33	0.0928902
$897$	0	0
$898$	16752.8	0.622548
$899$	−9053.76	−0.335884
$900$	0	0
$901$	−11684.3	−0.432030
$902$	65655.4	2.42360
$903$	0	0
$904$	−31994.5	−1.17712
$905$	24089.8	0.884832
$906$	0	0
$907$	−21719.2	−0.795122	−0.397561	−	0.917576i	$-0.630143\pi$
$907$	−21719.2	−0.795122	−0.397561	+	0.917576i	$0.630143\pi$
$908$	2590.38	0.0946747
$909$	0	0
$910$	0	0
$911$	39331.5	1.43042	0.715209	−	0.698911i	$-0.246331\pi$
$911$	39331.5	1.43042	0.715209	+	0.698911i	$0.246331\pi$
$912$	0	0
$913$	9569.59	0.346886
$914$	10769.0	0.389723
$915$	0	0
$916$	14894.2	0.537246
$917$	−3257.26	−0.117300
$918$	0	0
$919$	−33117.0	−1.18871	−0.594357	−	0.804202i	$-0.702593\pi$
$919$	−33117.0	−1.18871	−0.594357	+	0.804202i	$0.702593\pi$
$920$	−15675.4	−0.561742
$921$	0	0
$922$	29103.2	1.03955
$923$	0	0
$924$	0	0
$925$	−1244.61	−0.0442407
$926$	42538.6	1.50962
$927$	0	0
$928$	−6587.22	−0.233013
$929$	39917.7	1.40975	0.704874	−	0.709332i	$-0.251003\pi$
$929$	39917.7	1.40975	0.704874	+	0.709332i	$0.251003\pi$
$930$	0	0
$931$	35777.6	1.25947
$932$	10672.5	0.375095
$933$	0	0
$934$	−25238.9	−0.884201
$935$	−84792.1	−2.96577
$936$	0	0
$937$	−37219.5	−1.29766	−0.648830	−	0.760934i	$-0.724741\pi$
$937$	−37219.5	−1.29766	−0.648830	+	0.760934i	$0.724741\pi$
$938$	2659.57	0.0925779
$939$	0	0
$940$	−21505.6	−0.746208
$941$	−53360.4	−1.84856	−0.924282	−	0.381710i	$-0.875335\pi$
$941$	−53360.4	−1.84856	−0.924282	+	0.381710i	$0.875335\pi$
$942$	0	0
$943$	−23994.8	−0.828610
$944$	−54728.0	−1.88691
$945$	0	0
$946$	−7116.38	−0.244581
$947$	−29333.8	−1.00657	−0.503284	−	0.864121i	$-0.667875\pi$
$947$	−29333.8	−1.00657	−0.503284	+	0.864121i	$0.667875\pi$
$948$	0	0
$949$	0	0
$950$	−20192.9	−0.689626
$951$	0	0
$952$	−2924.29	−0.0995554
$953$	18645.0	0.633759	0.316879	−	0.948466i	$-0.397365\pi$
$953$	18645.0	0.633759	0.316879	+	0.948466i	$0.397365\pi$
$954$	0	0
$955$	−30577.0	−1.03607
$956$	−8270.88	−0.279811
$957$	0	0
$958$	23762.2	0.801378
$959$	1028.15	0.0346201
$960$	0	0
$961$	−7619.34	−0.255760
$962$	0	0
$963$	0	0
$964$	4904.01	0.163846
$965$	52648.6	1.75629
$966$	0	0
$967$	−43847.6	−1.45816	−0.729081	−	0.684427i	$-0.760052\pi$
$967$	−43847.6	−1.45816	−0.729081	+	0.684427i	$0.760052\pi$
$968$	−29444.3	−0.977659
$969$	0	0
$970$	53922.2	1.78489
$971$	−11602.4	−0.383459	−0.191729	−	0.981448i	$-0.561410\pi$
$971$	−11602.4	−0.383459	−0.191729	+	0.981448i	$0.561410\pi$
$972$	0	0
$973$	2540.85	0.0837161
$974$	8689.24	0.285853
$975$	0	0
$976$	63621.8	2.08656
$977$	−11324.2	−0.370823	−0.185411	−	0.982661i	$-0.559362\pi$
$977$	−11324.2	−0.370823	−0.185411	+	0.982661i	$0.559362\pi$
$978$	0	0
$979$	−1093.26	−0.0356903
$980$	−11460.6	−0.373565
$981$	0	0
$982$	29817.0	0.968939
$983$	−47443.8	−1.53939	−0.769696	−	0.638410i	$-0.779592\pi$
$983$	−47443.8	−1.53939	−0.769696	+	0.638410i	$0.779592\pi$
$984$	0	0
$985$	10277.3	0.332447
$986$	22515.3	0.727215
$987$	0	0
$988$	0	0
$989$	2600.80	0.0836203
$990$	0	0
$991$	51752.6	1.65891	0.829453	−	0.558577i	$-0.188653\pi$
$991$	51752.6	1.65891	0.829453	+	0.558577i	$0.188653\pi$
$992$	16131.4	0.516303
$993$	0	0
$994$	−1692.15	−0.0539957
$995$	−26977.3	−0.859537
$996$	0	0
$997$	16949.8	0.538420	0.269210	−	0.963082i	$-0.413237\pi$
$997$	16949.8	0.538420	0.269210	+	0.963082i	$0.413237\pi$
$998$	36902.3	1.17046
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.be.1.8		9
3.2	odd	2		507.4.a.q.1.2	yes	9
13.12	even	2		1521.4.a.bj.1.2		9
39.5	even	4		507.4.b.j.337.14		18
39.8	even	4		507.4.b.j.337.5		18
39.38	odd	2		507.4.a.n.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.8	✓	9	39.38	odd	2
507.4.a.q.1.2	yes	9	3.2	odd	2
507.4.b.j.337.5		18	39.8	even	4
507.4.b.j.337.14		18	39.5	even	4
1521.4.a.be.1.8		9	1.1	even	1	trivial
1521.4.a.bj.1.2		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)

\(f(q)\)	\(=\)	\(q+3.23649 q^{2} +2.47490 q^{4} +13.5815 q^{5} +1.42933 q^{7} -17.8820 q^{8} +O(q^{10})\)	\(q+3.23649 q^{2} +2.47490 q^{4} +13.5815 q^{5} +1.42933 q^{7} -17.8820 q^{8} +43.9566 q^{10} -54.5673 q^{11} +4.62601 q^{14} -77.6741 q^{16} +114.413 q^{17} -104.933 q^{19} +33.6129 q^{20} -176.607 q^{22} +64.5438 q^{23} +59.4583 q^{25} +3.53743 q^{28} +60.8037 q^{29} -148.902 q^{31} -108.336 q^{32} +370.296 q^{34} +19.4125 q^{35} -20.9326 q^{37} -339.615 q^{38} -242.865 q^{40} -371.761 q^{41} +40.2951 q^{43} -135.048 q^{44} +208.896 q^{46} -639.802 q^{47} -340.957 q^{49} +192.436 q^{50} -102.124 q^{53} -741.108 q^{55} -25.5592 q^{56} +196.791 q^{58} +704.586 q^{59} -819.087 q^{61} -481.919 q^{62} +270.764 q^{64} +574.918 q^{67} +283.159 q^{68} +62.8283 q^{70} -365.790 q^{71} -965.233 q^{73} -67.7482 q^{74} -259.698 q^{76} -77.9945 q^{77} +580.173 q^{79} -1054.93 q^{80} -1203.20 q^{82} -175.372 q^{83} +1553.90 q^{85} +130.415 q^{86} +975.770 q^{88} +20.0351 q^{89} +159.739 q^{92} -2070.72 q^{94} -1425.15 q^{95} +1226.72 q^{97} -1103.51 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8}+O(q^{10}) \) 9 * q - 8 * q^2 + 32 * q^4 - 41 * q^5 + q^7 - 111 * q^8	\( 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8} + 198 q^{10} - 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} - 72 q^{19} - 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} + 132 q^{28} + 547 q^{29} - 521 q^{31} - 721 q^{32} - 100 q^{34} - 138 q^{35} + 584 q^{37} + 416 q^{38} + 1342 q^{40} - 482 q^{41} + 158 q^{43} - 1453 q^{44} + 1537 q^{46} - 1500 q^{47} + 642 q^{49} - 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} + 1455 q^{58} - 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} + 252 q^{67} + 1579 q^{68} + 2492 q^{70} - 2352 q^{71} + 903 q^{73} - 1037 q^{74} - 485 q^{76} + 1686 q^{77} - 115 q^{79} - 5701 q^{80} - 5147 q^{82} - 1207 q^{83} + 4308 q^{85} - 5691 q^{86} - 484 q^{88} - 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} + 2155 q^{97} - 5593 q^{98}+O(q^{100}) \) 9 * q - 8 * q^2 + 32 * q^4 - 41 * q^5 + q^7 - 111 * q^8 + 198 * q^10 - 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 - 72 * q^19 - 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 + 132 * q^28 + 547 * q^29 - 521 * q^31 - 721 * q^32 - 100 * q^34 - 138 * q^35 + 584 * q^37 + 416 * q^38 + 1342 * q^40 - 482 * q^41 + 158 * q^43 - 1453 * q^44 + 1537 * q^46 - 1500 * q^47 + 642 * q^49 - 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 + 1455 * q^58 - 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 + 252 * q^67 + 1579 * q^68 + 2492 * q^70 - 2352 * q^71 + 903 * q^73 - 1037 * q^74 - 485 * q^76 + 1686 * q^77 - 115 * q^79 - 5701 * q^80 - 5147 * q^82 - 1207 * q^83 + 4308 * q^85 - 5691 * q^86 - 484 * q^88 - 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 + 2155 * q^97 - 5593 * q^98

Introduction

L-functions

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