LMFDB - Embedded newform 405.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 $ x^6 - 2x^5 - 38x^4 + 42x^3 + 393x^2 - 72*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.11734$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.11734 q^{2} +8.95250 q^{4} +5.00000 q^{5} -20.0229 q^{7} -3.92177 q^{8} +O(q^{10})$	\(q-4.11734 q^{2} +8.95250 q^{4} +5.00000 q^{5} -20.0229 q^{7} -3.92177 q^{8} -20.5867 q^{10} -1.67827 q^{11} +11.1123 q^{13} +82.4412 q^{14} -55.4727 q^{16} +8.98682 q^{17} -50.6419 q^{19} +44.7625 q^{20} +6.91000 q^{22} +214.491 q^{23} +25.0000 q^{25} -45.7531 q^{26} -179.255 q^{28} -76.9088 q^{29} -273.537 q^{31} +259.774 q^{32} -37.0018 q^{34} -100.115 q^{35} -137.283 q^{37} +208.510 q^{38} -19.6088 q^{40} -53.3457 q^{41} +295.270 q^{43} -15.0247 q^{44} -883.134 q^{46} -194.983 q^{47} +57.9173 q^{49} -102.934 q^{50} +99.4829 q^{52} -450.173 q^{53} -8.39133 q^{55} +78.5252 q^{56} +316.660 q^{58} +481.145 q^{59} +675.854 q^{61} +1126.25 q^{62} -625.798 q^{64} +55.5615 q^{65} +894.589 q^{67} +80.4545 q^{68} +412.206 q^{70} +721.947 q^{71} +915.175 q^{73} +565.241 q^{74} -453.372 q^{76} +33.6038 q^{77} +59.6642 q^{79} -277.364 q^{80} +219.643 q^{82} +742.875 q^{83} +44.9341 q^{85} -1215.73 q^{86} +6.58177 q^{88} +1540.72 q^{89} -222.501 q^{91} +1920.23 q^{92} +802.812 q^{94} -253.210 q^{95} -1121.57 q^{97} -238.465 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 34 q^{4} + 30 q^{5} + 40 q^{7} + 66 q^{8}+O(q^{10}) $ 6 * q + 4 * q^2 + 34 * q^4 + 30 * q^5 + 40 * q^7 + 66 * q^8	$ 6 q + 4 q^{2} + 34 q^{4} + 30 q^{5} + 40 q^{7} + 66 q^{8} + 20 q^{10} + 88 q^{11} + 20 q^{13} + 180 q^{14} + 58 q^{16} + 124 q^{17} - 46 q^{19} + 170 q^{20} - 74 q^{22} + 210 q^{23} + 150 q^{25} + 4 q^{26} + 352 q^{28} + 296 q^{29} - 104 q^{31} + 722 q^{32} - 428 q^{34} + 200 q^{35} - 204 q^{37} - 20 q^{38} + 330 q^{40} + 344 q^{41} + 512 q^{43} + 716 q^{44} - 186 q^{46} + 238 q^{47} + 68 q^{49} + 100 q^{50} - 468 q^{52} + 850 q^{53} + 440 q^{55} + 2316 q^{56} + 890 q^{58} + 1840 q^{59} - 364 q^{61} + 1038 q^{62} - 990 q^{64} + 100 q^{65} + 88 q^{67} + 236 q^{68} + 900 q^{70} + 1364 q^{71} + 836 q^{73} + 1316 q^{74} - 2106 q^{76} + 840 q^{77} - 680 q^{79} + 290 q^{80} + 1742 q^{82} + 2148 q^{83} + 620 q^{85} + 2872 q^{86} + 1296 q^{88} + 3000 q^{89} - 3058 q^{91} + 1002 q^{92} - 3662 q^{94} - 230 q^{95} - 612 q^{97} + 1982 q^{98}+O(q^{100}) $ 6 * q + 4 * q^2 + 34 * q^4 + 30 * q^5 + 40 * q^7 + 66 * q^8 + 20 * q^10 + 88 * q^11 + 20 * q^13 + 180 * q^14 + 58 * q^16 + 124 * q^17 - 46 * q^19 + 170 * q^20 - 74 * q^22 + 210 * q^23 + 150 * q^25 + 4 * q^26 + 352 * q^28 + 296 * q^29 - 104 * q^31 + 722 * q^32 - 428 * q^34 + 200 * q^35 - 204 * q^37 - 20 * q^38 + 330 * q^40 + 344 * q^41 + 512 * q^43 + 716 * q^44 - 186 * q^46 + 238 * q^47 + 68 * q^49 + 100 * q^50 - 468 * q^52 + 850 * q^53 + 440 * q^55 + 2316 * q^56 + 890 * q^58 + 1840 * q^59 - 364 * q^61 + 1038 * q^62 - 990 * q^64 + 100 * q^65 + 88 * q^67 + 236 * q^68 + 900 * q^70 + 1364 * q^71 + 836 * q^73 + 1316 * q^74 - 2106 * q^76 + 840 * q^77 - 680 * q^79 + 290 * q^80 + 1742 * q^82 + 2148 * q^83 + 620 * q^85 + 2872 * q^86 + 1296 * q^88 + 3000 * q^89 - 3058 * q^91 + 1002 * q^92 - 3662 * q^94 - 230 * q^95 - 612 * q^97 + 1982 * 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 $

$2$	−4.11734	−1.45570	−0.727850	−	0.685736i	$-0.759480\pi$
$2$	−4.11734	−1.45570	−0.727850	+	0.685736i	$0.759480\pi$
$3$	0	0
$3$	0	0
$4$	8.95250	1.11906
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−20.0229	−1.08114	−0.540568	−	0.841300i	$-0.681791\pi$
$7$	−20.0229	−1.08114	−0.540568	+	0.841300i	$0.681791\pi$
$8$	−3.92177	−0.173319
$9$	0	0
$10$	−20.5867	−0.651009
$11$	−1.67827	−0.0460015	−0.0230007	−	0.999735i	$-0.507322\pi$
$11$	−1.67827	−0.0460015	−0.0230007	+	0.999735i	$0.507322\pi$
$12$	0	0
$13$	11.1123	0.237077	0.118538	−	0.992949i	$-0.462179\pi$
$13$	11.1123	0.237077	0.118538	+	0.992949i	$0.462179\pi$
$14$	82.4412	1.57381
$15$	0	0
$16$	−55.4727	−0.866762
$17$	8.98682	0.128213	0.0641066	−	0.997943i	$-0.479580\pi$
$17$	8.98682	0.128213	0.0641066	+	0.997943i	$0.479580\pi$
$18$	0	0
$19$	−50.6419	−0.611477	−0.305738	−	0.952116i	$-0.598903\pi$
$19$	−50.6419	−0.611477	−0.305738	+	0.952116i	$0.598903\pi$
$20$	44.7625	0.500460
$21$	0	0
$22$	6.91000	0.0669644
$23$	214.491	1.94454	0.972272	−	0.233853i	$-0.0751334\pi$
$23$	214.491	1.94454	0.972272	+	0.233853i	$0.0751334\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−45.7531	−0.345113
$27$	0	0
$28$	−179.255	−1.20986
$29$	−76.9088	−0.492469	−0.246235	−	0.969210i	$-0.579193\pi$
$29$	−76.9088	−0.492469	−0.246235	+	0.969210i	$0.579193\pi$
$30$	0	0
$31$	−273.537	−1.58480	−0.792399	−	0.610003i	$-0.791168\pi$
$31$	−273.537	−1.58480	−0.792399	+	0.610003i	$0.791168\pi$
$32$	259.774	1.43506
$33$	0	0
$34$	−37.0018	−0.186640
$35$	−100.115	−0.483499
$36$	0	0
$37$	−137.283	−0.609978	−0.304989	−	0.952356i	$-0.598653\pi$
$37$	−137.283	−0.609978	−0.304989	+	0.952356i	$0.598653\pi$
$38$	208.510	0.890126
$39$	0	0
$40$	−19.6088	−0.0775107
$41$	−53.3457	−0.203200	−0.101600	−	0.994825i	$-0.532396\pi$
$41$	−53.3457	−0.203200	−0.101600	+	0.994825i	$0.532396\pi$
$42$	0	0
$43$	295.270	1.04717	0.523584	−	0.851974i	$-0.324595\pi$
$43$	295.270	1.04717	0.523584	+	0.851974i	$0.324595\pi$
$44$	−15.0247	−0.0514785
$45$	0	0
$46$	−883.134	−2.83067
$47$	−194.983	−0.605132	−0.302566	−	0.953128i	$-0.597843\pi$
$47$	−194.983	−0.605132	−0.302566	+	0.953128i	$0.597843\pi$
$48$	0	0
$49$	57.9173	0.168855
$50$	−102.934	−0.291140
$51$	0	0
$52$	99.4829	0.265304
$53$	−450.173	−1.16672	−0.583359	−	0.812214i	$-0.698262\pi$
$53$	−450.173	−1.16672	−0.583359	+	0.812214i	$0.698262\pi$
$54$	0	0
$55$	−8.39133	−0.0205725
$56$	78.5252	0.187382
$57$	0	0
$58$	316.660	0.716888
$59$	481.145	1.06169	0.530845	−	0.847469i	$-0.321875\pi$
$59$	481.145	1.06169	0.530845	+	0.847469i	$0.321875\pi$
$60$	0	0
$61$	675.854	1.41859	0.709297	−	0.704910i	$-0.249012\pi$
$61$	675.854	1.41859	0.709297	+	0.704910i	$0.249012\pi$
$62$	1126.25	2.30699
$63$	0	0
$64$	−625.798	−1.22226
$65$	55.5615	0.106024
$66$	0	0
$67$	894.589	1.63122	0.815608	−	0.578605i	$-0.196403\pi$
$67$	894.589	1.63122	0.815608	+	0.578605i	$0.196403\pi$
$68$	80.4545	0.143479
$69$	0	0
$70$	412.206	0.703829
$71$	721.947	1.20675	0.603376	−	0.797457i	$-0.293822\pi$
$71$	721.947	1.20675	0.603376	+	0.797457i	$0.293822\pi$
$72$	0	0
$73$	915.175	1.46730	0.733651	−	0.679526i	$-0.237815\pi$
$73$	915.175	1.46730	0.733651	+	0.679526i	$0.237815\pi$
$74$	565.241	0.887945
$75$	0	0
$76$	−453.372	−0.684280
$77$	33.6038	0.0497339
$78$	0	0
$79$	59.6642	0.0849715	0.0424858	−	0.999097i	$-0.486472\pi$
$79$	59.6642	0.0849715	0.0424858	+	0.999097i	$0.486472\pi$
$80$	−277.364	−0.387628
$81$	0	0
$82$	219.643	0.295798
$83$	742.875	0.982423	0.491212	−	0.871040i	$-0.336554\pi$
$83$	742.875	0.982423	0.491212	+	0.871040i	$0.336554\pi$
$84$	0	0
$85$	44.9341	0.0573387
$86$	−1215.73	−1.52436
$87$	0	0
$88$	6.58177	0.00797294
$89$	1540.72	1.83501	0.917505	−	0.397723i	$-0.130200\pi$
$89$	1540.72	1.83501	0.917505	+	0.397723i	$0.130200\pi$
$90$	0	0
$91$	−222.501	−0.256312
$92$	1920.23	2.17607
$93$	0	0
$94$	802.812	0.880891
$95$	−253.210	−0.273461
$96$	0	0
$97$	−1121.57	−1.17400	−0.586999	−	0.809588i	$-0.699691\pi$
$97$	−1121.57	−1.17400	−0.586999	+	0.809588i	$0.699691\pi$
$98$	−238.465	−0.245802
$99$	0	0
$100$	223.812	0.223812
$101$	−1758.47	−1.73242	−0.866211	−	0.499679i	$-0.833451\pi$
$101$	−1758.47	−1.73242	−0.866211	+	0.499679i	$0.833451\pi$
$102$	0	0
$103$	1056.48	1.01066	0.505332	−	0.862925i	$-0.331370\pi$
$103$	1056.48	1.01066	0.505332	+	0.862925i	$0.331370\pi$
$104$	−43.5799	−0.0410900
$105$	0	0
$106$	1853.52	1.69839
$107$	1470.03	1.32816	0.664081	−	0.747661i	$-0.268823\pi$
$107$	1470.03	1.32816	0.664081	+	0.747661i	$0.268823\pi$
$108$	0	0
$109$	918.889	0.807464	0.403732	−	0.914877i	$-0.367713\pi$
$109$	918.889	0.807464	0.403732	+	0.914877i	$0.367713\pi$
$110$	34.5500	0.0299474
$111$	0	0
$112$	1110.73	0.937087
$113$	987.217	0.821855	0.410927	−	0.911668i	$-0.365205\pi$
$113$	987.217	0.821855	0.410927	+	0.911668i	$0.365205\pi$
$114$	0	0
$115$	1072.46	0.869627
$116$	−688.526	−0.551104
$117$	0	0
$118$	−1981.04	−1.54550
$119$	−179.942	−0.138616
$120$	0	0
$121$	−1328.18	−0.997884
$122$	−2782.72	−2.06505
$123$	0	0
$124$	−2448.84	−1.77349
$125$	125.000	0.0894427
$126$	0	0
$127$	1480.10	1.03415	0.517077	−	0.855939i	$-0.327020\pi$
$127$	1480.10	1.03415	0.517077	+	0.855939i	$0.327020\pi$
$128$	498.428	0.344182
$129$	0	0
$130$	−228.766	−0.154339
$131$	−506.595	−0.337874	−0.168937	−	0.985627i	$-0.554033\pi$
$131$	−506.595	−0.337874	−0.168937	+	0.985627i	$0.554033\pi$
$132$	0	0
$133$	1014.00	0.661089
$134$	−3683.33	−2.37456
$135$	0	0
$136$	−35.2442	−0.0222218
$137$	975.723	0.608479	0.304240	−	0.952596i	$-0.401598\pi$
$137$	975.723	0.608479	0.304240	+	0.952596i	$0.401598\pi$
$138$	0	0
$139$	−1888.83	−1.15258	−0.576288	−	0.817247i	$-0.695499\pi$
$139$	−1888.83	−1.15258	−0.576288	+	0.817247i	$0.695499\pi$
$140$	−896.276	−0.541065
$141$	0	0
$142$	−2972.50	−1.75667
$143$	−18.6494	−0.0109059
$144$	0	0
$145$	−384.544	−0.220239
$146$	−3768.09	−2.13595
$147$	0	0
$148$	−1229.03	−0.682604
$149$	2582.29	1.41980	0.709898	−	0.704305i	$-0.248741\pi$
$149$	2582.29	1.41980	0.709898	+	0.704305i	$0.248741\pi$
$150$	0	0
$151$	2537.92	1.36777	0.683885	−	0.729590i	$-0.260289\pi$
$151$	2537.92	1.36777	0.683885	+	0.729590i	$0.260289\pi$
$152$	198.606	0.105981
$153$	0	0
$154$	−138.358	−0.0723976
$155$	−1367.69	−0.708743
$156$	0	0
$157$	105.035	0.0533930	0.0266965	−	0.999644i	$-0.491501\pi$
$157$	105.035	0.0533930	0.0266965	+	0.999644i	$0.491501\pi$
$158$	−245.658	−0.123693
$159$	0	0
$160$	1298.87	0.641780
$161$	−4294.74	−2.10232
$162$	0	0
$163$	2228.88	1.07104	0.535519	−	0.844523i	$-0.320116\pi$
$163$	2228.88	1.07104	0.535519	+	0.844523i	$0.320116\pi$
$164$	−477.578	−0.227394
$165$	0	0
$166$	−3058.67	−1.43011
$167$	2813.80	1.30382	0.651911	−	0.758296i	$-0.273968\pi$
$167$	2813.80	1.30382	0.651911	+	0.758296i	$0.273968\pi$
$168$	0	0
$169$	−2073.52	−0.943795
$170$	−185.009	−0.0834679
$171$	0	0
$172$	2643.40	1.17185
$173$	−3284.45	−1.44342	−0.721711	−	0.692195i	$-0.756644\pi$
$173$	−3284.45	−1.44342	−0.721711	+	0.692195i	$0.756644\pi$
$174$	0	0
$175$	−500.573	−0.216227
$176$	93.0981	0.0398723
$177$	0	0
$178$	−6343.67	−2.67123
$179$	1268.62	0.529726	0.264863	−	0.964286i	$-0.414673\pi$
$179$	1268.62	0.529726	0.264863	+	0.964286i	$0.414673\pi$
$180$	0	0
$181$	1080.10	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$181$	1080.10	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$182$	916.111	0.373114
$183$	0	0
$184$	−841.185	−0.337027
$185$	−686.415	−0.272790
$186$	0	0
$187$	−15.0823	−0.00589800
$188$	−1745.59	−0.677181
$189$	0	0
$190$	1042.55	0.398077
$191$	286.714	0.108617	0.0543087	−	0.998524i	$-0.482704\pi$
$191$	286.714	0.108617	0.0543087	+	0.998524i	$0.482704\pi$
$192$	0	0
$193$	−4650.70	−1.73453	−0.867267	−	0.497843i	$-0.834125\pi$
$193$	−4650.70	−1.73453	−0.867267	+	0.497843i	$0.834125\pi$
$194$	4617.87	1.70899
$195$	0	0
$196$	518.505	0.188959
$197$	1694.82	0.612947	0.306474	−	0.951879i	$-0.400851\pi$
$197$	1694.82	0.612947	0.306474	+	0.951879i	$0.400851\pi$
$198$	0	0
$199$	−3249.77	−1.15764	−0.578819	−	0.815456i	$-0.696486\pi$
$199$	−3249.77	−1.15764	−0.578819	+	0.815456i	$0.696486\pi$
$200$	−98.0442	−0.0346639
$201$	0	0
$202$	7240.23	2.52189
$203$	1539.94	0.532426
$204$	0	0
$205$	−266.729	−0.0908738
$206$	−4349.90	−1.47122
$207$	0	0
$208$	−616.430	−0.205489
$209$	84.9906	0.0281288
$210$	0	0
$211$	−1078.56	−0.351900	−0.175950	−	0.984399i	$-0.556300\pi$
$211$	−1078.56	−0.351900	−0.175950	+	0.984399i	$0.556300\pi$
$212$	−4030.18	−1.30563
$213$	0	0
$214$	−6052.62	−1.93340
$215$	1476.35	0.468308
$216$	0	0
$217$	5477.01	1.71338
$218$	−3783.38	−1.17543
$219$	0	0
$220$	−75.1234	−0.0230219
$221$	99.8643	0.0303964
$222$	0	0
$223$	−5143.28	−1.54448	−0.772241	−	0.635330i	$-0.780864\pi$
$223$	−5143.28	−1.54448	−0.772241	+	0.635330i	$0.780864\pi$
$224$	−5201.44	−1.55150
$225$	0	0
$226$	−4064.71	−1.19637
$227$	2909.66	0.850754	0.425377	−	0.905016i	$-0.360142\pi$
$227$	2909.66	0.850754	0.425377	+	0.905016i	$0.360142\pi$
$228$	0	0
$229$	1725.82	0.498015	0.249007	−	0.968502i	$-0.419896\pi$
$229$	1725.82	0.498015	0.249007	+	0.968502i	$0.419896\pi$
$230$	−4415.67	−1.26592
$231$	0	0
$232$	301.619	0.0853544
$233$	1087.48	0.305765	0.152882	−	0.988244i	$-0.451144\pi$
$233$	1087.48	0.305765	0.152882	+	0.988244i	$0.451144\pi$
$234$	0	0
$235$	−974.916	−0.270623
$236$	4307.45	1.18810
$237$	0	0
$238$	740.884	0.201783
$239$	−319.345	−0.0864297	−0.0432149	−	0.999066i	$-0.513760\pi$
$239$	−319.345	−0.0864297	−0.0432149	+	0.999066i	$0.513760\pi$
$240$	0	0
$241$	3645.83	0.974475	0.487238	−	0.873269i	$-0.338005\pi$
$241$	3645.83	0.974475	0.487238	+	0.873269i	$0.338005\pi$
$242$	5468.58	1.45262
$243$	0	0
$244$	6050.58	1.58750
$245$	289.587	0.0755143
$246$	0	0
$247$	−562.748	−0.144967
$248$	1072.75	0.274676
$249$	0	0
$250$	−514.668	−0.130202
$251$	−572.874	−0.144062	−0.0720309	−	0.997402i	$-0.522948\pi$
$251$	−572.874	−0.144062	−0.0720309	+	0.997402i	$0.522948\pi$
$252$	0	0
$253$	−359.973	−0.0894519
$254$	−6094.07	−1.50542
$255$	0	0
$256$	2954.18	0.721236
$257$	−3677.66	−0.892630	−0.446315	−	0.894876i	$-0.647264\pi$
$257$	−3677.66	−0.892630	−0.446315	+	0.894876i	$0.647264\pi$
$258$	0	0
$259$	2748.81	0.659469
$260$	497.414	0.118647
$261$	0	0
$262$	2085.83	0.491842
$263$	2001.71	0.469319	0.234659	−	0.972078i	$-0.424603\pi$
$263$	2001.71	0.469319	0.234659	+	0.972078i	$0.424603\pi$
$264$	0	0
$265$	−2250.87	−0.521772
$266$	−4174.98	−0.962348
$267$	0	0
$268$	8008.81	1.82543
$269$	−20.1629	−0.00457009	−0.00228504	−	0.999997i	$-0.500727\pi$
$269$	−20.1629	−0.00457009	−0.00228504	+	0.999997i	$0.500727\pi$
$270$	0	0
$271$	4733.12	1.06095	0.530474	−	0.847701i	$-0.322014\pi$
$271$	4733.12	1.06095	0.530474	+	0.847701i	$0.322014\pi$
$272$	−498.524	−0.111130
$273$	0	0
$274$	−4017.38	−0.885763
$275$	−41.9567	−0.00920030
$276$	0	0
$277$	5029.12	1.09087	0.545434	−	0.838154i	$-0.316365\pi$
$277$	5029.12	1.09087	0.545434	+	0.838154i	$0.316365\pi$
$278$	7776.94	1.67780
$279$	0	0
$280$	392.626	0.0837996
$281$	5808.31	1.23308	0.616539	−	0.787325i	$-0.288534\pi$
$281$	5808.31	1.23308	0.616539	+	0.787325i	$0.288534\pi$
$282$	0	0
$283$	213.023	0.0447453	0.0223726	−	0.999750i	$-0.492878\pi$
$283$	213.023	0.0447453	0.0223726	+	0.999750i	$0.492878\pi$
$284$	6463.23	1.35043
$285$	0	0
$286$	76.7859	0.0158757
$287$	1068.14	0.219687
$288$	0	0
$289$	−4832.24	−0.983561
$290$	1583.30	0.320602
$291$	0	0
$292$	8193.10	1.64200
$293$	−3593.09	−0.716419	−0.358210	−	0.933641i	$-0.616613\pi$
$293$	−3593.09	−0.716419	−0.358210	+	0.933641i	$0.616613\pi$
$294$	0	0
$295$	2405.72	0.474802
$296$	538.392	0.105721
$297$	0	0
$298$	−10632.2	−2.06680
$299$	2383.49	0.461006
$300$	0	0
$301$	−5912.17	−1.13213
$302$	−10449.5	−1.99106
$303$	0	0
$304$	2809.25	0.530004
$305$	3379.27	0.634415
$306$	0	0
$307$	2403.23	0.446774	0.223387	−	0.974730i	$-0.428289\pi$
$307$	2403.23	0.446774	0.223387	+	0.974730i	$0.428289\pi$
$308$	300.838	0.0556553
$309$	0	0
$310$	5631.23	1.03172
$311$	−1608.16	−0.293217	−0.146608	−	0.989195i	$-0.546836\pi$
$311$	−1608.16	−0.293217	−0.146608	+	0.989195i	$0.546836\pi$
$312$	0	0
$313$	2780.54	0.502127	0.251063	−	0.967971i	$-0.419220\pi$
$313$	2780.54	0.502127	0.251063	+	0.967971i	$0.419220\pi$
$314$	−432.465	−0.0777242
$315$	0	0
$316$	534.144	0.0950885
$317$	5535.70	0.980807	0.490403	−	0.871496i	$-0.336850\pi$
$317$	5535.70	0.980807	0.490403	+	0.871496i	$0.336850\pi$
$318$	0	0
$319$	129.074	0.0226543
$320$	−3128.99	−0.546612
$321$	0	0
$322$	17682.9	3.06034
$323$	−455.110	−0.0783994
$324$	0	0
$325$	277.807	0.0474153
$326$	−9177.06	−1.55911
$327$	0	0
$328$	209.209	0.0352185
$329$	3904.13	0.654230
$330$	0	0
$331$	−9616.75	−1.59693	−0.798466	−	0.602040i	$-0.794355\pi$
$331$	−9616.75	−1.59693	−0.798466	+	0.602040i	$0.794355\pi$
$332$	6650.59	1.09939
$333$	0	0
$334$	−11585.4	−1.89797
$335$	4472.95	0.729502
$336$	0	0
$337$	−2084.46	−0.336936	−0.168468	−	0.985707i	$-0.553882\pi$
$337$	−2084.46	−0.336936	−0.168468	+	0.985707i	$0.553882\pi$
$338$	8537.38	1.37388
$339$	0	0
$340$	402.273	0.0641656
$341$	459.068	0.0729030
$342$	0	0
$343$	5708.19	0.898581
$344$	−1157.98	−0.181494
$345$	0	0
$346$	13523.2	2.10119
$347$	−6113.42	−0.945780	−0.472890	−	0.881121i	$-0.656789\pi$
$347$	−6113.42	−0.945780	−0.472890	+	0.881121i	$0.656789\pi$
$348$	0	0
$349$	2189.06	0.335753	0.167877	−	0.985808i	$-0.446309\pi$
$349$	2189.06	0.335753	0.167877	+	0.985808i	$0.446309\pi$
$350$	2061.03	0.314762
$351$	0	0
$352$	−435.971	−0.0660151
$353$	−2500.20	−0.376976	−0.188488	−	0.982076i	$-0.560359\pi$
$353$	−2500.20	−0.376976	−0.188488	+	0.982076i	$0.560359\pi$
$354$	0	0
$355$	3609.74	0.539676
$356$	13793.3	2.05349
$357$	0	0
$358$	−5223.33	−0.771122
$359$	841.607	0.123728	0.0618639	−	0.998085i	$-0.480296\pi$
$359$	841.607	0.123728	0.0618639	+	0.998085i	$0.480296\pi$
$360$	0	0
$361$	−4294.40	−0.626096
$362$	−4447.14	−0.645681
$363$	0	0
$364$	−1991.94	−0.286829
$365$	4575.87	0.656198
$366$	0	0
$367$	−5614.42	−0.798557	−0.399278	−	0.916830i	$-0.630739\pi$
$367$	−5614.42	−0.798557	−0.399278	+	0.916830i	$0.630739\pi$
$368$	−11898.4	−1.68546
$369$	0	0
$370$	2826.21	0.397101
$371$	9013.78	1.26138
$372$	0	0
$373$	1319.17	0.183121	0.0915607	−	0.995799i	$-0.470814\pi$
$373$	1319.17	0.183121	0.0915607	+	0.995799i	$0.470814\pi$
$374$	62.0989	0.00858572
$375$	0	0
$376$	764.678	0.104881
$377$	−854.634	−0.116753
$378$	0	0
$379$	−5002.07	−0.677939	−0.338970	−	0.940797i	$-0.610078\pi$
$379$	−5002.07	−0.677939	−0.338970	+	0.940797i	$0.610078\pi$
$380$	−2266.86	−0.306020
$381$	0	0
$382$	−1180.50	−0.158114
$383$	2242.45	0.299174	0.149587	−	0.988749i	$-0.452206\pi$
$383$	2242.45	0.299174	0.149587	+	0.988749i	$0.452206\pi$
$384$	0	0
$385$	168.019	0.0222417
$386$	19148.5	2.52496
$387$	0	0
$388$	−10040.8	−1.31378
$389$	11393.8	1.48506	0.742529	−	0.669814i	$-0.233626\pi$
$389$	11393.8	1.48506	0.742529	+	0.669814i	$0.233626\pi$
$390$	0	0
$391$	1927.59	0.249316
$392$	−227.138	−0.0292659
$393$	0	0
$394$	−6978.13	−0.892267
$395$	298.321	0.0380004
$396$	0	0
$397$	−14926.0	−1.88694	−0.943472	−	0.331454i	$-0.892461\pi$
$397$	−14926.0	−1.88694	−0.943472	+	0.331454i	$0.892461\pi$
$398$	13380.4	1.68517
$399$	0	0
$400$	−1386.82	−0.173352
$401$	7165.63	0.892356	0.446178	−	0.894944i	$-0.352785\pi$
$401$	7165.63	0.892356	0.446178	+	0.894944i	$0.352785\pi$
$402$	0	0
$403$	−3039.63	−0.375719
$404$	−15742.7	−1.93869
$405$	0	0
$406$	−6340.46	−0.775053
$407$	230.397	0.0280599
$408$	0	0
$409$	−11324.1	−1.36905	−0.684524	−	0.728990i	$-0.739990\pi$
$409$	−11324.1	−1.36905	−0.684524	+	0.728990i	$0.739990\pi$
$410$	1098.21	0.132285
$411$	0	0
$412$	9458.16	1.13100
$413$	−9633.93	−1.14783
$414$	0	0
$415$	3714.38	0.439353
$416$	2886.69	0.340220
$417$	0	0
$418$	−349.936	−0.0409471
$419$	12699.8	1.48073	0.740365	−	0.672205i	$-0.234653\pi$
$419$	12699.8	1.48073	0.740365	+	0.672205i	$0.234653\pi$
$420$	0	0
$421$	−17162.8	−1.98685	−0.993424	−	0.114494i	$-0.963475\pi$
$421$	−17162.8	−1.98685	−0.993424	+	0.114494i	$0.963475\pi$
$422$	4440.79	0.512261
$423$	0	0
$424$	1765.48	0.202215
$425$	224.671	0.0256426
$426$	0	0
$427$	−13532.6	−1.53369
$428$	13160.5	1.48630
$429$	0	0
$430$	−6078.63	−0.681716
$431$	11130.9	1.24399	0.621994	−	0.783022i	$-0.286323\pi$
$431$	11130.9	1.24399	0.621994	+	0.783022i	$0.286323\pi$
$432$	0	0
$433$	7675.55	0.851878	0.425939	−	0.904752i	$-0.359944\pi$
$433$	7675.55	0.851878	0.425939	+	0.904752i	$0.359944\pi$
$434$	−22550.7	−2.49417
$435$	0	0
$436$	8226.35	0.903603
$437$	−10862.2	−1.18904
$438$	0	0
$439$	3363.80	0.365707	0.182853	−	0.983140i	$-0.441467\pi$
$439$	3363.80	0.365707	0.182853	+	0.983140i	$0.441467\pi$
$440$	32.9089	0.00356561
$441$	0	0
$442$	−411.175	−0.0442480
$443$	−7894.06	−0.846633	−0.423316	−	0.905982i	$-0.639134\pi$
$443$	−7894.06	−0.846633	−0.423316	+	0.905982i	$0.639134\pi$
$444$	0	0
$445$	7703.60	0.820642
$446$	21176.6	2.24830
$447$	0	0
$448$	12530.3	1.32143
$449$	−2244.52	−0.235914	−0.117957	−	0.993019i	$-0.537634\pi$
$449$	−2244.52	−0.235914	−0.117957	+	0.993019i	$0.537634\pi$
$450$	0	0
$451$	89.5283	0.00934750
$452$	8838.06	0.919707
$453$	0	0
$454$	−11980.1	−1.23844
$455$	−1112.50	−0.114626
$456$	0	0
$457$	14749.3	1.50972	0.754862	−	0.655884i	$-0.227704\pi$
$457$	14749.3	1.50972	0.754862	+	0.655884i	$0.227704\pi$
$458$	−7105.79	−0.724960
$459$	0	0
$460$	9601.16	0.973166
$461$	−12479.5	−1.26080	−0.630400	−	0.776271i	$-0.717109\pi$
$461$	−12479.5	−1.26080	−0.630400	+	0.776271i	$0.717109\pi$
$462$	0	0
$463$	−9679.12	−0.971549	−0.485774	−	0.874084i	$-0.661462\pi$
$463$	−9679.12	−0.971549	−0.485774	+	0.874084i	$0.661462\pi$
$464$	4266.34	0.426854
$465$	0	0
$466$	−4477.53	−0.445102
$467$	−6714.33	−0.665315	−0.332658	−	0.943048i	$-0.607945\pi$
$467$	−6714.33	−0.665315	−0.332658	+	0.943048i	$0.607945\pi$
$468$	0	0
$469$	−17912.3	−1.76357
$470$	4014.06	0.393946
$471$	0	0
$472$	−1886.94	−0.184011
$473$	−495.542	−0.0481713
$474$	0	0
$475$	−1266.05	−0.122295
$476$	−1610.93	−0.155120
$477$	0	0
$478$	1314.85	0.125816
$479$	−2127.48	−0.202937	−0.101469	−	0.994839i	$-0.532354\pi$
$479$	−2127.48	−0.202937	−0.101469	+	0.994839i	$0.532354\pi$
$480$	0	0
$481$	−1525.53	−0.144612
$482$	−15011.1	−1.41854
$483$	0	0
$484$	−11890.6	−1.11669
$485$	−5607.83	−0.525028
$486$	0	0
$487$	−6549.04	−0.609375	−0.304687	−	0.952452i	$-0.598552\pi$
$487$	−6549.04	−0.609375	−0.304687	+	0.952452i	$0.598552\pi$
$488$	−2650.54	−0.245870
$489$	0	0
$490$	−1192.33	−0.109926
$491$	−12933.7	−1.18878	−0.594389	−	0.804177i	$-0.702606\pi$
$491$	−12933.7	−1.18878	−0.594389	+	0.804177i	$0.702606\pi$
$492$	0	0
$493$	−691.166	−0.0631411
$494$	2317.03	0.211028
$495$	0	0
$496$	15173.9	1.37364
$497$	−14455.5	−1.30466
$498$	0	0
$499$	5120.66	0.459383	0.229692	−	0.973263i	$-0.426228\pi$
$499$	5120.66	0.459383	0.229692	+	0.973263i	$0.426228\pi$
$500$	1119.06	0.100092
$501$	0	0
$502$	2358.72	0.209711
$503$	−10809.6	−0.958204	−0.479102	−	0.877759i	$-0.659038\pi$
$503$	−10809.6	−0.958204	−0.479102	+	0.877759i	$0.659038\pi$
$504$	0	0
$505$	−8792.36	−0.774762
$506$	1482.13	0.130215
$507$	0	0
$508$	13250.6	1.15728
$509$	−14756.9	−1.28505	−0.642524	−	0.766266i	$-0.722113\pi$
$509$	−14756.9	−1.28505	−0.642524	+	0.766266i	$0.722113\pi$
$510$	0	0
$511$	−18324.5	−1.58635
$512$	−16150.8	−1.39409
$513$	0	0
$514$	15142.2	1.29940
$515$	5282.42	0.451983
$516$	0	0
$517$	327.234	0.0278370
$518$	−11317.8	−0.959989
$519$	0	0
$520$	−217.899	−0.0183760
$521$	1742.68	0.146542	0.0732709	−	0.997312i	$-0.476656\pi$
$521$	1742.68	0.146542	0.0732709	+	0.997312i	$0.476656\pi$
$522$	0	0
$523$	10304.8	0.861565	0.430782	−	0.902456i	$-0.358238\pi$
$523$	10304.8	0.861565	0.430782	+	0.902456i	$0.358238\pi$
$524$	−4535.29	−0.378102
$525$	0	0
$526$	−8241.73	−0.683187
$527$	−2458.23	−0.203192
$528$	0	0
$529$	33839.5	2.78125
$530$	9267.59	0.759544
$531$	0	0
$532$	9077.83	0.739800
$533$	−592.794	−0.0481740
$534$	0	0
$535$	7350.16	0.593972
$536$	−3508.37	−0.282721
$537$	0	0
$538$	83.0175	0.00665267
$539$	−97.2007	−0.00776759
$540$	0	0
$541$	3966.86	0.315247	0.157623	−	0.987499i	$-0.449617\pi$
$541$	3966.86	0.315247	0.157623	+	0.987499i	$0.449617\pi$
$542$	−19487.9	−1.54442
$543$	0	0
$544$	2334.55	0.183994
$545$	4594.44	0.361109
$546$	0	0
$547$	−5808.72	−0.454045	−0.227023	−	0.973889i	$-0.572899\pi$
$547$	−5808.72	−0.454045	−0.227023	+	0.973889i	$0.572899\pi$
$548$	8735.16	0.680926
$549$	0	0
$550$	172.750	0.0133929
$551$	3894.81	0.301133
$552$	0	0
$553$	−1194.65	−0.0918658
$554$	−20706.6	−1.58798
$555$	0	0
$556$	−16909.7	−1.28980
$557$	3563.91	0.271109	0.135554	−	0.990770i	$-0.456718\pi$
$557$	3563.91	0.271109	0.135554	+	0.990770i	$0.456718\pi$
$558$	0	0
$559$	3281.13	0.248259
$560$	5553.63	0.419078
$561$	0	0
$562$	−23914.8	−1.79499
$563$	−511.544	−0.0382931	−0.0191466	−	0.999817i	$-0.506095\pi$
$563$	−511.544	−0.0382931	−0.0191466	+	0.999817i	$0.506095\pi$
$564$	0	0
$565$	4936.09	0.367545
$566$	−877.089	−0.0651357
$567$	0	0
$568$	−2831.31	−0.209153
$569$	8116.54	0.598002	0.299001	−	0.954253i	$-0.403347\pi$
$569$	8116.54	0.598002	0.299001	+	0.954253i	$0.403347\pi$
$570$	0	0
$571$	15078.1	1.10508	0.552539	−	0.833487i	$-0.313659\pi$
$571$	15078.1	1.10508	0.552539	+	0.833487i	$0.313659\pi$
$572$	−166.959	−0.0122044
$573$	0	0
$574$	−4397.88	−0.319798
$575$	5362.28	0.388909
$576$	0	0
$577$	27101.5	1.95537	0.977685	−	0.210077i	$-0.0673716\pi$
$577$	27101.5	1.95537	0.977685	+	0.210077i	$0.0673716\pi$
$578$	19896.0	1.43177
$579$	0	0
$580$	−3442.63	−0.246461
$581$	−14874.5	−1.06213
$582$	0	0
$583$	755.511	0.0536708
$584$	−3589.10	−0.254312
$585$	0	0
$586$	14794.0	1.04289
$587$	102.082	0.00717782	0.00358891	−	0.999994i	$-0.498858\pi$
$587$	102.082	0.00717782	0.00358891	+	0.999994i	$0.498858\pi$
$588$	0	0
$589$	13852.4	0.969067
$590$	−9905.19	−0.691170
$591$	0	0
$592$	7615.47	0.528706
$593$	16467.2	1.14035	0.570174	−	0.821524i	$-0.306876\pi$
$593$	16467.2	1.14035	0.570174	+	0.821524i	$0.306876\pi$
$594$	0	0
$595$	−899.712	−0.0619909
$596$	23118.0	1.58884
$597$	0	0
$598$	−9813.65	−0.671087
$599$	23216.2	1.58362	0.791809	−	0.610769i	$-0.209140\pi$
$599$	23216.2	1.58362	0.791809	+	0.610769i	$0.209140\pi$
$600$	0	0
$601$	−1678.29	−0.113909	−0.0569543	−	0.998377i	$-0.518139\pi$
$601$	−1678.29	−0.113909	−0.0569543	+	0.998377i	$0.518139\pi$
$602$	24342.4	1.64804
$603$	0	0
$604$	22720.8	1.53062
$605$	−6640.92	−0.446267
$606$	0	0
$607$	19307.3	1.29103	0.645517	−	0.763746i	$-0.276642\pi$
$607$	19307.3	1.29103	0.645517	+	0.763746i	$0.276642\pi$
$608$	−13155.5	−0.877508
$609$	0	0
$610$	−13913.6	−0.923517
$611$	−2166.71	−0.143463
$612$	0	0
$613$	13659.9	0.900032	0.450016	−	0.893020i	$-0.351418\pi$
$613$	13659.9	0.900032	0.450016	+	0.893020i	$0.351418\pi$
$614$	−9894.93	−0.650369
$615$	0	0
$616$	−131.786	−0.00861984
$617$	−14603.7	−0.952872	−0.476436	−	0.879209i	$-0.658072\pi$
$617$	−14603.7	−0.952872	−0.476436	+	0.879209i	$0.658072\pi$
$618$	0	0
$619$	−26388.3	−1.71346	−0.856732	−	0.515762i	$-0.827509\pi$
$619$	−26388.3	−1.71346	−0.856732	+	0.515762i	$0.827509\pi$
$620$	−12244.2	−0.793128
$621$	0	0
$622$	6621.35	0.426836
$623$	−30849.7	−1.98390
$624$	0	0
$625$	625.000	0.0400000
$626$	−11448.5	−0.730946
$627$	0	0
$628$	940.326	0.0597501
$629$	−1233.74	−0.0782072
$630$	0	0
$631$	14241.1	0.898459	0.449229	−	0.893416i	$-0.351699\pi$
$631$	14241.1	0.898459	0.449229	+	0.893416i	$0.351699\pi$
$632$	−233.989	−0.0147272
$633$	0	0
$634$	−22792.4	−1.42776
$635$	7400.50	0.462488
$636$	0	0
$637$	643.595	0.0400316
$638$	−531.440	−0.0329779
$639$	0	0
$640$	2492.14	0.153923
$641$	18960.5	1.16832	0.584160	−	0.811638i	$-0.301424\pi$
$641$	18960.5	1.16832	0.584160	+	0.811638i	$0.301424\pi$
$642$	0	0
$643$	11054.3	0.677974	0.338987	−	0.940791i	$-0.389916\pi$
$643$	11054.3	0.677974	0.338987	+	0.940791i	$0.389916\pi$
$644$	−38448.7	−2.35262
$645$	0	0
$646$	1873.84	0.114126
$647$	−19310.1	−1.17335	−0.586677	−	0.809821i	$-0.699564\pi$
$647$	−19310.1	−1.17335	−0.586677	+	0.809821i	$0.699564\pi$
$648$	0	0
$649$	−807.489	−0.0488393
$650$	−1143.83	−0.0690225
$651$	0	0
$652$	19954.0	1.19856
$653$	−6934.26	−0.415557	−0.207778	−	0.978176i	$-0.566623\pi$
$653$	−6934.26	−0.415557	−0.207778	+	0.978176i	$0.566623\pi$
$654$	0	0
$655$	−2532.98	−0.151102
$656$	2959.23	0.176126
$657$	0	0
$658$	−16074.6	−0.952363
$659$	15878.7	0.938613	0.469307	−	0.883035i	$-0.344504\pi$
$659$	15878.7	0.938613	0.469307	+	0.883035i	$0.344504\pi$
$660$	0	0
$661$	−9418.32	−0.554206	−0.277103	−	0.960840i	$-0.589374\pi$
$661$	−9418.32	−0.554206	−0.277103	+	0.960840i	$0.589374\pi$
$662$	39595.4	2.32465
$663$	0	0
$664$	−2913.38	−0.170273
$665$	5070.00	0.295648
$666$	0	0
$667$	−16496.3	−0.957628
$668$	25190.5	1.45906
$669$	0	0
$670$	−18416.6	−1.06194
$671$	−1134.26	−0.0652574
$672$	0	0
$673$	−18392.3	−1.05345	−0.526726	−	0.850035i	$-0.676581\pi$
$673$	−18392.3	−1.05345	−0.526726	+	0.850035i	$0.676581\pi$
$674$	8582.41	0.490478
$675$	0	0
$676$	−18563.2	−1.05617
$677$	−10570.7	−0.600095	−0.300048	−	0.953924i	$-0.597003\pi$
$677$	−10570.7	−0.600095	−0.300048	+	0.953924i	$0.597003\pi$
$678$	0	0
$679$	22457.0	1.26925
$680$	−176.221	−0.00993790
$681$	0	0
$682$	−1890.14	−0.106125
$683$	23975.0	1.34316	0.671579	−	0.740933i	$-0.265616\pi$
$683$	23975.0	1.34316	0.671579	+	0.740933i	$0.265616\pi$
$684$	0	0
$685$	4878.61	0.272120
$686$	−23502.6	−1.30806
$687$	0	0
$688$	−16379.4	−0.907645
$689$	−5002.46	−0.276602
$690$	0	0
$691$	19641.0	1.08130	0.540650	−	0.841248i	$-0.318178\pi$
$691$	19641.0	1.08130	0.540650	+	0.841248i	$0.318178\pi$
$692$	−29404.0	−1.61528
$693$	0	0
$694$	25171.1	1.37677
$695$	−9444.13	−0.515448
$696$	0	0
$697$	−479.408	−0.0260529
$698$	−9013.12	−0.488756
$699$	0	0
$700$	−4481.38	−0.241972
$701$	−36098.2	−1.94495	−0.972475	−	0.233007i	$-0.925143\pi$
$701$	−36098.2	−1.94495	−0.972475	+	0.233007i	$0.925143\pi$
$702$	0	0
$703$	6952.28	0.372987
$704$	1050.26	0.0562258
$705$	0	0
$706$	10294.2	0.548764
$707$	35209.8	1.87298
$708$	0	0
$709$	35234.1	1.86635	0.933177	−	0.359418i	$-0.117025\pi$
$709$	35234.1	1.86635	0.933177	+	0.359418i	$0.117025\pi$
$710$	−14862.5	−0.785606
$711$	0	0
$712$	−6042.34	−0.318043
$713$	−58671.3	−3.08171
$714$	0	0
$715$	−93.2470	−0.00487726
$716$	11357.3	0.592797
$717$	0	0
$718$	−3465.18	−0.180111
$719$	22089.7	1.14577	0.572885	−	0.819636i	$-0.305824\pi$
$719$	22089.7	1.14577	0.572885	+	0.819636i	$0.305824\pi$
$720$	0	0
$721$	−21153.9	−1.09267
$722$	17681.5	0.911409
$723$	0	0
$724$	9669.60	0.496364
$725$	−1922.72	−0.0984939
$726$	0	0
$727$	−19435.2	−0.991488	−0.495744	−	0.868469i	$-0.665105\pi$
$727$	−19435.2	−0.991488	−0.495744	+	0.868469i	$0.665105\pi$
$728$	872.596	0.0444238
$729$	0	0
$730$	−18840.4	−0.955227
$731$	2653.54	0.134261
$732$	0	0
$733$	14974.2	0.754550	0.377275	−	0.926101i	$-0.376861\pi$
$733$	14974.2	0.754550	0.377275	+	0.926101i	$0.376861\pi$
$734$	23116.5	1.16246
$735$	0	0
$736$	55719.3	2.79055
$737$	−1501.36	−0.0750384
$738$	0	0
$739$	−30663.4	−1.52635	−0.763174	−	0.646193i	$-0.776360\pi$
$739$	−30663.4	−1.52635	−0.763174	+	0.646193i	$0.776360\pi$
$740$	−6145.13	−0.305270
$741$	0	0
$742$	−37112.8	−1.83619
$743$	−3414.68	−0.168604	−0.0843018	−	0.996440i	$-0.526866\pi$
$743$	−3414.68	−0.168604	−0.0843018	+	0.996440i	$0.526866\pi$
$744$	0	0
$745$	12911.5	0.634952
$746$	−5431.49	−0.266570
$747$	0	0
$748$	−135.024	−0.00660023
$749$	−29434.3	−1.43592
$750$	0	0
$751$	−746.259	−0.0362601	−0.0181301	−	0.999836i	$-0.505771\pi$
$751$	−746.259	−0.0362601	−0.0181301	+	0.999836i	$0.505771\pi$
$752$	10816.2	0.524505
$753$	0	0
$754$	3518.82	0.169957
$755$	12689.6	0.611685
$756$	0	0
$757$	−22929.5	−1.10091	−0.550454	−	0.834865i	$-0.685546\pi$
$757$	−22929.5	−1.10091	−0.550454	+	0.834865i	$0.685546\pi$
$758$	20595.2	0.986876
$759$	0	0
$760$	993.029	0.0473960
$761$	21176.6	1.00874	0.504370	−	0.863487i	$-0.331725\pi$
$761$	21176.6	1.00874	0.504370	+	0.863487i	$0.331725\pi$
$762$	0	0
$763$	−18398.8	−0.872978
$764$	2566.81	0.121550
$765$	0	0
$766$	−9232.92	−0.435508
$767$	5346.63	0.251702
$768$	0	0
$769$	2774.18	0.130091	0.0650453	−	0.997882i	$-0.479281\pi$
$769$	2774.18	0.130091	0.0650453	+	0.997882i	$0.479281\pi$
$770$	−691.792	−0.0323772
$771$	0	0
$772$	−41635.4	−1.94105
$773$	−28891.7	−1.34433	−0.672163	−	0.740403i	$-0.734635\pi$
$773$	−28891.7	−1.34433	−0.672163	+	0.740403i	$0.734635\pi$
$774$	0	0
$775$	−6838.43	−0.316960
$776$	4398.52	0.203476
$777$	0	0
$778$	−46912.1	−2.16180
$779$	2701.53	0.124252
$780$	0	0
$781$	−1211.62	−0.0555124
$782$	−7936.56	−0.362930
$783$	0	0
$784$	−3212.83	−0.146357
$785$	525.175	0.0238781
$786$	0	0
$787$	8808.75	0.398981	0.199490	−	0.979900i	$-0.436071\pi$
$787$	8808.75	0.398981	0.199490	+	0.979900i	$0.436071\pi$
$788$	15172.8	0.685926
$789$	0	0
$790$	−1228.29	−0.0553172
$791$	−19767.0	−0.888537
$792$	0	0
$793$	7510.29	0.336316
$794$	61455.6	2.74682
$795$	0	0
$796$	−29093.5	−1.29547
$797$	780.503	0.0346886	0.0173443	−	0.999850i	$-0.494479\pi$
$797$	780.503	0.0346886	0.0173443	+	0.999850i	$0.494479\pi$
$798$	0	0
$799$	−1752.28	−0.0775859
$800$	6494.36	0.287013
$801$	0	0
$802$	−29503.4	−1.29900
$803$	−1535.91	−0.0674981
$804$	0	0
$805$	−21473.7	−0.940185
$806$	12515.2	0.546934
$807$	0	0
$808$	6896.32	0.300262
$809$	−15234.0	−0.662051	−0.331025	−	0.943622i	$-0.607395\pi$
$809$	−15234.0	−0.662051	−0.331025	+	0.943622i	$0.607395\pi$
$810$	0	0
$811$	10825.2	0.468711	0.234356	−	0.972151i	$-0.424702\pi$
$811$	10825.2	0.468711	0.234356	+	0.972151i	$0.424702\pi$
$812$	13786.3	0.595818
$813$	0	0
$814$	−948.625	−0.0408468
$815$	11144.4	0.478983
$816$	0	0
$817$	−14953.0	−0.640319
$818$	46625.2	1.99292
$819$	0	0
$820$	−2387.89	−0.101693
$821$	−39971.2	−1.69915	−0.849577	−	0.527464i	$-0.823143\pi$
$821$	−39971.2	−1.69915	−0.849577	+	0.527464i	$0.823143\pi$
$822$	0	0
$823$	28620.6	1.21221	0.606107	−	0.795383i	$-0.292730\pi$
$823$	28620.6	1.21221	0.606107	+	0.795383i	$0.292730\pi$
$824$	−4143.28	−0.175168
$825$	0	0
$826$	39666.2	1.67090
$827$	−2265.62	−0.0952639	−0.0476320	−	0.998865i	$-0.515167\pi$
$827$	−2265.62	−0.0952639	−0.0476320	+	0.998865i	$0.515167\pi$
$828$	0	0
$829$	−14651.6	−0.613839	−0.306919	−	0.951736i	$-0.599298\pi$
$829$	−14651.6	−0.613839	−0.306919	+	0.951736i	$0.599298\pi$
$830$	−15293.4	−0.639566
$831$	0	0
$832$	−6954.05	−0.289770
$833$	520.493	0.0216495
$834$	0	0
$835$	14069.0	0.583087
$836$	760.879	0.0314779
$837$	0	0
$838$	−52289.4	−2.15550
$839$	22466.4	0.924466	0.462233	−	0.886758i	$-0.347048\pi$
$839$	22466.4	0.924466	0.462233	+	0.886758i	$0.347048\pi$
$840$	0	0
$841$	−18474.0	−0.757474
$842$	70665.0	2.89225
$843$	0	0
$844$	−9655.79	−0.393799
$845$	−10367.6	−0.422078
$846$	0	0
$847$	26594.1	1.07885
$848$	24972.4	1.01127
$849$	0	0
$850$	−925.045	−0.0373280
$851$	−29446.0	−1.18613
$852$	0	0
$853$	44471.0	1.78506	0.892531	−	0.450986i	$-0.148927\pi$
$853$	44471.0	1.78506	0.892531	+	0.450986i	$0.148927\pi$
$854$	55718.2	2.23260
$855$	0	0
$856$	−5765.12	−0.230196
$857$	31480.6	1.25479	0.627396	−	0.778700i	$-0.284121\pi$
$857$	31480.6	1.25479	0.627396	+	0.778700i	$0.284121\pi$
$858$	0	0
$859$	42983.1	1.70729	0.853646	−	0.520853i	$-0.174386\pi$
$859$	42983.1	1.70729	0.853646	+	0.520853i	$0.174386\pi$
$860$	13217.0	0.524066
$861$	0	0
$862$	−45829.9	−1.81087
$863$	25203.4	0.994131	0.497065	−	0.867713i	$-0.334411\pi$
$863$	25203.4	0.994131	0.497065	+	0.867713i	$0.334411\pi$
$864$	0	0
$865$	−16422.2	−0.645518
$866$	−31602.8	−1.24008
$867$	0	0
$868$	49033.0	1.91738
$869$	−100.132	−0.00390882
$870$	0	0
$871$	9940.95	0.386723
$872$	−3603.67	−0.139949
$873$	0	0
$874$	44723.6	1.73089
$875$	−2502.86	−0.0966997
$876$	0	0
$877$	3472.71	0.133712	0.0668558	−	0.997763i	$-0.478703\pi$
$877$	3472.71	0.133712	0.0668558	+	0.997763i	$0.478703\pi$
$878$	−13849.9	−0.532359
$879$	0	0
$880$	465.490	0.0178314
$881$	−26066.1	−0.996811	−0.498405	−	0.866944i	$-0.666081\pi$
$881$	−26066.1	−0.996811	−0.498405	+	0.866944i	$0.666081\pi$
$882$	0	0
$883$	−3032.93	−0.115590	−0.0577952	−	0.998328i	$-0.518407\pi$
$883$	−3032.93	−0.115590	−0.0577952	+	0.998328i	$0.518407\pi$
$884$	894.035	0.0340154
$885$	0	0
$886$	32502.5	1.23244
$887$	42339.4	1.60273	0.801363	−	0.598179i	$-0.204109\pi$
$887$	42339.4	1.60273	0.801363	+	0.598179i	$0.204109\pi$
$888$	0	0
$889$	−29635.9	−1.11806
$890$	−31718.3	−1.19461
$891$	0	0
$892$	−46045.2	−1.72837
$893$	9874.32	0.370024
$894$	0	0
$895$	6343.09	0.236901
$896$	−9979.99	−0.372107
$897$	0	0
$898$	9241.44	0.343419
$899$	21037.4	0.780464
$900$	0	0
$901$	−4045.63	−0.149589
$902$	−368.619	−0.0136072
$903$	0	0
$904$	−3871.64	−0.142443
$905$	5400.50	0.198363
$906$	0	0
$907$	−20209.8	−0.739861	−0.369931	−	0.929059i	$-0.620619\pi$
$907$	−20209.8	−0.739861	−0.369931	+	0.929059i	$0.620619\pi$
$908$	26048.8	0.952047
$909$	0	0
$910$	4580.56	0.166861
$911$	11254.6	0.409310	0.204655	−	0.978834i	$-0.434393\pi$
$911$	11254.6	0.409310	0.204655	+	0.978834i	$0.434393\pi$
$912$	0	0
$913$	−1246.74	−0.0451929
$914$	−60727.9	−2.19770
$915$	0	0
$916$	15450.4	0.557309
$917$	10143.5	0.365287
$918$	0	0
$919$	−17825.9	−0.639851	−0.319925	−	0.947443i	$-0.603658\pi$
$919$	−17825.9	−0.639851	−0.319925	+	0.947443i	$0.603658\pi$
$920$	−4205.92	−0.150723
$921$	0	0
$922$	51382.4	1.83535
$923$	8022.49	0.286093
$924$	0	0
$925$	−3432.08	−0.121996
$926$	39852.3	1.41428
$927$	0	0
$928$	−19978.9	−0.706725
$929$	−2084.24	−0.0736077	−0.0368039	−	0.999323i	$-0.511718\pi$
$929$	−2084.24	−0.0736077	−0.0368039	+	0.999323i	$0.511718\pi$
$930$	0	0
$931$	−2933.04	−0.103251
$932$	9735.67	0.342170
$933$	0	0
$934$	27645.2	0.968499
$935$	−75.4114	−0.00263767
$936$	0	0
$937$	−5419.51	−0.188952	−0.0944758	−	0.995527i	$-0.530117\pi$
$937$	−5419.51	−0.188952	−0.0944758	+	0.995527i	$0.530117\pi$
$938$	73751.0	2.56722
$939$	0	0
$940$	−8727.93	−0.302844
$941$	−46413.8	−1.60792	−0.803958	−	0.594687i	$-0.797276\pi$
$941$	−46413.8	−1.60792	−0.803958	+	0.594687i	$0.797276\pi$
$942$	0	0
$943$	−11442.2	−0.395131
$944$	−26690.4	−0.920232
$945$	0	0
$946$	2040.31	0.0701230
$947$	18783.1	0.644527	0.322264	−	0.946650i	$-0.395556\pi$
$947$	18783.1	0.644527	0.322264	+	0.946650i	$0.395556\pi$
$948$	0	0
$949$	10169.7	0.347863
$950$	5212.75	0.178025
$951$	0	0
$952$	705.692	0.0240248
$953$	−11796.7	−0.400979	−0.200489	−	0.979696i	$-0.564253\pi$
$953$	−11796.7	−0.400979	−0.200489	+	0.979696i	$0.564253\pi$
$954$	0	0
$955$	1433.57	0.0485752
$956$	−2858.94	−0.0967203
$957$	0	0
$958$	8759.55	0.295416
$959$	−19536.8	−0.657849
$960$	0	0
$961$	45031.6	1.51158
$962$	6281.13	0.210511
$963$	0	0
$964$	32639.3	1.09050
$965$	−23253.5	−0.775707
$966$	0	0
$967$	−10161.7	−0.337932	−0.168966	−	0.985622i	$-0.554043\pi$
$967$	−10161.7	−0.337932	−0.168966	+	0.985622i	$0.554043\pi$
$968$	5208.83	0.172953
$969$	0	0
$970$	23089.3	0.764283
$971$	36614.9	1.21012	0.605060	−	0.796180i	$-0.293149\pi$
$971$	36614.9	1.21012	0.605060	+	0.796180i	$0.293149\pi$
$972$	0	0
$973$	37819.8	1.24609
$974$	26964.6	0.887067
$975$	0	0
$976$	−37491.5	−1.22958
$977$	−50990.9	−1.66975	−0.834874	−	0.550441i	$-0.814459\pi$
$977$	−50990.9	−1.66975	−0.834874	+	0.550441i	$0.814459\pi$
$978$	0	0
$979$	−2585.74	−0.0844132
$980$	2592.52	0.0845052
$981$	0	0
$982$	53252.5	1.73051
$983$	−21783.8	−0.706812	−0.353406	−	0.935470i	$-0.614977\pi$
$983$	−21783.8	−0.706812	−0.353406	+	0.935470i	$0.614977\pi$
$984$	0	0
$985$	8474.08	0.274118
$986$	2845.77	0.0919145
$987$	0	0
$988$	−5038.00	−0.162227
$989$	63332.8	2.03626
$990$	0	0
$991$	−50955.6	−1.63336	−0.816679	−	0.577092i	$-0.804187\pi$
$991$	−50955.6	−1.63336	−0.816679	+	0.577092i	$0.804187\pi$
$992$	−71057.9	−2.27429
$993$	0	0
$994$	59518.2	1.89920
$995$	−16248.8	−0.517711
$996$	0	0
$997$	−19162.8	−0.608719	−0.304360	−	0.952557i	$-0.598442\pi$
$997$	−19162.8	−0.608719	−0.304360	+	0.952557i	$0.598442\pi$
$998$	−21083.5	−0.668724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.l.1.1	yes	6
3.2	odd	2		405.4.a.k.1.6	✓	6
5.4	even	2		2025.4.a.y.1.6		6
9.2	odd	6		405.4.e.x.271.1		12
9.4	even	3		405.4.e.w.136.6		12
9.5	odd	6		405.4.e.x.136.1		12
9.7	even	3		405.4.e.w.271.6		12
15.14	odd	2		2025.4.a.z.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.6	✓	6	3.2	odd	2
405.4.a.l.1.1	yes	6	1.1	even	1	trivial
405.4.e.w.136.6		12	9.4	even	3
405.4.e.w.271.6		12	9.7	even	3
405.4.e.x.136.1		12	9.5	odd	6
405.4.e.x.271.1		12	9.2	odd	6
2025.4.a.y.1.6		6	5.4	even	2
2025.4.a.z.1.1		6	15.14	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q-4.11734 q^{2} +8.95250 q^{4} +5.00000 q^{5} -20.0229 q^{7} -3.92177 q^{8} +O(q^{10})\)	\(q-4.11734 q^{2} +8.95250 q^{4} +5.00000 q^{5} -20.0229 q^{7} -3.92177 q^{8} -20.5867 q^{10} -1.67827 q^{11} +11.1123 q^{13} +82.4412 q^{14} -55.4727 q^{16} +8.98682 q^{17} -50.6419 q^{19} +44.7625 q^{20} +6.91000 q^{22} +214.491 q^{23} +25.0000 q^{25} -45.7531 q^{26} -179.255 q^{28} -76.9088 q^{29} -273.537 q^{31} +259.774 q^{32} -37.0018 q^{34} -100.115 q^{35} -137.283 q^{37} +208.510 q^{38} -19.6088 q^{40} -53.3457 q^{41} +295.270 q^{43} -15.0247 q^{44} -883.134 q^{46} -194.983 q^{47} +57.9173 q^{49} -102.934 q^{50} +99.4829 q^{52} -450.173 q^{53} -8.39133 q^{55} +78.5252 q^{56} +316.660 q^{58} +481.145 q^{59} +675.854 q^{61} +1126.25 q^{62} -625.798 q^{64} +55.5615 q^{65} +894.589 q^{67} +80.4545 q^{68} +412.206 q^{70} +721.947 q^{71} +915.175 q^{73} +565.241 q^{74} -453.372 q^{76} +33.6038 q^{77} +59.6642 q^{79} -277.364 q^{80} +219.643 q^{82} +742.875 q^{83} +44.9341 q^{85} -1215.73 q^{86} +6.58177 q^{88} +1540.72 q^{89} -222.501 q^{91} +1920.23 q^{92} +802.812 q^{94} -253.210 q^{95} -1121.57 q^{97} -238.465 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 34 q^{4} + 30 q^{5} + 40 q^{7} + 66 q^{8}+O(q^{10}) \) 6 * q + 4 * q^2 + 34 * q^4 + 30 * q^5 + 40 * q^7 + 66 * q^8	\( 6 q + 4 q^{2} + 34 q^{4} + 30 q^{5} + 40 q^{7} + 66 q^{8} + 20 q^{10} + 88 q^{11} + 20 q^{13} + 180 q^{14} + 58 q^{16} + 124 q^{17} - 46 q^{19} + 170 q^{20} - 74 q^{22} + 210 q^{23} + 150 q^{25} + 4 q^{26} + 352 q^{28} + 296 q^{29} - 104 q^{31} + 722 q^{32} - 428 q^{34} + 200 q^{35} - 204 q^{37} - 20 q^{38} + 330 q^{40} + 344 q^{41} + 512 q^{43} + 716 q^{44} - 186 q^{46} + 238 q^{47} + 68 q^{49} + 100 q^{50} - 468 q^{52} + 850 q^{53} + 440 q^{55} + 2316 q^{56} + 890 q^{58} + 1840 q^{59} - 364 q^{61} + 1038 q^{62} - 990 q^{64} + 100 q^{65} + 88 q^{67} + 236 q^{68} + 900 q^{70} + 1364 q^{71} + 836 q^{73} + 1316 q^{74} - 2106 q^{76} + 840 q^{77} - 680 q^{79} + 290 q^{80} + 1742 q^{82} + 2148 q^{83} + 620 q^{85} + 2872 q^{86} + 1296 q^{88} + 3000 q^{89} - 3058 q^{91} + 1002 q^{92} - 3662 q^{94} - 230 q^{95} - 612 q^{97} + 1982 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 + 34 * q^4 + 30 * q^5 + 40 * q^7 + 66 * q^8 + 20 * q^10 + 88 * q^11 + 20 * q^13 + 180 * q^14 + 58 * q^16 + 124 * q^17 - 46 * q^19 + 170 * q^20 - 74 * q^22 + 210 * q^23 + 150 * q^25 + 4 * q^26 + 352 * q^28 + 296 * q^29 - 104 * q^31 + 722 * q^32 - 428 * q^34 + 200 * q^35 - 204 * q^37 - 20 * q^38 + 330 * q^40 + 344 * q^41 + 512 * q^43 + 716 * q^44 - 186 * q^46 + 238 * q^47 + 68 * q^49 + 100 * q^50 - 468 * q^52 + 850 * q^53 + 440 * q^55 + 2316 * q^56 + 890 * q^58 + 1840 * q^59 - 364 * q^61 + 1038 * q^62 - 990 * q^64 + 100 * q^65 + 88 * q^67 + 236 * q^68 + 900 * q^70 + 1364 * q^71 + 836 * q^73 + 1316 * q^74 - 2106 * q^76 + 840 * q^77 - 680 * q^79 + 290 * q^80 + 1742 * q^82 + 2148 * q^83 + 620 * q^85 + 2872 * q^86 + 1296 * q^88 + 3000 * q^89 - 3058 * q^91 + 1002 * q^92 - 3662 * q^94 - 230 * q^95 - 612 * q^97 + 1982 * q^98

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