LMFDB - Embedded newform 1521.4.a.bc.1.6

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 52x^{6} + 805x^{4} - 4210x^{2} + 4992 $ x^8 - 52x^6 + 805x^4 - 4210*x^2 + 4992
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.75628$ 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)

$f(q)$	$=$	$q+2.75628 q^{2} -0.402937 q^{4} +0.313209 q^{5} +28.5660 q^{7} -23.1608 q^{8} +O(q^{10})$	$q+2.75628 q^{2} -0.402937 q^{4} +0.313209 q^{5} +28.5660 q^{7} -23.1608 q^{8} +0.863291 q^{10} +63.2173 q^{11} +78.7357 q^{14} -60.6141 q^{16} +98.8270 q^{17} -14.4897 q^{19} -0.126204 q^{20} +174.244 q^{22} -14.3900 q^{23} -124.902 q^{25} -11.5103 q^{28} -196.283 q^{29} +118.691 q^{31} +18.2172 q^{32} +272.395 q^{34} +8.94712 q^{35} +319.114 q^{37} -39.9377 q^{38} -7.25418 q^{40} +346.106 q^{41} -69.4854 q^{43} -25.4726 q^{44} -39.6628 q^{46} -101.875 q^{47} +473.014 q^{49} -344.264 q^{50} -594.823 q^{53} +19.8002 q^{55} -661.611 q^{56} -541.010 q^{58} -204.473 q^{59} -215.978 q^{61} +327.144 q^{62} +535.125 q^{64} +68.6049 q^{67} -39.8211 q^{68} +24.6607 q^{70} +946.241 q^{71} -779.872 q^{73} +879.567 q^{74} +5.83844 q^{76} +1805.86 q^{77} +240.022 q^{79} -18.9849 q^{80} +953.965 q^{82} +855.576 q^{83} +30.9535 q^{85} -191.521 q^{86} -1464.16 q^{88} +1264.06 q^{89} +5.79825 q^{92} -280.795 q^{94} -4.53831 q^{95} +662.290 q^{97} +1303.76 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 40 q^{4} - 22 q^{7}+O(q^{10}) $ 8 * q + 40 * q^4 - 22 * q^7	$ 8 q + 40 q^{4} - 22 q^{7} + 36 q^{10} + 204 q^{16} + 244 q^{19} + 136 q^{22} + 354 q^{25} - 452 q^{28} + 242 q^{31} - 1292 q^{34} + 1018 q^{37} + 1700 q^{40} + 74 q^{43} - 896 q^{46} + 298 q^{49} + 1300 q^{55} + 812 q^{58} + 1148 q^{61} + 3636 q^{64} - 2198 q^{67} + 2200 q^{70} - 2176 q^{73} + 6936 q^{76} + 1862 q^{79} + 5436 q^{82} - 890 q^{85} + 3528 q^{88} - 3104 q^{94} - 4370 q^{97}+O(q^{100}) $ 8 * q + 40 * q^4 - 22 * q^7 + 36 * q^10 + 204 * q^16 + 244 * q^19 + 136 * q^22 + 354 * q^25 - 452 * q^28 + 242 * q^31 - 1292 * q^34 + 1018 * q^37 + 1700 * q^40 + 74 * q^43 - 896 * q^46 + 298 * q^49 + 1300 * q^55 + 812 * q^58 + 1148 * q^61 + 3636 * q^64 - 2198 * q^67 + 2200 * q^70 - 2176 * q^73 + 6936 * q^76 + 1862 * q^79 + 5436 * q^82 - 890 * q^85 + 3528 * q^88 - 3104 * q^94 - 4370 * q^97

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 $

$2$	2.75628	0.974491	0.487246	−	0.873265i	$-0.338002\pi$
$2$	2.75628	0.974491	0.487246	+	0.873265i	$0.338002\pi$
$3$	0	0
$3$	0	0
$4$	−0.402937	−0.0503671
$5$	0.313209	0.0280143	0.0140071	−	0.999902i	$-0.495541\pi$
$5$	0.313209	0.0280143	0.0140071	+	0.999902i	$0.495541\pi$
$6$	0	0
$7$	28.5660	1.54242	0.771208	−	0.636583i	$-0.219653\pi$
$7$	28.5660	1.54242	0.771208	+	0.636583i	$0.219653\pi$
$8$	−23.1608	−1.02357
$9$	0	0
$10$	0.863291	0.0272997
$11$	63.2173	1.73279	0.866397	−	0.499356i	$-0.166430\pi$
$11$	63.2173	1.73279	0.866397	+	0.499356i	$0.166430\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	78.7357	1.50307
$15$	0	0
$16$	−60.6141	−0.947096
$17$	98.8270	1.40995	0.704973	−	0.709234i	$-0.250959\pi$
$17$	98.8270	1.40995	0.704973	+	0.709234i	$0.250959\pi$
$18$	0	0
$19$	−14.4897	−0.174956	−0.0874782	−	0.996166i	$-0.527881\pi$
$19$	−14.4897	−0.174956	−0.0874782	+	0.996166i	$0.527881\pi$
$20$	−0.126204	−0.00141100
$21$	0	0
$22$	174.244	1.68859
$23$	−14.3900	−0.130457	−0.0652286	−	0.997870i	$-0.520778\pi$
$23$	−14.3900	−0.130457	−0.0652286	+	0.997870i	$0.520778\pi$
$24$	0	0
$25$	−124.902	−0.999215
$26$	0	0
$27$	0	0
$28$	−11.5103	−0.0776871
$29$	−196.283	−1.25686	−0.628428	−	0.777868i	$-0.716301\pi$
$29$	−196.283	−1.25686	−0.628428	+	0.777868i	$0.716301\pi$
$30$	0	0
$31$	118.691	0.687661	0.343830	−	0.939032i	$-0.388276\pi$
$31$	118.691	0.687661	0.343830	+	0.939032i	$0.388276\pi$
$32$	18.2172	0.100637
$33$	0	0
$34$	272.395	1.37398
$35$	8.94712	0.0432097
$36$	0	0
$37$	319.114	1.41789	0.708947	−	0.705262i	$-0.249171\pi$
$37$	319.114	1.41789	0.708947	+	0.705262i	$0.249171\pi$
$38$	−39.9377	−0.170493
$39$	0	0
$40$	−7.25418	−0.0286747
$41$	346.106	1.31836	0.659180	−	0.751986i	$-0.270904\pi$
$41$	346.106	1.31836	0.659180	+	0.751986i	$0.270904\pi$
$42$	0	0
$43$	−69.4854	−0.246428	−0.123214	−	0.992380i	$-0.539320\pi$
$43$	−69.4854	−0.246428	−0.123214	+	0.992380i	$0.539320\pi$
$44$	−25.4726	−0.0872758
$45$	0	0
$46$	−39.6628	−0.127129
$47$	−101.875	−0.316170	−0.158085	−	0.987426i	$-0.550532\pi$
$47$	−101.875	−0.316170	−0.158085	+	0.987426i	$0.550532\pi$
$48$	0	0
$49$	473.014	1.37905
$50$	−344.264	−0.973726
$51$	0	0
$52$	0	0
$53$	−594.823	−1.54161	−0.770804	−	0.637072i	$-0.780145\pi$
$53$	−594.823	−1.54161	−0.770804	+	0.637072i	$0.780145\pi$
$54$	0	0
$55$	19.8002	0.0485430
$56$	−661.611	−1.57878
$57$	0	0
$58$	−541.010	−1.22480
$59$	−204.473	−0.451189	−0.225595	−	0.974221i	$-0.572432\pi$
$59$	−204.473	−0.451189	−0.225595	+	0.974221i	$0.572432\pi$
$60$	0	0
$61$	−215.978	−0.453331	−0.226666	−	0.973973i	$-0.572782\pi$
$61$	−215.978	−0.453331	−0.226666	+	0.973973i	$0.572782\pi$
$62$	327.144	0.670119
$63$	0	0
$64$	535.125	1.04517
$65$	0	0
$66$	0	0
$67$	68.6049	0.125096	0.0625480	−	0.998042i	$-0.480077\pi$
$67$	68.6049	0.125096	0.0625480	+	0.998042i	$0.480077\pi$
$68$	−39.8211	−0.0710149
$69$	0	0
$70$	24.6607	0.0421075
$71$	946.241	1.58166	0.790832	−	0.612033i	$-0.209648\pi$
$71$	946.241	1.58166	0.790832	+	0.612033i	$0.209648\pi$
$72$	0	0
$73$	−779.872	−1.25037	−0.625185	−	0.780476i	$-0.714977\pi$
$73$	−779.872	−1.25037	−0.625185	+	0.780476i	$0.714977\pi$
$74$	879.567	1.38172
$75$	0	0
$76$	5.83844	0.00881204
$77$	1805.86	2.67269
$78$	0	0
$79$	240.022	0.341831	0.170915	−	0.985286i	$-0.445328\pi$
$79$	240.022	0.341831	0.170915	+	0.985286i	$0.445328\pi$
$80$	−18.9849	−0.0265322
$81$	0	0
$82$	953.965	1.28473
$83$	855.576	1.13147	0.565733	−	0.824588i	$-0.308593\pi$
$83$	855.576	1.13147	0.565733	+	0.824588i	$0.308593\pi$
$84$	0	0
$85$	30.9535	0.0394986
$86$	−191.521	−0.240142
$87$	0	0
$88$	−1464.16	−1.77364
$89$	1264.06	1.50551	0.752753	−	0.658303i	$-0.228726\pi$
$89$	1264.06	1.50551	0.752753	+	0.658303i	$0.228726\pi$
$90$	0	0
$91$	0	0
$92$	5.79825	0.00657075
$93$	0	0
$94$	−280.795	−0.308105
$95$	−4.53831	−0.00490128
$96$	0	0
$97$	662.290	0.693251	0.346625	−	0.938004i	$-0.387328\pi$
$97$	662.290	0.693251	0.346625	+	0.938004i	$0.387328\pi$
$98$	1303.76	1.34387
$99$	0	0
$100$	50.3276	0.0503276
$101$	−793.571	−0.781814	−0.390907	−	0.920430i	$-0.627839\pi$
$101$	−793.571	−0.781814	−0.390907	+	0.920430i	$0.627839\pi$
$102$	0	0
$103$	980.791	0.938254	0.469127	−	0.883131i	$-0.344569\pi$
$103$	980.791	0.938254	0.469127	+	0.883131i	$0.344569\pi$
$104$	0	0
$105$	0	0
$106$	−1639.50	−1.50228
$107$	−805.944	−0.728164	−0.364082	−	0.931367i	$-0.618617\pi$
$107$	−805.944	−0.728164	−0.364082	+	0.931367i	$0.618617\pi$
$108$	0	0
$109$	845.184	0.742697	0.371348	−	0.928494i	$-0.378896\pi$
$109$	845.184	0.742697	0.371348	+	0.928494i	$0.378896\pi$
$110$	54.5749	0.0473047
$111$	0	0
$112$	−1731.50	−1.46082
$113$	56.3774	0.0469340	0.0234670	−	0.999725i	$-0.492530\pi$
$113$	56.3774	0.0469340	0.0234670	+	0.999725i	$0.492530\pi$
$114$	0	0
$115$	−4.50707	−0.00365467
$116$	79.0896	0.0633042
$117$	0	0
$118$	−563.585	−0.439680
$119$	2823.09	2.17472
$120$	0	0
$121$	2665.43	2.00257
$122$	−595.296	−0.441767
$123$	0	0
$124$	−47.8249	−0.0346355
$125$	−78.2716	−0.0560066
$126$	0	0
$127$	1666.65	1.16450	0.582248	−	0.813011i	$-0.302173\pi$
$127$	1666.65	1.16450	0.582248	+	0.813011i	$0.302173\pi$
$128$	1329.21	0.917868
$129$	0	0
$130$	0	0
$131$	−602.630	−0.401924	−0.200962	−	0.979599i	$-0.564407\pi$
$131$	−602.630	−0.401924	−0.200962	+	0.979599i	$0.564407\pi$
$132$	0	0
$133$	−413.913	−0.269856
$134$	189.094	0.121905
$135$	0	0
$136$	−2288.92	−1.44318
$137$	421.654	0.262951	0.131476	−	0.991319i	$-0.458029\pi$
$137$	421.654	0.262951	0.131476	+	0.991319i	$0.458029\pi$
$138$	0	0
$139$	29.1702	0.0177999	0.00889995	−	0.999960i	$-0.497167\pi$
$139$	29.1702	0.0177999	0.00889995	+	0.999960i	$0.497167\pi$
$140$	−3.60512	−0.00217635
$141$	0	0
$142$	2608.10	1.54132
$143$	0	0
$144$	0	0
$145$	−61.4776	−0.0352099
$146$	−2149.54	−1.21848
$147$	0	0
$148$	−128.583	−0.0714152
$149$	2141.80	1.17760	0.588802	−	0.808278i	$-0.299600\pi$
$149$	2141.80	1.17760	0.588802	+	0.808278i	$0.299600\pi$
$150$	0	0
$151$	1459.30	0.786462	0.393231	−	0.919440i	$-0.371357\pi$
$151$	1459.30	0.786462	0.393231	+	0.919440i	$0.371357\pi$
$152$	335.594	0.179081
$153$	0	0
$154$	4977.46	2.60451
$155$	37.1750	0.0192643
$156$	0	0
$157$	2008.20	1.02084	0.510419	−	0.859926i	$-0.329490\pi$
$157$	2008.20	1.02084	0.510419	+	0.859926i	$0.329490\pi$
$158$	661.568	0.333111
$159$	0	0
$160$	5.70579	0.00281927
$161$	−411.063	−0.201219
$162$	0	0
$163$	548.495	0.263567	0.131784	−	0.991279i	$-0.457930\pi$
$163$	548.495	0.263567	0.131784	+	0.991279i	$0.457930\pi$
$164$	−139.459	−0.0664019
$165$	0	0
$166$	2358.21	1.10260
$167$	−3962.58	−1.83613	−0.918064	−	0.396432i	$-0.870248\pi$
$167$	−3962.58	−1.83613	−0.918064	+	0.396432i	$0.870248\pi$
$168$	0	0
$169$	0	0
$170$	85.3165	0.0384910
$171$	0	0
$172$	27.9982	0.0124119
$173$	−1729.54	−0.760084	−0.380042	−	0.924969i	$-0.624091\pi$
$173$	−1729.54	−0.760084	−0.380042	+	0.924969i	$0.624091\pi$
$174$	0	0
$175$	−3567.94	−1.54121
$176$	−3831.86	−1.64112
$177$	0	0
$178$	3484.10	1.46710
$179$	1452.91	0.606680	0.303340	−	0.952882i	$-0.401898\pi$
$179$	1452.91	0.606680	0.303340	+	0.952882i	$0.401898\pi$
$180$	0	0
$181$	−550.329	−0.225998	−0.112999	−	0.993595i	$-0.536046\pi$
$181$	−550.329	−0.225998	−0.112999	+	0.993595i	$0.536046\pi$
$182$	0	0
$183$	0	0
$184$	333.284	0.133533
$185$	99.9495	0.0397213
$186$	0	0
$187$	6247.58	2.44314
$188$	41.0491	0.0159246
$189$	0	0
$190$	−12.5089	−0.00477625
$191$	1868.68	0.707921	0.353960	−	0.935260i	$-0.384835\pi$
$191$	1868.68	0.707921	0.353960	+	0.935260i	$0.384835\pi$
$192$	0	0
$193$	4156.52	1.55022	0.775112	−	0.631824i	$-0.217694\pi$
$193$	4156.52	1.55022	0.775112	+	0.631824i	$0.217694\pi$
$194$	1825.45	0.675567
$195$	0	0
$196$	−190.595	−0.0694587
$197$	−1522.18	−0.550513	−0.275257	−	0.961371i	$-0.588763\pi$
$197$	−1522.18	−0.550513	−0.275257	+	0.961371i	$0.588763\pi$
$198$	0	0
$199$	808.130	0.287873	0.143937	−	0.989587i	$-0.454024\pi$
$199$	808.130	0.287873	0.143937	+	0.989587i	$0.454024\pi$
$200$	2892.83	1.02277
$201$	0	0
$202$	−2187.30	−0.761871
$203$	−5607.01	−1.93860
$204$	0	0
$205$	108.404	0.0369329
$206$	2703.33	0.914320
$207$	0	0
$208$	0	0
$209$	−916.001	−0.303163
$210$	0	0
$211$	−5260.03	−1.71619	−0.858094	−	0.513493i	$-0.828351\pi$
$211$	−5260.03	−1.71619	−0.858094	+	0.513493i	$0.828351\pi$
$212$	239.676	0.0776464
$213$	0	0
$214$	−2221.41	−0.709590
$215$	−21.7635	−0.00690351
$216$	0	0
$217$	3390.51	1.06066
$218$	2329.56	0.723751
$219$	0	0
$220$	−7.97824	−0.00244497
$221$	0	0
$222$	0	0
$223$	−4690.71	−1.40858	−0.704290	−	0.709912i	$-0.748734\pi$
$223$	−4690.71	−1.40858	−0.704290	+	0.709912i	$0.748734\pi$
$224$	520.392	0.155224
$225$	0	0
$226$	155.392	0.0457368
$227$	−3560.61	−1.04108	−0.520542	−	0.853836i	$-0.674270\pi$
$227$	−3560.61	−1.04108	−0.520542	+	0.853836i	$0.674270\pi$
$228$	0	0
$229$	−3144.82	−0.907490	−0.453745	−	0.891131i	$-0.649912\pi$
$229$	−3144.82	−0.907490	−0.453745	+	0.891131i	$0.649912\pi$
$230$	−12.4227	−0.00356144
$231$	0	0
$232$	4546.07	1.28648
$233$	852.543	0.239708	0.119854	−	0.992792i	$-0.461757\pi$
$233$	852.543	0.239708	0.119854	+	0.992792i	$0.461757\pi$
$234$	0	0
$235$	−31.9081	−0.00885727
$236$	82.3898	0.0227251
$237$	0	0
$238$	7781.22	2.11925
$239$	−2189.33	−0.592534	−0.296267	−	0.955105i	$-0.595742\pi$
$239$	−2189.33	−0.592534	−0.296267	+	0.955105i	$0.595742\pi$
$240$	0	0
$241$	6329.98	1.69191	0.845954	−	0.533257i	$-0.179032\pi$
$241$	6329.98	1.69191	0.845954	+	0.533257i	$0.179032\pi$
$242$	7346.65	1.95149
$243$	0	0
$244$	87.0257	0.0228330
$245$	148.152	0.0386331
$246$	0	0
$247$	0	0
$248$	−2748.97	−0.703871
$249$	0	0
$250$	−215.738	−0.0545779
$251$	3602.97	0.906046	0.453023	−	0.891499i	$-0.350346\pi$
$251$	3602.97	0.906046	0.453023	+	0.891499i	$0.350346\pi$
$252$	0	0
$253$	−909.695	−0.226055
$254$	4593.74	1.13479
$255$	0	0
$256$	−617.315	−0.150712
$257$	−3072.64	−0.745781	−0.372891	−	0.927875i	$-0.621633\pi$
$257$	−3072.64	−0.745781	−0.372891	+	0.927875i	$0.621633\pi$
$258$	0	0
$259$	9115.80	2.18698
$260$	0	0
$261$	0	0
$262$	−1661.01	−0.391671
$263$	3543.20	0.830734	0.415367	−	0.909654i	$-0.363653\pi$
$263$	3543.20	0.830734	0.415367	+	0.909654i	$0.363653\pi$
$264$	0	0
$265$	−186.304	−0.0431871
$266$	−1140.86	−0.262972
$267$	0	0
$268$	−27.6435	−0.00630072
$269$	−5899.97	−1.33728	−0.668638	−	0.743588i	$-0.733122\pi$
$269$	−5899.97	−1.33728	−0.668638	+	0.743588i	$0.733122\pi$
$270$	0	0
$271$	−2109.52	−0.472858	−0.236429	−	0.971649i	$-0.575977\pi$
$271$	−2109.52	−0.472858	−0.236429	+	0.971649i	$0.575977\pi$
$272$	−5990.32	−1.33535
$273$	0	0
$274$	1162.19	0.256244
$275$	−7895.96	−1.73143
$276$	0	0
$277$	7265.00	1.57585	0.787927	−	0.615769i	$-0.211155\pi$
$277$	7265.00	1.57585	0.787927	+	0.615769i	$0.211155\pi$
$278$	80.4012	0.0173458
$279$	0	0
$280$	−207.223	−0.0442283
$281$	4771.36	1.01294	0.506469	−	0.862258i	$-0.330951\pi$
$281$	4771.36	1.01294	0.506469	+	0.862258i	$0.330951\pi$
$282$	0	0
$283$	2036.26	0.427714	0.213857	−	0.976865i	$-0.431397\pi$
$283$	2036.26	0.427714	0.213857	+	0.976865i	$0.431397\pi$
$284$	−381.275	−0.0796639
$285$	0	0
$286$	0	0
$287$	9886.86	2.03346
$288$	0	0
$289$	4853.78	0.987947
$290$	−169.449	−0.0343118
$291$	0	0
$292$	314.239	0.0629776
$293$	3719.02	0.741527	0.370764	−	0.928727i	$-0.379096\pi$
$293$	3719.02	0.741527	0.370764	+	0.928727i	$0.379096\pi$
$294$	0	0
$295$	−64.0429	−0.0126397
$296$	−7390.95	−1.45132
$297$	0	0
$298$	5903.39	1.14756
$299$	0	0
$300$	0	0
$301$	−1984.92	−0.380095
$302$	4022.22	0.766400
$303$	0	0
$304$	878.282	0.165700
$305$	−67.6464	−0.0126997
$306$	0	0
$307$	−7282.24	−1.35381	−0.676905	−	0.736070i	$-0.736679\pi$
$307$	−7282.24	−1.35381	−0.676905	+	0.736070i	$0.736679\pi$
$308$	−727.649	−0.134616
$309$	0	0
$310$	102.465	0.0187729
$311$	−4569.28	−0.833120	−0.416560	−	0.909108i	$-0.636764\pi$
$311$	−4569.28	−0.833120	−0.416560	+	0.909108i	$0.636764\pi$
$312$	0	0
$313$	21.0294	0.00379761	0.00189880	−	0.999998i	$-0.499396\pi$
$313$	21.0294	0.00379761	0.00189880	+	0.999998i	$0.499396\pi$
$314$	5535.15	0.994797
$315$	0	0
$316$	−96.7138	−0.0172170
$317$	−5159.17	−0.914095	−0.457047	−	0.889442i	$-0.651093\pi$
$317$	−5159.17	−0.914095	−0.457047	+	0.889442i	$0.651093\pi$
$318$	0	0
$319$	−12408.5	−2.17787
$320$	167.606	0.0292796
$321$	0	0
$322$	−1133.00	−0.196087
$323$	−1431.98	−0.246679
$324$	0	0
$325$	0	0
$326$	1511.80	0.256844
$327$	0	0
$328$	−8016.11	−1.34944
$329$	−2910.15	−0.487666
$330$	0	0
$331$	−7577.53	−1.25830	−0.629152	−	0.777282i	$-0.716598\pi$
$331$	−7577.53	−1.25830	−0.629152	+	0.777282i	$0.716598\pi$
$332$	−344.743	−0.0569887
$333$	0	0
$334$	−10922.0	−1.78929
$335$	21.4877	0.00350447
$336$	0	0
$337$	−2109.79	−0.341032	−0.170516	−	0.985355i	$-0.554543\pi$
$337$	−2109.79	−0.341032	−0.170516	+	0.985355i	$0.554543\pi$
$338$	0	0
$339$	0	0
$340$	−12.4723	−0.00198943
$341$	7503.30	1.19157
$342$	0	0
$343$	3713.98	0.584653
$344$	1609.34	0.252237
$345$	0	0
$346$	−4767.09	−0.740695
$347$	−8107.91	−1.25434	−0.627169	−	0.778883i	$-0.715786\pi$
$347$	−8107.91	−1.25434	−0.627169	+	0.778883i	$0.715786\pi$
$348$	0	0
$349$	8686.38	1.33230	0.666148	−	0.745820i	$-0.267942\pi$
$349$	8686.38	1.33230	0.666148	+	0.745820i	$0.267942\pi$
$350$	−9834.24	−1.50189
$351$	0	0
$352$	1151.64	0.174383
$353$	−9050.23	−1.36458	−0.682288	−	0.731083i	$-0.739015\pi$
$353$	−9050.23	−1.36458	−0.682288	+	0.731083i	$0.739015\pi$
$354$	0	0
$355$	296.371	0.0443092
$356$	−509.336	−0.0758279
$357$	0	0
$358$	4004.63	0.591204
$359$	−7043.80	−1.03554	−0.517768	−	0.855521i	$-0.673237\pi$
$359$	−7043.80	−1.03554	−0.517768	+	0.855521i	$0.673237\pi$
$360$	0	0
$361$	−6649.05	−0.969390
$362$	−1516.86	−0.220233
$363$	0	0
$364$	0	0
$365$	−244.263	−0.0350282
$366$	0	0
$367$	13825.1	1.96639	0.983194	−	0.182565i	$-0.0584401\pi$
$367$	13825.1	1.96639	0.983194	+	0.182565i	$0.0584401\pi$
$368$	872.236	0.123556
$369$	0	0
$370$	275.488	0.0387080
$371$	−16991.7	−2.37780
$372$	0	0
$373$	6368.19	0.884001	0.442000	−	0.897015i	$-0.354269\pi$
$373$	6368.19	0.884001	0.442000	+	0.897015i	$0.354269\pi$
$374$	17220.1	2.38082
$375$	0	0
$376$	2359.51	0.323623
$377$	0	0
$378$	0	0
$379$	2451.66	0.332278	0.166139	−	0.986102i	$-0.446870\pi$
$379$	2451.66	0.332278	0.166139	+	0.986102i	$0.446870\pi$
$380$	1.82865	0.000246863 0
$381$	0	0
$382$	5150.60	0.689863
$383$	−10830.9	−1.44500	−0.722500	−	0.691371i	$-0.757007\pi$
$383$	−10830.9	−1.44500	−0.722500	+	0.691371i	$0.757007\pi$
$384$	0	0
$385$	565.613	0.0748735
$386$	11456.5	1.51068
$387$	0	0
$388$	−266.861	−0.0349170
$389$	4978.07	0.648838	0.324419	−	0.945914i	$-0.394831\pi$
$389$	4978.07	0.648838	0.324419	+	0.945914i	$0.394831\pi$
$390$	0	0
$391$	−1422.12	−0.183938
$392$	−10955.4	−1.41156
$393$	0	0
$394$	−4195.56	−0.536470
$395$	75.1772	0.00957614
$396$	0	0
$397$	9450.67	1.19475	0.597375	−	0.801962i	$-0.296210\pi$
$397$	9450.67	1.19475	0.597375	+	0.801962i	$0.296210\pi$
$398$	2227.43	0.280530
$399$	0	0
$400$	7570.82	0.946353
$401$	−8776.77	−1.09300	−0.546498	−	0.837461i	$-0.684039\pi$
$401$	−8776.77	−1.09300	−0.546498	+	0.837461i	$0.684039\pi$
$402$	0	0
$403$	0	0
$404$	319.759	0.0393777
$405$	0	0
$406$	−15454.5	−1.88914
$407$	20173.5	2.45692
$408$	0	0
$409$	−4520.19	−0.546477	−0.273238	−	0.961946i	$-0.588095\pi$
$409$	−4520.19	−0.546477	−0.273238	+	0.961946i	$0.588095\pi$
$410$	298.791	0.0359908
$411$	0	0
$412$	−395.197	−0.0472571
$413$	−5840.98	−0.695922
$414$	0	0
$415$	267.974	0.0316972
$416$	0	0
$417$	0	0
$418$	−2524.75	−0.295430
$419$	−12006.4	−1.39989	−0.699944	−	0.714198i	$-0.746792\pi$
$419$	−12006.4	−1.39989	−0.699944	+	0.714198i	$0.746792\pi$
$420$	0	0
$421$	−9731.52	−1.12657	−0.563284	−	0.826263i	$-0.690462\pi$
$421$	−9731.52	−1.12657	−0.563284	+	0.826263i	$0.690462\pi$
$422$	−14498.1	−1.67241
$423$	0	0
$424$	13776.6	1.57795
$425$	−12343.7	−1.40884
$426$	0	0
$427$	−6169.63	−0.699226
$428$	324.745	0.0366755
$429$	0	0
$430$	−59.9861	−0.00672741
$431$	4111.03	0.459446	0.229723	−	0.973256i	$-0.426218\pi$
$431$	4111.03	0.459446	0.229723	+	0.973256i	$0.426218\pi$
$432$	0	0
$433$	2906.06	0.322532	0.161266	−	0.986911i	$-0.448442\pi$
$433$	2906.06	0.322532	0.161266	+	0.986911i	$0.448442\pi$
$434$	9345.19	1.03360
$435$	0	0
$436$	−340.556	−0.0374075
$437$	208.507	0.0228243
$438$	0	0
$439$	8874.72	0.964846	0.482423	−	0.875938i	$-0.339757\pi$
$439$	8874.72	0.964846	0.482423	+	0.875938i	$0.339757\pi$
$440$	−458.590	−0.0496873
$441$	0	0
$442$	0	0
$443$	−12640.5	−1.35569	−0.677843	−	0.735207i	$-0.737085\pi$
$443$	−12640.5	−1.35569	−0.677843	+	0.735207i	$0.737085\pi$
$444$	0	0
$445$	395.915	0.0421756
$446$	−12928.9	−1.37265
$447$	0	0
$448$	15286.4	1.61208
$449$	1774.29	0.186490	0.0932451	−	0.995643i	$-0.470276\pi$
$449$	1774.29	0.186490	0.0932451	+	0.995643i	$0.470276\pi$
$450$	0	0
$451$	21879.9	2.28444
$452$	−22.7165	−0.00236393
$453$	0	0
$454$	−9814.04	−1.01453
$455$	0	0
$456$	0	0
$457$	6168.82	0.631433	0.315717	−	0.948854i	$-0.397755\pi$
$457$	6168.82	0.631433	0.315717	+	0.948854i	$0.397755\pi$
$458$	−8667.98	−0.884341
$459$	0	0
$460$	1.81607	0.000184075 0
$461$	7455.38	0.753214	0.376607	−	0.926373i	$-0.377091\pi$
$461$	7455.38	0.753214	0.376607	+	0.926373i	$0.377091\pi$
$462$	0	0
$463$	5399.78	0.542006	0.271003	−	0.962578i	$-0.412645\pi$
$463$	5399.78	0.542006	0.271003	+	0.962578i	$0.412645\pi$
$464$	11897.5	1.19036
$465$	0	0
$466$	2349.85	0.233593
$467$	−3992.01	−0.395564	−0.197782	−	0.980246i	$-0.563374\pi$
$467$	−3992.01	−0.395564	−0.197782	+	0.980246i	$0.563374\pi$
$468$	0	0
$469$	1959.77	0.192950
$470$	−87.9477	−0.00863133
$471$	0	0
$472$	4735.77	0.461825
$473$	−4392.68	−0.427009
$474$	0	0
$475$	1809.79	0.174819
$476$	−1137.53	−0.109535
$477$	0	0
$478$	−6034.39	−0.577419
$479$	−3151.62	−0.300629	−0.150315	−	0.988638i	$-0.548029\pi$
$479$	−3151.62	−0.300629	−0.150315	+	0.988638i	$0.548029\pi$
$480$	0	0
$481$	0	0
$482$	17447.2	1.64875
$483$	0	0
$484$	−1074.00	−0.100864
$485$	207.435	0.0194209
$486$	0	0
$487$	−3909.18	−0.363741	−0.181870	−	0.983323i	$-0.558215\pi$
$487$	−3909.18	−0.363741	−0.181870	+	0.983323i	$0.558215\pi$
$488$	5002.24	0.464018
$489$	0	0
$490$	408.349	0.0376476
$491$	−6645.13	−0.610775	−0.305388	−	0.952228i	$-0.598786\pi$
$491$	−6645.13	−0.610775	−0.305388	+	0.952228i	$0.598786\pi$
$492$	0	0
$493$	−19398.1	−1.77210
$494$	0	0
$495$	0	0
$496$	−7194.33	−0.651281
$497$	27030.3	2.43959
$498$	0	0
$499$	6111.83	0.548303	0.274151	−	0.961687i	$-0.411603\pi$
$499$	6111.83	0.548303	0.274151	+	0.961687i	$0.411603\pi$
$500$	31.5385	0.00282089
$501$	0	0
$502$	9930.79	0.882934
$503$	−10954.1	−0.971010	−0.485505	−	0.874234i	$-0.661364\pi$
$503$	−10954.1	−0.971010	−0.485505	+	0.874234i	$0.661364\pi$
$504$	0	0
$505$	−248.554	−0.0219020
$506$	−2507.37	−0.220289
$507$	0	0
$508$	−671.553	−0.0586523
$509$	11121.1	0.968439	0.484220	−	0.874947i	$-0.339104\pi$
$509$	11121.1	0.968439	0.484220	+	0.874947i	$0.339104\pi$
$510$	0	0
$511$	−22277.8	−1.92859
$512$	−12335.2	−1.06473
$513$	0	0
$514$	−8469.04	−0.726757
$515$	307.193	0.0262845
$516$	0	0
$517$	−6440.25	−0.547857
$518$	25125.7	2.13119
$519$	0	0
$520$	0	0
$521$	−2403.33	−0.202096	−0.101048	−	0.994882i	$-0.532220\pi$
$521$	−2403.33	−0.202096	−0.101048	+	0.994882i	$0.532220\pi$
$522$	0	0
$523$	4251.72	0.355477	0.177739	−	0.984078i	$-0.443122\pi$
$523$	4251.72	0.355477	0.177739	+	0.984078i	$0.443122\pi$
$524$	242.822	0.0202437
$525$	0	0
$526$	9766.04	0.809543
$527$	11729.8	0.969564
$528$	0	0
$529$	−11959.9	−0.982981
$530$	−513.506	−0.0420854
$531$	0	0
$532$	166.781	0.0135918
$533$	0	0
$534$	0	0
$535$	−252.429	−0.0203990
$536$	−1588.95	−0.128045
$537$	0	0
$538$	−16261.9	−1.30316
$539$	29902.7	2.38961
$540$	0	0
$541$	−8924.44	−0.709227	−0.354613	−	0.935013i	$-0.615388\pi$
$541$	−8924.44	−0.709227	−0.354613	+	0.935013i	$0.615388\pi$
$542$	−5814.43	−0.460796
$543$	0	0
$544$	1800.35	0.141892
$545$	264.719	0.0208061
$546$	0	0
$547$	−14696.0	−1.14873	−0.574367	−	0.818598i	$-0.694752\pi$
$547$	−14696.0	−1.14873	−0.574367	+	0.818598i	$0.694752\pi$
$548$	−169.900	−0.0132441
$549$	0	0
$550$	−21763.5	−1.68727
$551$	2844.09	0.219895
$552$	0	0
$553$	6856.47	0.527245
$554$	20024.3	1.53566
$555$	0	0
$556$	−11.7538	−0.000896530 0
$557$	−2464.03	−0.187440	−0.0937202	−	0.995599i	$-0.529876\pi$
$557$	−2464.03	−0.187440	−0.0937202	+	0.995599i	$0.529876\pi$
$558$	0	0
$559$	0	0
$560$	−542.322	−0.0409237
$561$	0	0
$562$	13151.2	0.987098
$563$	−18029.2	−1.34963	−0.674815	−	0.737987i	$-0.735776\pi$
$563$	−18029.2	−1.34963	−0.674815	+	0.737987i	$0.735776\pi$
$564$	0	0
$565$	17.6579	0.00131482
$566$	5612.50	0.416804
$567$	0	0
$568$	−21915.7	−1.61895
$569$	−11924.6	−0.878572	−0.439286	−	0.898347i	$-0.644768\pi$
$569$	−11924.6	−0.878572	−0.439286	+	0.898347i	$0.644768\pi$
$570$	0	0
$571$	5834.77	0.427632	0.213816	−	0.976874i	$-0.431411\pi$
$571$	5834.77	0.427632	0.213816	+	0.976874i	$0.431411\pi$
$572$	0	0
$573$	0	0
$574$	27250.9	1.98159
$575$	1797.33	0.130355
$576$	0	0
$577$	15927.5	1.14917	0.574586	−	0.818444i	$-0.305163\pi$
$577$	15927.5	1.14917	0.574586	+	0.818444i	$0.305163\pi$
$578$	13378.4	0.962745
$579$	0	0
$580$	24.7716	0.00177342
$581$	24440.4	1.74519
$582$	0	0
$583$	−37603.1	−2.67129
$584$	18062.5	1.27985
$585$	0	0
$586$	10250.7	0.722612
$587$	13511.1	0.950022	0.475011	−	0.879980i	$-0.342444\pi$
$587$	13511.1	0.950022	0.475011	+	0.879980i	$0.342444\pi$
$588$	0	0
$589$	−1719.80	−0.120311
$590$	−176.520	−0.0123173
$591$	0	0
$592$	−19342.8	−1.34288
$593$	−15830.4	−1.09625	−0.548126	−	0.836396i	$-0.684659\pi$
$593$	−15830.4	−1.09625	−0.548126	+	0.836396i	$0.684659\pi$
$594$	0	0
$595$	884.217	0.0609233
$596$	−863.009	−0.0593125
$597$	0	0
$598$	0	0
$599$	5914.12	0.403413	0.201706	−	0.979446i	$-0.435351\pi$
$599$	5914.12	0.403413	0.201706	+	0.979446i	$0.435351\pi$
$600$	0	0
$601$	9892.19	0.671399	0.335700	−	0.941969i	$-0.391027\pi$
$601$	9892.19	0.671399	0.335700	+	0.941969i	$0.391027\pi$
$602$	−5470.98	−0.370399
$603$	0	0
$604$	−588.004	−0.0396118
$605$	834.836	0.0561007
$606$	0	0
$607$	−16782.3	−1.12220	−0.561098	−	0.827749i	$-0.689621\pi$
$607$	−16782.3	−1.12220	−0.561098	+	0.827749i	$0.689621\pi$
$608$	−263.962	−0.0176070
$609$	0	0
$610$	−186.452	−0.0123758
$611$	0	0
$612$	0	0
$613$	12245.7	0.806847	0.403424	−	0.915013i	$-0.367820\pi$
$613$	12245.7	0.806847	0.403424	+	0.915013i	$0.367820\pi$
$614$	−20071.9	−1.31928
$615$	0	0
$616$	−41825.3	−2.73569
$617$	−10043.7	−0.655341	−0.327670	−	0.944792i	$-0.606264\pi$
$617$	−10043.7	−0.655341	−0.327670	+	0.944792i	$0.606264\pi$
$618$	0	0
$619$	−9942.69	−0.645607	−0.322803	−	0.946466i	$-0.604625\pi$
$619$	−9942.69	−0.645607	−0.322803	+	0.946466i	$0.604625\pi$
$620$	−14.9792	−0.000970288 0
$621$	0	0
$622$	−12594.2	−0.811868
$623$	36109.1	2.32212
$624$	0	0
$625$	15588.2	0.997646
$626$	57.9628	0.00370074
$627$	0	0
$628$	−809.176	−0.0514166
$629$	31537.1	1.99915
$630$	0	0
$631$	10268.6	0.647841	0.323920	−	0.946084i	$-0.394999\pi$
$631$	10268.6	0.647841	0.323920	+	0.946084i	$0.394999\pi$
$632$	−5559.11	−0.349889
$633$	0	0
$634$	−14220.1	−0.890777
$635$	522.009	0.0326225
$636$	0	0
$637$	0	0
$638$	−34201.2	−2.12232
$639$	0	0
$640$	416.322	0.0257134
$641$	13657.4	0.841555	0.420777	−	0.907164i	$-0.361757\pi$
$641$	13657.4	0.841555	0.420777	+	0.907164i	$0.361757\pi$
$642$	0	0
$643$	24671.6	1.51314	0.756572	−	0.653910i	$-0.226873\pi$
$643$	24671.6	1.51314	0.756572	+	0.653910i	$0.226873\pi$
$644$	165.633	0.0101348
$645$	0	0
$646$	−3946.92	−0.240386
$647$	16512.3	1.00334	0.501672	−	0.865058i	$-0.332718\pi$
$647$	16512.3	1.00334	0.501672	+	0.865058i	$0.332718\pi$
$648$	0	0
$649$	−12926.3	−0.781818
$650$	0	0
$651$	0	0
$652$	−221.009	−0.0132751
$653$	−30594.8	−1.83348	−0.916742	−	0.399479i	$-0.869191\pi$
$653$	−30594.8	−1.83348	−0.916742	+	0.399479i	$0.869191\pi$
$654$	0	0
$655$	−188.749	−0.0112596
$656$	−20978.9	−1.24861
$657$	0	0
$658$	−8021.19	−0.475226
$659$	2307.79	0.136417	0.0682084	−	0.997671i	$-0.478272\pi$
$659$	2307.79	0.136417	0.0682084	+	0.997671i	$0.478272\pi$
$660$	0	0
$661$	−27622.5	−1.62540	−0.812700	−	0.582683i	$-0.802003\pi$
$661$	−27622.5	−1.62540	−0.812700	+	0.582683i	$0.802003\pi$
$662$	−20885.8	−1.22621
$663$	0	0
$664$	−19815.9	−1.15814
$665$	−129.641	−0.00755981
$666$	0	0
$667$	2824.51	0.163966
$668$	1596.67	0.0924805
$669$	0	0
$670$	59.2260	0.00341508
$671$	−13653.6	−0.785530
$672$	0	0
$673$	−11305.9	−0.647562	−0.323781	−	0.946132i	$-0.604954\pi$
$673$	−11305.9	−0.647562	−0.323781	+	0.946132i	$0.604954\pi$
$674$	−5815.17	−0.332332
$675$	0	0
$676$	0	0
$677$	−15455.8	−0.877419	−0.438710	−	0.898629i	$-0.644564\pi$
$677$	−15455.8	−0.877419	−0.438710	+	0.898629i	$0.644564\pi$
$678$	0	0
$679$	18918.9	1.06928
$680$	−716.909	−0.0404297
$681$	0	0
$682$	20681.2	1.16118
$683$	−16841.9	−0.943540	−0.471770	−	0.881722i	$-0.656385\pi$
$683$	−16841.9	−0.943540	−0.471770	+	0.881722i	$0.656385\pi$
$684$	0	0
$685$	132.066	0.00736639
$686$	10236.7	0.569739
$687$	0	0
$688$	4211.80	0.233391
$689$	0	0
$690$	0	0
$691$	−25041.6	−1.37862	−0.689310	−	0.724467i	$-0.742086\pi$
$691$	−25041.6	−1.37862	−0.689310	+	0.724467i	$0.742086\pi$
$692$	696.896	0.0382832
$693$	0	0
$694$	−22347.6	−1.22234
$695$	9.13638	0.000498652 0
$696$	0	0
$697$	34204.7	1.85881
$698$	23942.1	1.29831
$699$	0	0
$700$	1437.66	0.0776261
$701$	28309.0	1.52527	0.762635	−	0.646829i	$-0.223905\pi$
$701$	28309.0	1.52527	0.762635	+	0.646829i	$0.223905\pi$
$702$	0	0
$703$	−4623.88	−0.248069
$704$	33829.1	1.81106
$705$	0	0
$706$	−24945.0	−1.32977
$707$	−22669.1	−1.20588
$708$	0	0
$709$	−5041.17	−0.267031	−0.133516	−	0.991047i	$-0.542627\pi$
$709$	−5041.17	−0.267031	−0.133516	+	0.991047i	$0.542627\pi$
$710$	816.882	0.0431789
$711$	0	0
$712$	−29276.6	−1.54100
$713$	−1707.96	−0.0897103
$714$	0	0
$715$	0	0
$716$	−585.431	−0.0305567
$717$	0	0
$718$	−19414.7	−1.00912
$719$	−11839.2	−0.614088	−0.307044	−	0.951695i	$-0.599340\pi$
$719$	−11839.2	−0.614088	−0.307044	+	0.951695i	$0.599340\pi$
$720$	0	0
$721$	28017.2	1.44718
$722$	−18326.6	−0.944662
$723$	0	0
$724$	221.748	0.0113829
$725$	24516.1	1.25587
$726$	0	0
$727$	−31004.9	−1.58172	−0.790858	−	0.611999i	$-0.790365\pi$
$727$	−31004.9	−1.58172	−0.790858	+	0.611999i	$0.790365\pi$
$728$	0	0
$729$	0	0
$730$	−673.256	−0.0341347
$731$	−6867.03	−0.347451
$732$	0	0
$733$	15666.0	0.789411	0.394706	−	0.918808i	$-0.370847\pi$
$733$	15666.0	0.789411	0.394706	+	0.918808i	$0.370847\pi$
$734$	38105.8	1.91623
$735$	0	0
$736$	−262.145	−0.0131288
$737$	4337.02	0.216765
$738$	0	0
$739$	−33339.4	−1.65955	−0.829777	−	0.558095i	$-0.811533\pi$
$739$	−33339.4	−1.65955	−0.829777	+	0.558095i	$0.811533\pi$
$740$	−40.2733	−0.00200064
$741$	0	0
$742$	−46833.8	−2.31715
$743$	−2026.14	−0.100043	−0.0500215	−	0.998748i	$-0.515929\pi$
$743$	−2026.14	−0.100043	−0.0500215	+	0.998748i	$0.515929\pi$
$744$	0	0
$745$	670.831	0.0329897
$746$	17552.5	0.861451
$747$	0	0
$748$	−2517.38	−0.123054
$749$	−23022.6	−1.12313
$750$	0	0
$751$	35474.8	1.72369	0.861846	−	0.507170i	$-0.169309\pi$
$751$	35474.8	1.72369	0.861846	+	0.507170i	$0.169309\pi$
$752$	6175.06	0.299443
$753$	0	0
$754$	0	0
$755$	457.065	0.0220322
$756$	0	0
$757$	2398.58	0.115162	0.0575811	−	0.998341i	$-0.481661\pi$
$757$	2398.58	0.115162	0.0575811	+	0.998341i	$0.481661\pi$
$758$	6757.46	0.323802
$759$	0	0
$760$	105.111	0.00501682
$761$	5005.07	0.238415	0.119207	−	0.992869i	$-0.461965\pi$
$761$	5005.07	0.238415	0.119207	+	0.992869i	$0.461965\pi$
$762$	0	0
$763$	24143.5	1.14555
$764$	−752.960	−0.0356559
$765$	0	0
$766$	−29853.1	−1.40814
$767$	0	0
$768$	0	0
$769$	14109.6	0.661644	0.330822	−	0.943693i	$-0.392674\pi$
$769$	14109.6	0.661644	0.330822	+	0.943693i	$0.392674\pi$
$770$	1558.99	0.0729635
$771$	0	0
$772$	−1674.82	−0.0780803
$773$	12733.7	0.592496	0.296248	−	0.955111i	$-0.404265\pi$
$773$	12733.7	0.592496	0.296248	+	0.955111i	$0.404265\pi$
$774$	0	0
$775$	−14824.7	−0.687121
$776$	−15339.2	−0.709593
$777$	0	0
$778$	13720.9	0.632287
$779$	−5014.98	−0.230655
$780$	0	0
$781$	59818.8	2.74070
$782$	−3919.75	−0.179246
$783$	0	0
$784$	−28671.3	−1.30609
$785$	628.986	0.0285980
$786$	0	0
$787$	28429.3	1.28767	0.643834	−	0.765165i	$-0.277343\pi$
$787$	28429.3	1.28767	0.643834	+	0.765165i	$0.277343\pi$
$788$	613.344	0.0277278
$789$	0	0
$790$	207.209	0.00933186
$791$	1610.48	0.0723918
$792$	0	0
$793$	0	0
$794$	26048.7	1.16427
$795$	0	0
$796$	−325.625	−0.0144993
$797$	−11926.7	−0.530070	−0.265035	−	0.964239i	$-0.585384\pi$
$797$	−11926.7	−0.530070	−0.265035	+	0.964239i	$0.585384\pi$
$798$	0	0
$799$	−10068.0	−0.445782
$800$	−2275.36	−0.100558
$801$	0	0
$802$	−24191.2	−1.06511
$803$	−49301.4	−2.16663
$804$	0	0
$805$	−128.749	−0.00563702
$806$	0	0
$807$	0	0
$808$	18379.7	0.800244
$809$	−20005.7	−0.869421	−0.434710	−	0.900570i	$-0.643149\pi$
$809$	−20005.7	−0.869421	−0.434710	+	0.900570i	$0.643149\pi$
$810$	0	0
$811$	−5933.84	−0.256924	−0.128462	−	0.991714i	$-0.541004\pi$
$811$	−5933.84	−0.256924	−0.128462	+	0.991714i	$0.541004\pi$
$812$	2259.27	0.0976415
$813$	0	0
$814$	55603.8	2.39424
$815$	171.794	0.00738364
$816$	0	0
$817$	1006.82	0.0431142
$818$	−12458.9	−0.532537
$819$	0	0
$820$	−43.6798	−0.00186020
$821$	18457.0	0.784597	0.392299	−	0.919838i	$-0.371680\pi$
$821$	18457.0	0.784597	0.392299	+	0.919838i	$0.371680\pi$
$822$	0	0
$823$	−26520.3	−1.12325	−0.561627	−	0.827390i	$-0.689825\pi$
$823$	−26520.3	−1.12325	−0.561627	+	0.827390i	$0.689825\pi$
$824$	−22715.9	−0.960372
$825$	0	0
$826$	−16099.4	−0.678170
$827$	25664.8	1.07915	0.539573	−	0.841939i	$-0.318586\pi$
$827$	25664.8	1.07915	0.539573	+	0.841939i	$0.318586\pi$
$828$	0	0
$829$	4362.69	0.182777	0.0913887	−	0.995815i	$-0.470869\pi$
$829$	4362.69	0.182777	0.0913887	+	0.995815i	$0.470869\pi$
$830$	738.612	0.0308887
$831$	0	0
$832$	0	0
$833$	46746.6	1.94439
$834$	0	0
$835$	−1241.12	−0.0514378
$836$	369.091	0.0152695
$837$	0	0
$838$	−33093.1	−1.36418
$839$	35020.2	1.44104	0.720520	−	0.693435i	$-0.243903\pi$
$839$	35020.2	1.44104	0.720520	+	0.693435i	$0.243903\pi$
$840$	0	0
$841$	14138.0	0.579687
$842$	−26822.8	−1.09783
$843$	0	0
$844$	2119.46	0.0864394
$845$	0	0
$846$	0	0
$847$	76140.4	3.08880
$848$	36054.7	1.46005
$849$	0	0
$850$	−34022.6	−1.37290
$851$	−4592.04	−0.184974
$852$	0	0
$853$	8318.36	0.333898	0.166949	−	0.985966i	$-0.446608\pi$
$853$	8318.36	0.333898	0.166949	+	0.985966i	$0.446608\pi$
$854$	−17005.2	−0.681389
$855$	0	0
$856$	18666.3	0.745330
$857$	−34832.3	−1.38839	−0.694193	−	0.719789i	$-0.744239\pi$
$857$	−34832.3	−1.38839	−0.694193	+	0.719789i	$0.744239\pi$
$858$	0	0
$859$	−12571.8	−0.499353	−0.249676	−	0.968329i	$-0.580324\pi$
$859$	−12571.8	−0.499353	−0.249676	+	0.968329i	$0.580324\pi$
$860$	8.76930	0.000347710 0
$861$	0	0
$862$	11331.1	0.447726
$863$	−22474.2	−0.886480	−0.443240	−	0.896403i	$-0.646171\pi$
$863$	−22474.2	−0.886480	−0.443240	+	0.896403i	$0.646171\pi$
$864$	0	0
$865$	−541.708	−0.0212932
$866$	8009.90	0.314304
$867$	0	0
$868$	−1366.16	−0.0534223
$869$	15173.6	0.592322
$870$	0	0
$871$	0	0
$872$	−19575.2	−0.760205
$873$	0	0
$874$	574.702	0.0222421
$875$	−2235.90	−0.0863855
$876$	0	0
$877$	5385.19	0.207349	0.103675	−	0.994611i	$-0.466940\pi$
$877$	5385.19	0.207349	0.103675	+	0.994611i	$0.466940\pi$
$878$	24461.2	0.940233
$879$	0	0
$880$	−1200.17	−0.0459749
$881$	29772.7	1.13856	0.569278	−	0.822145i	$-0.307223\pi$
$881$	29772.7	1.13856	0.569278	+	0.822145i	$0.307223\pi$
$882$	0	0
$883$	−27309.6	−1.04082	−0.520409	−	0.853917i	$-0.674220\pi$
$883$	−27309.6	−1.04082	−0.520409	+	0.853917i	$0.674220\pi$
$884$	0	0
$885$	0	0
$886$	−34840.7	−1.32110
$887$	6529.97	0.247187	0.123594	−	0.992333i	$-0.460558\pi$
$887$	6529.97	0.247187	0.123594	+	0.992333i	$0.460558\pi$
$888$	0	0
$889$	47609.3	1.79614
$890$	1091.25	0.0410998
$891$	0	0
$892$	1890.06	0.0709461
$893$	1476.14	0.0553159
$894$	0	0
$895$	455.065	0.0169957
$896$	37970.3	1.41573
$897$	0	0
$898$	4890.45	0.181733
$899$	−23297.0	−0.864290
$900$	0	0
$901$	−58784.6	−2.17358
$902$	60307.1	2.22617
$903$	0	0
$904$	−1305.75	−0.0480404
$905$	−172.368	−0.00633116
$906$	0	0
$907$	19437.4	0.711586	0.355793	−	0.934565i	$-0.384211\pi$
$907$	19437.4	0.711586	0.355793	+	0.934565i	$0.384211\pi$
$908$	1434.70	0.0524364
$909$	0	0
$910$	0	0
$911$	−50234.6	−1.82694	−0.913472	−	0.406901i	$-0.866609\pi$
$911$	−50234.6	−1.82694	−0.913472	+	0.406901i	$0.866609\pi$
$912$	0	0
$913$	54087.2	1.96060
$914$	17003.0	0.615326
$915$	0	0
$916$	1267.16	0.0457077
$917$	−17214.7	−0.619934
$918$	0	0
$919$	12285.2	0.440971	0.220486	−	0.975390i	$-0.429236\pi$
$919$	12285.2	0.440971	0.220486	+	0.975390i	$0.429236\pi$
$920$	104.387	0.00374082
$921$	0	0
$922$	20549.1	0.734000
$923$	0	0
$924$	0	0
$925$	−39858.0	−1.41678
$926$	14883.3	0.528180
$927$	0	0
$928$	−3575.72	−0.126486
$929$	−25610.0	−0.904454	−0.452227	−	0.891903i	$-0.649370\pi$
$929$	−25610.0	−0.904454	−0.452227	+	0.891903i	$0.649370\pi$
$930$	0	0
$931$	−6853.84	−0.241273
$932$	−343.521	−0.0120734
$933$	0	0
$934$	−11003.1	−0.385474
$935$	1956.80	0.0684429
$936$	0	0
$937$	−29893.2	−1.04223	−0.521114	−	0.853487i	$-0.674483\pi$
$937$	−29893.2	−1.04223	−0.521114	+	0.853487i	$0.674483\pi$
$938$	5401.66	0.188028
$939$	0	0
$940$	12.8570	0.000446115 0
$941$	−44960.3	−1.55756	−0.778780	−	0.627297i	$-0.784161\pi$
$941$	−44960.3	−1.55756	−0.778780	+	0.627297i	$0.784161\pi$
$942$	0	0
$943$	−4980.46	−0.171989
$944$	12394.0	0.427320
$945$	0	0
$946$	−12107.4	−0.416117
$947$	33597.6	1.15288	0.576439	−	0.817140i	$-0.304442\pi$
$947$	33597.6	1.15288	0.576439	+	0.817140i	$0.304442\pi$
$948$	0	0
$949$	0	0
$950$	4988.29	0.170360
$951$	0	0
$952$	−65385.1	−2.22599
$953$	25850.3	0.878670	0.439335	−	0.898323i	$-0.355214\pi$
$953$	25850.3	0.878670	0.439335	+	0.898323i	$0.355214\pi$
$954$	0	0
$955$	585.288	0.0198319
$956$	882.160	0.0298442
$957$	0	0
$958$	−8686.75	−0.292960
$959$	12044.9	0.405580
$960$	0	0
$961$	−15703.5	−0.527123
$962$	0	0
$963$	0	0
$964$	−2550.58	−0.0852165
$965$	1301.86	0.0434284
$966$	0	0
$967$	6296.49	0.209392	0.104696	−	0.994504i	$-0.466613\pi$
$967$	6296.49	0.209392	0.104696	+	0.994504i	$0.466613\pi$
$968$	−61733.5	−2.04978
$969$	0	0
$970$	571.749	0.0189255
$971$	7806.43	0.258002	0.129001	−	0.991644i	$-0.458823\pi$
$971$	7806.43	0.258002	0.129001	+	0.991644i	$0.458823\pi$
$972$	0	0
$973$	833.276	0.0274549
$974$	−10774.8	−0.354462
$975$	0	0
$976$	13091.3	0.429348
$977$	−55692.8	−1.82372	−0.911858	−	0.410507i	$-0.865352\pi$
$977$	−55692.8	−1.82372	−0.911858	+	0.410507i	$0.865352\pi$
$978$	0	0
$979$	79910.4	2.60873
$980$	−59.6960	−0.00194584
$981$	0	0
$982$	−18315.8	−0.595195
$983$	44635.5	1.44827	0.724135	−	0.689658i	$-0.242239\pi$
$983$	44635.5	1.44827	0.724135	+	0.689658i	$0.242239\pi$
$984$	0	0
$985$	−476.762	−0.0154222
$986$	−53466.4	−1.72689
$987$	0	0
$988$	0	0
$989$	999.892	0.0321484
$990$	0	0
$991$	3884.32	0.124510	0.0622551	−	0.998060i	$-0.480171\pi$
$991$	3884.32	0.124510	0.0622551	+	0.998060i	$0.480171\pi$
$992$	2162.21	0.0692039
$993$	0	0
$994$	74503.0	2.37735
$995$	253.114	0.00806456
$996$	0	0
$997$	−18021.4	−0.572462	−0.286231	−	0.958161i	$-0.592402\pi$
$997$	−18021.4	−0.572462	−0.286231	+	0.958161i	$0.592402\pi$
$998$	16845.9	0.534316
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bc.1.6		8
3.2	odd	2	inner	1521.4.a.bc.1.3		8
13.3	even	3		117.4.g.f.100.3	yes	16
13.9	even	3		117.4.g.f.55.3	✓	16
13.12	even	2		1521.4.a.bd.1.3		8
39.29	odd	6		117.4.g.f.100.6	yes	16
39.35	odd	6		117.4.g.f.55.6	yes	16
39.38	odd	2		1521.4.a.bd.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.g.f.55.3	✓	16	13.9	even	3
117.4.g.f.55.6	yes	16	39.35	odd	6
117.4.g.f.100.3	yes	16	13.3	even	3
117.4.g.f.100.6	yes	16	39.29	odd	6
1521.4.a.bc.1.3		8	3.2	odd	2	inner
1521.4.a.bc.1.6		8	1.1	even	1	trivial
1521.4.a.bd.1.3		8	13.12	even	2
1521.4.a.bd.1.6		8	39.38	odd	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)

\(f(q)\)	\(=\)	\(q+2.75628 q^{2} -0.402937 q^{4} +0.313209 q^{5} +28.5660 q^{7} -23.1608 q^{8} +O(q^{10})\)	\(q+2.75628 q^{2} -0.402937 q^{4} +0.313209 q^{5} +28.5660 q^{7} -23.1608 q^{8} +0.863291 q^{10} +63.2173 q^{11} +78.7357 q^{14} -60.6141 q^{16} +98.8270 q^{17} -14.4897 q^{19} -0.126204 q^{20} +174.244 q^{22} -14.3900 q^{23} -124.902 q^{25} -11.5103 q^{28} -196.283 q^{29} +118.691 q^{31} +18.2172 q^{32} +272.395 q^{34} +8.94712 q^{35} +319.114 q^{37} -39.9377 q^{38} -7.25418 q^{40} +346.106 q^{41} -69.4854 q^{43} -25.4726 q^{44} -39.6628 q^{46} -101.875 q^{47} +473.014 q^{49} -344.264 q^{50} -594.823 q^{53} +19.8002 q^{55} -661.611 q^{56} -541.010 q^{58} -204.473 q^{59} -215.978 q^{61} +327.144 q^{62} +535.125 q^{64} +68.6049 q^{67} -39.8211 q^{68} +24.6607 q^{70} +946.241 q^{71} -779.872 q^{73} +879.567 q^{74} +5.83844 q^{76} +1805.86 q^{77} +240.022 q^{79} -18.9849 q^{80} +953.965 q^{82} +855.576 q^{83} +30.9535 q^{85} -191.521 q^{86} -1464.16 q^{88} +1264.06 q^{89} +5.79825 q^{92} -280.795 q^{94} -4.53831 q^{95} +662.290 q^{97} +1303.76 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{4} - 22 q^{7}+O(q^{10}) \) 8 * q + 40 * q^4 - 22 * q^7	\( 8 q + 40 q^{4} - 22 q^{7} + 36 q^{10} + 204 q^{16} + 244 q^{19} + 136 q^{22} + 354 q^{25} - 452 q^{28} + 242 q^{31} - 1292 q^{34} + 1018 q^{37} + 1700 q^{40} + 74 q^{43} - 896 q^{46} + 298 q^{49} + 1300 q^{55} + 812 q^{58} + 1148 q^{61} + 3636 q^{64} - 2198 q^{67} + 2200 q^{70} - 2176 q^{73} + 6936 q^{76} + 1862 q^{79} + 5436 q^{82} - 890 q^{85} + 3528 q^{88} - 3104 q^{94} - 4370 q^{97}+O(q^{100}) \) 8 * q + 40 * q^4 - 22 * q^7 + 36 * q^10 + 204 * q^16 + 244 * q^19 + 136 * q^22 + 354 * q^25 - 452 * q^28 + 242 * q^31 - 1292 * q^34 + 1018 * q^37 + 1700 * q^40 + 74 * q^43 - 896 * q^46 + 298 * q^49 + 1300 * q^55 + 812 * q^58 + 1148 * q^61 + 3636 * q^64 - 2198 * q^67 + 2200 * q^70 - 2176 * q^73 + 6936 * q^76 + 1862 * q^79 + 5436 * q^82 - 890 * q^85 + 3528 * q^88 - 3104 * q^94 - 4370 * q^97

Introduction

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