LMFDB - Embedded newform 1521.4.a.bj.1.7

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} - 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			1.7
Root			$-2.37739$ 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)

$f(q)$	$=$	$q+3.37739 q^{2} +3.40677 q^{4} -15.7127 q^{5} +17.1681 q^{7} -15.5131 q^{8} +O(q^{10})$	$q+3.37739 q^{2} +3.40677 q^{4} -15.7127 q^{5} +17.1681 q^{7} -15.5131 q^{8} -53.0679 q^{10} +52.8187 q^{11} +57.9835 q^{14} -79.6481 q^{16} +71.0654 q^{17} -92.6916 q^{19} -53.5295 q^{20} +178.390 q^{22} -190.712 q^{23} +121.888 q^{25} +58.4878 q^{28} +128.204 q^{29} -3.29674 q^{31} -144.898 q^{32} +240.016 q^{34} -269.757 q^{35} -241.546 q^{37} -313.056 q^{38} +243.753 q^{40} -97.1824 q^{41} +376.151 q^{43} +179.941 q^{44} -644.109 q^{46} +577.354 q^{47} -48.2555 q^{49} +411.664 q^{50} +307.686 q^{53} -829.924 q^{55} -266.331 q^{56} +432.995 q^{58} +349.914 q^{59} +127.467 q^{61} -11.1344 q^{62} +147.809 q^{64} +903.564 q^{67} +242.104 q^{68} -911.076 q^{70} +826.106 q^{71} +131.760 q^{73} -815.796 q^{74} -315.779 q^{76} +906.799 q^{77} -556.244 q^{79} +1251.48 q^{80} -328.223 q^{82} -254.664 q^{83} -1116.63 q^{85} +1270.41 q^{86} -819.384 q^{88} +183.410 q^{89} -649.712 q^{92} +1949.95 q^{94} +1456.43 q^{95} +780.498 q^{97} -162.978 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.37739	1.19409	0.597044	−	0.802208i	$-0.296342\pi$
$2$	3.37739	1.19409	0.597044	+	0.802208i	$0.296342\pi$
$3$	0	0
$3$	0	0
$4$	3.40677	0.425846
$5$	−15.7127	−1.40538	−0.702692	−	0.711494i	$-0.748019\pi$
$5$	−15.7127	−1.40538	−0.702692	+	0.711494i	$0.748019\pi$
$6$	0	0
$7$	17.1681	0.926992	0.463496	−	0.886099i	$-0.346595\pi$
$7$	17.1681	0.926992	0.463496	+	0.886099i	$0.346595\pi$
$8$	−15.5131	−0.685590
$9$	0	0
$10$	−53.0679	−1.67815
$11$	52.8187	1.44777	0.723884	−	0.689922i	$-0.242355\pi$
$11$	52.8187	1.44777	0.723884	+	0.689922i	$0.242355\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	57.9835	1.10691
$15$	0	0
$16$	−79.6481	−1.24450
$17$	71.0654	1.01388	0.506938	−	0.861982i	$-0.330777\pi$
$17$	71.0654	1.01388	0.506938	+	0.861982i	$0.330777\pi$
$18$	0	0
$19$	−92.6916	−1.11921	−0.559603	−	0.828761i	$-0.689046\pi$
$19$	−92.6916	−1.11921	−0.559603	+	0.828761i	$0.689046\pi$
$20$	−53.5295	−0.598478
$21$	0	0
$22$	178.390	1.72876
$23$	−190.712	−1.72896	−0.864482	−	0.502663i	$-0.832354\pi$
$23$	−190.712	−1.72896	−0.864482	+	0.502663i	$0.832354\pi$
$24$	0	0
$25$	121.888	0.975106
$26$	0	0
$27$	0	0
$28$	58.4878	0.394756
$29$	128.204	0.820927	0.410463	−	0.911877i	$-0.365367\pi$
$29$	128.204	0.820927	0.410463	+	0.911877i	$0.365367\pi$
$30$	0	0
$31$	−3.29674	−0.0191004	−0.00955020	−	0.999954i	$-0.503040\pi$
$31$	−3.29674	−0.0191004	−0.00955020	+	0.999954i	$0.503040\pi$
$32$	−144.898	−0.800454
$33$	0	0
$34$	240.016	1.21066
$35$	−269.757	−1.30278
$36$	0	0
$37$	−241.546	−1.07324	−0.536621	−	0.843823i	$-0.680300\pi$
$37$	−241.546	−1.07324	−0.536621	+	0.843823i	$0.680300\pi$
$38$	−313.056	−1.33643
$39$	0	0
$40$	243.753	0.963518
$41$	−97.1824	−0.370179	−0.185090	−	0.982722i	$-0.559258\pi$
$41$	−97.1824	−0.370179	−0.185090	+	0.982722i	$0.559258\pi$
$42$	0	0
$43$	376.151	1.33401	0.667006	−	0.745052i	$-0.267575\pi$
$43$	376.151	1.33401	0.667006	+	0.745052i	$0.267575\pi$
$44$	179.941	0.616527
$45$	0	0
$46$	−644.109	−2.06454
$47$	577.354	1.79182	0.895912	−	0.444231i	$-0.146523\pi$
$47$	577.354	1.79182	0.895912	+	0.444231i	$0.146523\pi$
$48$	0	0
$49$	−48.2555	−0.140686
$50$	411.664	1.16436
$51$	0	0
$52$	0	0
$53$	307.686	0.797433	0.398716	−	0.917074i	$-0.369456\pi$
$53$	307.686	0.797433	0.398716	+	0.917074i	$0.369456\pi$
$54$	0	0
$55$	−829.924	−2.03467
$56$	−266.331	−0.635536
$57$	0	0
$58$	432.995	0.980259
$59$	349.914	0.772117	0.386059	−	0.922474i	$-0.373836\pi$
$59$	349.914	0.772117	0.386059	+	0.922474i	$0.373836\pi$
$60$	0	0
$61$	127.467	0.267548	0.133774	−	0.991012i	$-0.457290\pi$
$61$	127.467	0.267548	0.133774	+	0.991012i	$0.457290\pi$
$62$	−11.1344	−0.0228076
$63$	0	0
$64$	147.809	0.288689
$65$	0	0
$66$	0	0
$67$	903.564	1.64758	0.823791	−	0.566894i	$-0.191855\pi$
$67$	903.564	1.64758	0.823791	+	0.566894i	$0.191855\pi$
$68$	242.104	0.431755
$69$	0	0
$70$	−911.076	−1.55563
$71$	826.106	1.38086	0.690428	−	0.723401i	$-0.257422\pi$
$71$	826.106	1.38086	0.690428	+	0.723401i	$0.257422\pi$
$72$	0	0
$73$	131.760	0.211252	0.105626	−	0.994406i	$-0.466315\pi$
$73$	131.760	0.211252	0.105626	+	0.994406i	$0.466315\pi$
$74$	−815.796	−1.28155
$75$	0	0
$76$	−315.779	−0.476610
$77$	906.799	1.34207
$78$	0	0
$79$	−556.244	−0.792182	−0.396091	−	0.918211i	$-0.629633\pi$
$79$	−556.244	−0.792182	−0.396091	+	0.918211i	$0.629633\pi$
$80$	1251.48	1.74900
$81$	0	0
$82$	−328.223	−0.442026
$83$	−254.664	−0.336783	−0.168391	−	0.985720i	$-0.553857\pi$
$83$	−254.664	−0.336783	−0.168391	+	0.985720i	$0.553857\pi$
$84$	0	0
$85$	−1116.63	−1.42489
$86$	1270.41	1.59293
$87$	0	0
$88$	−819.384	−0.992576
$89$	183.410	0.218443	0.109222	−	0.994017i	$-0.465164\pi$
$89$	183.410	0.218443	0.109222	+	0.994017i	$0.465164\pi$
$90$	0	0
$91$	0	0
$92$	−649.712	−0.736273
$93$	0	0
$94$	1949.95	2.13960
$95$	1456.43	1.57292
$96$	0	0
$97$	780.498	0.816986	0.408493	−	0.912762i	$-0.366054\pi$
$97$	780.498	0.816986	0.408493	+	0.912762i	$0.366054\pi$
$98$	−162.978	−0.167992
$99$	0	0
$100$	415.245	0.415245
$101$	1807.67	1.78089	0.890444	−	0.455093i	$-0.150394\pi$
$101$	1807.67	1.78089	0.890444	+	0.455093i	$0.150394\pi$
$102$	0	0
$103$	1560.78	1.49308	0.746542	−	0.665338i	$-0.231712\pi$
$103$	1560.78	1.49308	0.746542	+	0.665338i	$0.231712\pi$
$104$	0	0
$105$	0	0
$106$	1039.18	0.952205
$107$	−322.266	−0.291165	−0.145582	−	0.989346i	$-0.546506\pi$
$107$	−322.266	−0.291165	−0.145582	+	0.989346i	$0.546506\pi$
$108$	0	0
$109$	−1607.72	−1.41277	−0.706386	−	0.707827i	$-0.749676\pi$
$109$	−1607.72	−1.41277	−0.706386	+	0.707827i	$0.749676\pi$
$110$	−2802.98	−2.42958
$111$	0	0
$112$	−1367.41	−1.15364
$113$	429.905	0.357895	0.178947	−	0.983859i	$-0.442731\pi$
$113$	429.905	0.357895	0.178947	+	0.983859i	$0.442731\pi$
$114$	0	0
$115$	2996.60	2.42986
$116$	436.761	0.349589
$117$	0	0
$118$	1181.80	0.921976
$119$	1220.06	0.939855
$120$	0	0
$121$	1458.82	1.09603
$122$	430.505	0.319476
$123$	0	0
$124$	−11.2312	−0.00813383
$125$	48.8932	0.0349851
$126$	0	0
$127$	2371.36	1.65688	0.828442	−	0.560076i	$-0.189228\pi$
$127$	2371.36	1.65688	0.828442	+	0.560076i	$0.189228\pi$
$128$	1658.39	1.14517
$129$	0	0
$130$	0	0
$131$	−169.200	−0.112848	−0.0564239	−	0.998407i	$-0.517970\pi$
$131$	−169.200	−0.112848	−0.0564239	+	0.998407i	$0.517970\pi$
$132$	0	0
$133$	−1591.34	−1.03749
$134$	3051.69	1.96736
$135$	0	0
$136$	−1102.45	−0.695104
$137$	2976.21	1.85602	0.928009	−	0.372558i	$-0.121519\pi$
$137$	2976.21	1.85602	0.928009	+	0.372558i	$0.121519\pi$
$138$	0	0
$139$	−2555.15	−1.55917	−0.779585	−	0.626296i	$-0.784570\pi$
$139$	−2555.15	−1.55917	−0.779585	+	0.626296i	$0.784570\pi$
$140$	−919.001	−0.554784
$141$	0	0
$142$	2790.08	1.64886
$143$	0	0
$144$	0	0
$145$	−2014.43	−1.15372
$146$	445.006	0.252253
$147$	0	0
$148$	−822.892	−0.457036
$149$	424.529	0.233415	0.116707	−	0.993166i	$-0.462766\pi$
$149$	424.529	0.233415	0.116707	+	0.993166i	$0.462766\pi$
$150$	0	0
$151$	−42.3232	−0.0228093	−0.0114047	−	0.999935i	$-0.503630\pi$
$151$	−42.3232	−0.0228093	−0.0114047	+	0.999935i	$0.503630\pi$
$152$	1437.94	0.767317
$153$	0	0
$154$	3062.61	1.60255
$155$	51.8007	0.0268434
$156$	0	0
$157$	−990.187	−0.503347	−0.251674	−	0.967812i	$-0.580981\pi$
$157$	−990.187	−0.503347	−0.251674	+	0.967812i	$0.580981\pi$
$158$	−1878.65	−0.945935
$159$	0	0
$160$	2276.73	1.12495
$161$	−3274.17	−1.60274
$162$	0	0
$163$	−2645.82	−1.27139	−0.635695	−	0.771940i	$-0.719286\pi$
$163$	−2645.82	−1.27139	−0.635695	+	0.771940i	$0.719286\pi$
$164$	−331.078	−0.157639
$165$	0	0
$166$	−860.099	−0.402148
$167$	295.428	0.136892	0.0684458	−	0.997655i	$-0.478196\pi$
$167$	295.428	0.136892	0.0684458	+	0.997655i	$0.478196\pi$
$168$	0	0
$169$	0	0
$170$	−3771.29	−1.70144
$171$	0	0
$172$	1281.46	0.568084
$173$	−1495.46	−0.657213	−0.328606	−	0.944467i	$-0.606579\pi$
$173$	−1495.46	−0.657213	−0.328606	+	0.944467i	$0.606579\pi$
$174$	0	0
$175$	2092.59	0.903915
$176$	−4206.91	−1.80175
$177$	0	0
$178$	619.448	0.260840
$179$	−785.097	−0.327826	−0.163913	−	0.986475i	$-0.552412\pi$
$179$	−785.097	−0.327826	−0.163913	+	0.986475i	$0.552412\pi$
$180$	0	0
$181$	−1287.01	−0.528522	−0.264261	−	0.964451i	$-0.585128\pi$
$181$	−1287.01	−0.528522	−0.264261	+	0.964451i	$0.585128\pi$
$182$	0	0
$183$	0	0
$184$	2958.54	1.18536
$185$	3795.34	1.50832
$186$	0	0
$187$	3753.59	1.46786
$188$	1966.91	0.763042
$189$	0	0
$190$	4918.95	1.87820
$191$	−466.452	−0.176708	−0.0883542	−	0.996089i	$-0.528161\pi$
$191$	−466.452	−0.176708	−0.0883542	+	0.996089i	$0.528161\pi$
$192$	0	0
$193$	−2636.99	−0.983495	−0.491747	−	0.870738i	$-0.663642\pi$
$193$	−2636.99	−0.983495	−0.491747	+	0.870738i	$0.663642\pi$
$194$	2636.05	0.975553
$195$	0	0
$196$	−164.395	−0.0599108
$197$	3676.90	1.32979	0.664894	−	0.746937i	$-0.268477\pi$
$197$	3676.90	1.32979	0.664894	+	0.746937i	$0.268477\pi$
$198$	0	0
$199$	220.557	0.0785670	0.0392835	−	0.999228i	$-0.487492\pi$
$199$	220.557	0.0785670	0.0392835	+	0.999228i	$0.487492\pi$
$200$	−1890.87	−0.668523
$201$	0	0
$202$	6105.20	2.12654
$203$	2201.02	0.760992
$204$	0	0
$205$	1527.00	0.520244
$206$	5271.35	1.78287
$207$	0	0
$208$	0	0
$209$	−4895.85	−1.62035
$210$	0	0
$211$	1385.93	0.452186	0.226093	−	0.974106i	$-0.427405\pi$
$211$	1385.93	0.452186	0.226093	+	0.974106i	$0.427405\pi$
$212$	1048.22	0.339584
$213$	0	0
$214$	−1088.42	−0.347676
$215$	−5910.35	−1.87480
$216$	0	0
$217$	−56.5989	−0.0177059
$218$	−5429.91	−1.68697
$219$	0	0
$220$	−2827.36	−0.866457
$221$	0	0
$222$	0	0
$223$	111.847	0.0335867	0.0167933	−	0.999859i	$-0.494654\pi$
$223$	111.847	0.0335867	0.0167933	+	0.999859i	$0.494654\pi$
$224$	−2487.62	−0.742014
$225$	0	0
$226$	1451.96	0.427358
$227$	4990.96	1.45930	0.729651	−	0.683819i	$-0.239682\pi$
$227$	4990.96	1.45930	0.729651	+	0.683819i	$0.239682\pi$
$228$	0	0
$229$	5787.56	1.67010	0.835050	−	0.550174i	$-0.185439\pi$
$229$	5787.56	1.67010	0.835050	+	0.550174i	$0.185439\pi$
$230$	10120.7	2.90147
$231$	0	0
$232$	−1988.85	−0.562819
$233$	723.707	0.203483	0.101742	−	0.994811i	$-0.467559\pi$
$233$	723.707	0.203483	0.101742	+	0.994811i	$0.467559\pi$
$234$	0	0
$235$	−9071.78	−2.51820
$236$	1192.08	0.328803
$237$	0	0
$238$	4120.62	1.12227
$239$	−1239.74	−0.335531	−0.167766	−	0.985827i	$-0.553655\pi$
$239$	−1239.74	−0.335531	−0.167766	+	0.985827i	$0.553655\pi$
$240$	0	0
$241$	755.393	0.201905	0.100953	−	0.994891i	$-0.467811\pi$
$241$	755.393	0.201905	0.100953	+	0.994891i	$0.467811\pi$
$242$	4927.00	1.30876
$243$	0	0
$244$	434.249	0.113934
$245$	758.222	0.197719
$246$	0	0
$247$	0	0
$248$	51.1428	0.0130950
$249$	0	0
$250$	165.131	0.0417753
$251$	−2252.06	−0.566330	−0.283165	−	0.959071i	$-0.591384\pi$
$251$	−2252.06	−0.566330	−0.283165	+	0.959071i	$0.591384\pi$
$252$	0	0
$253$	−10073.2	−2.50314
$254$	8009.01	1.97846
$255$	0	0
$256$	4418.56	1.07875
$257$	−3716.75	−0.902118	−0.451059	−	0.892494i	$-0.648954\pi$
$257$	−3716.75	−0.902118	−0.451059	+	0.892494i	$0.648954\pi$
$258$	0	0
$259$	−4146.90	−0.994887
$260$	0	0
$261$	0	0
$262$	−571.454	−0.134750
$263$	49.9521	0.0117117	0.00585585	−	0.999983i	$-0.498136\pi$
$263$	49.9521	0.0117117	0.00585585	+	0.999983i	$0.498136\pi$
$264$	0	0
$265$	−4834.57	−1.12070
$266$	−5374.58	−1.23886
$267$	0	0
$268$	3078.23	0.701616
$269$	−3421.44	−0.775497	−0.387748	−	0.921765i	$-0.626747\pi$
$269$	−3421.44	−0.775497	−0.387748	+	0.921765i	$0.626747\pi$
$270$	0	0
$271$	2958.89	0.663247	0.331624	−	0.943412i	$-0.392404\pi$
$271$	2958.89	0.663247	0.331624	+	0.943412i	$0.392404\pi$
$272$	−5660.23	−1.26177
$273$	0	0
$274$	10051.8	2.21625
$275$	6437.99	1.41173
$276$	0	0
$277$	−7461.08	−1.61839	−0.809193	−	0.587543i	$-0.800095\pi$
$277$	−7461.08	−1.61839	−0.809193	+	0.587543i	$0.800095\pi$
$278$	−8629.73	−1.86179
$279$	0	0
$280$	4184.78	0.893173
$281$	−4728.04	−1.00374	−0.501871	−	0.864943i	$-0.667355\pi$
$281$	−4728.04	−1.00374	−0.501871	+	0.864943i	$0.667355\pi$
$282$	0	0
$283$	−2879.69	−0.604876	−0.302438	−	0.953169i	$-0.597801\pi$
$283$	−2879.69	−0.604876	−0.302438	+	0.953169i	$0.597801\pi$
$284$	2814.35	0.588032
$285$	0	0
$286$	0	0
$287$	−1668.44	−0.343153
$288$	0	0
$289$	137.298	0.0279458
$290$	−6803.51	−1.37764
$291$	0	0
$292$	448.877	0.0899607
$293$	8200.94	1.63517	0.817583	−	0.575810i	$-0.195313\pi$
$293$	8200.94	1.63517	0.817583	+	0.575810i	$0.195313\pi$
$294$	0	0
$295$	−5498.09	−1.08512
$296$	3747.14	0.735804
$297$	0	0
$298$	1433.80	0.278718
$299$	0	0
$300$	0	0
$301$	6457.81	1.23662
$302$	−142.942	−0.0272364
$303$	0	0
$304$	7382.71	1.39285
$305$	−2002.84	−0.376008
$306$	0	0
$307$	−2058.82	−0.382747	−0.191374	−	0.981517i	$-0.561294\pi$
$307$	−2058.82	−0.382747	−0.191374	+	0.981517i	$0.561294\pi$
$308$	3089.25	0.571515
$309$	0	0
$310$	174.951	0.0320534
$311$	−1862.08	−0.339514	−0.169757	−	0.985486i	$-0.554298\pi$
$311$	−1862.08	−0.339514	−0.169757	+	0.985486i	$0.554298\pi$
$312$	0	0
$313$	−255.896	−0.0462112	−0.0231056	−	0.999733i	$-0.507355\pi$
$313$	−255.896	−0.0462112	−0.0231056	+	0.999733i	$0.507355\pi$
$314$	−3344.25	−0.601041
$315$	0	0
$316$	−1895.00	−0.337348
$317$	6265.68	1.11014	0.555072	−	0.831802i	$-0.312691\pi$
$317$	6265.68	1.11014	0.555072	+	0.831802i	$0.312691\pi$
$318$	0	0
$319$	6771.57	1.18851
$320$	−2322.47	−0.405719
$321$	0	0
$322$	−11058.1	−1.91381
$323$	−6587.17	−1.13474
$324$	0	0
$325$	0	0
$326$	−8935.96	−1.51815
$327$	0	0
$328$	1507.60	0.253791
$329$	9912.09	1.66101
$330$	0	0
$331$	−1071.93	−0.178002	−0.0890011	−	0.996032i	$-0.528367\pi$
$331$	−1071.93	−0.178002	−0.0890011	+	0.996032i	$0.528367\pi$
$332$	−867.581	−0.143418
$333$	0	0
$334$	997.775	0.163461
$335$	−14197.4	−2.31549
$336$	0	0
$337$	−5572.02	−0.900675	−0.450337	−	0.892858i	$-0.648696\pi$
$337$	−5572.02	−0.900675	−0.450337	+	0.892858i	$0.648696\pi$
$338$	0	0
$339$	0	0
$340$	−3804.10	−0.606783
$341$	−174.130	−0.0276530
$342$	0	0
$343$	−6717.12	−1.05741
$344$	−5835.29	−0.914586
$345$	0	0
$346$	−5050.76	−0.784770
$347$	1903.44	0.294472	0.147236	−	0.989101i	$-0.452962\pi$
$347$	1903.44	0.294472	0.147236	+	0.989101i	$0.452962\pi$
$348$	0	0
$349$	−1097.17	−0.168282	−0.0841409	−	0.996454i	$-0.526815\pi$
$349$	−1097.17	−0.168282	−0.0841409	+	0.996454i	$0.526815\pi$
$350$	7067.51	1.07935
$351$	0	0
$352$	−7653.31	−1.15887
$353$	5420.90	0.817352	0.408676	−	0.912679i	$-0.365991\pi$
$353$	5420.90	0.817352	0.408676	+	0.912679i	$0.365991\pi$
$354$	0	0
$355$	−12980.3	−1.94063
$356$	624.836	0.0930232
$357$	0	0
$358$	−2651.58	−0.391453
$359$	3885.40	0.571208	0.285604	−	0.958348i	$-0.407806\pi$
$359$	3885.40	0.571208	0.285604	+	0.958348i	$0.407806\pi$
$360$	0	0
$361$	1732.74	0.252622
$362$	−4346.72	−0.631102
$363$	0	0
$364$	0	0
$365$	−2070.31	−0.296890
$366$	0	0
$367$	−3888.26	−0.553039	−0.276520	−	0.961008i	$-0.589181\pi$
$367$	−3888.26	−0.553039	−0.276520	+	0.961008i	$0.589181\pi$
$368$	15189.8	2.15170
$369$	0	0
$370$	12818.3	1.80106
$371$	5282.39	0.739213
$372$	0	0
$373$	9960.88	1.38272	0.691361	−	0.722510i	$-0.257012\pi$
$373$	9960.88	1.38272	0.691361	+	0.722510i	$0.257012\pi$
$374$	12677.3	1.75275
$375$	0	0
$376$	−8956.57	−1.22846
$377$	0	0
$378$	0	0
$379$	−5288.25	−0.716726	−0.358363	−	0.933582i	$-0.616665\pi$
$379$	−5288.25	−0.716726	−0.358363	+	0.933582i	$0.616665\pi$
$380$	4961.73	0.669820
$381$	0	0
$382$	−1575.39	−0.211005
$383$	688.944	0.0919149	0.0459575	−	0.998943i	$-0.485366\pi$
$383$	688.944	0.0919149	0.0459575	+	0.998943i	$0.485366\pi$
$384$	0	0
$385$	−14248.2	−1.88612
$386$	−8906.14	−1.17438
$387$	0	0
$388$	2658.98	0.347910
$389$	−2102.57	−0.274048	−0.137024	−	0.990568i	$-0.543754\pi$
$389$	−2102.57	−0.274048	−0.137024	+	0.990568i	$0.543754\pi$
$390$	0	0
$391$	−13553.0	−1.75296
$392$	748.593	0.0964533
$393$	0	0
$394$	12418.3	1.58788
$395$	8740.09	1.11332
$396$	0	0
$397$	11254.8	1.42283	0.711413	−	0.702774i	$-0.248056\pi$
$397$	11254.8	1.42283	0.711413	+	0.702774i	$0.248056\pi$
$398$	744.906	0.0938159
$399$	0	0
$400$	−9708.17	−1.21352
$401$	−3523.22	−0.438756	−0.219378	−	0.975640i	$-0.570403\pi$
$401$	−3523.22	−0.438756	−0.219378	+	0.975640i	$0.570403\pi$
$402$	0	0
$403$	0	0
$404$	6158.31	0.758384
$405$	0	0
$406$	7433.71	0.908692
$407$	−12758.2	−1.55381
$408$	0	0
$409$	6366.65	0.769708	0.384854	−	0.922978i	$-0.374252\pi$
$409$	6366.65	0.769708	0.384854	+	0.922978i	$0.374252\pi$
$410$	5157.26	0.621217
$411$	0	0
$412$	5317.20	0.635824
$413$	6007.37	0.715746
$414$	0	0
$415$	4001.45	0.473310
$416$	0	0
$417$	0	0
$418$	−16535.2	−1.93484
$419$	−8919.77	−1.04000	−0.519999	−	0.854167i	$-0.674068\pi$
$419$	−8919.77	−1.04000	−0.519999	+	0.854167i	$0.674068\pi$
$420$	0	0
$421$	1090.71	0.126266	0.0631328	−	0.998005i	$-0.479891\pi$
$421$	1090.71	0.126266	0.0631328	+	0.998005i	$0.479891\pi$
$422$	4680.82	0.539949
$423$	0	0
$424$	−4773.17	−0.546712
$425$	8662.05	0.988638
$426$	0	0
$427$	2188.36	0.248015
$428$	−1097.89	−0.123991
$429$	0	0
$430$	−19961.5	−2.23868
$431$	−3760.43	−0.420263	−0.210132	−	0.977673i	$-0.567389\pi$
$431$	−3760.43	−0.420263	−0.210132	+	0.977673i	$0.567389\pi$
$432$	0	0
$433$	−4085.06	−0.453384	−0.226692	−	0.973967i	$-0.572791\pi$
$433$	−4085.06	−0.453384	−0.226692	+	0.973967i	$0.572791\pi$
$434$	−191.157	−0.0211424
$435$	0	0
$436$	−5477.15	−0.601623
$437$	17677.4	1.93507
$438$	0	0
$439$	9385.35	1.02036	0.510180	−	0.860068i	$-0.329579\pi$
$439$	9385.35	1.02036	0.510180	+	0.860068i	$0.329579\pi$
$440$	12874.7	1.39495
$441$	0	0
$442$	0	0
$443$	−5755.41	−0.617264	−0.308632	−	0.951182i	$-0.599871\pi$
$443$	−5755.41	−0.617264	−0.308632	+	0.951182i	$0.599871\pi$
$444$	0	0
$445$	−2881.87	−0.306997
$446$	377.751	0.0401055
$447$	0	0
$448$	2537.60	0.267612
$449$	1104.67	0.116108	0.0580541	−	0.998313i	$-0.481510\pi$
$449$	1104.67	0.116108	0.0580541	+	0.998313i	$0.481510\pi$
$450$	0	0
$451$	−5133.05	−0.535934
$452$	1464.59	0.152408
$453$	0	0
$454$	16856.4	1.74254
$455$	0	0
$456$	0	0
$457$	13548.5	1.38681	0.693405	−	0.720548i	$-0.256110\pi$
$457$	13548.5	1.38681	0.693405	+	0.720548i	$0.256110\pi$
$458$	19546.9	1.99425
$459$	0	0
$460$	10208.7	1.03475
$461$	2996.03	0.302688	0.151344	−	0.988481i	$-0.451640\pi$
$461$	2996.03	0.302688	0.151344	+	0.988481i	$0.451640\pi$
$462$	0	0
$463$	2026.10	0.203371	0.101686	−	0.994817i	$-0.467576\pi$
$463$	2026.10	0.203371	0.101686	+	0.994817i	$0.467576\pi$
$464$	−10211.2	−1.02164
$465$	0	0
$466$	2444.24	0.242977
$467$	3284.20	0.325428	0.162714	−	0.986673i	$-0.447975\pi$
$467$	3284.20	0.325428	0.162714	+	0.986673i	$0.447975\pi$
$468$	0	0
$469$	15512.5	1.52729
$470$	−30638.9	−3.00696
$471$	0	0
$472$	−5428.26	−0.529356
$473$	19867.8	1.93134
$474$	0	0
$475$	−11298.0	−1.09134
$476$	4156.46	0.400234
$477$	0	0
$478$	−4187.08	−0.400654
$479$	13208.5	1.25994	0.629969	−	0.776621i	$-0.283068\pi$
$479$	13208.5	1.25994	0.629969	+	0.776621i	$0.283068\pi$
$480$	0	0
$481$	0	0
$482$	2551.26	0.241092
$483$	0	0
$484$	4969.86	0.466741
$485$	−12263.7	−1.14818
$486$	0	0
$487$	−8974.09	−0.835020	−0.417510	−	0.908672i	$-0.637097\pi$
$487$	−8974.09	−0.835020	−0.417510	+	0.908672i	$0.637097\pi$
$488$	−1977.41	−0.183428
$489$	0	0
$490$	2560.81	0.236093
$491$	12889.8	1.18474	0.592370	−	0.805666i	$-0.298192\pi$
$491$	12889.8	1.18474	0.592370	+	0.805666i	$0.298192\pi$
$492$	0	0
$493$	9110.87	0.832319
$494$	0	0
$495$	0	0
$496$	262.579	0.0237705
$497$	14182.7	1.28004
$498$	0	0
$499$	−7371.92	−0.661348	−0.330674	−	0.943745i	$-0.607276\pi$
$499$	−7371.92	−0.661348	−0.330674	+	0.943745i	$0.607276\pi$
$500$	166.568	0.0148983
$501$	0	0
$502$	−7606.09	−0.676248
$503$	−19788.2	−1.75410	−0.877049	−	0.480400i	$-0.840491\pi$
$503$	−19788.2	−1.75410	−0.877049	+	0.480400i	$0.840491\pi$
$504$	0	0
$505$	−28403.3	−2.50283
$506$	−34021.0	−2.98897
$507$	0	0
$508$	8078.67	0.705577
$509$	11682.4	1.01732	0.508658	−	0.860968i	$-0.330142\pi$
$509$	11682.4	1.01732	0.508658	+	0.860968i	$0.330142\pi$
$510$	0	0
$511$	2262.08	0.195829
$512$	1656.08	0.142948
$513$	0	0
$514$	−12552.9	−1.07721
$515$	−24524.0	−2.09836
$516$	0	0
$517$	30495.1	2.59415
$518$	−14005.7	−1.18798
$519$	0	0
$520$	0	0
$521$	19025.1	1.59982	0.799908	−	0.600122i	$-0.204881\pi$
$521$	19025.1	1.59982	0.799908	+	0.600122i	$0.204881\pi$
$522$	0	0
$523$	5345.69	0.446942	0.223471	−	0.974711i	$-0.428261\pi$
$523$	5345.69	0.446942	0.223471	+	0.974711i	$0.428261\pi$
$524$	−576.425	−0.0480558
$525$	0	0
$526$	168.708	0.0139848
$527$	−234.284	−0.0193655
$528$	0	0
$529$	24204.0	1.98932
$530$	−16328.2	−1.33821
$531$	0	0
$532$	−5421.33	−0.441813
$533$	0	0
$534$	0	0
$535$	5063.66	0.409199
$536$	−14017.1	−1.12957
$537$	0	0
$538$	−11555.5	−0.926012
$539$	−2548.79	−0.203681
$540$	0	0
$541$	3058.01	0.243021	0.121510	−	0.992590i	$-0.461226\pi$
$541$	3058.01	0.243021	0.121510	+	0.992590i	$0.461226\pi$
$542$	9993.34	0.791975
$543$	0	0
$544$	−10297.2	−0.811561
$545$	25261.7	1.98549
$546$	0	0
$547$	17921.1	1.40082	0.700410	−	0.713740i	$-0.253000\pi$
$547$	17921.1	1.40082	0.700410	+	0.713740i	$0.253000\pi$
$548$	10139.2	0.790378
$549$	0	0
$550$	21743.6	1.68573
$551$	−11883.4	−0.918786
$552$	0	0
$553$	−9549.67	−0.734346
$554$	−25199.0	−1.93250
$555$	0	0
$556$	−8704.79	−0.663967
$557$	9106.04	0.692702	0.346351	−	0.938105i	$-0.387421\pi$
$557$	9106.04	0.692702	0.346351	+	0.938105i	$0.387421\pi$
$558$	0	0
$559$	0	0
$560$	21485.6	1.62131
$561$	0	0
$562$	−15968.4	−1.19856
$563$	−11327.9	−0.847985	−0.423992	−	0.905666i	$-0.639372\pi$
$563$	−11327.9	−0.847985	−0.423992	+	0.905666i	$0.639372\pi$
$564$	0	0
$565$	−6754.97	−0.502980
$566$	−9725.84	−0.722275
$567$	0	0
$568$	−12815.5	−0.946701
$569$	−1963.27	−0.144647	−0.0723237	−	0.997381i	$-0.523041\pi$
$569$	−1963.27	−0.144647	−0.0723237	+	0.997381i	$0.523041\pi$
$570$	0	0
$571$	−2270.35	−0.166394	−0.0831971	−	0.996533i	$-0.526513\pi$
$571$	−2270.35	−0.166394	−0.0831971	+	0.996533i	$0.526513\pi$
$572$	0	0
$573$	0	0
$574$	−5634.97	−0.409755
$575$	−23245.6	−1.68592
$576$	0	0
$577$	13949.6	1.00646	0.503231	−	0.864152i	$-0.332145\pi$
$577$	13949.6	1.00646	0.503231	+	0.864152i	$0.332145\pi$
$578$	463.708	0.0333697
$579$	0	0
$580$	−6862.69	−0.491306
$581$	−4372.10	−0.312195
$582$	0	0
$583$	16251.6	1.15450
$584$	−2044.02	−0.144832
$585$	0	0
$586$	27697.8	1.95253
$587$	−26731.4	−1.87960	−0.939799	−	0.341727i	$-0.888988\pi$
$587$	−26731.4	−1.87960	−0.939799	+	0.341727i	$0.888988\pi$
$588$	0	0
$589$	305.580	0.0213773
$590$	−18569.2	−1.29573
$591$	0	0
$592$	19238.7	1.33565
$593$	10508.6	0.727715	0.363858	−	0.931455i	$-0.381459\pi$
$593$	10508.6	0.727715	0.363858	+	0.931455i	$0.381459\pi$
$594$	0	0
$595$	−19170.4	−1.32086
$596$	1446.27	0.0993987
$597$	0	0
$598$	0	0
$599$	1935.87	0.132049	0.0660246	−	0.997818i	$-0.478968\pi$
$599$	1935.87	0.132049	0.0660246	+	0.997818i	$0.478968\pi$
$600$	0	0
$601$	−16155.5	−1.09650	−0.548249	−	0.836315i	$-0.684705\pi$
$601$	−16155.5	−1.09650	−0.548249	+	0.836315i	$0.684705\pi$
$602$	21810.6	1.47663
$603$	0	0
$604$	−144.185	−0.00971327
$605$	−22922.0	−1.54035
$606$	0	0
$607$	−1698.75	−0.113592	−0.0567959	−	0.998386i	$-0.518088\pi$
$607$	−1698.75	−0.113592	−0.0567959	+	0.998386i	$0.518088\pi$
$608$	13430.8	0.895873
$609$	0	0
$610$	−6764.38	−0.448987
$611$	0	0
$612$	0	0
$613$	8627.42	0.568448	0.284224	−	0.958758i	$-0.408264\pi$
$613$	8627.42	0.568448	0.284224	+	0.958758i	$0.408264\pi$
$614$	−6953.46	−0.457034
$615$	0	0
$616$	−14067.3	−0.920109
$617$	−21415.4	−1.39733	−0.698664	−	0.715450i	$-0.746222\pi$
$617$	−21415.4	−1.39733	−0.698664	+	0.715450i	$0.746222\pi$
$618$	0	0
$619$	17396.0	1.12957	0.564784	−	0.825238i	$-0.308959\pi$
$619$	17396.0	1.12957	0.564784	+	0.825238i	$0.308959\pi$
$620$	176.473	0.0114312
$621$	0	0
$622$	−6288.96	−0.405409
$623$	3148.81	0.202495
$624$	0	0
$625$	−16004.3	−1.02427
$626$	−864.261	−0.0551802
$627$	0	0
$628$	−3373.34	−0.214349
$629$	−17165.6	−1.08814
$630$	0	0
$631$	13059.2	0.823898	0.411949	−	0.911207i	$-0.364848\pi$
$631$	13059.2	0.823898	0.411949	+	0.911207i	$0.364848\pi$
$632$	8629.09	0.543112
$633$	0	0
$634$	21161.6	1.32561
$635$	−37260.4	−2.32856
$636$	0	0
$637$	0	0
$638$	22870.2	1.41919
$639$	0	0
$640$	−26057.7	−1.60941
$641$	−22397.7	−1.38012	−0.690060	−	0.723752i	$-0.742416\pi$
$641$	−22397.7	−1.38012	−0.690060	+	0.723752i	$0.742416\pi$
$642$	0	0
$643$	−5878.08	−0.360511	−0.180256	−	0.983620i	$-0.557693\pi$
$643$	−5878.08	−0.360511	−0.180256	+	0.983620i	$0.557693\pi$
$644$	−11154.3	−0.682519
$645$	0	0
$646$	−22247.5	−1.35498
$647$	20768.3	1.26196	0.630979	−	0.775800i	$-0.282653\pi$
$647$	20768.3	1.26196	0.630979	+	0.775800i	$0.282653\pi$
$648$	0	0
$649$	18482.0	1.11785
$650$	0	0
$651$	0	0
$652$	−9013.69	−0.541416
$653$	−15516.4	−0.929865	−0.464933	−	0.885346i	$-0.653921\pi$
$653$	−15516.4	−0.929865	−0.464933	+	0.885346i	$0.653921\pi$
$654$	0	0
$655$	2658.58	0.158594
$656$	7740.39	0.460688
$657$	0	0
$658$	33477.0	1.98339
$659$	−24025.0	−1.42015	−0.710077	−	0.704124i	$-0.751340\pi$
$659$	−24025.0	−1.42015	−0.710077	+	0.704124i	$0.751340\pi$
$660$	0	0
$661$	10778.8	0.634262	0.317131	−	0.948382i	$-0.397281\pi$
$661$	10778.8	0.634262	0.317131	+	0.948382i	$0.397281\pi$
$662$	−3620.33	−0.212550
$663$	0	0
$664$	3950.63	0.230895
$665$	25004.2	1.45808
$666$	0	0
$667$	−24450.0	−1.41935
$668$	1006.45	0.0582948
$669$	0	0
$670$	−47950.2	−2.76489
$671$	6732.63	0.387348
$672$	0	0
$673$	22345.9	1.27990	0.639950	−	0.768417i	$-0.278955\pi$
$673$	22345.9	1.27990	0.639950	+	0.768417i	$0.278955\pi$
$674$	−18818.9	−1.07549
$675$	0	0
$676$	0	0
$677$	−12326.7	−0.699782	−0.349891	−	0.936790i	$-0.613781\pi$
$677$	−12326.7	−0.699782	−0.349891	+	0.936790i	$0.613781\pi$
$678$	0	0
$679$	13399.7	0.757339
$680$	17322.4	0.976888
$681$	0	0
$682$	−588.104	−0.0330201
$683$	−27452.5	−1.53798	−0.768989	−	0.639262i	$-0.779240\pi$
$683$	−27452.5	−1.53798	−0.768989	+	0.639262i	$0.779240\pi$
$684$	0	0
$685$	−46764.2	−2.60842
$686$	−22686.3	−1.26264
$687$	0	0
$688$	−29959.7	−1.66018
$689$	0	0
$690$	0	0
$691$	18663.6	1.02749	0.513746	−	0.857942i	$-0.328257\pi$
$691$	18663.6	1.02749	0.513746	+	0.857942i	$0.328257\pi$
$692$	−5094.69	−0.279871
$693$	0	0
$694$	6428.65	0.351626
$695$	40148.2	2.19123
$696$	0	0
$697$	−6906.31	−0.375316
$698$	−3705.59	−0.200943
$699$	0	0
$700$	7128.98	0.384929
$701$	4131.00	0.222576	0.111288	−	0.993788i	$-0.464502\pi$
$701$	4131.00	0.222576	0.111288	+	0.993788i	$0.464502\pi$
$702$	0	0
$703$	22389.3	1.20118
$704$	7807.07	0.417955
$705$	0	0
$706$	18308.5	0.975990
$707$	31034.3	1.65087
$708$	0	0
$709$	−33578.1	−1.77863	−0.889317	−	0.457291i	$-0.848820\pi$
$709$	−33578.1	−1.77863	−0.889317	+	0.457291i	$0.848820\pi$
$710$	−43839.7	−2.31729
$711$	0	0
$712$	−2845.27	−0.149762
$713$	628.728	0.0330239
$714$	0	0
$715$	0	0
$716$	−2674.64	−0.139603
$717$	0	0
$718$	13122.5	0.682073
$719$	−22527.2	−1.16846	−0.584229	−	0.811589i	$-0.698603\pi$
$719$	−22527.2	−1.16846	−0.584229	+	0.811589i	$0.698603\pi$
$720$	0	0
$721$	26795.6	1.38408
$722$	5852.12	0.301653
$723$	0	0
$724$	−4384.53	−0.225069
$725$	15626.6	0.800491
$726$	0	0
$727$	241.718	0.0123312	0.00616562	−	0.999981i	$-0.498037\pi$
$727$	241.718	0.0123312	0.00616562	+	0.999981i	$0.498037\pi$
$728$	0	0
$729$	0	0
$730$	−6992.24	−0.354513
$731$	26731.4	1.35252
$732$	0	0
$733$	17235.4	0.868490	0.434245	−	0.900795i	$-0.357015\pi$
$733$	17235.4	0.868490	0.434245	+	0.900795i	$0.357015\pi$
$734$	−13132.2	−0.660377
$735$	0	0
$736$	27633.7	1.38396
$737$	47725.1	2.38532
$738$	0	0
$739$	34304.3	1.70758	0.853792	−	0.520615i	$-0.174297\pi$
$739$	34304.3	1.70758	0.853792	+	0.520615i	$0.174297\pi$
$740$	12929.8	0.642312
$741$	0	0
$742$	17840.7	0.882686
$743$	2114.59	0.104410	0.0522052	−	0.998636i	$-0.483375\pi$
$743$	2114.59	0.104410	0.0522052	+	0.998636i	$0.483375\pi$
$744$	0	0
$745$	−6670.49	−0.328037
$746$	33641.8	1.65109
$747$	0	0
$748$	12787.6	0.625082
$749$	−5532.70	−0.269907
$750$	0	0
$751$	−40958.7	−1.99015	−0.995077	−	0.0991024i	$-0.968403\pi$
$751$	−40958.7	−1.99015	−0.995077	+	0.0991024i	$0.968403\pi$
$752$	−45985.1	−2.22993
$753$	0	0
$754$	0	0
$755$	665.010	0.0320559
$756$	0	0
$757$	10530.8	0.505613	0.252807	−	0.967517i	$-0.418646\pi$
$757$	10530.8	0.505613	0.252807	+	0.967517i	$0.418646\pi$
$758$	−17860.5	−0.855834
$759$	0	0
$760$	−22593.9	−1.07838
$761$	−787.955	−0.0375340	−0.0187670	−	0.999824i	$-0.505974\pi$
$761$	−787.955	−0.0375340	−0.0187670	+	0.999824i	$0.505974\pi$
$762$	0	0
$763$	−27601.6	−1.30963
$764$	−1589.10	−0.0752506
$765$	0	0
$766$	2326.83	0.109755
$767$	0	0
$768$	0	0
$769$	−5227.62	−0.245140	−0.122570	−	0.992460i	$-0.539114\pi$
$769$	−5227.62	−0.245140	−0.122570	+	0.992460i	$0.539114\pi$
$770$	−48121.9	−2.25220
$771$	0	0
$772$	−8983.61	−0.418818
$773$	4516.79	0.210165	0.105082	−	0.994464i	$-0.466489\pi$
$773$	4516.79	0.210165	0.105082	+	0.994464i	$0.466489\pi$
$774$	0	0
$775$	−401.834	−0.0186249
$776$	−12108.0	−0.560117
$777$	0	0
$778$	−7101.21	−0.327237
$779$	9008.00	0.414307
$780$	0	0
$781$	43633.9	1.99916
$782$	−45773.9	−2.09318
$783$	0	0
$784$	3843.45	0.175084
$785$	15558.5	0.707397
$786$	0	0
$787$	−6800.09	−0.308001	−0.154001	−	0.988071i	$-0.549216\pi$
$787$	−6800.09	−0.308001	−0.154001	+	0.988071i	$0.549216\pi$
$788$	12526.4	0.566285
$789$	0	0
$790$	29518.7	1.32940
$791$	7380.67	0.331765
$792$	0	0
$793$	0	0
$794$	38011.8	1.69898
$795$	0	0
$796$	751.385	0.0334575
$797$	40553.3	1.80235	0.901174	−	0.433457i	$-0.142707\pi$
$797$	40553.3	1.80235	0.901174	+	0.433457i	$0.142707\pi$
$798$	0	0
$799$	41029.9	1.81669
$800$	−17661.3	−0.780528
$801$	0	0
$802$	−11899.3	−0.523913
$803$	6959.41	0.305844
$804$	0	0
$805$	51445.9	2.25246
$806$	0	0
$807$	0	0
$808$	−28042.6	−1.22096
$809$	6517.83	0.283257	0.141628	−	0.989920i	$-0.454766\pi$
$809$	6517.83	0.283257	0.141628	+	0.989920i	$0.454766\pi$
$810$	0	0
$811$	−2898.99	−0.125521	−0.0627603	−	0.998029i	$-0.519990\pi$
$811$	−2898.99	−0.125521	−0.0627603	+	0.998029i	$0.519990\pi$
$812$	7498.37	0.324066
$813$	0	0
$814$	−43089.3	−1.85538
$815$	41572.9	1.78679
$816$	0	0
$817$	−34866.1	−1.49303
$818$	21502.7	0.919099
$819$	0	0
$820$	5202.12	0.221544
$821$	716.621	0.0304632	0.0152316	−	0.999884i	$-0.495151\pi$
$821$	716.621	0.0304632	0.0152316	+	0.999884i	$0.495151\pi$
$822$	0	0
$823$	15510.0	0.656920	0.328460	−	0.944518i	$-0.393470\pi$
$823$	15510.0	0.656920	0.328460	+	0.944518i	$0.393470\pi$
$824$	−24212.5	−1.02364
$825$	0	0
$826$	20289.2	0.854664
$827$	−24063.2	−1.01180	−0.505901	−	0.862591i	$-0.668840\pi$
$827$	−24063.2	−1.01180	−0.505901	+	0.862591i	$0.668840\pi$
$828$	0	0
$829$	10037.8	0.420540	0.210270	−	0.977643i	$-0.432566\pi$
$829$	10037.8	0.420540	0.210270	+	0.977643i	$0.432566\pi$
$830$	13514.5	0.565173
$831$	0	0
$832$	0	0
$833$	−3429.30	−0.142639
$834$	0	0
$835$	−4641.96	−0.192385
$836$	−16679.0	−0.690020
$837$	0	0
$838$	−30125.6	−1.24185
$839$	−4005.19	−0.164809	−0.0824044	−	0.996599i	$-0.526260\pi$
$839$	−4005.19	−0.164809	−0.0824044	+	0.996599i	$0.526260\pi$
$840$	0	0
$841$	−7952.74	−0.326079
$842$	3683.74	0.150772
$843$	0	0
$844$	4721.53	0.192562
$845$	0	0
$846$	0	0
$847$	25045.2	1.01601
$848$	−24506.6	−0.992406
$849$	0	0
$850$	29255.1	1.18052
$851$	46065.8	1.85560
$852$	0	0
$853$	−44062.6	−1.76867	−0.884334	−	0.466856i	$-0.845387\pi$
$853$	−44062.6	−1.76867	−0.884334	+	0.466856i	$0.845387\pi$
$854$	7390.96	0.296152
$855$	0	0
$856$	4999.36	0.199620
$857$	−3365.44	−0.134144	−0.0670718	−	0.997748i	$-0.521366\pi$
$857$	−3365.44	−0.134144	−0.0670718	+	0.997748i	$0.521366\pi$
$858$	0	0
$859$	−40969.3	−1.62731	−0.813653	−	0.581351i	$-0.802524\pi$
$859$	−40969.3	−1.62731	−0.813653	+	0.581351i	$0.802524\pi$
$860$	−20135.2	−0.798377
$861$	0	0
$862$	−12700.4	−0.501831
$863$	−18132.8	−0.715236	−0.357618	−	0.933868i	$-0.616411\pi$
$863$	−18132.8	−0.715236	−0.357618	+	0.933868i	$0.616411\pi$
$864$	0	0
$865$	23497.7	0.923637
$866$	−13796.8	−0.541380
$867$	0	0
$868$	−192.819	−0.00753999
$869$	−29380.1	−1.14690
$870$	0	0
$871$	0	0
$872$	24940.9	0.968582
$873$	0	0
$874$	59703.5	2.31064
$875$	839.404	0.0324309
$876$	0	0
$877$	−33047.6	−1.27245	−0.636226	−	0.771503i	$-0.719505\pi$
$877$	−33047.6	−1.27245	−0.636226	+	0.771503i	$0.719505\pi$
$878$	31698.0	1.21840
$879$	0	0
$880$	66101.9	2.53215
$881$	−1349.22	−0.0515965	−0.0257982	−	0.999667i	$-0.508213\pi$
$881$	−1349.22	−0.0515965	−0.0257982	+	0.999667i	$0.508213\pi$
$882$	0	0
$883$	−33934.5	−1.29330	−0.646651	−	0.762786i	$-0.723831\pi$
$883$	−33934.5	−1.29330	−0.646651	+	0.762786i	$0.723831\pi$
$884$	0	0
$885$	0	0
$886$	−19438.3	−0.737067
$887$	45609.5	1.72651	0.863256	−	0.504766i	$-0.168421\pi$
$887$	45609.5	1.72651	0.863256	+	0.504766i	$0.168421\pi$
$888$	0	0
$889$	40711.8	1.53592
$890$	−9733.19	−0.366581
$891$	0	0
$892$	381.037	0.0143028
$893$	−53515.9	−2.00542
$894$	0	0
$895$	12336.0	0.460722
$896$	28471.4	1.06157
$897$	0	0
$898$	3730.90	0.138643
$899$	−422.655	−0.0156800
$900$	0	0
$901$	21865.8	0.808498
$902$	−17336.3	−0.639952
$903$	0	0
$904$	−6669.18	−0.245369
$905$	20222.3	0.742777
$906$	0	0
$907$	8620.99	0.315607	0.157803	−	0.987471i	$-0.449559\pi$
$907$	8620.99	0.315607	0.157803	+	0.987471i	$0.449559\pi$
$908$	17003.1	0.621438
$909$	0	0
$910$	0	0
$911$	43993.6	1.59997	0.799986	−	0.600019i	$-0.204840\pi$
$911$	43993.6	1.59997	0.799986	+	0.600019i	$0.204840\pi$
$912$	0	0
$913$	−13451.0	−0.487584
$914$	45758.6	1.65597
$915$	0	0
$916$	19716.9	0.711205
$917$	−2904.84	−0.104609
$918$	0	0
$919$	15176.0	0.544733	0.272366	−	0.962194i	$-0.412194\pi$
$919$	15176.0	0.544733	0.272366	+	0.962194i	$0.412194\pi$
$920$	−46486.6	−1.66589
$921$	0	0
$922$	10118.8	0.361436
$923$	0	0
$924$	0	0
$925$	−29441.7	−1.04653
$926$	6842.94	0.242843
$927$	0	0
$928$	−18576.4	−0.657114
$929$	−29749.7	−1.05065	−0.525326	−	0.850901i	$-0.676057\pi$
$929$	−29749.7	−1.05065	−0.525326	+	0.850901i	$0.676057\pi$
$930$	0	0
$931$	4472.88	0.157457
$932$	2465.50	0.0866526
$933$	0	0
$934$	11092.0	0.388589
$935$	−58978.9	−2.06291
$936$	0	0
$937$	−46456.0	−1.61969	−0.809846	−	0.586643i	$-0.800449\pi$
$937$	−46456.0	−1.61969	−0.809846	+	0.586643i	$0.800449\pi$
$938$	52391.8	1.82372
$939$	0	0
$940$	−30905.5	−1.07237
$941$	3243.36	0.112360	0.0561799	−	0.998421i	$-0.482108\pi$
$941$	3243.36	0.112360	0.0561799	+	0.998421i	$0.482108\pi$
$942$	0	0
$943$	18533.9	0.640027
$944$	−27870.0	−0.960901
$945$	0	0
$946$	67101.5	2.30619
$947$	53344.1	1.83047	0.915233	−	0.402926i	$-0.132007\pi$
$947$	53344.1	1.83047	0.915233	+	0.402926i	$0.132007\pi$
$948$	0	0
$949$	0	0
$950$	−38157.8	−1.30316
$951$	0	0
$952$	−18927.0	−0.644356
$953$	−53496.9	−1.81840	−0.909200	−	0.416360i	$-0.863306\pi$
$953$	−53496.9	−1.81840	−0.909200	+	0.416360i	$0.863306\pi$
$954$	0	0
$955$	7329.22	0.248343
$956$	−4223.50	−0.142885
$957$	0	0
$958$	44610.1	1.50448
$959$	51095.9	1.72051
$960$	0	0
$961$	−29780.1	−0.999635
$962$	0	0
$963$	0	0
$964$	2573.45	0.0859805
$965$	41434.1	1.38219
$966$	0	0
$967$	−48822.0	−1.62359	−0.811794	−	0.583944i	$-0.801509\pi$
$967$	−48822.0	−1.62359	−0.811794	+	0.583944i	$0.801509\pi$
$968$	−22630.9	−0.751429
$969$	0	0
$970$	−41419.4	−1.37103
$971$	−36228.1	−1.19734	−0.598670	−	0.800996i	$-0.704304\pi$
$971$	−36228.1	−1.19734	−0.598670	+	0.800996i	$0.704304\pi$
$972$	0	0
$973$	−43867.1	−1.44534
$974$	−30309.0	−0.997087
$975$	0	0
$976$	−10152.5	−0.332964
$977$	−46689.8	−1.52890	−0.764451	−	0.644681i	$-0.776990\pi$
$977$	−46689.8	−1.52890	−0.764451	+	0.644681i	$0.776990\pi$
$978$	0	0
$979$	9687.50	0.316255
$980$	2583.09	0.0841977
$981$	0	0
$982$	43533.8	1.41468
$983$	−11501.9	−0.373197	−0.186598	−	0.982436i	$-0.559746\pi$
$983$	−11501.9	−0.373197	−0.186598	+	0.982436i	$0.559746\pi$
$984$	0	0
$985$	−57774.0	−1.86886
$986$	30771.0	0.993862
$987$	0	0
$988$	0	0
$989$	−71736.6	−2.30646
$990$	0	0
$991$	7137.46	0.228788	0.114394	−	0.993435i	$-0.463507\pi$
$991$	7137.46	0.228788	0.114394	+	0.993435i	$0.463507\pi$
$992$	477.690	0.0152890
$993$	0	0
$994$	47900.5	1.52848
$995$	−3465.53	−0.110417
$996$	0	0
$997$	−18389.5	−0.584152	−0.292076	−	0.956395i	$-0.594346\pi$
$997$	−18389.5	−0.584152	−0.292076	+	0.956395i	$0.594346\pi$
$998$	−24897.9	−0.789708
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bj.1.7		9
3.2	odd	2		507.4.a.n.1.3	✓	9
13.12	even	2		1521.4.a.be.1.3		9
39.5	even	4		507.4.b.j.337.15		18
39.8	even	4		507.4.b.j.337.4		18
39.38	odd	2		507.4.a.q.1.7	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.3	✓	9	3.2	odd	2
507.4.a.q.1.7	yes	9	39.38	odd	2
507.4.b.j.337.4		18	39.8	even	4
507.4.b.j.337.15		18	39.5	even	4
1521.4.a.be.1.3		9	13.12	even	2
1521.4.a.bj.1.7		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+3.37739 q^{2} +3.40677 q^{4} -15.7127 q^{5} +17.1681 q^{7} -15.5131 q^{8} +O(q^{10})\)	\(q+3.37739 q^{2} +3.40677 q^{4} -15.7127 q^{5} +17.1681 q^{7} -15.5131 q^{8} -53.0679 q^{10} +52.8187 q^{11} +57.9835 q^{14} -79.6481 q^{16} +71.0654 q^{17} -92.6916 q^{19} -53.5295 q^{20} +178.390 q^{22} -190.712 q^{23} +121.888 q^{25} +58.4878 q^{28} +128.204 q^{29} -3.29674 q^{31} -144.898 q^{32} +240.016 q^{34} -269.757 q^{35} -241.546 q^{37} -313.056 q^{38} +243.753 q^{40} -97.1824 q^{41} +376.151 q^{43} +179.941 q^{44} -644.109 q^{46} +577.354 q^{47} -48.2555 q^{49} +411.664 q^{50} +307.686 q^{53} -829.924 q^{55} -266.331 q^{56} +432.995 q^{58} +349.914 q^{59} +127.467 q^{61} -11.1344 q^{62} +147.809 q^{64} +903.564 q^{67} +242.104 q^{68} -911.076 q^{70} +826.106 q^{71} +131.760 q^{73} -815.796 q^{74} -315.779 q^{76} +906.799 q^{77} -556.244 q^{79} +1251.48 q^{80} -328.223 q^{82} -254.664 q^{83} -1116.63 q^{85} +1270.41 q^{86} -819.384 q^{88} +183.410 q^{89} -649.712 q^{92} +1949.95 q^{94} +1456.43 q^{95} +780.498 q^{97} -162.978 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

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Database

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