LMFDB - Embedded newform 1521.4.a.bk.1.10

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 $ x^10 - 70x^8 + 1645x^6 - 14700x^4 + 44100x^2 - 27648
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$5.36472$ 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)

$f(q)$	$=$	$q+5.36472 q^{2} +20.7803 q^{4} -2.69631 q^{5} +15.2025 q^{7} +68.5626 q^{8} +O(q^{10})$	$q+5.36472 q^{2} +20.7803 q^{4} -2.69631 q^{5} +15.2025 q^{7} +68.5626 q^{8} -14.4650 q^{10} +66.8848 q^{11} +81.5570 q^{14} +201.577 q^{16} -4.16354 q^{17} +26.0850 q^{19} -56.0301 q^{20} +358.819 q^{22} -47.3242 q^{23} -117.730 q^{25} +315.911 q^{28} -257.007 q^{29} +206.242 q^{31} +532.906 q^{32} -22.3362 q^{34} -40.9906 q^{35} +175.686 q^{37} +139.939 q^{38} -184.866 q^{40} -156.463 q^{41} +51.9845 q^{43} +1389.88 q^{44} -253.881 q^{46} +354.222 q^{47} -111.885 q^{49} -631.588 q^{50} +10.4723 q^{53} -180.342 q^{55} +1042.32 q^{56} -1378.77 q^{58} -445.114 q^{59} +119.696 q^{61} +1106.43 q^{62} +1246.28 q^{64} -22.4078 q^{67} -86.5195 q^{68} -219.903 q^{70} -285.207 q^{71} +740.989 q^{73} +942.507 q^{74} +542.053 q^{76} +1016.81 q^{77} -547.679 q^{79} -543.516 q^{80} -839.378 q^{82} -603.056 q^{83} +11.2262 q^{85} +278.882 q^{86} +4585.80 q^{88} -215.668 q^{89} -983.409 q^{92} +1900.31 q^{94} -70.3333 q^{95} +1447.50 q^{97} -600.233 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 60 q^{4}+O(q^{10}) $ 10 * q + 60 * q^4	$ 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) $ 10 * q + 60 * q^4 + 80 * q^10 + 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 + 960 * q^25 - 990 * q^29 + 120 * q^35 - 1380 * q^38 + 2000 * q^40 - 740 * q^43 + 1550 * q^49 - 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 + 3140 * q^64 - 1200 * q^68 + 4380 * q^74 - 4320 * q^77 + 1100 * q^79 - 4780 * q^82 + 6340 * q^88 + 1740 * q^92 + 6460 * q^94 + 2760 * q^95

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 $

$2$	5.36472	1.89672	0.948358	−	0.317201i	$-0.102743\pi$
$2$	5.36472	1.89672	0.948358	+	0.317201i	$0.102743\pi$
$3$	0	0
$3$	0	0
$4$	20.7803	2.59753
$5$	−2.69631	−0.241165	−0.120583	−	0.992703i	$-0.538476\pi$
$5$	−2.69631	−0.241165	−0.120583	+	0.992703i	$0.538476\pi$
$6$	0	0
$7$	15.2025	0.820856	0.410428	−	0.911893i	$-0.365379\pi$
$7$	15.2025	0.820856	0.410428	+	0.911893i	$0.365379\pi$
$8$	68.5626	3.03007
$9$	0	0
$10$	−14.4650	−0.457423
$11$	66.8848	1.83332	0.916661	−	0.399666i	$-0.130874\pi$
$11$	66.8848	1.83332	0.916661	+	0.399666i	$0.130874\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	81.5570	1.55693
$15$	0	0
$16$	201.577	3.14965
$17$	−4.16354	−0.0594004	−0.0297002	−	0.999559i	$-0.509455\pi$
$17$	−4.16354	−0.0594004	−0.0297002	+	0.999559i	$0.509455\pi$
$18$	0	0
$19$	26.0850	0.314963	0.157482	−	0.987522i	$-0.449662\pi$
$19$	26.0850	0.314963	0.157482	+	0.987522i	$0.449662\pi$
$20$	−56.0301	−0.626436
$21$	0	0
$22$	358.819	3.47729
$23$	−47.3242	−0.429034	−0.214517	−	0.976720i	$-0.568818\pi$
$23$	−47.3242	−0.429034	−0.214517	+	0.976720i	$0.568818\pi$
$24$	0	0
$25$	−117.730	−0.941839
$26$	0	0
$27$	0	0
$28$	315.911	2.13220
$29$	−257.007	−1.64569	−0.822845	−	0.568266i	$-0.807614\pi$
$29$	−257.007	−1.64569	−0.822845	+	0.568266i	$0.807614\pi$
$30$	0	0
$31$	206.242	1.19491	0.597455	−	0.801903i	$-0.296179\pi$
$31$	206.242	1.19491	0.597455	+	0.801903i	$0.296179\pi$
$32$	532.906	2.94392
$33$	0	0
$34$	−22.3362	−0.112666
$35$	−40.9906	−0.197962
$36$	0	0
$37$	175.686	0.780611	0.390305	−	0.920685i	$-0.372369\pi$
$37$	175.686	0.780611	0.390305	+	0.920685i	$0.372369\pi$
$38$	139.939	0.597396
$39$	0	0
$40$	−184.866	−0.730748
$41$	−156.463	−0.595984	−0.297992	−	0.954568i	$-0.596317\pi$
$41$	−156.463	−0.595984	−0.297992	+	0.954568i	$0.596317\pi$
$42$	0	0
$43$	51.9845	0.184362	0.0921809	−	0.995742i	$-0.470616\pi$
$43$	51.9845	0.184362	0.0921809	+	0.995742i	$0.470616\pi$
$44$	1389.88	4.76211
$45$	0	0
$46$	−253.881	−0.813755
$47$	354.222	1.09933	0.549666	−	0.835384i	$-0.314755\pi$
$47$	354.222	1.09933	0.549666	+	0.835384i	$0.314755\pi$
$48$	0	0
$49$	−111.885	−0.326196
$50$	−631.588	−1.78640
$51$	0	0
$52$	0	0
$53$	10.4723	0.0271412	0.0135706	−	0.999908i	$-0.495680\pi$
$53$	10.4723	0.0271412	0.0135706	+	0.999908i	$0.495680\pi$
$54$	0	0
$55$	−180.342	−0.442134
$56$	1042.32	2.48725
$57$	0	0
$58$	−1378.77	−3.12141
$59$	−445.114	−0.982185	−0.491092	−	0.871107i	$-0.663402\pi$
$59$	−445.114	−0.982185	−0.491092	+	0.871107i	$0.663402\pi$
$60$	0	0
$61$	119.696	0.251238	0.125619	−	0.992079i	$-0.459908\pi$
$61$	119.696	0.251238	0.125619	+	0.992079i	$0.459908\pi$
$62$	1106.43	2.26640
$63$	0	0
$64$	1246.28	2.43413
$65$	0	0
$66$	0	0
$67$	−22.4078	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$67$	−22.4078	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$68$	−86.5195	−0.154295
$69$	0	0
$70$	−219.903	−0.375478
$71$	−285.207	−0.476731	−0.238365	−	0.971176i	$-0.576612\pi$
$71$	−285.207	−0.476731	−0.238365	+	0.971176i	$0.576612\pi$
$72$	0	0
$73$	740.989	1.18803	0.594015	−	0.804454i	$-0.297542\pi$
$73$	740.989	1.18803	0.594015	+	0.804454i	$0.297542\pi$
$74$	942.507	1.48060
$75$	0	0
$76$	542.053	0.818128
$77$	1016.81	1.50489
$78$	0	0
$79$	−547.679	−0.779983	−0.389992	−	0.920818i	$-0.627522\pi$
$79$	−547.679	−0.779983	−0.389992	+	0.920818i	$0.627522\pi$
$80$	−543.516	−0.759586
$81$	0	0
$82$	−839.378	−1.13041
$83$	−603.056	−0.797518	−0.398759	−	0.917056i	$-0.630559\pi$
$83$	−603.056	−0.797518	−0.398759	+	0.917056i	$0.630559\pi$
$84$	0	0
$85$	11.2262	0.0143253
$86$	278.882	0.349682
$87$	0	0
$88$	4585.80	5.55509
$89$	−215.668	−0.256863	−0.128431	−	0.991718i	$-0.540994\pi$
$89$	−215.668	−0.256863	−0.128431	+	0.991718i	$0.540994\pi$
$90$	0	0
$91$	0	0
$92$	−983.409	−1.11443
$93$	0	0
$94$	1900.31	2.08512
$95$	−70.3333	−0.0759583
$96$	0	0
$97$	1447.50	1.51517	0.757586	−	0.652735i	$-0.226378\pi$
$97$	1447.50	1.51517	0.757586	+	0.652735i	$0.226378\pi$
$98$	−600.233	−0.618700
$99$	0	0
$100$	−2446.46	−2.44646
$101$	883.450	0.870362	0.435181	−	0.900343i	$-0.356684\pi$
$101$	883.450	0.870362	0.435181	+	0.900343i	$0.356684\pi$
$102$	0	0
$103$	1251.74	1.19745	0.598726	−	0.800954i	$-0.295674\pi$
$103$	1251.74	1.19745	0.598726	+	0.800954i	$0.295674\pi$
$104$	0	0
$105$	0	0
$106$	56.1812	0.0514792
$107$	341.614	0.308645	0.154323	−	0.988021i	$-0.450681\pi$
$107$	341.614	0.308645	0.154323	+	0.988021i	$0.450681\pi$
$108$	0	0
$109$	775.177	0.681179	0.340589	−	0.940212i	$-0.389373\pi$
$109$	775.177	0.681179	0.340589	+	0.940212i	$0.389373\pi$
$110$	−967.487	−0.838603
$111$	0	0
$112$	3064.47	2.58541
$113$	−1279.05	−1.06480	−0.532402	−	0.846492i	$-0.678710\pi$
$113$	−1279.05	−1.06480	−0.532402	+	0.846492i	$0.678710\pi$
$114$	0	0
$115$	127.601	0.103468
$116$	−5340.67	−4.27473
$117$	0	0
$118$	−2387.91	−1.86293
$119$	−63.2961	−0.0487592
$120$	0	0
$121$	3142.58	2.36107
$122$	642.137	0.476528
$123$	0	0
$124$	4285.77	3.10382
$125$	654.476	0.468305
$126$	0	0
$127$	−1113.82	−0.778233	−0.389117	−	0.921188i	$-0.627220\pi$
$127$	−1113.82	−0.778233	−0.389117	+	0.921188i	$0.627220\pi$
$128$	2422.68	1.67294
$129$	0	0
$130$	0	0
$131$	−2100.12	−1.40068	−0.700339	−	0.713811i	$-0.746968\pi$
$131$	−2100.12	−1.40068	−0.700339	+	0.713811i	$0.746968\pi$
$132$	0	0
$133$	396.556	0.258540
$134$	−120.211	−0.0774977
$135$	0	0
$136$	−285.463	−0.179987
$137$	1205.02	0.751471	0.375736	−	0.926727i	$-0.377390\pi$
$137$	1205.02	0.751471	0.375736	+	0.926727i	$0.377390\pi$
$138$	0	0
$139$	322.890	0.197030	0.0985149	−	0.995136i	$-0.468591\pi$
$139$	322.890	0.197030	0.0985149	+	0.995136i	$0.468591\pi$
$140$	−851.796	−0.514213
$141$	0	0
$142$	−1530.06	−0.904223
$143$	0	0
$144$	0	0
$145$	692.971	0.396884
$146$	3975.20	2.25336
$147$	0	0
$148$	3650.80	2.02766
$149$	−1128.86	−0.620669	−0.310335	−	0.950627i	$-0.600441\pi$
$149$	−1128.86	−0.620669	−0.310335	+	0.950627i	$0.600441\pi$
$150$	0	0
$151$	−2940.44	−1.58470	−0.792350	−	0.610066i	$-0.791143\pi$
$151$	−2940.44	−1.58470	−0.792350	+	0.610066i	$0.791143\pi$
$152$	1788.46	0.954361
$153$	0	0
$154$	5454.93	2.85436
$155$	−556.093	−0.288171
$156$	0	0
$157$	629.388	0.319940	0.159970	−	0.987122i	$-0.448860\pi$
$157$	629.388	0.319940	0.159970	+	0.987122i	$0.448860\pi$
$158$	−2938.15	−1.47941
$159$	0	0
$160$	−1436.88	−0.709972
$161$	−719.444	−0.352175
$162$	0	0
$163$	−394.912	−0.189766	−0.0948832	−	0.995488i	$-0.530248\pi$
$163$	−394.912	−0.189766	−0.0948832	+	0.995488i	$0.530248\pi$
$164$	−3251.33	−1.54809
$165$	0	0
$166$	−3235.23	−1.51267
$167$	151.860	0.0703669	0.0351834	−	0.999381i	$-0.488798\pi$
$167$	151.860	0.0703669	0.0351834	+	0.999381i	$0.488798\pi$
$168$	0	0
$169$	0	0
$170$	60.2255	0.0271711
$171$	0	0
$172$	1080.25	0.478886
$173$	−538.813	−0.236793	−0.118397	−	0.992966i	$-0.537775\pi$
$173$	−538.813	−0.236793	−0.118397	+	0.992966i	$0.537775\pi$
$174$	0	0
$175$	−1789.78	−0.773114
$176$	13482.5	5.77432
$177$	0	0
$178$	−1157.00	−0.487196
$179$	2220.80	0.927319	0.463659	−	0.886014i	$-0.346536\pi$
$179$	2220.80	0.927319	0.463659	+	0.886014i	$0.346536\pi$
$180$	0	0
$181$	−3822.78	−1.56986	−0.784932	−	0.619582i	$-0.787302\pi$
$181$	−3822.78	−1.56986	−0.784932	+	0.619582i	$0.787302\pi$
$182$	0	0
$183$	0	0
$184$	−3244.67	−1.30000
$185$	−473.704	−0.188256
$186$	0	0
$187$	−278.478	−0.108900
$188$	7360.84	2.85555
$189$	0	0
$190$	−377.319	−0.144071
$191$	−3464.19	−1.31236	−0.656178	−	0.754606i	$-0.727828\pi$
$191$	−3464.19	−1.31236	−0.656178	+	0.754606i	$0.727828\pi$
$192$	0	0
$193$	4697.40	1.75195	0.875975	−	0.482357i	$-0.160219\pi$
$193$	4697.40	1.75195	0.875975	+	0.482357i	$0.160219\pi$
$194$	7765.46	2.87385
$195$	0	0
$196$	−2325.00	−0.847304
$197$	2887.89	1.04443	0.522217	−	0.852813i	$-0.325105\pi$
$197$	2887.89	1.04443	0.522217	+	0.852813i	$0.325105\pi$
$198$	0	0
$199$	63.0092	0.0224453	0.0112226	−	0.999937i	$-0.496428\pi$
$199$	63.0092	0.0224453	0.0112226	+	0.999937i	$0.496428\pi$
$200$	−8071.87	−2.85384
$201$	0	0
$202$	4739.47	1.65083
$203$	−3907.14	−1.35087
$204$	0	0
$205$	421.872	0.143731
$206$	6715.24	2.27123
$207$	0	0
$208$	0	0
$209$	1744.69	0.577429
$210$	0	0
$211$	−1049.70	−0.342484	−0.171242	−	0.985229i	$-0.554778\pi$
$211$	−1049.70	−0.342484	−0.171242	+	0.985229i	$0.554778\pi$
$212$	217.618	0.0705002
$213$	0	0
$214$	1832.66	0.585412
$215$	−140.166	−0.0444617
$216$	0	0
$217$	3135.39	0.980848
$218$	4158.61	1.29200
$219$	0	0
$220$	−3747.56	−1.14846
$221$	0	0
$222$	0	0
$223$	2313.49	0.694722	0.347361	−	0.937732i	$-0.387078\pi$
$223$	2313.49	0.694722	0.347361	+	0.937732i	$0.387078\pi$
$224$	8101.49	2.41653
$225$	0	0
$226$	−6861.74	−2.01963
$227$	−3799.46	−1.11092	−0.555460	−	0.831543i	$-0.687458\pi$
$227$	−3799.46	−1.11092	−0.555460	+	0.831543i	$0.687458\pi$
$228$	0	0
$229$	−4321.07	−1.24692	−0.623459	−	0.781856i	$-0.714273\pi$
$229$	−4321.07	−1.24692	−0.623459	+	0.781856i	$0.714273\pi$
$230$	684.543	0.196250
$231$	0	0
$232$	−17621.1	−4.98655
$233$	−5279.77	−1.48450	−0.742251	−	0.670122i	$-0.766242\pi$
$233$	−5279.77	−1.48450	−0.742251	+	0.670122i	$0.766242\pi$
$234$	0	0
$235$	−955.094	−0.265121
$236$	−9249.59	−2.55126
$237$	0	0
$238$	−339.566	−0.0924823
$239$	−1547.92	−0.418939	−0.209469	−	0.977815i	$-0.567174\pi$
$239$	−1547.92	−0.418939	−0.209469	+	0.977815i	$0.567174\pi$
$240$	0	0
$241$	−4918.01	−1.31451	−0.657255	−	0.753669i	$-0.728282\pi$
$241$	−4918.01	−1.31451	−0.657255	+	0.753669i	$0.728282\pi$
$242$	16859.1	4.47828
$243$	0	0
$244$	2487.32	0.652600
$245$	301.677	0.0786671
$246$	0	0
$247$	0	0
$248$	14140.5	3.62066
$249$	0	0
$250$	3511.08	0.888241
$251$	−1155.78	−0.290646	−0.145323	−	0.989384i	$-0.546422\pi$
$251$	−1155.78	−0.290646	−0.145323	+	0.989384i	$0.546422\pi$
$252$	0	0
$253$	−3165.27	−0.786556
$254$	−5975.34	−1.47609
$255$	0	0
$256$	3026.80	0.738965
$257$	−2351.95	−0.570859	−0.285429	−	0.958400i	$-0.592136\pi$
$257$	−2351.95	−0.570859	−0.285429	+	0.958400i	$0.592136\pi$
$258$	0	0
$259$	2670.86	0.640769
$260$	0	0
$261$	0	0
$262$	−11266.6	−2.65669
$263$	−5521.88	−1.29465	−0.647326	−	0.762213i	$-0.724113\pi$
$263$	−5521.88	−1.29465	−0.647326	+	0.762213i	$0.724113\pi$
$264$	0	0
$265$	−28.2367	−0.00654553
$266$	2127.41	0.490376
$267$	0	0
$268$	−465.639	−0.106132
$269$	−3916.95	−0.887810	−0.443905	−	0.896074i	$-0.646407\pi$
$269$	−3916.95	−0.887810	−0.443905	+	0.896074i	$0.646407\pi$
$270$	0	0
$271$	−2777.53	−0.622593	−0.311297	−	0.950313i	$-0.600763\pi$
$271$	−2777.53	−0.622593	−0.311297	+	0.950313i	$0.600763\pi$
$272$	−839.276	−0.187090
$273$	0	0
$274$	6464.58	1.42533
$275$	−7874.34	−1.72669
$276$	0	0
$277$	6583.08	1.42794	0.713969	−	0.700177i	$-0.246896\pi$
$277$	6583.08	1.42794	0.713969	+	0.700177i	$0.246896\pi$
$278$	1732.21	0.373710
$279$	0	0
$280$	−2810.42	−0.599839
$281$	−2871.66	−0.609640	−0.304820	−	0.952410i	$-0.598596\pi$
$281$	−2871.66	−0.609640	−0.304820	+	0.952410i	$0.598596\pi$
$282$	0	0
$283$	7518.04	1.57916	0.789578	−	0.613651i	$-0.210300\pi$
$283$	7518.04	1.57916	0.789578	+	0.613651i	$0.210300\pi$
$284$	−5926.69	−1.23832
$285$	0	0
$286$	0	0
$287$	−2378.62	−0.489217
$288$	0	0
$289$	−4895.66	−0.996472
$290$	3717.60	0.752776
$291$	0	0
$292$	15397.9	3.08595
$293$	−4506.57	−0.898555	−0.449278	−	0.893392i	$-0.648319\pi$
$293$	−4506.57	−0.898555	−0.449278	+	0.893392i	$0.648319\pi$
$294$	0	0
$295$	1200.17	0.236869
$296$	12045.5	2.36530
$297$	0	0
$298$	−6056.02	−1.17723
$299$	0	0
$300$	0	0
$301$	790.292	0.151335
$302$	−15774.7	−3.00573
$303$	0	0
$304$	5258.15	0.992024
$305$	−322.738	−0.0605900
$306$	0	0
$307$	9538.89	1.77333	0.886667	−	0.462409i	$-0.153015\pi$
$307$	9538.89	1.77333	0.886667	+	0.462409i	$0.153015\pi$
$308$	21129.7	3.90901
$309$	0	0
$310$	−2983.29	−0.546578
$311$	7466.28	1.36133	0.680666	−	0.732594i	$-0.261691\pi$
$311$	7466.28	1.36133	0.680666	+	0.732594i	$0.261691\pi$
$312$	0	0
$313$	−1821.65	−0.328964	−0.164482	−	0.986380i	$-0.552595\pi$
$313$	−1821.65	−0.328964	−0.164482	+	0.986380i	$0.552595\pi$
$314$	3376.49	0.606836
$315$	0	0
$316$	−11380.9	−2.02603
$317$	−3125.14	−0.553708	−0.276854	−	0.960912i	$-0.589292\pi$
$317$	−3125.14	−0.553708	−0.276854	+	0.960912i	$0.589292\pi$
$318$	0	0
$319$	−17189.9	−3.01708
$320$	−3360.35	−0.587029
$321$	0	0
$322$	−3859.62	−0.667976
$323$	−108.606	−0.0187090
$324$	0	0
$325$	0	0
$326$	−2118.60	−0.359933
$327$	0	0
$328$	−10727.5	−1.80587
$329$	5385.05	0.902394
$330$	0	0
$331$	−1553.67	−0.257999	−0.128999	−	0.991645i	$-0.541177\pi$
$331$	−1553.67	−0.257999	−0.128999	+	0.991645i	$0.541177\pi$
$332$	−12531.7	−2.07158
$333$	0	0
$334$	814.686	0.133466
$335$	60.4183	0.00985375
$336$	0	0
$337$	−3190.43	−0.515709	−0.257855	−	0.966184i	$-0.583016\pi$
$337$	−3190.43	−0.515709	−0.257855	+	0.966184i	$0.583016\pi$
$338$	0	0
$339$	0	0
$340$	233.283	0.0372105
$341$	13794.5	2.19065
$342$	0	0
$343$	−6915.37	−1.08862
$344$	3564.19	0.558629
$345$	0	0
$346$	−2890.59	−0.449130
$347$	5718.68	0.884712	0.442356	−	0.896840i	$-0.354143\pi$
$347$	5718.68	0.884712	0.442356	+	0.896840i	$0.354143\pi$
$348$	0	0
$349$	−3328.46	−0.510511	−0.255256	−	0.966874i	$-0.582160\pi$
$349$	−3328.46	−0.510511	−0.255256	+	0.966874i	$0.582160\pi$
$350$	−9601.70	−1.46638
$351$	0	0
$352$	35643.4	5.39715
$353$	12306.5	1.85555	0.927774	−	0.373142i	$-0.121719\pi$
$353$	12306.5	1.85555	0.927774	+	0.373142i	$0.121719\pi$
$354$	0	0
$355$	769.008	0.114971
$356$	−4481.64	−0.667210
$357$	0	0
$358$	11914.0	1.75886
$359$	−8539.97	−1.25549	−0.627747	−	0.778418i	$-0.716023\pi$
$359$	−8539.97	−1.25549	−0.627747	+	0.778418i	$0.716023\pi$
$360$	0	0
$361$	−6178.57	−0.900798
$362$	−20508.2	−2.97759
$363$	0	0
$364$	0	0
$365$	−1997.94	−0.286512
$366$	0	0
$367$	2496.65	0.355107	0.177553	−	0.984111i	$-0.443182\pi$
$367$	2496.65	0.355107	0.177553	+	0.984111i	$0.443182\pi$
$368$	−9539.49	−1.35130
$369$	0	0
$370$	−2541.29	−0.357069
$371$	159.205	0.0222790
$372$	0	0
$373$	−1142.91	−0.158653	−0.0793264	−	0.996849i	$-0.525277\pi$
$373$	−1142.91	−0.158653	−0.0793264	+	0.996849i	$0.525277\pi$
$374$	−1493.96	−0.206552
$375$	0	0
$376$	24286.4	3.33105
$377$	0	0
$378$	0	0
$379$	−12181.8	−1.65102	−0.825512	−	0.564384i	$-0.809114\pi$
$379$	−12181.8	−1.65102	−0.825512	+	0.564384i	$0.809114\pi$
$380$	−1461.54	−0.197304
$381$	0	0
$382$	−18584.4	−2.48917
$383$	10180.6	1.35824	0.679118	−	0.734029i	$-0.262362\pi$
$383$	10180.6	1.35824	0.679118	+	0.734029i	$0.262362\pi$
$384$	0	0
$385$	−2741.65	−0.362928
$386$	25200.3	3.32295
$387$	0	0
$388$	30079.5	3.93571
$389$	−5845.83	−0.761941	−0.380971	−	0.924587i	$-0.624410\pi$
$389$	−5845.83	−0.761941	−0.380971	+	0.924587i	$0.624410\pi$
$390$	0	0
$391$	197.036	0.0254848
$392$	−7671.13	−0.988395
$393$	0	0
$394$	15492.7	1.98100
$395$	1476.71	0.188105
$396$	0	0
$397$	−2500.92	−0.316166	−0.158083	−	0.987426i	$-0.550531\pi$
$397$	−2500.92	−0.316166	−0.158083	+	0.987426i	$0.550531\pi$
$398$	338.027	0.0425723
$399$	0	0
$400$	−23731.7	−2.96646
$401$	9189.25	1.14436	0.572181	−	0.820127i	$-0.306098\pi$
$401$	9189.25	1.14436	0.572181	+	0.820127i	$0.306098\pi$
$402$	0	0
$403$	0	0
$404$	18358.3	2.26080
$405$	0	0
$406$	−20960.7	−2.56223
$407$	11750.7	1.43111
$408$	0	0
$409$	9214.38	1.11399	0.556995	−	0.830516i	$-0.311954\pi$
$409$	9214.38	1.11399	0.556995	+	0.830516i	$0.311954\pi$
$410$	2263.23	0.272617
$411$	0	0
$412$	26011.5	3.11042
$413$	−6766.83	−0.806232
$414$	0	0
$415$	1626.03	0.192334
$416$	0	0
$417$	0	0
$418$	9359.78	1.09522
$419$	6494.41	0.757214	0.378607	−	0.925558i	$-0.376403\pi$
$419$	6494.41	0.757214	0.378607	+	0.925558i	$0.376403\pi$
$420$	0	0
$421$	3059.56	0.354190	0.177095	−	0.984194i	$-0.443330\pi$
$421$	3059.56	0.354190	0.177095	+	0.984194i	$0.443330\pi$
$422$	−5631.33	−0.649595
$423$	0	0
$424$	718.010	0.0822398
$425$	490.173	0.0559456
$426$	0	0
$427$	1819.68	0.206230
$428$	7098.82	0.801716
$429$	0	0
$430$	−751.954	−0.0843313
$431$	−7937.05	−0.887040	−0.443520	−	0.896264i	$-0.646271\pi$
$431$	−7937.05	−0.887040	−0.443520	+	0.896264i	$0.646271\pi$
$432$	0	0
$433$	7294.37	0.809573	0.404786	−	0.914411i	$-0.367346\pi$
$433$	7294.37	0.809573	0.404786	+	0.914411i	$0.367346\pi$
$434$	16820.5	1.86039
$435$	0	0
$436$	16108.4	1.76938
$437$	−1234.45	−0.135130
$438$	0	0
$439$	15214.7	1.65412	0.827059	−	0.562115i	$-0.190012\pi$
$439$	15214.7	1.65412	0.827059	+	0.562115i	$0.190012\pi$
$440$	−12364.7	−1.33970
$441$	0	0
$442$	0	0
$443$	−1517.05	−0.162703	−0.0813515	−	0.996685i	$-0.525924\pi$
$443$	−1517.05	−0.162703	−0.0813515	+	0.996685i	$0.525924\pi$
$444$	0	0
$445$	581.509	0.0619464
$446$	12411.3	1.31769
$447$	0	0
$448$	18946.5	1.99807
$449$	705.247	0.0741262	0.0370631	−	0.999313i	$-0.488200\pi$
$449$	705.247	0.0741262	0.0370631	+	0.999313i	$0.488200\pi$
$450$	0	0
$451$	−10465.0	−1.09263
$452$	−26579.0	−2.76586
$453$	0	0
$454$	−20383.1	−2.10710
$455$	0	0
$456$	0	0
$457$	−7277.73	−0.744940	−0.372470	−	0.928044i	$-0.621489\pi$
$457$	−7277.73	−0.744940	−0.372470	+	0.928044i	$0.621489\pi$
$458$	−23181.4	−2.36505
$459$	0	0
$460$	2651.58	0.268762
$461$	−1961.88	−0.198208	−0.0991041	−	0.995077i	$-0.531598\pi$
$461$	−1961.88	−0.198208	−0.0991041	+	0.995077i	$0.531598\pi$
$462$	0	0
$463$	−10374.1	−1.04131	−0.520653	−	0.853768i	$-0.674311\pi$
$463$	−10374.1	−1.04131	−0.520653	+	0.853768i	$0.674311\pi$
$464$	−51806.8	−5.18334
$465$	0	0
$466$	−28324.5	−2.81568
$467$	−8788.92	−0.870883	−0.435442	−	0.900217i	$-0.643408\pi$
$467$	−8788.92	−0.870883	−0.435442	+	0.900217i	$0.643408\pi$
$468$	0	0
$469$	−340.653	−0.0335392
$470$	−5123.82	−0.502860
$471$	0	0
$472$	−30518.2	−2.97609
$473$	3476.97	0.337995
$474$	0	0
$475$	−3070.98	−0.296645
$476$	−1315.31	−0.126654
$477$	0	0
$478$	−8304.14	−0.794608
$479$	11141.7	1.06279	0.531394	−	0.847125i	$-0.321668\pi$
$479$	11141.7	1.06279	0.531394	+	0.847125i	$0.321668\pi$
$480$	0	0
$481$	0	0
$482$	−26383.8	−2.49325
$483$	0	0
$484$	65303.7	6.13295
$485$	−3902.92	−0.365407
$486$	0	0
$487$	−19640.5	−1.82750	−0.913752	−	0.406273i	$-0.866828\pi$
$487$	−19640.5	−1.82750	−0.913752	+	0.406273i	$0.866828\pi$
$488$	8206.69	0.761269
$489$	0	0
$490$	1618.41	0.149209
$491$	3410.31	0.313453	0.156726	−	0.987642i	$-0.449906\pi$
$491$	3410.31	0.313453	0.156726	+	0.987642i	$0.449906\pi$
$492$	0	0
$493$	1070.06	0.0977546
$494$	0	0
$495$	0	0
$496$	41573.8	3.76354
$497$	−4335.86	−0.391327
$498$	0	0
$499$	−5032.44	−0.451469	−0.225735	−	0.974189i	$-0.572478\pi$
$499$	−5032.44	−0.451469	−0.225735	+	0.974189i	$0.572478\pi$
$500$	13600.2	1.21644
$501$	0	0
$502$	−6200.44	−0.551274
$503$	−17189.4	−1.52373	−0.761866	−	0.647735i	$-0.775716\pi$
$503$	−17189.4	−1.52373	−0.761866	+	0.647735i	$0.775716\pi$
$504$	0	0
$505$	−2382.06	−0.209901
$506$	−16980.8	−1.49187
$507$	0	0
$508$	−23145.5	−2.02149
$509$	930.560	0.0810341	0.0405170	−	0.999179i	$-0.487099\pi$
$509$	930.560	0.0810341	0.0405170	+	0.999179i	$0.487099\pi$
$510$	0	0
$511$	11264.9	0.975201
$512$	−3143.50	−0.271337
$513$	0	0
$514$	−12617.6	−1.08276
$515$	−3375.08	−0.288784
$516$	0	0
$517$	23692.1	2.01543
$518$	14328.4	1.21536
$519$	0	0
$520$	0	0
$521$	9869.60	0.829933	0.414966	−	0.909837i	$-0.363793\pi$
$521$	9869.60	0.829933	0.414966	+	0.909837i	$0.363793\pi$
$522$	0	0
$523$	−21420.6	−1.79093	−0.895466	−	0.445129i	$-0.853158\pi$
$523$	−21420.6	−1.79093	−0.895466	+	0.445129i	$0.853158\pi$
$524$	−43641.1	−3.63831
$525$	0	0
$526$	−29623.4	−2.45559
$527$	−858.697	−0.0709781
$528$	0	0
$529$	−9927.42	−0.815930
$530$	−151.482	−0.0124150
$531$	0	0
$532$	8240.54	0.671565
$533$	0	0
$534$	0	0
$535$	−921.097	−0.0744345
$536$	−1536.34	−0.123805
$537$	0	0
$538$	−21013.4	−1.68392
$539$	−7483.41	−0.598021
$540$	0	0
$541$	7771.50	0.617602	0.308801	−	0.951127i	$-0.400072\pi$
$541$	7771.50	0.617602	0.308801	+	0.951127i	$0.400072\pi$
$542$	−14900.7	−1.18088
$543$	0	0
$544$	−2218.78	−0.174870
$545$	−2090.12	−0.164277
$546$	0	0
$547$	−15577.5	−1.21763	−0.608817	−	0.793310i	$-0.708356\pi$
$547$	−15577.5	−1.21763	−0.608817	+	0.793310i	$0.708356\pi$
$548$	25040.6	1.95197
$549$	0	0
$550$	−42243.7	−3.27505
$551$	−6704.02	−0.518332
$552$	0	0
$553$	−8326.07	−0.640254
$554$	35316.4	2.70840
$555$	0	0
$556$	6709.74	0.511792
$557$	22804.5	1.73475	0.867377	−	0.497652i	$-0.165804\pi$
$557$	22804.5	1.73475	0.867377	+	0.497652i	$0.165804\pi$
$558$	0	0
$559$	0	0
$560$	−8262.78	−0.623511
$561$	0	0
$562$	−15405.7	−1.15631
$563$	3517.41	0.263306	0.131653	−	0.991296i	$-0.457972\pi$
$563$	3517.41	0.263306	0.131653	+	0.991296i	$0.457972\pi$
$564$	0	0
$565$	3448.71	0.256794
$566$	40332.2	2.99521
$567$	0	0
$568$	−19554.6	−1.44453
$569$	6093.44	0.448946	0.224473	−	0.974480i	$-0.427934\pi$
$569$	6093.44	0.448946	0.224473	+	0.974480i	$0.427934\pi$
$570$	0	0
$571$	10460.2	0.766630	0.383315	−	0.923618i	$-0.374782\pi$
$571$	10460.2	0.766630	0.383315	+	0.923618i	$0.374782\pi$
$572$	0	0
$573$	0	0
$574$	−12760.6	−0.927906
$575$	5571.47	0.404081
$576$	0	0
$577$	−9648.19	−0.696117	−0.348058	−	0.937473i	$-0.613159\pi$
$577$	−9648.19	−0.696117	−0.348058	+	0.937473i	$0.613159\pi$
$578$	−26263.9	−1.89002
$579$	0	0
$580$	14400.1	1.03092
$581$	−9167.93	−0.654647
$582$	0	0
$583$	700.440	0.0497586
$584$	50804.1	3.59981
$585$	0	0
$586$	−24176.5	−1.70430
$587$	−2170.73	−0.152633	−0.0763166	−	0.997084i	$-0.524316\pi$
$587$	−2170.73	−0.152633	−0.0763166	+	0.997084i	$0.524316\pi$
$588$	0	0
$589$	5379.82	0.376353
$590$	6438.56	0.449274
$591$	0	0
$592$	35414.3	2.45865
$593$	−22885.9	−1.58484	−0.792421	−	0.609975i	$-0.791180\pi$
$593$	−22885.9	−1.58484	−0.792421	+	0.609975i	$0.791180\pi$
$594$	0	0
$595$	170.666	0.0117590
$596$	−23458.0	−1.61221
$597$	0	0
$598$	0	0
$599$	23978.7	1.63563	0.817815	−	0.575482i	$-0.195185\pi$
$599$	23978.7	1.63563	0.817815	+	0.575482i	$0.195185\pi$
$600$	0	0
$601$	12873.2	0.873728	0.436864	−	0.899528i	$-0.356089\pi$
$601$	12873.2	0.873728	0.436864	+	0.899528i	$0.356089\pi$
$602$	4239.70	0.287039
$603$	0	0
$604$	−61103.2	−4.11631
$605$	−8473.38	−0.569408
$606$	0	0
$607$	−7117.15	−0.475908	−0.237954	−	0.971276i	$-0.576477\pi$
$607$	−7117.15	−0.475908	−0.237954	+	0.971276i	$0.576477\pi$
$608$	13900.9	0.927227
$609$	0	0
$610$	−1731.40	−0.114922
$611$	0	0
$612$	0	0
$613$	173.297	0.0114183	0.00570913	−	0.999984i	$-0.498183\pi$
$613$	173.297	0.0114183	0.00570913	+	0.999984i	$0.498183\pi$
$614$	51173.5	3.36351
$615$	0	0
$616$	69715.5	4.55993
$617$	6102.75	0.398197	0.199099	−	0.979979i	$-0.436199\pi$
$617$	6102.75	0.398197	0.199099	+	0.979979i	$0.436199\pi$
$618$	0	0
$619$	−14867.8	−0.965409	−0.482705	−	0.875783i	$-0.660346\pi$
$619$	−14867.8	−0.965409	−0.482705	+	0.875783i	$0.660346\pi$
$620$	−11555.8	−0.748534
$621$	0	0
$622$	40054.6	2.58206
$623$	−3278.69	−0.210847
$624$	0	0
$625$	12951.6	0.828900
$626$	−9772.64	−0.623951
$627$	0	0
$628$	13078.9	0.831056
$629$	−731.475	−0.0463686
$630$	0	0
$631$	−17210.6	−1.08580	−0.542902	−	0.839796i	$-0.682675\pi$
$631$	−17210.6	−1.08580	−0.542902	+	0.839796i	$0.682675\pi$
$632$	−37550.3	−2.36340
$633$	0	0
$634$	−16765.5	−1.05023
$635$	3003.21	0.187683
$636$	0	0
$637$	0	0
$638$	−92218.9	−5.72254
$639$	0	0
$640$	−6532.30	−0.403456
$641$	12636.0	0.778616	0.389308	−	0.921108i	$-0.372714\pi$
$641$	12636.0	0.778616	0.389308	+	0.921108i	$0.372714\pi$
$642$	0	0
$643$	−9586.37	−0.587947	−0.293973	−	0.955814i	$-0.594978\pi$
$643$	−9586.37	−0.587947	−0.293973	+	0.955814i	$0.594978\pi$
$644$	−14950.2	−0.914786
$645$	0	0
$646$	−582.641	−0.0354856
$647$	−5244.32	−0.318664	−0.159332	−	0.987225i	$-0.550934\pi$
$647$	−5244.32	−0.318664	−0.159332	+	0.987225i	$0.550934\pi$
$648$	0	0
$649$	−29771.4	−1.80066
$650$	0	0
$651$	0	0
$652$	−8206.39	−0.492925
$653$	−18869.0	−1.13078	−0.565392	−	0.824822i	$-0.691275\pi$
$653$	−18869.0	−1.13078	−0.565392	+	0.824822i	$0.691275\pi$
$654$	0	0
$655$	5662.59	0.337795
$656$	−31539.3	−1.87714
$657$	0	0
$658$	28889.3	1.71159
$659$	24299.8	1.43639	0.718197	−	0.695839i	$-0.244967\pi$
$659$	24299.8	1.43639	0.718197	+	0.695839i	$0.244967\pi$
$660$	0	0
$661$	29915.0	1.76030	0.880151	−	0.474694i	$-0.157441\pi$
$661$	29915.0	1.76030	0.880151	+	0.474694i	$0.157441\pi$
$662$	−8335.03	−0.489350
$663$	0	0
$664$	−41347.1	−2.41653
$665$	−1069.24	−0.0623508
$666$	0	0
$667$	12162.6	0.706056
$668$	3155.69	0.182780
$669$	0	0
$670$	324.128	0.0186898
$671$	8005.86	0.460600
$672$	0	0
$673$	15493.7	0.887426	0.443713	−	0.896169i	$-0.353661\pi$
$673$	15493.7	0.887426	0.443713	+	0.896169i	$0.353661\pi$
$674$	−17115.8	−0.978154
$675$	0	0
$676$	0	0
$677$	−11729.7	−0.665891	−0.332945	−	0.942946i	$-0.608042\pi$
$677$	−11729.7	−0.665891	−0.332945	+	0.942946i	$0.608042\pi$
$678$	0	0
$679$	22005.6	1.24374
$680$	769.698	0.0434067
$681$	0	0
$682$	74003.5	4.15505
$683$	1168.42	0.0654587	0.0327294	−	0.999464i	$-0.489580\pi$
$683$	1168.42	0.0654587	0.0327294	+	0.999464i	$0.489580\pi$
$684$	0	0
$685$	−3249.10	−0.181229
$686$	−37099.1	−2.06480
$687$	0	0
$688$	10478.9	0.580675
$689$	0	0
$690$	0	0
$691$	32992.9	1.81637	0.908183	−	0.418574i	$-0.137470\pi$
$691$	32992.9	1.81637	0.908183	+	0.418574i	$0.137470\pi$
$692$	−11196.7	−0.615078
$693$	0	0
$694$	30679.2	1.67805
$695$	−870.612	−0.0475168
$696$	0	0
$697$	651.438	0.0354017
$698$	−17856.3	−0.968295
$699$	0	0
$700$	−37192.2	−2.00819
$701$	14785.8	0.796651	0.398326	−	0.917244i	$-0.369591\pi$
$701$	14785.8	0.796651	0.398326	+	0.917244i	$0.369591\pi$
$702$	0	0
$703$	4582.77	0.245864
$704$	83357.0	4.46255
$705$	0	0
$706$	66021.0	3.51945
$707$	13430.6	0.714442
$708$	0	0
$709$	12634.0	0.669225	0.334612	−	0.942356i	$-0.391395\pi$
$709$	12634.0	0.669225	0.334612	+	0.942356i	$0.391395\pi$
$710$	4125.52	0.218067
$711$	0	0
$712$	−14786.8	−0.778312
$713$	−9760.24	−0.512656
$714$	0	0
$715$	0	0
$716$	46148.7	2.40874
$717$	0	0
$718$	−45814.6	−2.38132
$719$	27296.5	1.41584	0.707920	−	0.706292i	$-0.249634\pi$
$719$	27296.5	1.41584	0.707920	+	0.706292i	$0.249634\pi$
$720$	0	0
$721$	19029.5	0.982936
$722$	−33146.3	−1.70856
$723$	0	0
$724$	−79438.5	−4.07777
$725$	30257.4	1.54997
$726$	0	0
$727$	4658.21	0.237639	0.118819	−	0.992916i	$-0.462089\pi$
$727$	4658.21	0.237639	0.118819	+	0.992916i	$0.462089\pi$
$728$	0	0
$729$	0	0
$730$	−10718.4	−0.543432
$731$	−216.439	−0.0109512
$732$	0	0
$733$	−166.474	−0.00838864	−0.00419432	−	0.999991i	$-0.501335\pi$
$733$	−166.474	−0.00838864	−0.00419432	+	0.999991i	$0.501335\pi$
$734$	13393.9	0.673537
$735$	0	0
$736$	−25219.4	−1.26304
$737$	−1498.74	−0.0749074
$738$	0	0
$739$	12738.1	0.634069	0.317035	−	0.948414i	$-0.397313\pi$
$739$	12738.1	0.634069	0.317035	+	0.948414i	$0.397313\pi$
$740$	−9843.70	−0.489002
$741$	0	0
$742$	854.092	0.0422570
$743$	30724.5	1.51705	0.758527	−	0.651641i	$-0.225919\pi$
$743$	30724.5	1.51705	0.758527	+	0.651641i	$0.225919\pi$
$744$	0	0
$745$	3043.76	0.149684
$746$	−6131.38	−0.300919
$747$	0	0
$748$	−5786.84	−0.282871
$749$	5193.37	0.253353
$750$	0	0
$751$	−39538.6	−1.92115	−0.960575	−	0.278021i	$-0.910322\pi$
$751$	−39538.6	−1.92115	−0.960575	+	0.278021i	$0.910322\pi$
$752$	71403.2	3.46251
$753$	0	0
$754$	0	0
$755$	7928.35	0.382175
$756$	0	0
$757$	23035.1	1.10598	0.552990	−	0.833188i	$-0.313487\pi$
$757$	23035.1	1.10598	0.552990	+	0.833188i	$0.313487\pi$
$758$	−65352.1	−3.13153
$759$	0	0
$760$	−4822.23	−0.230159
$761$	−32454.0	−1.54594	−0.772968	−	0.634445i	$-0.781229\pi$
$761$	−32454.0	−1.54594	−0.772968	+	0.634445i	$0.781229\pi$
$762$	0	0
$763$	11784.6	0.559150
$764$	−71986.8	−3.40889
$765$	0	0
$766$	54616.1	2.57619
$767$	0	0
$768$	0	0
$769$	−32216.2	−1.51072	−0.755362	−	0.655307i	$-0.772539\pi$
$769$	−32216.2	−1.51072	−0.755362	+	0.655307i	$0.772539\pi$
$770$	−14708.2	−0.688372
$771$	0	0
$772$	97613.2	4.55075
$773$	−2924.60	−0.136081	−0.0680404	−	0.997683i	$-0.521675\pi$
$773$	−2924.60	−0.136081	−0.0680404	+	0.997683i	$0.521675\pi$
$774$	0	0
$775$	−24280.9	−1.12541
$776$	99244.7	4.59108
$777$	0	0
$778$	−31361.2	−1.44519
$779$	−4081.32	−0.187713
$780$	0	0
$781$	−19076.1	−0.874001
$782$	1057.04	0.0483374
$783$	0	0
$784$	−22553.5	−1.02740
$785$	−1697.03	−0.0771586
$786$	0	0
$787$	25507.1	1.15531	0.577656	−	0.816280i	$-0.303967\pi$
$787$	25507.1	1.15531	0.577656	+	0.816280i	$0.303967\pi$
$788$	60011.1	2.71295
$789$	0	0
$790$	7922.16	0.356782
$791$	−19444.7	−0.874050
$792$	0	0
$793$	0	0
$794$	−13416.8	−0.599677
$795$	0	0
$796$	1309.35	0.0583023
$797$	−5448.98	−0.242174	−0.121087	−	0.992642i	$-0.538638\pi$
$797$	−5448.98	−0.242174	−0.121087	+	0.992642i	$0.538638\pi$
$798$	0	0
$799$	−1474.82	−0.0653008
$800$	−62739.0	−2.77270
$801$	0	0
$802$	49297.8	2.17053
$803$	49560.9	2.17804
$804$	0	0
$805$	1939.85	0.0849324
$806$	0	0
$807$	0	0
$808$	60571.7	2.63726
$809$	2453.12	0.106610	0.0533048	−	0.998578i	$-0.483025\pi$
$809$	2453.12	0.106610	0.0533048	+	0.998578i	$0.483025\pi$
$810$	0	0
$811$	−5133.85	−0.222286	−0.111143	−	0.993804i	$-0.535451\pi$
$811$	−5133.85	−0.222286	−0.111143	+	0.993804i	$0.535451\pi$
$812$	−81191.4	−3.50894
$813$	0	0
$814$	63039.4	2.71441
$815$	1064.81	0.0457651
$816$	0	0
$817$	1356.01	0.0580673
$818$	49432.6	2.11292
$819$	0	0
$820$	8766.61	0.373345
$821$	23854.0	1.01402	0.507009	−	0.861941i	$-0.330751\pi$
$821$	23854.0	1.01402	0.507009	+	0.861941i	$0.330751\pi$
$822$	0	0
$823$	757.156	0.0320690	0.0160345	−	0.999871i	$-0.494896\pi$
$823$	757.156	0.0320690	0.0160345	+	0.999871i	$0.494896\pi$
$824$	85822.6	3.62836
$825$	0	0
$826$	−36302.2	−1.52919
$827$	−28621.2	−1.20345	−0.601726	−	0.798702i	$-0.705520\pi$
$827$	−28621.2	−1.20345	−0.601726	+	0.798702i	$0.705520\pi$
$828$	0	0
$829$	27429.2	1.14916	0.574582	−	0.818447i	$-0.305165\pi$
$829$	27429.2	1.14916	0.574582	+	0.818447i	$0.305165\pi$
$830$	8723.19	0.364803
$831$	0	0
$832$	0	0
$833$	465.838	0.0193761
$834$	0	0
$835$	−409.462	−0.0169701
$836$	36255.1	1.49989
$837$	0	0
$838$	34840.7	1.43622
$839$	4633.62	0.190668	0.0953339	−	0.995445i	$-0.469608\pi$
$839$	4633.62	0.190668	0.0953339	+	0.995445i	$0.469608\pi$
$840$	0	0
$841$	41663.6	1.70829
$842$	16413.7	0.671798
$843$	0	0
$844$	−21813.0	−0.889614
$845$	0	0
$846$	0	0
$847$	47775.0	1.93810
$848$	2110.99	0.0854853
$849$	0	0
$850$	2629.64	0.106113
$851$	−8314.19	−0.334908
$852$	0	0
$853$	−14854.6	−0.596261	−0.298131	−	0.954525i	$-0.596363\pi$
$853$	−14854.6	−0.596261	−0.298131	+	0.954525i	$0.596363\pi$
$854$	9762.07	0.391161
$855$	0	0
$856$	23421.9	0.935216
$857$	42799.5	1.70595	0.852977	−	0.521948i	$-0.174794\pi$
$857$	42799.5	1.70595	0.852977	+	0.521948i	$0.174794\pi$
$858$	0	0
$859$	−8246.47	−0.327551	−0.163775	−	0.986498i	$-0.552367\pi$
$859$	−8246.47	−0.327551	−0.163775	+	0.986498i	$0.552367\pi$
$860$	−2912.70	−0.115491
$861$	0	0
$862$	−42580.1	−1.68246
$863$	−17695.4	−0.697983	−0.348991	−	0.937126i	$-0.613476\pi$
$863$	−17695.4	−0.697983	−0.348991	+	0.937126i	$0.613476\pi$
$864$	0	0
$865$	1452.81	0.0571064
$866$	39132.3	1.53553
$867$	0	0
$868$	65154.2	2.54779
$869$	−36631.4	−1.42996
$870$	0	0
$871$	0	0
$872$	53148.2	2.06402
$873$	0	0
$874$	−6622.49	−0.256303
$875$	9949.64	0.384411
$876$	0	0
$877$	−14346.8	−0.552401	−0.276200	−	0.961100i	$-0.589075\pi$
$877$	−14346.8	−0.552401	−0.276200	+	0.961100i	$0.589075\pi$
$878$	81622.7	3.13739
$879$	0	0
$880$	−36353.0	−1.39257
$881$	2063.51	0.0789121	0.0394560	−	0.999221i	$-0.487437\pi$
$881$	2063.51	0.0789121	0.0394560	+	0.999221i	$0.487437\pi$
$882$	0	0
$883$	−34137.1	−1.30103	−0.650513	−	0.759495i	$-0.725446\pi$
$883$	−34137.1	−1.30103	−0.650513	+	0.759495i	$0.725446\pi$
$884$	0	0
$885$	0	0
$886$	−8138.58	−0.308602
$887$	22238.5	0.841821	0.420910	−	0.907102i	$-0.361711\pi$
$887$	22238.5	0.841821	0.420910	+	0.907102i	$0.361711\pi$
$888$	0	0
$889$	−16932.8	−0.638817
$890$	3119.63	0.117495
$891$	0	0
$892$	48075.0	1.80456
$893$	9239.89	0.346250
$894$	0	0
$895$	−5987.96	−0.223637
$896$	36830.7	1.37325
$897$	0	0
$898$	3783.45	0.140596
$899$	−53005.7	−1.96645
$900$	0	0
$901$	−43.6019	−0.00161220
$902$	−56141.7	−2.07241
$903$	0	0
$904$	−87694.9	−3.22643
$905$	10307.4	0.378597
$906$	0	0
$907$	−44160.3	−1.61667	−0.808335	−	0.588723i	$-0.799631\pi$
$907$	−44160.3	−1.61667	−0.808335	+	0.588723i	$0.799631\pi$
$908$	−78953.8	−2.88565
$909$	0	0
$910$	0	0
$911$	−11916.3	−0.433376	−0.216688	−	0.976241i	$-0.569525\pi$
$911$	−11916.3	−0.433376	−0.216688	+	0.976241i	$0.569525\pi$
$912$	0	0
$913$	−40335.3	−1.46211
$914$	−39043.0	−1.41294
$915$	0	0
$916$	−89793.0	−3.23891
$917$	−31927.1	−1.14975
$918$	0	0
$919$	−20128.9	−0.722516	−0.361258	−	0.932466i	$-0.617653\pi$
$919$	−20128.9	−0.722516	−0.361258	+	0.932466i	$0.617653\pi$
$920$	8748.64	0.313515
$921$	0	0
$922$	−10525.0	−0.375945
$923$	0	0
$924$	0	0
$925$	−20683.5	−0.735210
$926$	−55654.1	−1.97506
$927$	0	0
$928$	−136961.	−4.84478
$929$	31428.4	1.10994	0.554969	−	0.831871i	$-0.312730\pi$
$929$	31428.4	1.10994	0.554969	+	0.831871i	$0.312730\pi$
$930$	0	0
$931$	−2918.52	−0.102740
$932$	−109715.	−3.85605
$933$	0	0
$934$	−47150.1	−1.65182
$935$	750.863	0.0262629
$936$	0	0
$937$	42473.2	1.48083	0.740416	−	0.672149i	$-0.234629\pi$
$937$	42473.2	1.48083	0.740416	+	0.672149i	$0.234629\pi$
$938$	−1827.51	−0.0636144
$939$	0	0
$940$	−19847.1	−0.688661
$941$	42644.0	1.47732	0.738659	−	0.674079i	$-0.235459\pi$
$941$	42644.0	1.47732	0.738659	+	0.674079i	$0.235459\pi$
$942$	0	0
$943$	7404.46	0.255697
$944$	−89725.0	−3.09354
$945$	0	0
$946$	18653.0	0.641080
$947$	−4282.95	−0.146966	−0.0734831	−	0.997296i	$-0.523412\pi$
$947$	−4282.95	−0.146966	−0.0734831	+	0.997296i	$0.523412\pi$
$948$	0	0
$949$	0	0
$950$	−16475.0	−0.562651
$951$	0	0
$952$	−4339.74	−0.147744
$953$	−40593.9	−1.37982	−0.689908	−	0.723897i	$-0.742349\pi$
$953$	−40593.9	−1.37982	−0.689908	+	0.723897i	$0.742349\pi$
$954$	0	0
$955$	9340.54	0.316495
$956$	−32166.1	−1.08821
$957$	0	0
$958$	59771.9	2.01581
$959$	18319.2	0.616850
$960$	0	0
$961$	12744.8	0.427808
$962$	0	0
$963$	0	0
$964$	−102198.	−3.41448
$965$	−12665.7	−0.422510
$966$	0	0
$967$	17709.4	0.588931	0.294466	−	0.955662i	$-0.404858\pi$
$967$	17709.4	0.588931	0.294466	+	0.955662i	$0.404858\pi$
$968$	215464.	7.15420
$969$	0	0
$970$	−20938.1	−0.693074
$971$	−4038.97	−0.133488	−0.0667440	−	0.997770i	$-0.521261\pi$
$971$	−4038.97	−0.133488	−0.0667440	+	0.997770i	$0.521261\pi$
$972$	0	0
$973$	4908.72	0.161733
$974$	−105366.	−3.46626
$975$	0	0
$976$	24128.1	0.791312
$977$	2764.30	0.0905197	0.0452598	−	0.998975i	$-0.485588\pi$
$977$	2764.30	0.0905197	0.0452598	+	0.998975i	$0.485588\pi$
$978$	0	0
$979$	−14424.9	−0.470912
$980$	6268.93	0.204340
$981$	0	0
$982$	18295.4	0.594531
$983$	17804.5	0.577695	0.288848	−	0.957375i	$-0.406728\pi$
$983$	17804.5	0.577695	0.288848	+	0.957375i	$0.406728\pi$
$984$	0	0
$985$	−7786.65	−0.251881
$986$	5740.57	0.185413
$987$	0	0
$988$	0	0
$989$	−2460.12	−0.0790974
$990$	0	0
$991$	−8129.69	−0.260593	−0.130297	−	0.991475i	$-0.541593\pi$
$991$	−8129.69	−0.260593	−0.130297	+	0.991475i	$0.541593\pi$
$992$	109908.	3.51772
$993$	0	0
$994$	−23260.7	−0.742237
$995$	−169.893	−0.00541302
$996$	0	0
$997$	−29452.9	−0.935589	−0.467795	−	0.883837i	$-0.654951\pi$
$997$	−29452.9	−0.935589	−0.467795	+	0.883837i	$0.654951\pi$
$998$	−26997.7	−0.856309
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bk.1.10		10
3.2	odd	2		507.4.a.r.1.1		10
13.2	odd	12		117.4.q.e.82.5		10
13.7	odd	12		117.4.q.e.10.5		10
13.12	even	2	inner	1521.4.a.bk.1.1		10
39.2	even	12		39.4.j.c.4.1	✓	10
39.5	even	4		507.4.b.i.337.10		10
39.8	even	4		507.4.b.i.337.1		10
39.20	even	12		39.4.j.c.10.1	yes	10
39.38	odd	2		507.4.a.r.1.10		10
156.59	odd	12		624.4.bv.h.49.3		10
156.119	odd	12		624.4.bv.h.433.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.1	✓	10	39.2	even	12
39.4.j.c.10.1	yes	10	39.20	even	12
117.4.q.e.10.5		10	13.7	odd	12
117.4.q.e.82.5		10	13.2	odd	12
507.4.a.r.1.1		10	3.2	odd	2
507.4.a.r.1.10		10	39.38	odd	2
507.4.b.i.337.1		10	39.8	even	4
507.4.b.i.337.10		10	39.5	even	4
624.4.bv.h.49.3		10	156.59	odd	12
624.4.bv.h.433.3		10	156.119	odd	12
1521.4.a.bk.1.1		10	13.12	even	2	inner
1521.4.a.bk.1.10		10	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)

\(f(q)\)	\(=\)	\(q+5.36472 q^{2} +20.7803 q^{4} -2.69631 q^{5} +15.2025 q^{7} +68.5626 q^{8} +O(q^{10})\)	\(q+5.36472 q^{2} +20.7803 q^{4} -2.69631 q^{5} +15.2025 q^{7} +68.5626 q^{8} -14.4650 q^{10} +66.8848 q^{11} +81.5570 q^{14} +201.577 q^{16} -4.16354 q^{17} +26.0850 q^{19} -56.0301 q^{20} +358.819 q^{22} -47.3242 q^{23} -117.730 q^{25} +315.911 q^{28} -257.007 q^{29} +206.242 q^{31} +532.906 q^{32} -22.3362 q^{34} -40.9906 q^{35} +175.686 q^{37} +139.939 q^{38} -184.866 q^{40} -156.463 q^{41} +51.9845 q^{43} +1389.88 q^{44} -253.881 q^{46} +354.222 q^{47} -111.885 q^{49} -631.588 q^{50} +10.4723 q^{53} -180.342 q^{55} +1042.32 q^{56} -1378.77 q^{58} -445.114 q^{59} +119.696 q^{61} +1106.43 q^{62} +1246.28 q^{64} -22.4078 q^{67} -86.5195 q^{68} -219.903 q^{70} -285.207 q^{71} +740.989 q^{73} +942.507 q^{74} +542.053 q^{76} +1016.81 q^{77} -547.679 q^{79} -543.516 q^{80} -839.378 q^{82} -603.056 q^{83} +11.2262 q^{85} +278.882 q^{86} +4585.80 q^{88} -215.668 q^{89} -983.409 q^{92} +1900.31 q^{94} -70.3333 q^{95} +1447.50 q^{97} -600.233 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 60 q^{4}+O(q^{10}) \) 10 * q + 60 * q^4	\( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55} + 5340 q^{56} + 2750 q^{61} + 1560 q^{62} + 3140 q^{64} - 1200 q^{68} + 4380 q^{74} - 4320 q^{77} + 1100 q^{79} - 4780 q^{82} + 6340 q^{88} + 1740 q^{92} + 6460 q^{94} + 2760 q^{95}+O(q^{100}) \) 10 * q + 60 * q^4 + 80 * q^10 + 60 * q^14 + 500 * q^16 - 210 * q^17 + 580 * q^22 + 120 * q^23 + 960 * q^25 - 990 * q^29 + 120 * q^35 - 1380 * q^38 + 2000 * q^40 - 740 * q^43 + 1550 * q^49 - 330 * q^53 + 520 * q^55 + 5340 * q^56 + 2750 * q^61 + 1560 * q^62 + 3140 * q^64 - 1200 * q^68 + 4380 * q^74 - 4320 * q^77 + 1100 * q^79 - 4780 * q^82 + 6340 * q^88 + 1740 * q^92 + 6460 * q^94 + 2760 * q^95

Introduction

L-functions

Modular forms

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