LMFDB - Embedded newform 1521.4.a.be.1.1

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			1.1
Root			$-4.48584$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.48584 q^{2} +22.0945 q^{4} -13.3185 q^{5} +21.4234 q^{7} -77.3200 q^{8} +O(q^{10})$	$q-5.48584 q^{2} +22.0945 q^{4} -13.3185 q^{5} +21.4234 q^{7} -77.3200 q^{8} +73.0635 q^{10} +19.0520 q^{11} -117.525 q^{14} +247.410 q^{16} +71.7906 q^{17} -102.134 q^{19} -294.266 q^{20} -104.516 q^{22} +37.8302 q^{23} +52.3838 q^{25} +473.338 q^{28} -40.8605 q^{29} -6.05542 q^{31} -738.690 q^{32} -393.832 q^{34} -285.328 q^{35} -285.682 q^{37} +560.293 q^{38} +1029.79 q^{40} +342.705 q^{41} +306.458 q^{43} +420.943 q^{44} -207.530 q^{46} -346.863 q^{47} +115.961 q^{49} -287.369 q^{50} -398.219 q^{53} -253.745 q^{55} -1656.46 q^{56} +224.155 q^{58} +208.497 q^{59} +546.936 q^{61} +33.2191 q^{62} +2073.06 q^{64} -678.268 q^{67} +1586.18 q^{68} +1565.27 q^{70} -957.777 q^{71} +270.360 q^{73} +1567.21 q^{74} -2256.60 q^{76} +408.157 q^{77} -1032.86 q^{79} -3295.14 q^{80} -1880.03 q^{82} +1065.90 q^{83} -956.147 q^{85} -1681.18 q^{86} -1473.10 q^{88} -427.185 q^{89} +835.837 q^{92} +1902.83 q^{94} +1360.28 q^{95} +698.084 q^{97} -636.144 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.48584	−1.93954	−0.969769	−	0.244025i	$-0.921532\pi$
$2$	−5.48584	−1.93954	−0.969769	+	0.244025i	$0.921532\pi$
$3$	0	0
$3$	0	0
$4$	22.0945	2.76181
$5$	−13.3185	−1.19125	−0.595624	−	0.803264i	$-0.703095\pi$
$5$	−13.3185	−1.19125	−0.595624	+	0.803264i	$0.703095\pi$
$6$	0	0
$7$	21.4234	1.15675	0.578377	−	0.815770i	$-0.303686\pi$
$7$	21.4234	1.15675	0.578377	+	0.815770i	$0.303686\pi$
$8$	−77.3200	−3.41709
$9$	0	0
$10$	73.0635	2.31047
$11$	19.0520	0.522217	0.261108	−	0.965310i	$-0.415912\pi$
$11$	19.0520	0.522217	0.261108	+	0.965310i	$0.415912\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−117.525	−2.24357
$15$	0	0
$16$	247.410	3.86578
$17$	71.7906	1.02422	0.512111	−	0.858919i	$-0.328864\pi$
$17$	71.7906	1.02422	0.512111	+	0.858919i	$0.328864\pi$
$18$	0	0
$19$	−102.134	−1.23322	−0.616611	−	0.787268i	$-0.711495\pi$
$19$	−102.134	−1.23322	−0.616611	+	0.787268i	$0.711495\pi$
$20$	−294.266	−3.29000
$21$	0	0
$22$	−104.516	−1.01286
$23$	37.8302	0.342962	0.171481	−	0.985187i	$-0.445145\pi$
$23$	37.8302	0.342962	0.171481	+	0.985187i	$0.445145\pi$
$24$	0	0
$25$	52.3838	0.419070
$26$	0	0
$27$	0	0
$28$	473.338	3.19473
$29$	−40.8605	−0.261642	−0.130821	−	0.991406i	$-0.541761\pi$
$29$	−40.8605	−0.261642	−0.130821	+	0.991406i	$0.541761\pi$
$30$	0	0
$31$	−6.05542	−0.0350834	−0.0175417	−	0.999846i	$-0.505584\pi$
$31$	−6.05542	−0.0350834	−0.0175417	+	0.999846i	$0.505584\pi$
$32$	−738.690	−4.08073
$33$	0	0
$34$	−393.832	−1.98652
$35$	−285.328	−1.37798
$36$	0	0
$37$	−285.682	−1.26935	−0.634673	−	0.772781i	$-0.718865\pi$
$37$	−285.682	−1.26935	−0.634673	+	0.772781i	$0.718865\pi$
$38$	560.293	2.39188
$39$	0	0
$40$	1029.79	4.07060
$41$	342.705	1.30540	0.652702	−	0.757615i	$-0.273636\pi$
$41$	342.705	1.30540	0.652702	+	0.757615i	$0.273636\pi$
$42$	0	0
$43$	306.458	1.08685	0.543424	−	0.839458i	$-0.317128\pi$
$43$	306.458	1.08685	0.543424	+	0.839458i	$0.317128\pi$
$44$	420.943	1.44226
$45$	0	0
$46$	−207.530	−0.665188
$47$	−346.863	−1.07649	−0.538246	−	0.842788i	$-0.680913\pi$
$47$	−346.863	−1.07649	−0.538246	+	0.842788i	$0.680913\pi$
$48$	0	0
$49$	115.961	0.338079
$50$	−287.369	−0.812802
$51$	0	0
$52$	0	0
$53$	−398.219	−1.03207	−0.516034	−	0.856568i	$-0.672592\pi$
$53$	−398.219	−1.03207	−0.516034	+	0.856568i	$0.672592\pi$
$54$	0	0
$55$	−253.745	−0.622089
$56$	−1656.46	−3.95274
$57$	0	0
$58$	224.155	0.507464
$59$	208.497	0.460067	0.230034	−	0.973183i	$-0.426116\pi$
$59$	208.497	0.460067	0.230034	+	0.973183i	$0.426116\pi$
$60$	0	0
$61$	546.936	1.14800	0.574000	−	0.818855i	$-0.305391\pi$
$61$	546.936	1.14800	0.574000	+	0.818855i	$0.305391\pi$
$62$	33.2191	0.0680456
$63$	0	0
$64$	2073.06	4.04895
$65$	0	0
$66$	0	0
$67$	−678.268	−1.23677	−0.618386	−	0.785875i	$-0.712213\pi$
$67$	−678.268	−1.23677	−0.618386	+	0.785875i	$0.712213\pi$
$68$	1586.18	2.82871
$69$	0	0
$70$	1565.27	2.67264
$71$	−957.777	−1.60095	−0.800474	−	0.599368i	$-0.795419\pi$
$71$	−957.777	−1.60095	−0.800474	+	0.599368i	$0.795419\pi$
$72$	0	0
$73$	270.360	0.433469	0.216734	−	0.976231i	$-0.430459\pi$
$73$	270.360	0.433469	0.216734	+	0.976231i	$0.430459\pi$
$74$	1567.21	2.46195
$75$	0	0
$76$	−2256.60	−3.40592
$77$	408.157	0.604076
$78$	0	0
$79$	−1032.86	−1.47096	−0.735482	−	0.677544i	$-0.763044\pi$
$79$	−1032.86	−1.47096	−0.735482	+	0.677544i	$0.763044\pi$
$80$	−3295.14	−4.60510
$81$	0	0
$82$	−1880.03	−2.53188
$83$	1065.90	1.40961	0.704806	−	0.709400i	$-0.251034\pi$
$83$	1065.90	1.40961	0.704806	+	0.709400i	$0.251034\pi$
$84$	0	0
$85$	−956.147	−1.22010
$86$	−1681.18	−2.10798
$87$	0	0
$88$	−1473.10	−1.78446
$89$	−427.185	−0.508782	−0.254391	−	0.967101i	$-0.581875\pi$
$89$	−427.185	−0.508782	−0.254391	+	0.967101i	$0.581875\pi$
$90$	0	0
$91$	0	0
$92$	835.837	0.947196
$93$	0	0
$94$	1902.83	2.08790
$95$	1360.28	1.46907
$96$	0	0
$97$	698.084	0.730718	0.365359	−	0.930867i	$-0.380946\pi$
$97$	698.084	0.730718	0.365359	+	0.930867i	$0.380946\pi$
$98$	−636.144	−0.655717
$99$	0	0
$100$	1157.39	1.15739
$101$	−88.7364	−0.0874218	−0.0437109	−	0.999044i	$-0.513918\pi$
$101$	−88.7364	−0.0874218	−0.0437109	+	0.999044i	$0.513918\pi$
$102$	0	0
$103$	−1427.61	−1.36570	−0.682849	−	0.730560i	$-0.739259\pi$
$103$	−1427.61	−1.36570	−0.682849	+	0.730560i	$0.739259\pi$
$104$	0	0
$105$	0	0
$106$	2184.57	2.00173
$107$	15.0233	0.0135735	0.00678673	−	0.999977i	$-0.497840\pi$
$107$	15.0233	0.0135735	0.00678673	+	0.999977i	$0.497840\pi$
$108$	0	0
$109$	2053.56	1.80455	0.902274	−	0.431163i	$-0.141896\pi$
$109$	2053.56	1.80455	0.902274	+	0.431163i	$0.141896\pi$
$110$	1392.00	1.20657
$111$	0	0
$112$	5300.35	4.47175
$113$	−717.456	−0.597280	−0.298640	−	0.954366i	$-0.596533\pi$
$113$	−717.456	−0.597280	−0.298640	+	0.954366i	$0.596533\pi$
$114$	0	0
$115$	−503.843	−0.408553
$116$	−902.792	−0.722605
$117$	0	0
$118$	−1143.78	−0.892318
$119$	1538.00	1.18477
$120$	0	0
$121$	−968.023	−0.727290
$122$	−3000.41	−2.22659
$123$	0	0
$124$	−133.791	−0.0968937
$125$	967.143	0.692031
$126$	0	0
$127$	−117.640	−0.0821955	−0.0410978	−	0.999155i	$-0.513086\pi$
$127$	−117.640	−0.0821955	−0.0410978	+	0.999155i	$0.513086\pi$
$128$	−5462.96	−3.77236
$129$	0	0
$130$	0	0
$131$	262.376	0.174991	0.0874957	−	0.996165i	$-0.472114\pi$
$131$	262.376	0.174991	0.0874957	+	0.996165i	$0.472114\pi$
$132$	0	0
$133$	−2188.06	−1.42653
$134$	3720.87	2.39876
$135$	0	0
$136$	−5550.85	−3.49986
$137$	1317.41	0.821563	0.410782	−	0.911734i	$-0.365256\pi$
$137$	1317.41	0.821563	0.410782	+	0.911734i	$0.365256\pi$
$138$	0	0
$139$	−6.43478	−0.00392656	−0.00196328	−	0.999998i	$-0.500625\pi$
$139$	−6.43478	−0.00392656	−0.00196328	+	0.999998i	$0.500625\pi$
$140$	−6304.18	−3.80572
$141$	0	0
$142$	5254.22	3.10510
$143$	0	0
$144$	0	0
$145$	544.203	0.311680
$146$	−1483.15	−0.840729
$147$	0	0
$148$	−6311.99	−3.50569
$149$	−548.312	−0.301473	−0.150736	−	0.988574i	$-0.548164\pi$
$149$	−548.312	−0.301473	−0.150736	+	0.988574i	$0.548164\pi$
$150$	0	0
$151$	1485.90	0.800799	0.400399	−	0.916341i	$-0.368871\pi$
$151$	1485.90	0.800799	0.400399	+	0.916341i	$0.368871\pi$
$152$	7897.03	4.21404
$153$	0	0
$154$	−2239.09	−1.17163
$155$	80.6494	0.0417930
$156$	0	0
$157$	2132.36	1.08396	0.541978	−	0.840393i	$-0.317676\pi$
$157$	2132.36	1.08396	0.541978	+	0.840393i	$0.317676\pi$
$158$	5666.13	2.85299
$159$	0	0
$160$	9838.28	4.86115
$161$	810.450	0.396723
$162$	0	0
$163$	480.633	0.230957	0.115479	−	0.993310i	$-0.463160\pi$
$163$	480.633	0.230957	0.115479	+	0.993310i	$0.463160\pi$
$164$	7571.88	3.60527
$165$	0	0
$166$	−5847.36	−2.73400
$167$	2919.71	1.35290	0.676448	−	0.736490i	$-0.263518\pi$
$167$	2919.71	1.35290	0.676448	+	0.736490i	$0.263518\pi$
$168$	0	0
$169$	0	0
$170$	5245.27	2.36643
$171$	0	0
$172$	6771.04	3.00167
$173$	2339.62	1.02820	0.514098	−	0.857731i	$-0.328127\pi$
$173$	2339.62	1.02820	0.514098	+	0.857731i	$0.328127\pi$
$174$	0	0
$175$	1122.24	0.484761
$176$	4713.64	2.01877
$177$	0	0
$178$	2343.47	0.986802
$179$	4558.71	1.90354	0.951772	−	0.306808i	$-0.0992608\pi$
$179$	4558.71	1.90354	0.951772	+	0.306808i	$0.0992608\pi$
$180$	0	0
$181$	−3524.56	−1.44739	−0.723696	−	0.690118i	$-0.757558\pi$
$181$	−3524.56	−1.44739	−0.723696	+	0.690118i	$0.757558\pi$
$182$	0	0
$183$	0	0
$184$	−2925.03	−1.17193
$185$	3804.87	1.51211
$186$	0	0
$187$	1367.75	0.534866
$188$	−7663.75	−2.97306
$189$	0	0
$190$	−7462.29	−2.84932
$191$	−3343.27	−1.26655	−0.633273	−	0.773929i	$-0.718289\pi$
$191$	−3343.27	−1.26655	−0.633273	+	0.773929i	$0.718289\pi$
$192$	0	0
$193$	3341.33	1.24619	0.623094	−	0.782147i	$-0.285876\pi$
$193$	3341.33	1.24619	0.623094	+	0.782147i	$0.285876\pi$
$194$	−3829.58	−1.41726
$195$	0	0
$196$	2562.10	0.933709
$197$	−3653.95	−1.32149	−0.660745	−	0.750611i	$-0.729759\pi$
$197$	−3653.95	−1.32149	−0.660745	+	0.750611i	$0.729759\pi$
$198$	0	0
$199$	1126.55	0.401301	0.200651	−	0.979663i	$-0.435694\pi$
$199$	1126.55	0.401301	0.200651	+	0.979663i	$0.435694\pi$
$200$	−4050.31	−1.43200
$201$	0	0
$202$	486.794	0.169558
$203$	−875.371	−0.302655
$204$	0	0
$205$	−4564.33	−1.55506
$206$	7831.66	2.64882
$207$	0	0
$208$	0	0
$209$	−1945.86	−0.644009
$210$	0	0
$211$	−74.5243	−0.0243150	−0.0121575	−	0.999926i	$-0.503870\pi$
$211$	−74.5243	−0.0243150	−0.0121575	+	0.999926i	$0.503870\pi$
$212$	−8798.44	−2.85037
$213$	0	0
$214$	−82.4156	−0.0263262
$215$	−4081.58	−1.29471
$216$	0	0
$217$	−129.728	−0.0405829
$218$	−11265.5	−3.49999
$219$	0	0
$220$	−5606.35	−1.71809
$221$	0	0
$222$	0	0
$223$	−3178.62	−0.954512	−0.477256	−	0.878764i	$-0.658369\pi$
$223$	−3178.62	−0.954512	−0.477256	+	0.878764i	$0.658369\pi$
$224$	−15825.2	−4.72039
$225$	0	0
$226$	3935.85	1.15845
$227$	−1349.76	−0.394655	−0.197327	−	0.980338i	$-0.563226\pi$
$227$	−1349.76	−0.394655	−0.197327	+	0.980338i	$0.563226\pi$
$228$	0	0
$229$	4821.46	1.39131	0.695657	−	0.718374i	$-0.255113\pi$
$229$	4821.46	1.39131	0.695657	+	0.718374i	$0.255113\pi$
$230$	2764.00	0.792404
$231$	0	0
$232$	3159.34	0.894055
$233$	2400.88	0.675050	0.337525	−	0.941317i	$-0.390410\pi$
$233$	2400.88	0.675050	0.337525	+	0.941317i	$0.390410\pi$
$234$	0	0
$235$	4619.71	1.28237
$236$	4606.63	1.27062
$237$	0	0
$238$	−8437.21	−2.29791
$239$	−1880.85	−0.509047	−0.254523	−	0.967067i	$-0.581919\pi$
$239$	−1880.85	−0.509047	−0.254523	+	0.967067i	$0.581919\pi$
$240$	0	0
$241$	−5435.01	−1.45270	−0.726349	−	0.687327i	$-0.758784\pi$
$241$	−5435.01	−1.45270	−0.726349	+	0.687327i	$0.758784\pi$
$242$	5310.42	1.41061
$243$	0	0
$244$	12084.3	3.17056
$245$	−1544.43	−0.402736
$246$	0	0
$247$	0	0
$248$	468.205	0.119883
$249$	0	0
$250$	−5305.59	−1.34222
$251$	−1256.70	−0.316024	−0.158012	−	0.987437i	$-0.550508\pi$
$251$	−1256.70	−0.316024	−0.158012	+	0.987437i	$0.550508\pi$
$252$	0	0
$253$	720.739	0.179101
$254$	645.353	0.159421
$255$	0	0
$256$	13384.5	3.26769
$257$	−5504.87	−1.33612	−0.668062	−	0.744105i	$-0.732876\pi$
$257$	−5504.87	−1.33612	−0.668062	+	0.744105i	$0.732876\pi$
$258$	0	0
$259$	−6120.27	−1.46832
$260$	0	0
$261$	0	0
$262$	−1439.35	−0.339402
$263$	−2032.44	−0.476522	−0.238261	−	0.971201i	$-0.576577\pi$
$263$	−2032.44	−0.476522	−0.238261	+	0.971201i	$0.576577\pi$
$264$	0	0
$265$	5303.70	1.22945
$266$	12003.4	2.76682
$267$	0	0
$268$	−14986.0	−3.41572
$269$	−5523.82	−1.25202	−0.626009	−	0.779816i	$-0.715313\pi$
$269$	−5523.82	−1.25202	−0.626009	+	0.779816i	$0.715313\pi$
$270$	0	0
$271$	−3918.69	−0.878389	−0.439194	−	0.898392i	$-0.644736\pi$
$271$	−3918.69	−0.878389	−0.439194	+	0.898392i	$0.644736\pi$
$272$	17761.7	3.95941
$273$	0	0
$274$	−7227.12	−1.59345
$275$	998.013	0.218845
$276$	0	0
$277$	−2944.56	−0.638705	−0.319353	−	0.947636i	$-0.603465\pi$
$277$	−2944.56	−0.638705	−0.319353	+	0.947636i	$0.603465\pi$
$278$	35.3002	0.00761570
$279$	0	0
$280$	22061.6	4.70869
$281$	−5048.76	−1.07183	−0.535914	−	0.844273i	$-0.680033\pi$
$281$	−5048.76	−1.07183	−0.535914	+	0.844273i	$0.680033\pi$
$282$	0	0
$283$	4491.66	0.943469	0.471734	−	0.881741i	$-0.343628\pi$
$283$	4491.66	0.943469	0.471734	+	0.881741i	$0.343628\pi$
$284$	−21161.6	−4.42151
$285$	0	0
$286$	0	0
$287$	7341.90	1.51003
$288$	0	0
$289$	240.893	0.0490318
$290$	−2985.41	−0.604515
$291$	0	0
$292$	5973.45	1.19716
$293$	−4754.86	−0.948061	−0.474031	−	0.880508i	$-0.657201\pi$
$293$	−4754.86	−0.948061	−0.474031	+	0.880508i	$0.657201\pi$
$294$	0	0
$295$	−2776.88	−0.548054
$296$	22088.9	4.33747
$297$	0	0
$298$	3007.95	0.584718
$299$	0	0
$300$	0	0
$301$	6565.38	1.25722
$302$	−8151.40	−1.55318
$303$	0	0
$304$	−25269.0	−4.76736
$305$	−7284.40	−1.36755
$306$	0	0
$307$	1623.31	0.301783	0.150891	−	0.988550i	$-0.451786\pi$
$307$	1623.31	0.301783	0.150891	+	0.988550i	$0.451786\pi$
$308$	9018.02	1.66834
$309$	0	0
$310$	−442.430	−0.0810592
$311$	−6683.49	−1.21860	−0.609302	−	0.792938i	$-0.708550\pi$
$311$	−6683.49	−1.21860	−0.609302	+	0.792938i	$0.708550\pi$
$312$	0	0
$313$	1584.55	0.286147	0.143073	−	0.989712i	$-0.454301\pi$
$313$	1584.55	0.286147	0.143073	+	0.989712i	$0.454301\pi$
$314$	−11697.8	−2.10237
$315$	0	0
$316$	−22820.6	−4.06252
$317$	−787.932	−0.139605	−0.0698023	−	0.997561i	$-0.522237\pi$
$317$	−787.932	−0.139605	−0.0698023	+	0.997561i	$0.522237\pi$
$318$	0	0
$319$	−778.474	−0.136634
$320$	−27610.2	−4.82330
$321$	0	0
$322$	−4446.00	−0.769459
$323$	−7332.28	−1.26309
$324$	0	0
$325$	0	0
$326$	−2636.67	−0.447951
$327$	0	0
$328$	−26498.0	−4.46069
$329$	−7430.97	−1.24524
$330$	0	0
$331$	−9493.56	−1.57647	−0.788237	−	0.615371i	$-0.789006\pi$
$331$	−9493.56	−1.57647	−0.788237	+	0.615371i	$0.789006\pi$
$332$	23550.5	3.89308
$333$	0	0
$334$	−16017.1	−2.62399
$335$	9033.55	1.47330
$336$	0	0
$337$	−4123.06	−0.666461	−0.333230	−	0.942845i	$-0.608139\pi$
$337$	−4123.06	−0.666461	−0.333230	+	0.942845i	$0.608139\pi$
$338$	0	0
$339$	0	0
$340$	−21125.6	−3.36969
$341$	−115.368	−0.0183211
$342$	0	0
$343$	−4863.94	−0.765680
$344$	−23695.4	−3.71386
$345$	0	0
$346$	−12834.8	−1.99423
$347$	−2320.76	−0.359034	−0.179517	−	0.983755i	$-0.557454\pi$
$347$	−2320.76	−0.359034	−0.179517	+	0.983755i	$0.557454\pi$
$348$	0	0
$349$	3818.85	0.585726	0.292863	−	0.956154i	$-0.405392\pi$
$349$	3818.85	0.585726	0.292863	+	0.956154i	$0.405392\pi$
$350$	−6156.41	−0.940212
$351$	0	0
$352$	−14073.5	−2.13102
$353$	6065.53	0.914549	0.457274	−	0.889326i	$-0.348826\pi$
$353$	6065.53	0.914549	0.457274	+	0.889326i	$0.348826\pi$
$354$	0	0
$355$	12756.2	1.90712
$356$	−9438.43	−1.40516
$357$	0	0
$358$	−25008.4	−3.69199
$359$	−7406.02	−1.08879	−0.544394	−	0.838830i	$-0.683240\pi$
$359$	−7406.02	−1.08879	−0.544394	+	0.838830i	$0.683240\pi$
$360$	0	0
$361$	3572.42	0.520836
$362$	19335.2	2.80727
$363$	0	0
$364$	0	0
$365$	−3600.80	−0.516368
$366$	0	0
$367$	−4754.23	−0.676209	−0.338105	−	0.941109i	$-0.609786\pi$
$367$	−4754.23	−0.676209	−0.338105	+	0.941109i	$0.609786\pi$
$368$	9359.55	1.32582
$369$	0	0
$370$	−20872.9	−2.93279
$371$	−8531.20	−1.19385
$372$	0	0
$373$	−4684.13	−0.650229	−0.325114	−	0.945675i	$-0.605403\pi$
$373$	−4684.13	−0.650229	−0.325114	+	0.945675i	$0.605403\pi$
$374$	−7503.27	−1.03739
$375$	0	0
$376$	26819.4	3.67847
$377$	0	0
$378$	0	0
$379$	12103.9	1.64047	0.820233	−	0.572030i	$-0.193844\pi$
$379$	12103.9	1.64047	0.820233	+	0.572030i	$0.193844\pi$
$380$	30054.7	4.05730
$381$	0	0
$382$	18340.6	2.45651
$383$	5601.56	0.747327	0.373663	−	0.927564i	$-0.378102\pi$
$383$	5601.56	0.747327	0.373663	+	0.927564i	$0.378102\pi$
$384$	0	0
$385$	−5436.06	−0.719604
$386$	−18330.0	−2.41703
$387$	0	0
$388$	15423.8	2.01810
$389$	−9450.46	−1.23177	−0.615883	−	0.787837i	$-0.711201\pi$
$389$	−9450.46	−1.23177	−0.615883	+	0.787837i	$0.711201\pi$
$390$	0	0
$391$	2715.85	0.351270
$392$	−8966.11	−1.15525
$393$	0	0
$394$	20045.0	2.56308
$395$	13756.2	1.75228
$396$	0	0
$397$	−2723.98	−0.344364	−0.172182	−	0.985065i	$-0.555082\pi$
$397$	−2723.98	−0.344364	−0.172182	+	0.985065i	$0.555082\pi$
$398$	−6180.07	−0.778339
$399$	0	0
$400$	12960.2	1.62003
$401$	−4680.46	−0.582870	−0.291435	−	0.956591i	$-0.594133\pi$
$401$	−4680.46	−0.582870	−0.291435	+	0.956591i	$0.594133\pi$
$402$	0	0
$403$	0	0
$404$	−1960.58	−0.241442
$405$	0	0
$406$	4802.15	0.587011
$407$	−5442.80	−0.662874
$408$	0	0
$409$	−3272.70	−0.395659	−0.197830	−	0.980236i	$-0.563389\pi$
$409$	−3272.70	−0.395659	−0.197830	+	0.980236i	$0.563389\pi$
$410$	25039.2	3.01609
$411$	0	0
$412$	−31542.3	−3.77179
$413$	4466.71	0.532185
$414$	0	0
$415$	−14196.2	−1.67920
$416$	0	0
$417$	0	0
$418$	10674.7	1.24908
$419$	−1501.48	−0.175065	−0.0875323	−	0.996162i	$-0.527898\pi$
$419$	−1501.48	−0.175065	−0.0875323	+	0.996162i	$0.527898\pi$
$420$	0	0
$421$	−16578.1	−1.91916	−0.959580	−	0.281436i	$-0.909189\pi$
$421$	−16578.1	−1.91916	−0.959580	+	0.281436i	$0.909189\pi$
$422$	408.828	0.0471598
$423$	0	0
$424$	30790.3	3.52667
$425$	3760.66	0.429221
$426$	0	0
$427$	11717.2	1.32795
$428$	331.933	0.0374873
$429$	0	0
$430$	22390.9	2.51113
$431$	3776.55	0.422065	0.211032	−	0.977479i	$-0.432317\pi$
$431$	3776.55	0.422065	0.211032	+	0.977479i	$0.432317\pi$
$432$	0	0
$433$	−709.953	−0.0787948	−0.0393974	−	0.999224i	$-0.512544\pi$
$433$	−709.953	−0.0787948	−0.0393974	+	0.999224i	$0.512544\pi$
$434$	711.665	0.0787120
$435$	0	0
$436$	45372.4	4.98382
$437$	−3863.76	−0.422949
$438$	0	0
$439$	−16262.2	−1.76800	−0.884002	−	0.467484i	$-0.845161\pi$
$439$	−16262.2	−1.76800	−0.884002	+	0.467484i	$0.845161\pi$
$440$	19619.5	2.12574
$441$	0	0
$442$	0	0
$443$	−2899.45	−0.310964	−0.155482	−	0.987839i	$-0.549693\pi$
$443$	−2899.45	−0.310964	−0.155482	+	0.987839i	$0.549693\pi$
$444$	0	0
$445$	5689.49	0.606085
$446$	17437.4	1.85131
$447$	0	0
$448$	44412.0	4.68363
$449$	−7817.78	−0.821701	−0.410851	−	0.911703i	$-0.634768\pi$
$449$	−7817.78	−0.821701	−0.410851	+	0.911703i	$0.634768\pi$
$450$	0	0
$451$	6529.20	0.681703
$452$	−15851.8	−1.64957
$453$	0	0
$454$	7404.57	0.765448
$455$	0	0
$456$	0	0
$457$	14452.9	1.47939	0.739694	−	0.672943i	$-0.234970\pi$
$457$	14452.9	1.47939	0.739694	+	0.672943i	$0.234970\pi$
$458$	−26449.8	−2.69851
$459$	0	0
$460$	−11132.1	−1.12834
$461$	2891.46	0.292123	0.146061	−	0.989276i	$-0.453340\pi$
$461$	2891.46	0.292123	0.146061	+	0.989276i	$0.453340\pi$
$462$	0	0
$463$	−9223.83	−0.925848	−0.462924	−	0.886398i	$-0.653200\pi$
$463$	−9223.83	−0.925848	−0.462924	+	0.886398i	$0.653200\pi$
$464$	−10109.3	−1.01145
$465$	0	0
$466$	−13170.8	−1.30929
$467$	7988.25	0.791546	0.395773	−	0.918348i	$-0.370477\pi$
$467$	7988.25	0.791546	0.395773	+	0.918348i	$0.370477\pi$
$468$	0	0
$469$	−14530.8	−1.43064
$470$	−25343.0	−2.48720
$471$	0	0
$472$	−16121.0	−1.57209
$473$	5838.64	0.567570
$474$	0	0
$475$	−5350.18	−0.516806
$476$	33981.2	3.27212
$477$	0	0
$478$	10318.1	0.987315
$479$	−6908.64	−0.659006	−0.329503	−	0.944155i	$-0.606881\pi$
$479$	−6908.64	−0.659006	−0.329503	+	0.944155i	$0.606881\pi$
$480$	0	0
$481$	0	0
$482$	29815.6	2.81756
$483$	0	0
$484$	−21387.9	−2.00863
$485$	−9297.47	−0.870466
$486$	0	0
$487$	−13455.0	−1.25196	−0.625978	−	0.779841i	$-0.715300\pi$
$487$	−13455.0	−1.25196	−0.625978	+	0.779841i	$0.715300\pi$
$488$	−42289.1	−3.92282
$489$	0	0
$490$	8472.52	0.781121
$491$	1044.02	0.0959589	0.0479795	−	0.998848i	$-0.484722\pi$
$491$	1044.02	0.0959589	0.0479795	+	0.998848i	$0.484722\pi$
$492$	0	0
$493$	−2933.40	−0.267979
$494$	0	0
$495$	0	0
$496$	−1498.17	−0.135625
$497$	−20518.8	−1.85190
$498$	0	0
$499$	16190.7	1.45249	0.726246	−	0.687435i	$-0.241263\pi$
$499$	16190.7	1.45249	0.726246	+	0.687435i	$0.241263\pi$
$500$	21368.5	1.91126
$501$	0	0
$502$	6894.04	0.612940
$503$	11854.8	1.05086	0.525429	−	0.850838i	$-0.323905\pi$
$503$	11854.8	1.05086	0.525429	+	0.850838i	$0.323905\pi$
$504$	0	0
$505$	1181.84	0.104141
$506$	−3953.86	−0.347372
$507$	0	0
$508$	−2599.19	−0.227008
$509$	6647.07	0.578834	0.289417	−	0.957203i	$-0.406539\pi$
$509$	6647.07	0.578834	0.289417	+	0.957203i	$0.406539\pi$
$510$	0	0
$511$	5792.02	0.501416
$512$	−29721.4	−2.56545
$513$	0	0
$514$	30198.8	2.59146
$515$	19013.7	1.62688
$516$	0	0
$517$	−6608.42	−0.562162
$518$	33574.8	2.84786
$519$	0	0
$520$	0	0
$521$	−16839.6	−1.41604	−0.708019	−	0.706193i	$-0.750411\pi$
$521$	−16839.6	−1.41604	−0.708019	+	0.706193i	$0.750411\pi$
$522$	0	0
$523$	10414.1	0.870701	0.435351	−	0.900261i	$-0.356624\pi$
$523$	10414.1	0.870701	0.435351	+	0.900261i	$0.356624\pi$
$524$	5797.05	0.483293
$525$	0	0
$526$	11149.6	0.924233
$527$	−434.723	−0.0359332
$528$	0	0
$529$	−10735.9	−0.882377
$530$	−29095.3	−2.38456
$531$	0	0
$532$	−48344.1	−3.93981
$533$	0	0
$534$	0	0
$535$	−200.089	−0.0161694
$536$	52443.7	4.22616
$537$	0	0
$538$	30302.8	2.42834
$539$	2209.29	0.176550
$540$	0	0
$541$	5464.03	0.434227	0.217114	−	0.976146i	$-0.430336\pi$
$541$	5464.03	0.434227	0.217114	+	0.976146i	$0.430336\pi$
$542$	21497.3	1.70367
$543$	0	0
$544$	−53031.0	−4.17957
$545$	−27350.5	−2.14966
$546$	0	0
$547$	24902.3	1.94652	0.973261	−	0.229702i	$-0.0737753\pi$
$547$	24902.3	1.94652	0.973261	+	0.229702i	$0.0737753\pi$
$548$	29107.5	2.26900
$549$	0	0
$550$	−5474.94	−0.424459
$551$	4173.26	0.322662
$552$	0	0
$553$	−22127.4	−1.70154
$554$	16153.4	1.23879
$555$	0	0
$556$	−142.173	−0.0108444
$557$	−9817.71	−0.746840	−0.373420	−	0.927662i	$-0.621815\pi$
$557$	−9817.71	−0.746840	−0.373420	+	0.927662i	$0.621815\pi$
$558$	0	0
$559$	0	0
$560$	−70593.0	−5.32696
$561$	0	0
$562$	27696.7	2.07885
$563$	23333.7	1.74671	0.873356	−	0.487083i	$-0.161939\pi$
$563$	23333.7	1.74671	0.873356	+	0.487083i	$0.161939\pi$
$564$	0	0
$565$	9555.48	0.711508
$566$	−24640.6	−1.82989
$567$	0	0
$568$	74055.4	5.47059
$569$	−24543.9	−1.80832	−0.904160	−	0.427194i	$-0.859502\pi$
$569$	−24543.9	−1.80832	−0.904160	+	0.427194i	$0.859502\pi$
$570$	0	0
$571$	−10562.5	−0.774126	−0.387063	−	0.922053i	$-0.626510\pi$
$571$	−10562.5	−0.774126	−0.387063	+	0.922053i	$0.626510\pi$
$572$	0	0
$573$	0	0
$574$	−40276.5	−2.92876
$575$	1981.69	0.143725
$576$	0	0
$577$	18922.5	1.36526	0.682629	−	0.730765i	$-0.260837\pi$
$577$	18922.5	1.36526	0.682629	+	0.730765i	$0.260837\pi$
$578$	−1321.50	−0.0950991
$579$	0	0
$580$	12023.9	0.860801
$581$	22835.2	1.63057
$582$	0	0
$583$	−7586.85	−0.538963
$584$	−20904.2	−1.48120
$585$	0	0
$586$	26084.4	1.83880
$587$	−7989.65	−0.561786	−0.280893	−	0.959739i	$-0.590631\pi$
$587$	−7989.65	−0.561786	−0.280893	+	0.959739i	$0.590631\pi$
$588$	0	0
$589$	618.466	0.0432656
$590$	15233.5	1.06297
$591$	0	0
$592$	−70680.4	−4.90701
$593$	−25239.2	−1.74781	−0.873905	−	0.486097i	$-0.838420\pi$
$593$	−25239.2	−1.74781	−0.873905	+	0.486097i	$0.838420\pi$
$594$	0	0
$595$	−20483.9	−1.41136
$596$	−12114.7	−0.832610
$597$	0	0
$598$	0	0
$599$	7412.19	0.505599	0.252800	−	0.967519i	$-0.418649\pi$
$599$	7412.19	0.505599	0.252800	+	0.967519i	$0.418649\pi$
$600$	0	0
$601$	−21459.2	−1.45647	−0.728236	−	0.685327i	$-0.759659\pi$
$601$	−21459.2	−1.45647	−0.728236	+	0.685327i	$0.759659\pi$
$602$	−36016.6	−2.43842
$603$	0	0
$604$	32830.1	2.21165
$605$	12892.7	0.866382
$606$	0	0
$607$	−18246.7	−1.22011	−0.610057	−	0.792358i	$-0.708853\pi$
$607$	−18246.7	−1.22011	−0.610057	+	0.792358i	$0.708853\pi$
$608$	75445.6	5.03244
$609$	0	0
$610$	39961.1	2.65242
$611$	0	0
$612$	0	0
$613$	24087.5	1.58709	0.793545	−	0.608512i	$-0.208233\pi$
$613$	24087.5	1.58709	0.793545	+	0.608512i	$0.208233\pi$
$614$	−8905.23	−0.585319
$615$	0	0
$616$	−31558.7	−2.06418
$617$	−14400.6	−0.939621	−0.469811	−	0.882767i	$-0.655678\pi$
$617$	−14400.6	−0.939621	−0.469811	+	0.882767i	$0.655678\pi$
$618$	0	0
$619$	−25635.8	−1.66460	−0.832302	−	0.554322i	$-0.812978\pi$
$619$	−25635.8	−1.66460	−0.832302	+	0.554322i	$0.812978\pi$
$620$	1781.91	0.115424
$621$	0	0
$622$	36664.6	2.36353
$623$	−9151.76	−0.588535
$624$	0	0
$625$	−19428.9	−1.24345
$626$	−8692.58	−0.554993
$627$	0	0
$628$	47113.4	2.99368
$629$	−20509.3	−1.30009
$630$	0	0
$631$	2757.17	0.173948	0.0869741	−	0.996211i	$-0.472280\pi$
$631$	2757.17	0.173948	0.0869741	+	0.996211i	$0.472280\pi$
$632$	79861.0	5.02643
$633$	0	0
$634$	4322.47	0.270768
$635$	1566.79	0.0979152
$636$	0	0
$637$	0	0
$638$	4270.58	0.265006
$639$	0	0
$640$	72758.7	4.49382
$641$	−14317.0	−0.882196	−0.441098	−	0.897459i	$-0.645411\pi$
$641$	−14317.0	−0.882196	−0.441098	+	0.897459i	$0.645411\pi$
$642$	0	0
$643$	51.9171	0.00318415	0.00159208	−	0.999999i	$-0.499493\pi$
$643$	51.9171	0.00318415	0.00159208	+	0.999999i	$0.499493\pi$
$644$	17906.5	1.09567
$645$	0	0
$646$	40223.8	2.44982
$647$	9735.07	0.591538	0.295769	−	0.955260i	$-0.404424\pi$
$647$	9735.07	0.591538	0.295769	+	0.955260i	$0.404424\pi$
$648$	0	0
$649$	3972.27	0.240255
$650$	0	0
$651$	0	0
$652$	10619.3	0.637860
$653$	−10842.6	−0.649778	−0.324889	−	0.945752i	$-0.605327\pi$
$653$	−10842.6	−0.649778	−0.324889	+	0.945752i	$0.605327\pi$
$654$	0	0
$655$	−3494.46	−0.208458
$656$	84788.5	5.04640
$657$	0	0
$658$	40765.1	2.41518
$659$	−28940.9	−1.71074	−0.855369	−	0.518020i	$-0.826669\pi$
$659$	−28940.9	−1.71074	−0.855369	+	0.518020i	$0.826669\pi$
$660$	0	0
$661$	3108.58	0.182919	0.0914597	−	0.995809i	$-0.470847\pi$
$661$	3108.58	0.182919	0.0914597	+	0.995809i	$0.470847\pi$
$662$	52080.2	3.05763
$663$	0	0
$664$	−82415.4	−4.81677
$665$	29141.8	1.69935
$666$	0	0
$667$	−1545.76	−0.0897333
$668$	64509.4	3.73644
$669$	0	0
$670$	−49556.6	−2.85752
$671$	10420.2	0.599505
$672$	0	0
$673$	−9950.09	−0.569908	−0.284954	−	0.958541i	$-0.591978\pi$
$673$	−9950.09	−0.569908	−0.284954	+	0.958541i	$0.591978\pi$
$674$	22618.4	1.29263
$675$	0	0
$676$	0	0
$677$	3483.60	0.197763	0.0988816	−	0.995099i	$-0.468473\pi$
$677$	3483.60	0.197763	0.0988816	+	0.995099i	$0.468473\pi$
$678$	0	0
$679$	14955.3	0.845261
$680$	73929.3	4.16920
$681$	0	0
$682$	632.889	0.0355346
$683$	16811.4	0.941829	0.470915	−	0.882179i	$-0.343924\pi$
$683$	16811.4	0.941829	0.470915	+	0.882179i	$0.343924\pi$
$684$	0	0
$685$	−17546.0	−0.978685
$686$	26682.8	1.48506
$687$	0	0
$688$	75820.8	4.20151
$689$	0	0
$690$	0	0
$691$	−3078.78	−0.169497	−0.0847484	−	0.996402i	$-0.527009\pi$
$691$	−3078.78	−0.169497	−0.0847484	+	0.996402i	$0.527009\pi$
$692$	51692.6	2.83968
$693$	0	0
$694$	12731.3	0.696361
$695$	85.7020	0.00467750
$696$	0	0
$697$	24603.0	1.33702
$698$	−20949.6	−1.13604
$699$	0	0
$700$	24795.2	1.33882
$701$	18608.6	1.00262	0.501311	−	0.865267i	$-0.332851\pi$
$701$	18608.6	1.00262	0.501311	+	0.865267i	$0.332851\pi$
$702$	0	0
$703$	29177.9	1.56539
$704$	39495.9	2.11443
$705$	0	0
$706$	−33274.5	−1.77380
$707$	−1901.03	−0.101126
$708$	0	0
$709$	19046.2	1.00888	0.504439	−	0.863448i	$-0.331699\pi$
$709$	19046.2	1.00888	0.504439	+	0.863448i	$0.331699\pi$
$710$	−69978.5	−3.69894
$711$	0	0
$712$	33030.0	1.73855
$713$	−229.078	−0.0120323
$714$	0	0
$715$	0	0
$716$	100722.	5.25722
$717$	0	0
$718$	40628.2	2.11174
$719$	14013.4	0.726859	0.363430	−	0.931622i	$-0.381606\pi$
$719$	14013.4	0.726859	0.363430	+	0.931622i	$0.381606\pi$
$720$	0	0
$721$	−30584.3	−1.57978
$722$	−19597.7	−1.01018
$723$	0	0
$724$	−77873.2	−3.99742
$725$	−2140.43	−0.109646
$726$	0	0
$727$	2578.98	0.131567	0.0657834	−	0.997834i	$-0.479045\pi$
$727$	2578.98	0.131567	0.0657834	+	0.997834i	$0.479045\pi$
$728$	0	0
$729$	0	0
$730$	19753.4	1.00152
$731$	22000.8	1.11317
$732$	0	0
$733$	5029.12	0.253417	0.126709	−	0.991940i	$-0.459559\pi$
$733$	5029.12	0.253417	0.126709	+	0.991940i	$0.459559\pi$
$734$	26081.0	1.31153
$735$	0	0
$736$	−27944.8	−1.39954
$737$	−12922.3	−0.645863
$738$	0	0
$739$	−23856.4	−1.18751	−0.593757	−	0.804644i	$-0.702356\pi$
$739$	−23856.4	−1.18751	−0.593757	+	0.804644i	$0.702356\pi$
$740$	84066.5	4.17614
$741$	0	0
$742$	46800.8	2.31551
$743$	−4887.62	−0.241332	−0.120666	−	0.992693i	$-0.538503\pi$
$743$	−4887.62	−0.241332	−0.120666	+	0.992693i	$0.538503\pi$
$744$	0	0
$745$	7302.72	0.359129
$746$	25696.4	1.26114
$747$	0	0
$748$	30219.8	1.47720
$749$	321.851	0.0157012
$750$	0	0
$751$	24814.3	1.20571	0.602854	−	0.797851i	$-0.294030\pi$
$751$	24814.3	1.20571	0.602854	+	0.797851i	$0.294030\pi$
$752$	−85817.2	−4.16148
$753$	0	0
$754$	0	0
$755$	−19790.0	−0.953949
$756$	0	0
$757$	−6552.91	−0.314623	−0.157311	−	0.987549i	$-0.550283\pi$
$757$	−6552.91	−0.314623	−0.157311	+	0.987549i	$0.550283\pi$
$758$	−66400.2	−3.18175
$759$	0	0
$760$	−105177.	−5.01996
$761$	−20351.4	−0.969431	−0.484715	−	0.874672i	$-0.661077\pi$
$761$	−20351.4	−0.969431	−0.484715	+	0.874672i	$0.661077\pi$
$762$	0	0
$763$	43994.3	2.08742
$764$	−73867.7	−3.49796
$765$	0	0
$766$	−30729.3	−1.44947
$767$	0	0
$768$	0	0
$769$	−22209.7	−1.04148	−0.520742	−	0.853714i	$-0.674345\pi$
$769$	−22209.7	−1.04148	−0.520742	+	0.853714i	$0.674345\pi$
$770$	29821.4	1.39570
$771$	0	0
$772$	73824.9	3.44173
$773$	−28496.6	−1.32594	−0.662971	−	0.748645i	$-0.730705\pi$
$773$	−28496.6	−1.32594	−0.662971	+	0.748645i	$0.730705\pi$
$774$	0	0
$775$	−317.206	−0.0147024
$776$	−53975.9	−2.49693
$777$	0	0
$778$	51843.7	2.38906
$779$	−35001.9	−1.60985
$780$	0	0
$781$	−18247.5	−0.836041
$782$	−14898.7	−0.681301
$783$	0	0
$784$	28689.9	1.30694
$785$	−28400.0	−1.29126
$786$	0	0
$787$	−2394.07	−0.108436	−0.0542181	−	0.998529i	$-0.517267\pi$
$787$	−2394.07	−0.108436	−0.0542181	+	0.998529i	$0.517267\pi$
$788$	−80732.2	−3.64970
$789$	0	0
$790$	−75464.6	−3.39862
$791$	−15370.3	−0.690906
$792$	0	0
$793$	0	0
$794$	14943.3	0.667907
$795$	0	0
$796$	24890.5	1.10832
$797$	37996.7	1.68872	0.844362	−	0.535773i	$-0.179980\pi$
$797$	37996.7	1.68872	0.844362	+	0.535773i	$0.179980\pi$
$798$	0	0
$799$	−24901.5	−1.10257
$800$	−38695.4	−1.71011
$801$	0	0
$802$	25676.3	1.13050
$803$	5150.88	0.226364
$804$	0	0
$805$	−10794.0	−0.472595
$806$	0	0
$807$	0	0
$808$	6861.10	0.298729
$809$	25993.4	1.12964	0.564821	−	0.825213i	$-0.308945\pi$
$809$	25993.4	1.12964	0.564821	+	0.825213i	$0.308945\pi$
$810$	0	0
$811$	−15524.4	−0.672176	−0.336088	−	0.941831i	$-0.609104\pi$
$811$	−15524.4	−0.672176	−0.336088	+	0.941831i	$0.609104\pi$
$812$	−19340.9	−0.835875
$813$	0	0
$814$	29858.3	1.28567
$815$	−6401.33	−0.275127
$816$	0	0
$817$	−31299.9	−1.34033
$818$	17953.5	0.767396
$819$	0	0
$820$	−100847.	−4.29477
$821$	2029.40	0.0862684	0.0431342	−	0.999069i	$-0.486266\pi$
$821$	2029.40	0.0862684	0.0431342	+	0.999069i	$0.486266\pi$
$822$	0	0
$823$	−42010.4	−1.77933	−0.889667	−	0.456610i	$-0.849064\pi$
$823$	−42010.4	−1.77933	−0.889667	+	0.456610i	$0.849064\pi$
$824$	110383.	4.66672
$825$	0	0
$826$	−24503.6	−1.03219
$827$	−3941.24	−0.165720	−0.0828599	−	0.996561i	$-0.526405\pi$
$827$	−3941.24	−0.165720	−0.0828599	+	0.996561i	$0.526405\pi$
$828$	0	0
$829$	11264.7	0.471941	0.235970	−	0.971760i	$-0.424173\pi$
$829$	11264.7	0.471941	0.235970	+	0.971760i	$0.424173\pi$
$830$	77878.4	3.25686
$831$	0	0
$832$	0	0
$833$	8324.92	0.346268
$834$	0	0
$835$	−38886.3	−1.61163
$836$	−42992.7	−1.77863
$837$	0	0
$838$	8236.88	0.339544
$839$	−35221.1	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$839$	−35221.1	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$840$	0	0
$841$	−22719.4	−0.931544
$842$	90944.8	3.72228
$843$	0	0
$844$	−1646.57	−0.0671533
$845$	0	0
$846$	0	0
$847$	−20738.3	−0.841295
$848$	−98523.2	−3.98974
$849$	0	0
$850$	−20630.4	−0.832490
$851$	−10807.4	−0.435338
$852$	0	0
$853$	−37662.7	−1.51178	−0.755888	−	0.654701i	$-0.772795\pi$
$853$	−37662.7	−1.51178	−0.755888	+	0.654701i	$0.772795\pi$
$854$	−64278.8	−2.57562
$855$	0	0
$856$	−1161.60	−0.0463818
$857$	−800.368	−0.0319020	−0.0159510	−	0.999873i	$-0.505078\pi$
$857$	−800.368	−0.0319020	−0.0159510	+	0.999873i	$0.505078\pi$
$858$	0	0
$859$	−8800.69	−0.349564	−0.174782	−	0.984607i	$-0.555922\pi$
$859$	−8800.69	−0.349564	−0.174782	+	0.984607i	$0.555922\pi$
$860$	−90180.4	−3.57573
$861$	0	0
$862$	−20717.6	−0.818611
$863$	−17991.0	−0.709641	−0.354821	−	0.934934i	$-0.615458\pi$
$863$	−17991.0	−0.709641	−0.354821	+	0.934934i	$0.615458\pi$
$864$	0	0
$865$	−31160.3	−1.22484
$866$	3894.69	0.152826
$867$	0	0
$868$	−2866.26	−0.112082
$869$	−19678.1	−0.768162
$870$	0	0
$871$	0	0
$872$	−158782.	−6.16631
$873$	0	0
$874$	21196.0	0.820325
$875$	20719.5	0.800510
$876$	0	0
$877$	44304.5	1.70588	0.852940	−	0.522008i	$-0.174817\pi$
$877$	44304.5	1.70588	0.852940	+	0.522008i	$0.174817\pi$
$878$	89212.0	3.42911
$879$	0	0
$880$	−62778.8	−2.40486
$881$	−10814.6	−0.413567	−0.206783	−	0.978387i	$-0.566300\pi$
$881$	−10814.6	−0.413567	−0.206783	+	0.978387i	$0.566300\pi$
$882$	0	0
$883$	14530.8	0.553793	0.276897	−	0.960900i	$-0.410694\pi$
$883$	14530.8	0.553793	0.276897	+	0.960900i	$0.410694\pi$
$884$	0	0
$885$	0	0
$886$	15905.9	0.603126
$887$	3385.82	0.128168	0.0640838	−	0.997945i	$-0.479588\pi$
$887$	3385.82	0.128168	0.0640838	+	0.997945i	$0.479588\pi$
$888$	0	0
$889$	−2520.24	−0.0950800
$890$	−31211.7	−1.17552
$891$	0	0
$892$	−70229.9	−2.63618
$893$	35426.6	1.32755
$894$	0	0
$895$	−60715.4	−2.26759
$896$	−117035.	−4.36369
$897$	0	0
$898$	42887.1	1.59372
$899$	247.428	0.00917929
$900$	0	0
$901$	−28588.4	−1.05707
$902$	−35818.2	−1.32219
$903$	0	0
$904$	55473.7	2.04096
$905$	46942.0	1.72420
$906$	0	0
$907$	38174.2	1.39752	0.698762	−	0.715354i	$-0.253735\pi$
$907$	38174.2	1.39752	0.698762	+	0.715354i	$0.253735\pi$
$908$	−29822.2	−1.08996
$909$	0	0
$910$	0	0
$911$	11699.5	0.425489	0.212745	−	0.977108i	$-0.431760\pi$
$911$	11699.5	0.425489	0.212745	+	0.977108i	$0.431760\pi$
$912$	0	0
$913$	20307.5	0.736123
$914$	−79286.6	−2.86933
$915$	0	0
$916$	106528.	3.84254
$917$	5620.97	0.202422
$918$	0	0
$919$	21615.6	0.775879	0.387939	−	0.921685i	$-0.373187\pi$
$919$	21615.6	0.775879	0.387939	+	0.921685i	$0.373187\pi$
$920$	38957.1	1.39606
$921$	0	0
$922$	−15862.1	−0.566583
$923$	0	0
$924$	0	0
$925$	−14965.1	−0.531945
$926$	50600.5	1.79572
$927$	0	0
$928$	30183.3	1.06769
$929$	−22325.9	−0.788471	−0.394236	−	0.919009i	$-0.628991\pi$
$929$	−22325.9	−0.788471	−0.394236	+	0.919009i	$0.628991\pi$
$930$	0	0
$931$	−11843.6	−0.416926
$932$	53046.1	1.86436
$933$	0	0
$934$	−43822.3	−1.53523
$935$	−18216.5	−0.637158
$936$	0	0
$937$	−29401.5	−1.02509	−0.512543	−	0.858661i	$-0.671297\pi$
$937$	−29401.5	−1.02509	−0.512543	+	0.858661i	$0.671297\pi$
$938$	79713.7	2.77478
$939$	0	0
$940$	102070.	3.54166
$941$	−30280.7	−1.04902	−0.524508	−	0.851406i	$-0.675751\pi$
$941$	−30280.7	−1.04902	−0.524508	+	0.851406i	$0.675751\pi$
$942$	0	0
$943$	12964.6	0.447704
$944$	51584.1	1.77852
$945$	0	0
$946$	−32029.8	−1.10082
$947$	−25226.1	−0.865617	−0.432809	−	0.901486i	$-0.642477\pi$
$947$	−25226.1	−0.865617	−0.432809	+	0.901486i	$0.642477\pi$
$948$	0	0
$949$	0	0
$950$	29350.2	1.00237
$951$	0	0
$952$	−118918.	−4.04848
$953$	11893.7	0.404275	0.202137	−	0.979357i	$-0.435211\pi$
$953$	11893.7	0.404275	0.202137	+	0.979357i	$0.435211\pi$
$954$	0	0
$955$	44527.5	1.50877
$956$	−41556.4	−1.40589
$957$	0	0
$958$	37899.7	1.27817
$959$	28223.4	0.950346
$960$	0	0
$961$	−29754.3	−0.998769
$962$	0	0
$963$	0	0
$964$	−120084.	−4.01207
$965$	−44501.7	−1.48452
$966$	0	0
$967$	8534.72	0.283824	0.141912	−	0.989879i	$-0.454675\pi$
$967$	8534.72	0.283824	0.141912	+	0.989879i	$0.454675\pi$
$968$	74847.5	2.48522
$969$	0	0
$970$	51004.4	1.68830
$971$	−25615.3	−0.846585	−0.423292	−	0.905993i	$-0.639126\pi$
$971$	−25615.3	−0.846585	−0.423292	+	0.905993i	$0.639126\pi$
$972$	0	0
$973$	−137.855	−0.00454206
$974$	73811.8	2.42822
$975$	0	0
$976$	135317.	4.43791
$977$	22995.5	0.753011	0.376506	−	0.926414i	$-0.377126\pi$
$977$	22995.5	0.753011	0.376506	+	0.926414i	$0.377126\pi$
$978$	0	0
$979$	−8138.72	−0.265694
$980$	−34123.4	−1.11228
$981$	0	0
$982$	−5727.31	−0.186116
$983$	20592.1	0.668144	0.334072	−	0.942548i	$-0.391577\pi$
$983$	20592.1	0.668144	0.334072	+	0.942548i	$0.391577\pi$
$984$	0	0
$985$	48665.4	1.57422
$986$	16092.2	0.519756
$987$	0	0
$988$	0	0
$989$	11593.4	0.372748
$990$	0	0
$991$	−11557.9	−0.370485	−0.185242	−	0.982693i	$-0.559307\pi$
$991$	−11557.9	−0.370485	−0.185242	+	0.982693i	$0.559307\pi$
$992$	4473.08	0.143166
$993$	0	0
$994$	112563.	3.59183
$995$	−15004.0	−0.478049
$996$	0	0
$997$	−15481.5	−0.491778	−0.245889	−	0.969298i	$-0.579080\pi$
$997$	−15481.5	−0.491778	−0.245889	+	0.969298i	$0.579080\pi$
$998$	−88819.4	−2.81716
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.be.1.1		9
3.2	odd	2		507.4.a.q.1.9	yes	9
13.12	even	2		1521.4.a.bj.1.9		9
39.5	even	4		507.4.b.j.337.1		18
39.8	even	4		507.4.b.j.337.18		18
39.38	odd	2		507.4.a.n.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.1	✓	9	39.38	odd	2
507.4.a.q.1.9	yes	9	3.2	odd	2
507.4.b.j.337.1		18	39.5	even	4
507.4.b.j.337.18		18	39.8	even	4
1521.4.a.be.1.1		9	1.1	even	1	trivial
1521.4.a.bj.1.9		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)

\(f(q)\)	\(=\)	\(q-5.48584 q^{2} +22.0945 q^{4} -13.3185 q^{5} +21.4234 q^{7} -77.3200 q^{8} +O(q^{10})\)	\(q-5.48584 q^{2} +22.0945 q^{4} -13.3185 q^{5} +21.4234 q^{7} -77.3200 q^{8} +73.0635 q^{10} +19.0520 q^{11} -117.525 q^{14} +247.410 q^{16} +71.7906 q^{17} -102.134 q^{19} -294.266 q^{20} -104.516 q^{22} +37.8302 q^{23} +52.3838 q^{25} +473.338 q^{28} -40.8605 q^{29} -6.05542 q^{31} -738.690 q^{32} -393.832 q^{34} -285.328 q^{35} -285.682 q^{37} +560.293 q^{38} +1029.79 q^{40} +342.705 q^{41} +306.458 q^{43} +420.943 q^{44} -207.530 q^{46} -346.863 q^{47} +115.961 q^{49} -287.369 q^{50} -398.219 q^{53} -253.745 q^{55} -1656.46 q^{56} +224.155 q^{58} +208.497 q^{59} +546.936 q^{61} +33.2191 q^{62} +2073.06 q^{64} -678.268 q^{67} +1586.18 q^{68} +1565.27 q^{70} -957.777 q^{71} +270.360 q^{73} +1567.21 q^{74} -2256.60 q^{76} +408.157 q^{77} -1032.86 q^{79} -3295.14 q^{80} -1880.03 q^{82} +1065.90 q^{83} -956.147 q^{85} -1681.18 q^{86} -1473.10 q^{88} -427.185 q^{89} +835.837 q^{92} +1902.83 q^{94} +1360.28 q^{95} +698.084 q^{97} -636.144 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8}+O(q^{10}) \) 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

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