LMFDB - Embedded newform 405.4.a.n.1.5

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 $ x^7 - 2x^6 - 44x^5 + 74x^4 + 479x^3 - 460x^2 - 1200x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.19444$ 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)

$f(q)$	$=$	$q+2.19444 q^{2} -3.18442 q^{4} +5.00000 q^{5} +2.76605 q^{7} -24.5436 q^{8} +O(q^{10})$	\(q+2.19444 q^{2} -3.18442 q^{4} +5.00000 q^{5} +2.76605 q^{7} -24.5436 q^{8} +10.9722 q^{10} +52.6590 q^{11} +20.4535 q^{13} +6.06994 q^{14} -28.3842 q^{16} -3.66084 q^{17} -95.6705 q^{19} -15.9221 q^{20} +115.557 q^{22} +89.8411 q^{23} +25.0000 q^{25} +44.8840 q^{26} -8.80825 q^{28} +227.780 q^{29} +279.139 q^{31} +134.061 q^{32} -8.03351 q^{34} +13.8302 q^{35} +273.725 q^{37} -209.944 q^{38} -122.718 q^{40} -64.8647 q^{41} +418.762 q^{43} -167.688 q^{44} +197.151 q^{46} -138.709 q^{47} -335.349 q^{49} +54.8611 q^{50} -65.1323 q^{52} +197.063 q^{53} +263.295 q^{55} -67.8887 q^{56} +499.852 q^{58} -741.103 q^{59} +488.468 q^{61} +612.554 q^{62} +521.263 q^{64} +102.267 q^{65} -411.468 q^{67} +11.6576 q^{68} +30.3497 q^{70} +310.343 q^{71} -51.0260 q^{73} +600.675 q^{74} +304.655 q^{76} +145.657 q^{77} +1208.00 q^{79} -141.921 q^{80} -142.342 q^{82} -905.221 q^{83} -18.3042 q^{85} +918.949 q^{86} -1292.44 q^{88} -663.633 q^{89} +56.5752 q^{91} -286.092 q^{92} -304.389 q^{94} -478.353 q^{95} -725.336 q^{97} -735.905 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 36 * q^4 + 35 * q^5 + 22 * q^7 + 18 * q^8	$ 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + 10 q^{10} + 23 q^{11} + 96 q^{13} - 21 q^{14} + 324 q^{16} + 161 q^{17} + 279 q^{19} + 180 q^{20} + 311 q^{22} + 96 q^{23} + 175 q^{25} - 358 q^{26} + 337 q^{28} - 296 q^{29} + 244 q^{31} - 314 q^{32} + 125 q^{34} + 110 q^{35} + 404 q^{37} + 305 q^{38} + 90 q^{40} - 47 q^{41} + 525 q^{43} + 55 q^{44} + 717 q^{46} + 164 q^{47} + 1225 q^{49} + 50 q^{50} + 1682 q^{52} + 506 q^{53} + 115 q^{55} - 981 q^{56} + 1183 q^{58} - 85 q^{59} + 828 q^{61} - 786 q^{62} + 2236 q^{64} + 480 q^{65} + 1093 q^{67} + 2473 q^{68} - 105 q^{70} + 328 q^{71} + 2085 q^{73} - 1316 q^{74} + 2789 q^{76} + 24 q^{77} + 2110 q^{79} + 1620 q^{80} - 62 q^{82} + 1290 q^{83} + 805 q^{85} - 2569 q^{86} + 2271 q^{88} - 3048 q^{89} + 3338 q^{91} + 2763 q^{92} - 517 q^{94} + 1395 q^{95} + 1787 q^{97} + 1279 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 36 * q^4 + 35 * q^5 + 22 * q^7 + 18 * q^8 + 10 * q^10 + 23 * q^11 + 96 * q^13 - 21 * q^14 + 324 * q^16 + 161 * q^17 + 279 * q^19 + 180 * q^20 + 311 * q^22 + 96 * q^23 + 175 * q^25 - 358 * q^26 + 337 * q^28 - 296 * q^29 + 244 * q^31 - 314 * q^32 + 125 * q^34 + 110 * q^35 + 404 * q^37 + 305 * q^38 + 90 * q^40 - 47 * q^41 + 525 * q^43 + 55 * q^44 + 717 * q^46 + 164 * q^47 + 1225 * q^49 + 50 * q^50 + 1682 * q^52 + 506 * q^53 + 115 * q^55 - 981 * q^56 + 1183 * q^58 - 85 * q^59 + 828 * q^61 - 786 * q^62 + 2236 * q^64 + 480 * q^65 + 1093 * q^67 + 2473 * q^68 - 105 * q^70 + 328 * q^71 + 2085 * q^73 - 1316 * q^74 + 2789 * q^76 + 24 * q^77 + 2110 * q^79 + 1620 * q^80 - 62 * q^82 + 1290 * q^83 + 805 * q^85 - 2569 * q^86 + 2271 * q^88 - 3048 * q^89 + 3338 * q^91 + 2763 * q^92 - 517 * q^94 + 1395 * q^95 + 1787 * q^97 + 1279 * 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 $

$2$	2.19444	0.775853	0.387927	−	0.921690i	$-0.373191\pi$
$2$	2.19444	0.775853	0.387927	+	0.921690i	$0.373191\pi$
$3$	0	0
$3$	0	0
$4$	−3.18442	−0.398052
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	2.76605	0.149353	0.0746763	−	0.997208i	$-0.476208\pi$
$7$	2.76605	0.149353	0.0746763	+	0.997208i	$0.476208\pi$
$8$	−24.5436	−1.08468
$9$	0	0
$10$	10.9722	0.346972
$11$	52.6590	1.44339	0.721694	−	0.692212i	$-0.243364\pi$
$11$	52.6590	1.44339	0.721694	+	0.692212i	$0.243364\pi$
$12$	0	0
$13$	20.4535	0.436367	0.218183	−	0.975908i	$-0.429987\pi$
$13$	20.4535	0.436367	0.218183	+	0.975908i	$0.429987\pi$
$14$	6.06994	0.115876
$15$	0	0
$16$	−28.3842	−0.443503
$17$	−3.66084	−0.0522285	−0.0261142	−	0.999659i	$-0.508313\pi$
$17$	−3.66084	−0.0522285	−0.0261142	+	0.999659i	$0.508313\pi$
$18$	0	0
$19$	−95.6705	−1.15517	−0.577587	−	0.816329i	$-0.696006\pi$
$19$	−95.6705	−1.15517	−0.577587	+	0.816329i	$0.696006\pi$
$20$	−15.9221	−0.178014
$21$	0	0
$22$	115.557	1.11986
$23$	89.8411	0.814486	0.407243	−	0.913320i	$-0.366490\pi$
$23$	89.8411	0.814486	0.407243	+	0.913320i	$0.366490\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	44.8840	0.338556
$27$	0	0
$28$	−8.80825	−0.0594501
$29$	227.780	1.45854	0.729272	−	0.684224i	$-0.239859\pi$
$29$	227.780	1.45854	0.729272	+	0.684224i	$0.239859\pi$
$30$	0	0
$31$	279.139	1.61725	0.808625	−	0.588324i	$-0.200212\pi$
$31$	279.139	1.61725	0.808625	+	0.588324i	$0.200212\pi$
$32$	134.061	0.740590
$33$	0	0
$34$	−8.03351	−0.0405216
$35$	13.8302	0.0667925
$36$	0	0
$37$	273.725	1.21622	0.608110	−	0.793852i	$-0.291928\pi$
$37$	273.725	1.21622	0.608110	+	0.793852i	$0.291928\pi$
$38$	−209.944	−0.896246
$39$	0	0
$40$	−122.718	−0.485085
$41$	−64.8647	−0.247077	−0.123539	−	0.992340i	$-0.539424\pi$
$41$	−64.8647	−0.247077	−0.123539	+	0.992340i	$0.539424\pi$
$42$	0	0
$43$	418.762	1.48513	0.742565	−	0.669774i	$-0.233609\pi$
$43$	418.762	1.48513	0.742565	+	0.669774i	$0.233609\pi$
$44$	−167.688	−0.574544
$45$	0	0
$46$	197.151	0.631921
$47$	−138.709	−0.430484	−0.215242	−	0.976561i	$-0.569054\pi$
$47$	−138.709	−0.430484	−0.215242	+	0.976561i	$0.569054\pi$
$48$	0	0
$49$	−335.349	−0.977694
$50$	54.8611	0.155171
$51$	0	0
$52$	−65.1323	−0.173697
$53$	197.063	0.510730	0.255365	−	0.966845i	$-0.417804\pi$
$53$	197.063	0.510730	0.255365	+	0.966845i	$0.417804\pi$
$54$	0	0
$55$	263.295	0.645503
$56$	−67.8887	−0.162000
$57$	0	0
$58$	499.852	1.13162
$59$	−741.103	−1.63531	−0.817656	−	0.575707i	$-0.804727\pi$
$59$	−741.103	−1.63531	−0.817656	+	0.575707i	$0.804727\pi$
$60$	0	0
$61$	488.468	1.02528	0.512639	−	0.858604i	$-0.328668\pi$
$61$	488.468	1.02528	0.512639	+	0.858604i	$0.328668\pi$
$62$	612.554	1.25475
$63$	0	0
$64$	521.263	1.01809
$65$	102.267	0.195149
$66$	0	0
$67$	−411.468	−0.750280	−0.375140	−	0.926968i	$-0.622405\pi$
$67$	−411.468	−0.750280	−0.375140	+	0.926968i	$0.622405\pi$
$68$	11.6576	0.0207896
$69$	0	0
$70$	30.3497	0.0518212
$71$	310.343	0.518746	0.259373	−	0.965777i	$-0.416484\pi$
$71$	310.343	0.518746	0.259373	+	0.965777i	$0.416484\pi$
$72$	0	0
$73$	−51.0260	−0.0818101	−0.0409051	−	0.999163i	$-0.513024\pi$
$73$	−51.0260	−0.0818101	−0.0409051	+	0.999163i	$0.513024\pi$
$74$	600.675	0.943609
$75$	0	0
$76$	304.655	0.459820
$77$	145.657	0.215574
$78$	0	0
$79$	1208.00	1.72039	0.860193	−	0.509969i	$-0.170343\pi$
$79$	1208.00	1.72039	0.860193	+	0.509969i	$0.170343\pi$
$80$	−141.921	−0.198340
$81$	0	0
$82$	−142.342	−0.191695
$83$	−905.221	−1.19712	−0.598560	−	0.801078i	$-0.704260\pi$
$83$	−905.221	−1.19712	−0.598560	+	0.801078i	$0.704260\pi$
$84$	0	0
$85$	−18.3042	−0.0233573
$86$	918.949	1.15224
$87$	0	0
$88$	−1292.44	−1.56562
$89$	−663.633	−0.790393	−0.395197	−	0.918597i	$-0.629324\pi$
$89$	−663.633	−0.790393	−0.395197	+	0.918597i	$0.629324\pi$
$90$	0	0
$91$	56.5752	0.0651725
$92$	−286.092	−0.324208
$93$	0	0
$94$	−304.389	−0.333992
$95$	−478.353	−0.516610
$96$	0	0
$97$	−725.336	−0.759244	−0.379622	−	0.925142i	$-0.623946\pi$
$97$	−725.336	−0.759244	−0.379622	+	0.925142i	$0.623946\pi$
$98$	−735.905	−0.758547
$99$	0	0
$100$	−79.6104	−0.0796104
$101$	−977.782	−0.963296	−0.481648	−	0.876365i	$-0.659962\pi$
$101$	−977.782	−0.963296	−0.481648	+	0.876365i	$0.659962\pi$
$102$	0	0
$103$	1587.70	1.51885	0.759423	−	0.650597i	$-0.225481\pi$
$103$	1587.70	1.51885	0.759423	+	0.650597i	$0.225481\pi$
$104$	−502.001	−0.473319
$105$	0	0
$106$	432.444	0.396252
$107$	−897.731	−0.811093	−0.405546	−	0.914074i	$-0.632919\pi$
$107$	−897.731	−0.811093	−0.405546	+	0.914074i	$0.632919\pi$
$108$	0	0
$109$	855.492	0.751754	0.375877	−	0.926669i	$-0.377341\pi$
$109$	855.492	0.751754	0.375877	+	0.926669i	$0.377341\pi$
$110$	577.786	0.500816
$111$	0	0
$112$	−78.5120	−0.0662383
$113$	−910.241	−0.757772	−0.378886	−	0.925443i	$-0.623693\pi$
$113$	−910.241	−0.757772	−0.378886	+	0.925443i	$0.623693\pi$
$114$	0	0
$115$	449.206	0.364249
$116$	−725.348	−0.580576
$117$	0	0
$118$	−1626.31	−1.26876
$119$	−10.1261	−0.00780046
$120$	0	0
$121$	1441.97	1.08337
$122$	1071.92	0.795465
$123$	0	0
$124$	−888.893	−0.643750
$125$	125.000	0.0894427
$126$	0	0
$127$	−2038.25	−1.42414	−0.712068	−	0.702111i	$-0.752241\pi$
$127$	−2038.25	−1.42414	−0.712068	+	0.702111i	$0.752241\pi$
$128$	71.3936	0.0492997
$129$	0	0
$130$	224.420	0.151407
$131$	−235.645	−0.157163	−0.0785817	−	0.996908i	$-0.525039\pi$
$131$	−235.645	−0.157163	−0.0785817	+	0.996908i	$0.525039\pi$
$132$	0	0
$133$	−264.629	−0.172528
$134$	−902.942	−0.582107
$135$	0	0
$136$	89.8501	0.0566513
$137$	2972.89	1.85395	0.926975	−	0.375123i	$-0.122400\pi$
$137$	2972.89	1.85395	0.926975	+	0.375123i	$0.122400\pi$
$138$	0	0
$139$	−2094.63	−1.27816	−0.639080	−	0.769140i	$-0.720685\pi$
$139$	−2094.63	−1.27816	−0.639080	+	0.769140i	$0.720685\pi$
$140$	−44.0412	−0.0265869
$141$	0	0
$142$	681.031	0.402471
$143$	1077.06	0.629847
$144$	0	0
$145$	1138.90	0.652281
$146$	−111.974	−0.0634726
$147$	0	0
$148$	−871.655	−0.484119
$149$	−273.937	−0.150616	−0.0753081	−	0.997160i	$-0.523994\pi$
$149$	−273.937	−0.150616	−0.0753081	+	0.997160i	$0.523994\pi$
$150$	0	0
$151$	−936.844	−0.504896	−0.252448	−	0.967610i	$-0.581236\pi$
$151$	−936.844	−0.504896	−0.252448	+	0.967610i	$0.581236\pi$
$152$	2348.10	1.25300
$153$	0	0
$154$	319.637	0.167254
$155$	1395.69	0.723256
$156$	0	0
$157$	398.436	0.202539	0.101269	−	0.994859i	$-0.467710\pi$
$157$	398.436	0.202539	0.101269	+	0.994859i	$0.467710\pi$
$158$	2650.89	1.33477
$159$	0	0
$160$	670.306	0.331202
$161$	248.505	0.121646
$162$	0	0
$163$	478.154	0.229766	0.114883	−	0.993379i	$-0.463351\pi$
$163$	478.154	0.229766	0.114883	+	0.993379i	$0.463351\pi$
$164$	206.556	0.0983495
$165$	0	0
$166$	−1986.46	−0.928789
$167$	340.362	0.157713	0.0788564	−	0.996886i	$-0.474873\pi$
$167$	340.362	0.157713	0.0788564	+	0.996886i	$0.474873\pi$
$168$	0	0
$169$	−1778.66	−0.809584
$170$	−40.1675	−0.0181218
$171$	0	0
$172$	−1333.51	−0.591159
$173$	−3776.07	−1.65948	−0.829738	−	0.558154i	$-0.811510\pi$
$173$	−3776.07	−1.65948	−0.829738	+	0.558154i	$0.811510\pi$
$174$	0	0
$175$	69.1512	0.0298705
$176$	−1494.68	−0.640147
$177$	0	0
$178$	−1456.31	−0.613229
$179$	−186.652	−0.0779385	−0.0389693	−	0.999240i	$-0.512407\pi$
$179$	−186.652	−0.0779385	−0.0389693	+	0.999240i	$0.512407\pi$
$180$	0	0
$181$	1438.75	0.590837	0.295418	−	0.955368i	$-0.404541\pi$
$181$	1438.75	0.590837	0.295418	+	0.955368i	$0.404541\pi$
$182$	124.151	0.0505643
$183$	0	0
$184$	−2205.02	−0.883459
$185$	1368.63	0.543911
$186$	0	0
$187$	−192.776	−0.0753860
$188$	441.706	0.171355
$189$	0	0
$190$	−1049.72	−0.400813
$191$	390.435	0.147910	0.0739552	−	0.997262i	$-0.476438\pi$
$191$	390.435	0.147910	0.0739552	+	0.997262i	$0.476438\pi$
$192$	0	0
$193$	−3915.05	−1.46016	−0.730081	−	0.683361i	$-0.760518\pi$
$193$	−3915.05	−1.46016	−0.730081	+	0.683361i	$0.760518\pi$
$194$	−1591.71	−0.589062
$195$	0	0
$196$	1067.89	0.389173
$197$	892.680	0.322847	0.161423	−	0.986885i	$-0.448392\pi$
$197$	892.680	0.322847	0.161423	+	0.986885i	$0.448392\pi$
$198$	0	0
$199$	−2770.50	−0.986913	−0.493457	−	0.869770i	$-0.664267\pi$
$199$	−2770.50	−0.986913	−0.493457	+	0.869770i	$0.664267\pi$
$200$	−613.589	−0.216937
$201$	0	0
$202$	−2145.69	−0.747376
$203$	630.052	0.217837
$204$	0	0
$205$	−324.323	−0.110496
$206$	3484.13	1.17840
$207$	0	0
$208$	−580.554	−0.193530
$209$	−5037.91	−1.66737
$210$	0	0
$211$	4582.51	1.49513	0.747566	−	0.664187i	$-0.231222\pi$
$211$	4582.51	1.49513	0.747566	+	0.664187i	$0.231222\pi$
$212$	−627.531	−0.203297
$213$	0	0
$214$	−1970.02	−0.629289
$215$	2093.81	0.664170
$216$	0	0
$217$	772.111	0.241541
$218$	1877.33	0.583251
$219$	0	0
$220$	−838.440	−0.256944
$221$	−74.8768	−0.0227908
$222$	0	0
$223$	4095.34	1.22979	0.614897	−	0.788608i	$-0.289198\pi$
$223$	4095.34	1.22979	0.614897	+	0.788608i	$0.289198\pi$
$224$	370.820	0.110609
$225$	0	0
$226$	−1997.47	−0.587920
$227$	3869.35	1.13136	0.565678	−	0.824626i	$-0.308614\pi$
$227$	3869.35	1.13136	0.565678	+	0.824626i	$0.308614\pi$
$228$	0	0
$229$	1236.01	0.356672	0.178336	−	0.983970i	$-0.442929\pi$
$229$	1236.01	0.356672	0.178336	+	0.983970i	$0.442929\pi$
$230$	985.757	0.282604
$231$	0	0
$232$	−5590.55	−1.58206
$233$	−2207.05	−0.620552	−0.310276	−	0.950647i	$-0.600421\pi$
$233$	−2207.05	−0.620552	−0.310276	+	0.950647i	$0.600421\pi$
$234$	0	0
$235$	−693.544	−0.192518
$236$	2359.98	0.650939
$237$	0	0
$238$	−22.2211	−0.00605201
$239$	−876.648	−0.237262	−0.118631	−	0.992938i	$-0.537851\pi$
$239$	−876.648	−0.237262	−0.118631	+	0.992938i	$0.537851\pi$
$240$	0	0
$241$	−477.861	−0.127725	−0.0638626	−	0.997959i	$-0.520342\pi$
$241$	−477.861	−0.127725	−0.0638626	+	0.997959i	$0.520342\pi$
$242$	3164.32	0.840537
$243$	0	0
$244$	−1555.49	−0.408114
$245$	−1676.74	−0.437238
$246$	0	0
$247$	−1956.79	−0.504080
$248$	−6851.06	−1.75420
$249$	0	0
$250$	274.306	0.0693944
$251$	−6892.28	−1.73322	−0.866608	−	0.498990i	$-0.833704\pi$
$251$	−6892.28	−1.73322	−0.866608	+	0.498990i	$0.833704\pi$
$252$	0	0
$253$	4730.94	1.17562
$254$	−4472.82	−1.10492
$255$	0	0
$256$	−4013.43	−0.979842
$257$	7256.71	1.76133	0.880664	−	0.473742i	$-0.157097\pi$
$257$	7256.71	1.76133	0.880664	+	0.473742i	$0.157097\pi$
$258$	0	0
$259$	757.138	0.181646
$260$	−325.661	−0.0776795
$261$	0	0
$262$	−517.110	−0.121936
$263$	6317.55	1.48121	0.740603	−	0.671943i	$-0.234540\pi$
$263$	6317.55	1.48121	0.740603	+	0.671943i	$0.234540\pi$
$264$	0	0
$265$	985.316	0.228406
$266$	−580.714	−0.133857
$267$	0	0
$268$	1310.28	0.298650
$269$	−5746.22	−1.30243	−0.651214	−	0.758894i	$-0.725740\pi$
$269$	−5746.22	−1.30243	−0.651214	+	0.758894i	$0.725740\pi$
$270$	0	0
$271$	4925.20	1.10400	0.552001	−	0.833844i	$-0.313865\pi$
$271$	4925.20	1.10400	0.552001	+	0.833844i	$0.313865\pi$
$272$	103.910	0.0231635
$273$	0	0
$274$	6523.84	1.43839
$275$	1316.47	0.288678
$276$	0	0
$277$	2325.88	0.504508	0.252254	−	0.967661i	$-0.418828\pi$
$277$	2325.88	0.504508	0.252254	+	0.967661i	$0.418828\pi$
$278$	−4596.55	−0.991665
$279$	0	0
$280$	−339.444	−0.0724487
$281$	−3283.42	−0.697055	−0.348527	−	0.937299i	$-0.613318\pi$
$281$	−3283.42	−0.697055	−0.348527	+	0.937299i	$0.613318\pi$
$282$	0	0
$283$	2014.21	0.423083	0.211541	−	0.977369i	$-0.432152\pi$
$283$	2014.21	0.423083	0.211541	+	0.977369i	$0.432152\pi$
$284$	−988.262	−0.206488
$285$	0	0
$286$	2363.54	0.488669
$287$	−179.419	−0.0369016
$288$	0	0
$289$	−4899.60	−0.997272
$290$	2499.26	0.506074
$291$	0	0
$292$	162.488	0.0325647
$293$	−480.697	−0.0958450	−0.0479225	−	0.998851i	$-0.515260\pi$
$293$	−480.697	−0.0958450	−0.0479225	+	0.998851i	$0.515260\pi$
$294$	0	0
$295$	−3705.52	−0.731334
$296$	−6718.20	−1.31921
$297$	0	0
$298$	−601.140	−0.116856
$299$	1837.56	0.355414
$300$	0	0
$301$	1158.32	0.221808
$302$	−2055.85	−0.391725
$303$	0	0
$304$	2715.53	0.512323
$305$	2442.34	0.458518
$306$	0	0
$307$	3222.21	0.599026	0.299513	−	0.954092i	$-0.403176\pi$
$307$	3222.21	0.599026	0.299513	+	0.954092i	$0.403176\pi$
$308$	−463.833	−0.0858096
$309$	0	0
$310$	3062.77	0.561141
$311$	−2414.01	−0.440147	−0.220074	−	0.975483i	$-0.570630\pi$
$311$	−2414.01	−0.440147	−0.220074	+	0.975483i	$0.570630\pi$
$312$	0	0
$313$	−2506.94	−0.452717	−0.226359	−	0.974044i	$-0.572682\pi$
$313$	−2506.94	−0.452717	−0.226359	+	0.974044i	$0.572682\pi$
$314$	874.344	0.157140
$315$	0	0
$316$	−3846.77	−0.684803
$317$	−3707.12	−0.656822	−0.328411	−	0.944535i	$-0.606513\pi$
$317$	−3707.12	−0.656822	−0.328411	+	0.944535i	$0.606513\pi$
$318$	0	0
$319$	11994.7	2.10525
$320$	2606.31	0.455304
$321$	0	0
$322$	545.330	0.0943791
$323$	350.234	0.0603330
$324$	0	0
$325$	511.336	0.0872733
$326$	1049.28	0.178265
$327$	0	0
$328$	1592.01	0.268000
$329$	−383.675	−0.0642939
$330$	0	0
$331$	2553.45	0.424018	0.212009	−	0.977268i	$-0.431999\pi$
$331$	2553.45	0.424018	0.212009	+	0.977268i	$0.431999\pi$
$332$	2882.60	0.476516
$333$	0	0
$334$	746.906	0.122362
$335$	−2057.34	−0.335535
$336$	0	0
$337$	3552.89	0.574297	0.287148	−	0.957886i	$-0.407293\pi$
$337$	3552.89	0.574297	0.287148	+	0.957886i	$0.407293\pi$
$338$	−3903.16	−0.628118
$339$	0	0
$340$	58.2882	0.00929741
$341$	14699.2	2.33432
$342$	0	0
$343$	−1876.35	−0.295374
$344$	−10277.9	−1.61089
$345$	0	0
$346$	−8286.37	−1.28751
$347$	7768.02	1.20176	0.600878	−	0.799341i	$-0.294818\pi$
$347$	7768.02	1.20176	0.600878	+	0.799341i	$0.294818\pi$
$348$	0	0
$349$	696.007	0.106752	0.0533759	−	0.998574i	$-0.483002\pi$
$349$	696.007	0.106752	0.0533759	+	0.998574i	$0.483002\pi$
$350$	151.748	0.0231751
$351$	0	0
$352$	7059.52	1.06896
$353$	5451.87	0.822022	0.411011	−	0.911630i	$-0.365176\pi$
$353$	5451.87	0.822022	0.411011	+	0.911630i	$0.365176\pi$
$354$	0	0
$355$	1551.72	0.231990
$356$	2113.28	0.314618
$357$	0	0
$358$	−409.596	−0.0604688
$359$	−4036.41	−0.593408	−0.296704	−	0.954969i	$-0.595887\pi$
$359$	−4036.41	−0.593408	−0.296704	+	0.954969i	$0.595887\pi$
$360$	0	0
$361$	2293.85	0.334429
$362$	3157.26	0.458403
$363$	0	0
$364$	−180.159	−0.0259420
$365$	−255.130	−0.0365866
$366$	0	0
$367$	11239.9	1.59869	0.799345	−	0.600872i	$-0.205180\pi$
$367$	11239.9	1.59869	0.799345	+	0.600872i	$0.205180\pi$
$368$	−2550.07	−0.361227
$369$	0	0
$370$	3003.38	0.421995
$371$	545.086	0.0762789
$372$	0	0
$373$	−6320.72	−0.877411	−0.438706	−	0.898631i	$-0.644563\pi$
$373$	−6320.72	−0.877411	−0.438706	+	0.898631i	$0.644563\pi$
$374$	−423.036	−0.0584885
$375$	0	0
$376$	3404.41	0.466939
$377$	4658.90	0.636460
$378$	0	0
$379$	9325.49	1.26390	0.631950	−	0.775009i	$-0.282255\pi$
$379$	9325.49	1.26390	0.631950	+	0.775009i	$0.282255\pi$
$380$	1523.27	0.205638
$381$	0	0
$382$	856.788	0.114757
$383$	−11148.0	−1.48730	−0.743652	−	0.668567i	$-0.766908\pi$
$383$	−11148.0	−1.48730	−0.743652	+	0.668567i	$0.766908\pi$
$384$	0	0
$385$	728.286	0.0964076
$386$	−8591.35	−1.13287
$387$	0	0
$388$	2309.77	0.302219
$389$	−6570.81	−0.856436	−0.428218	−	0.903676i	$-0.640858\pi$
$389$	−6570.81	−0.856436	−0.428218	+	0.903676i	$0.640858\pi$
$390$	0	0
$391$	−328.894	−0.0425394
$392$	8230.66	1.06049
$393$	0	0
$394$	1958.94	0.250482
$395$	6039.99	0.769380
$396$	0	0
$397$	3969.33	0.501800	0.250900	−	0.968013i	$-0.419273\pi$
$397$	3969.33	0.501800	0.250900	+	0.968013i	$0.419273\pi$
$398$	−6079.71	−0.765700
$399$	0	0
$400$	−709.604	−0.0887005
$401$	−4374.70	−0.544793	−0.272396	−	0.962185i	$-0.587816\pi$
$401$	−4374.70	−0.544793	−0.272396	+	0.962185i	$0.587816\pi$
$402$	0	0
$403$	5709.35	0.705714
$404$	3113.66	0.383442
$405$	0	0
$406$	1382.61	0.169010
$407$	14414.1	1.75548
$408$	0	0
$409$	−8872.33	−1.07264	−0.536318	−	0.844016i	$-0.680185\pi$
$409$	−8872.33	−1.07264	−0.536318	+	0.844016i	$0.680185\pi$
$410$	−711.709	−0.0857288
$411$	0	0
$412$	−5055.91	−0.604580
$413$	−2049.93	−0.244238
$414$	0	0
$415$	−4526.11	−0.535368
$416$	2742.01	0.323169
$417$	0	0
$418$	−11055.4	−1.29363
$419$	1014.17	0.118247	0.0591236	−	0.998251i	$-0.481169\pi$
$419$	1014.17	0.118247	0.0591236	+	0.998251i	$0.481169\pi$
$420$	0	0
$421$	−14893.1	−1.72410	−0.862051	−	0.506822i	$-0.830820\pi$
$421$	−14893.1	−1.72410	−0.862051	+	0.506822i	$0.830820\pi$
$422$	10056.1	1.16000
$423$	0	0
$424$	−4836.63	−0.553980
$425$	−91.5210	−0.0104457
$426$	0	0
$427$	1351.13	0.153128
$428$	2858.75	0.322857
$429$	0	0
$430$	4594.75	0.515298
$431$	4363.90	0.487707	0.243853	−	0.969812i	$-0.421588\pi$
$431$	4363.90	0.487707	0.243853	+	0.969812i	$0.421588\pi$
$432$	0	0
$433$	9301.59	1.03235	0.516173	−	0.856484i	$-0.327356\pi$
$433$	9301.59	1.03235	0.516173	+	0.856484i	$0.327356\pi$
$434$	1694.35	0.187400
$435$	0	0
$436$	−2724.24	−0.299237
$437$	−8595.15	−0.940873
$438$	0	0
$439$	1520.15	0.165269	0.0826343	−	0.996580i	$-0.473667\pi$
$439$	1520.15	0.165269	0.0826343	+	0.996580i	$0.473667\pi$
$440$	−6462.20	−0.700166
$441$	0	0
$442$	−164.313	−0.0176823
$443$	−4462.48	−0.478598	−0.239299	−	0.970946i	$-0.576918\pi$
$443$	−4462.48	−0.478598	−0.239299	+	0.970946i	$0.576918\pi$
$444$	0	0
$445$	−3318.17	−0.353475
$446$	8986.98	0.954139
$447$	0	0
$448$	1441.84	0.152055
$449$	5371.66	0.564598	0.282299	−	0.959326i	$-0.408903\pi$
$449$	5371.66	0.564598	0.282299	+	0.959326i	$0.408903\pi$
$450$	0	0
$451$	−3415.71	−0.356628
$452$	2898.59	0.301633
$453$	0	0
$454$	8491.08	0.877767
$455$	282.876	0.0291460
$456$	0	0
$457$	15525.4	1.58917	0.794583	−	0.607155i	$-0.207689\pi$
$457$	15525.4	1.58917	0.794583	+	0.607155i	$0.207689\pi$
$458$	2712.36	0.276725
$459$	0	0
$460$	−1430.46	−0.144990
$461$	56.4179	0.00569988	0.00284994	−	0.999996i	$-0.499093\pi$
$461$	56.4179	0.00569988	0.00284994	+	0.999996i	$0.499093\pi$
$462$	0	0
$463$	13373.8	1.34240	0.671201	−	0.741275i	$-0.265779\pi$
$463$	13373.8	1.34240	0.671201	+	0.741275i	$0.265779\pi$
$464$	−6465.36	−0.646868
$465$	0	0
$466$	−4843.25	−0.481457
$467$	−2677.46	−0.265306	−0.132653	−	0.991163i	$-0.542350\pi$
$467$	−2677.46	−0.265306	−0.132653	+	0.991163i	$0.542350\pi$
$468$	0	0
$469$	−1138.14	−0.112056
$470$	−1521.94	−0.149366
$471$	0	0
$472$	18189.3	1.77380
$473$	22051.6	2.14362
$474$	0	0
$475$	−2391.76	−0.231035
$476$	32.2456	0.00310499
$477$	0	0
$478$	−1923.75	−0.184080
$479$	3469.93	0.330992	0.165496	−	0.986210i	$-0.447078\pi$
$479$	3469.93	0.330992	0.165496	+	0.986210i	$0.447078\pi$
$480$	0	0
$481$	5598.63	0.530718
$482$	−1048.64	−0.0990960
$483$	0	0
$484$	−4591.82	−0.431238
$485$	−3626.68	−0.339544
$486$	0	0
$487$	−14040.6	−1.30645	−0.653224	−	0.757165i	$-0.726584\pi$
$487$	−14040.6	−1.30645	−0.653224	+	0.757165i	$0.726584\pi$
$488$	−11988.8	−1.11210
$489$	0	0
$490$	−3679.52	−0.339232
$491$	−9815.44	−0.902169	−0.451084	−	0.892481i	$-0.648963\pi$
$491$	−9815.44	−0.902169	−0.451084	+	0.892481i	$0.648963\pi$
$492$	0	0
$493$	−833.868	−0.0761775
$494$	−4294.07	−0.391092
$495$	0	0
$496$	−7923.12	−0.717255
$497$	858.425	0.0774761
$498$	0	0
$499$	−12052.0	−1.08121	−0.540603	−	0.841278i	$-0.681804\pi$
$499$	−12052.0	−1.08121	−0.540603	+	0.841278i	$0.681804\pi$
$500$	−398.052	−0.0356028
$501$	0	0
$502$	−15124.7	−1.34472
$503$	4695.09	0.416191	0.208095	−	0.978109i	$-0.433274\pi$
$503$	4695.09	0.416191	0.208095	+	0.978109i	$0.433274\pi$
$504$	0	0
$505$	−4888.91	−0.430799
$506$	10381.8	0.912108
$507$	0	0
$508$	6490.62	0.566880
$509$	819.813	0.0713902	0.0356951	−	0.999363i	$-0.488635\pi$
$509$	819.813	0.0713902	0.0356951	+	0.999363i	$0.488635\pi$
$510$	0	0
$511$	−141.140	−0.0122186
$512$	−9378.41	−0.809514
$513$	0	0
$514$	15924.4	1.36653
$515$	7938.52	0.679249
$516$	0	0
$517$	−7304.26	−0.621356
$518$	1661.50	0.140930
$519$	0	0
$520$	−2510.00	−0.211675
$521$	−3282.80	−0.276050	−0.138025	−	0.990429i	$-0.544075\pi$
$521$	−3282.80	−0.276050	−0.138025	+	0.990429i	$0.544075\pi$
$522$	0	0
$523$	−10768.1	−0.900300	−0.450150	−	0.892953i	$-0.648629\pi$
$523$	−10768.1	−0.900300	−0.450150	+	0.892953i	$0.648629\pi$
$524$	750.392	0.0625592
$525$	0	0
$526$	13863.5	1.14920
$527$	−1021.88	−0.0844665
$528$	0	0
$529$	−4095.57	−0.336613
$530$	2162.22	0.177209
$531$	0	0
$532$	842.690	0.0686752
$533$	−1326.71	−0.107816
$534$	0	0
$535$	−4488.65	−0.362732
$536$	10098.9	0.813816
$537$	0	0
$538$	−12609.8	−1.01049
$539$	−17659.1	−1.41119
$540$	0	0
$541$	−16037.9	−1.27453	−0.637266	−	0.770644i	$-0.719935\pi$
$541$	−16037.9	−1.27453	−0.637266	+	0.770644i	$0.719935\pi$
$542$	10808.1	0.856543
$543$	0	0
$544$	−490.776	−0.0386799
$545$	4277.46	0.336195
$546$	0	0
$547$	2049.12	0.160172	0.0800862	−	0.996788i	$-0.474480\pi$
$547$	2049.12	0.160172	0.0800862	+	0.996788i	$0.474480\pi$
$548$	−9466.92	−0.737968
$549$	0	0
$550$	2888.93	0.223972
$551$	−21791.9	−1.68487
$552$	0	0
$553$	3341.38	0.256944
$554$	5104.02	0.391424
$555$	0	0
$556$	6670.18	0.508774
$557$	−3644.07	−0.277207	−0.138603	−	0.990348i	$-0.544261\pi$
$557$	−3644.07	−0.277207	−0.138603	+	0.990348i	$0.544261\pi$
$558$	0	0
$559$	8565.12	0.648061
$560$	−392.560	−0.0296227
$561$	0	0
$562$	−7205.28	−0.540812
$563$	−350.736	−0.0262553	−0.0131277	−	0.999914i	$-0.504179\pi$
$563$	−350.736	−0.0262553	−0.0131277	+	0.999914i	$0.504179\pi$
$564$	0	0
$565$	−4551.21	−0.338886
$566$	4420.07	0.328250
$567$	0	0
$568$	−7616.93	−0.562675
$569$	19448.7	1.43292	0.716460	−	0.697628i	$-0.245761\pi$
$569$	19448.7	1.43292	0.716460	+	0.697628i	$0.245761\pi$
$570$	0	0
$571$	−15145.9	−1.11005	−0.555024	−	0.831834i	$-0.687291\pi$
$571$	−15145.9	−1.11005	−0.555024	+	0.831834i	$0.687291\pi$
$572$	−3429.80	−0.250712
$573$	0	0
$574$	−393.725	−0.0286302
$575$	2246.03	0.162897
$576$	0	0
$577$	−6365.11	−0.459243	−0.229621	−	0.973280i	$-0.573749\pi$
$577$	−6365.11	−0.459243	−0.229621	+	0.973280i	$0.573749\pi$
$578$	−10751.9	−0.773737
$579$	0	0
$580$	−3626.74	−0.259642
$581$	−2503.89	−0.178793
$582$	0	0
$583$	10377.1	0.737182
$584$	1252.36	0.0887381
$585$	0	0
$586$	−1054.86	−0.0743617
$587$	−10284.4	−0.723136	−0.361568	−	0.932346i	$-0.617759\pi$
$587$	−10284.4	−0.723136	−0.361568	+	0.932346i	$0.617759\pi$
$588$	0	0
$589$	−26705.3	−1.86821
$590$	−8131.55	−0.567408
$591$	0	0
$592$	−7769.47	−0.539397
$593$	−666.566	−0.0461595	−0.0230798	−	0.999734i	$-0.507347\pi$
$593$	−666.566	−0.0461595	−0.0230798	+	0.999734i	$0.507347\pi$
$594$	0	0
$595$	−50.6303	−0.00348847
$596$	872.330	0.0599530
$597$	0	0
$598$	4032.43	0.275749
$599$	−25213.3	−1.71984	−0.859922	−	0.510426i	$-0.829488\pi$
$599$	−25213.3	−1.71984	−0.859922	+	0.510426i	$0.829488\pi$
$600$	0	0
$601$	−20618.7	−1.39942	−0.699712	−	0.714426i	$-0.746688\pi$
$601$	−20618.7	−1.39942	−0.699712	+	0.714426i	$0.746688\pi$
$602$	2541.86	0.172090
$603$	0	0
$604$	2983.30	0.200975
$605$	7209.84	0.484499
$606$	0	0
$607$	−5083.52	−0.339923	−0.169962	−	0.985451i	$-0.554364\pi$
$607$	−5083.52	−0.339923	−0.169962	+	0.985451i	$0.554364\pi$
$608$	−12825.7	−0.855511
$609$	0	0
$610$	5359.58	0.355743
$611$	−2837.07	−0.187849
$612$	0	0
$613$	2625.18	0.172969	0.0864845	−	0.996253i	$-0.472437\pi$
$613$	2625.18	0.172969	0.0864845	+	0.996253i	$0.472437\pi$
$614$	7070.95	0.464756
$615$	0	0
$616$	−3574.95	−0.233829
$617$	1631.32	0.106442	0.0532208	−	0.998583i	$-0.483051\pi$
$617$	1631.32	0.106442	0.0532208	+	0.998583i	$0.483051\pi$
$618$	0	0
$619$	3184.67	0.206789	0.103395	−	0.994640i	$-0.467030\pi$
$619$	3184.67	0.206789	0.103395	+	0.994640i	$0.467030\pi$
$620$	−4444.47	−0.287894
$621$	0	0
$622$	−5297.40	−0.341489
$623$	−1835.64	−0.118047
$624$	0	0
$625$	625.000	0.0400000
$626$	−5501.33	−0.351242
$627$	0	0
$628$	−1268.78	−0.0806210
$629$	−1002.06	−0.0635214
$630$	0	0
$631$	10436.7	0.658447	0.329223	−	0.944252i	$-0.393213\pi$
$631$	10436.7	0.658447	0.329223	+	0.944252i	$0.393213\pi$
$632$	−29648.6	−1.86607
$633$	0	0
$634$	−8135.07	−0.509598
$635$	−10191.2	−0.636893
$636$	0	0
$637$	−6859.04	−0.426633
$638$	26321.7	1.63336
$639$	0	0
$640$	356.968	0.0220475
$641$	−9083.45	−0.559711	−0.279856	−	0.960042i	$-0.590287\pi$
$641$	−9083.45	−0.559711	−0.279856	+	0.960042i	$0.590287\pi$
$642$	0	0
$643$	17187.6	1.05414	0.527071	−	0.849821i	$-0.323290\pi$
$643$	17187.6	1.05414	0.527071	+	0.849821i	$0.323290\pi$
$644$	−791.343	−0.0484213
$645$	0	0
$646$	768.570	0.0468096
$647$	−2359.17	−0.143352	−0.0716758	−	0.997428i	$-0.522835\pi$
$647$	−2359.17	−0.143352	−0.0716758	+	0.997428i	$0.522835\pi$
$648$	0	0
$649$	−39025.7	−2.36039
$650$	1122.10	0.0677113
$651$	0	0
$652$	−1522.64	−0.0914589
$653$	−5436.11	−0.325776	−0.162888	−	0.986645i	$-0.552081\pi$
$653$	−5436.11	−0.325776	−0.162888	+	0.986645i	$0.552081\pi$
$654$	0	0
$655$	−1178.23	−0.0702856
$656$	1841.13	0.109579
$657$	0	0
$658$	−841.954	−0.0498826
$659$	−3088.91	−0.182590	−0.0912950	−	0.995824i	$-0.529101\pi$
$659$	−3088.91	−0.182590	−0.0912950	+	0.995824i	$0.529101\pi$
$660$	0	0
$661$	−19053.9	−1.12120	−0.560598	−	0.828088i	$-0.689429\pi$
$661$	−19053.9	−1.12120	−0.560598	+	0.828088i	$0.689429\pi$
$662$	5603.39	0.328976
$663$	0	0
$664$	22217.4	1.29850
$665$	−1323.15	−0.0771570
$666$	0	0
$667$	20464.1	1.18796
$668$	−1083.86	−0.0627779
$669$	0	0
$670$	−4514.71	−0.260326
$671$	25722.2	1.47987
$672$	0	0
$673$	−29237.5	−1.67462	−0.837312	−	0.546725i	$-0.815874\pi$
$673$	−29237.5	−1.67462	−0.837312	+	0.546725i	$0.815874\pi$
$674$	7796.61	0.445570
$675$	0	0
$676$	5663.98	0.322257
$677$	−14471.2	−0.821529	−0.410764	−	0.911742i	$-0.634738\pi$
$677$	−14471.2	−0.821529	−0.410764	+	0.911742i	$0.634738\pi$
$678$	0	0
$679$	−2006.31	−0.113395
$680$	449.250	0.0253352
$681$	0	0
$682$	32256.5	1.81109
$683$	−5782.94	−0.323980	−0.161990	−	0.986792i	$-0.551791\pi$
$683$	−5782.94	−0.323980	−0.161990	+	0.986792i	$0.551791\pi$
$684$	0	0
$685$	14864.5	0.829112
$686$	−4117.54	−0.229167
$687$	0	0
$688$	−11886.2	−0.658659
$689$	4030.62	0.222866
$690$	0	0
$691$	11055.1	0.608620	0.304310	−	0.952573i	$-0.401574\pi$
$691$	11055.1	0.608620	0.304310	+	0.952573i	$0.401574\pi$
$692$	12024.6	0.660557
$693$	0	0
$694$	17046.5	0.932386
$695$	−10473.2	−0.571611
$696$	0	0
$697$	237.459	0.0129045
$698$	1527.35	0.0828238
$699$	0	0
$700$	−220.206	−0.0118900
$701$	13554.2	0.730294	0.365147	−	0.930950i	$-0.381019\pi$
$701$	13554.2	0.730294	0.365147	+	0.930950i	$0.381019\pi$
$702$	0	0
$703$	−26187.4	−1.40495
$704$	27449.2	1.46950
$705$	0	0
$706$	11963.8	0.637768
$707$	−2704.59	−0.143871
$708$	0	0
$709$	−9389.17	−0.497345	−0.248672	−	0.968588i	$-0.579994\pi$
$709$	−9389.17	−0.497345	−0.248672	+	0.968588i	$0.579994\pi$
$710$	3405.15	0.179990
$711$	0	0
$712$	16287.9	0.857326
$713$	25078.1	1.31723
$714$	0	0
$715$	5385.29	0.281676
$716$	594.376	0.0310236
$717$	0	0
$718$	−8857.67	−0.460398
$719$	26301.8	1.36424	0.682122	−	0.731238i	$-0.261057\pi$
$719$	26301.8	1.36424	0.682122	+	0.731238i	$0.261057\pi$
$720$	0	0
$721$	4391.67	0.226844
$722$	5033.72	0.259467
$723$	0	0
$724$	−4581.58	−0.235184
$725$	5694.51	0.291709
$726$	0	0
$727$	−25538.6	−1.30285	−0.651427	−	0.758711i	$-0.725829\pi$
$727$	−25538.6	−1.30285	−0.651427	+	0.758711i	$0.725829\pi$
$728$	−1388.56	−0.0706915
$729$	0	0
$730$	−559.868	−0.0283858
$731$	−1533.02	−0.0775660
$732$	0	0
$733$	6982.12	0.351829	0.175914	−	0.984405i	$-0.443712\pi$
$733$	6982.12	0.351829	0.175914	+	0.984405i	$0.443712\pi$
$734$	24665.4	1.24035
$735$	0	0
$736$	12044.2	0.603200
$737$	−21667.5	−1.08295
$738$	0	0
$739$	8863.91	0.441224	0.220612	−	0.975362i	$-0.429195\pi$
$739$	8863.91	0.441224	0.220612	+	0.975362i	$0.429195\pi$
$740$	−4358.28	−0.216505
$741$	0	0
$742$	1196.16	0.0591812
$743$	38944.9	1.92295	0.961473	−	0.274899i	$-0.0886443\pi$
$743$	38944.9	1.92295	0.961473	+	0.274899i	$0.0886443\pi$
$744$	0	0
$745$	−1369.69	−0.0673576
$746$	−13870.5	−0.680742
$747$	0	0
$748$	613.879	0.0300075
$749$	−2483.17	−0.121139
$750$	0	0
$751$	17590.3	0.854700	0.427350	−	0.904086i	$-0.359447\pi$
$751$	17590.3	0.854700	0.427350	+	0.904086i	$0.359447\pi$
$752$	3937.13	0.190921
$753$	0	0
$754$	10223.7	0.493799
$755$	−4684.22	−0.225796
$756$	0	0
$757$	−4075.85	−0.195693	−0.0978463	−	0.995202i	$-0.531195\pi$
$757$	−4075.85	−0.195693	−0.0978463	+	0.995202i	$0.531195\pi$
$758$	20464.3	0.980601
$759$	0	0
$760$	11740.5	0.560358
$761$	−40537.0	−1.93097	−0.965483	−	0.260466i	$-0.916124\pi$
$761$	−40537.0	−1.93097	−0.965483	+	0.260466i	$0.916124\pi$
$762$	0	0
$763$	2366.33	0.112276
$764$	−1243.31	−0.0588761
$765$	0	0
$766$	−24463.7	−1.15393
$767$	−15158.1	−0.713596
$768$	0	0
$769$	24032.7	1.12697	0.563485	−	0.826126i	$-0.309460\pi$
$769$	24032.7	1.12697	0.563485	+	0.826126i	$0.309460\pi$
$770$	1598.18	0.0747981
$771$	0	0
$772$	12467.1	0.581220
$773$	20881.5	0.971611	0.485805	−	0.874067i	$-0.338526\pi$
$773$	20881.5	0.971611	0.485805	+	0.874067i	$0.338526\pi$
$774$	0	0
$775$	6978.46	0.323450
$776$	17802.3	0.823539
$777$	0	0
$778$	−14419.3	−0.664468
$779$	6205.64	0.285417
$780$	0	0
$781$	16342.4	0.748752
$782$	−721.740	−0.0330043
$783$	0	0
$784$	9518.60	0.433610
$785$	1992.18	0.0905782
$786$	0	0
$787$	−5136.39	−0.232646	−0.116323	−	0.993211i	$-0.537111\pi$
$787$	−5136.39	−0.232646	−0.116323	+	0.993211i	$0.537111\pi$
$788$	−2842.67	−0.128510
$789$	0	0
$790$	13254.4	0.596926
$791$	−2517.77	−0.113175
$792$	0	0
$793$	9990.86	0.447397
$794$	8710.46	0.389323
$795$	0	0
$796$	8822.43	0.392843
$797$	−21996.0	−0.977587	−0.488794	−	0.872400i	$-0.662563\pi$
$797$	−21996.0	−0.977587	−0.488794	+	0.872400i	$0.662563\pi$
$798$	0	0
$799$	507.791	0.0224835
$800$	3351.53	0.148118
$801$	0	0
$802$	−9600.03	−0.422679
$803$	−2686.98	−0.118084
$804$	0	0
$805$	1242.52	0.0544016
$806$	12528.8	0.547531
$807$	0	0
$808$	23998.3	1.04487
$809$	−31094.2	−1.35132	−0.675658	−	0.737215i	$-0.736140\pi$
$809$	−31094.2	−1.35132	−0.675658	+	0.737215i	$0.736140\pi$
$810$	0	0
$811$	19130.6	0.828320	0.414160	−	0.910204i	$-0.364075\pi$
$811$	19130.6	0.828320	0.414160	+	0.910204i	$0.364075\pi$
$812$	−2006.35	−0.0867106
$813$	0	0
$814$	31630.9	1.36199
$815$	2390.77	0.102755
$816$	0	0
$817$	−40063.1	−1.71558
$818$	−19469.8	−0.832208
$819$	0	0
$820$	1032.78	0.0439832
$821$	3557.62	0.151232	0.0756162	−	0.997137i	$-0.475908\pi$
$821$	3557.62	0.151232	0.0756162	+	0.997137i	$0.475908\pi$
$822$	0	0
$823$	6149.38	0.260454	0.130227	−	0.991484i	$-0.458429\pi$
$823$	6149.38	0.260454	0.130227	+	0.991484i	$0.458429\pi$
$824$	−38967.9	−1.64747
$825$	0	0
$826$	−4498.45	−0.189493
$827$	−21152.8	−0.889425	−0.444713	−	0.895673i	$-0.646694\pi$
$827$	−21152.8	−0.889425	−0.444713	+	0.895673i	$0.646694\pi$
$828$	0	0
$829$	−17402.4	−0.729083	−0.364541	−	0.931187i	$-0.618774\pi$
$829$	−17402.4	−0.729083	−0.364541	+	0.931187i	$0.618774\pi$
$830$	−9932.29	−0.415367
$831$	0	0
$832$	10661.6	0.444261
$833$	1227.66	0.0510635
$834$	0	0
$835$	1701.81	0.0705313
$836$	16042.8	0.663698
$837$	0	0
$838$	2225.55	0.0917425
$839$	18074.2	0.743733	0.371867	−	0.928286i	$-0.378718\pi$
$839$	18074.2	0.743733	0.371867	+	0.928286i	$0.378718\pi$
$840$	0	0
$841$	27495.0	1.12735
$842$	−32682.1	−1.33765
$843$	0	0
$844$	−14592.6	−0.595140
$845$	−8893.28	−0.362057
$846$	0	0
$847$	3988.55	0.161804
$848$	−5593.47	−0.226510
$849$	0	0
$850$	−200.838	−0.00810432
$851$	24591.8	0.990595
$852$	0	0
$853$	40792.3	1.63740	0.818699	−	0.574222i	$-0.194696\pi$
$853$	40792.3	1.63740	0.818699	+	0.574222i	$0.194696\pi$
$854$	2964.97	0.118805
$855$	0	0
$856$	22033.5	0.879778
$857$	11057.9	0.440758	0.220379	−	0.975414i	$-0.429271\pi$
$857$	11057.9	0.440758	0.220379	+	0.975414i	$0.429271\pi$
$858$	0	0
$859$	−1753.63	−0.0696543	−0.0348272	−	0.999393i	$-0.511088\pi$
$859$	−1753.63	−0.0696543	−0.0348272	+	0.999393i	$0.511088\pi$
$860$	−6667.56	−0.264374
$861$	0	0
$862$	9576.33	0.378389
$863$	19186.5	0.756796	0.378398	−	0.925643i	$-0.376475\pi$
$863$	19186.5	0.756796	0.378398	+	0.925643i	$0.376475\pi$
$864$	0	0
$865$	−18880.3	−0.742140
$866$	20411.8	0.800949
$867$	0	0
$868$	−2458.72	−0.0961457
$869$	63612.0	2.48319
$870$	0	0
$871$	−8415.93	−0.327397
$872$	−20996.8	−0.815415
$873$	0	0
$874$	−18861.6	−0.729980
$875$	345.756	0.0133585
$876$	0	0
$877$	−8514.60	−0.327842	−0.163921	−	0.986473i	$-0.552414\pi$
$877$	−8514.60	−0.327842	−0.163921	+	0.986473i	$0.552414\pi$
$878$	3335.89	0.128224
$879$	0	0
$880$	−7473.41	−0.286282
$881$	−41177.0	−1.57467	−0.787337	−	0.616522i	$-0.788541\pi$
$881$	−41177.0	−1.57467	−0.787337	+	0.616522i	$0.788541\pi$
$882$	0	0
$883$	32540.4	1.24017	0.620086	−	0.784533i	$-0.287098\pi$
$883$	32540.4	1.24017	0.620086	+	0.784533i	$0.287098\pi$
$884$	238.439	0.00907191
$885$	0	0
$886$	−9792.67	−0.371322
$887$	−2166.10	−0.0819960	−0.0409980	−	0.999159i	$-0.513054\pi$
$887$	−2166.10	−0.0819960	−0.0409980	+	0.999159i	$0.513054\pi$
$888$	0	0
$889$	−5637.89	−0.212698
$890$	−7281.53	−0.274244
$891$	0	0
$892$	−13041.2	−0.489522
$893$	13270.3	0.497284
$894$	0	0
$895$	−933.258	−0.0348552
$896$	197.478	0.00736304
$897$	0	0
$898$	11787.8	0.438045
$899$	63582.3	2.35883
$900$	0	0
$901$	−721.416	−0.0266747
$902$	−7495.58	−0.276691
$903$	0	0
$904$	22340.6	0.821943
$905$	7193.75	0.264230
$906$	0	0
$907$	−706.062	−0.0258483	−0.0129241	−	0.999916i	$-0.504114\pi$
$907$	−706.062	−0.0258483	−0.0129241	+	0.999916i	$0.504114\pi$
$908$	−12321.6	−0.450339
$909$	0	0
$910$	620.756	0.0226130
$911$	495.485	0.0180199	0.00900997	−	0.999959i	$-0.497132\pi$
$911$	495.485	0.0180199	0.00900997	+	0.999959i	$0.497132\pi$
$912$	0	0
$913$	−47668.0	−1.72791
$914$	34069.7	1.23296
$915$	0	0
$916$	−3935.98	−0.141974
$917$	−651.806	−0.0234728
$918$	0	0
$919$	20473.6	0.734889	0.367445	−	0.930045i	$-0.380233\pi$
$919$	20473.6	0.734889	0.367445	+	0.930045i	$0.380233\pi$
$920$	−11025.1	−0.395095
$921$	0	0
$922$	123.806	0.00442227
$923$	6347.59	0.226363
$924$	0	0
$925$	6843.13	0.243244
$926$	29348.0	1.04151
$927$	0	0
$928$	30536.5	1.08018
$929$	−49715.3	−1.75577	−0.877883	−	0.478875i	$-0.841045\pi$
$929$	−49715.3	−1.75577	−0.877883	+	0.478875i	$0.841045\pi$
$930$	0	0
$931$	32083.0	1.12941
$932$	7028.16	0.247012
$933$	0	0
$934$	−5875.53	−0.205839
$935$	−963.880	−0.0337136
$936$	0	0
$937$	−31524.9	−1.09912	−0.549560	−	0.835454i	$-0.685204\pi$
$937$	−31524.9	−1.09912	−0.549560	+	0.835454i	$0.685204\pi$
$938$	−2497.58	−0.0869392
$939$	0	0
$940$	2208.53	0.0766323
$941$	−37528.5	−1.30010	−0.650050	−	0.759892i	$-0.725252\pi$
$941$	−37528.5	−1.30010	−0.650050	+	0.759892i	$0.725252\pi$
$942$	0	0
$943$	−5827.52	−0.201241
$944$	21035.6	0.725265
$945$	0	0
$946$	48390.9	1.66313
$947$	−25863.4	−0.887483	−0.443741	−	0.896155i	$-0.646349\pi$
$947$	−25863.4	−0.887483	−0.443741	+	0.896155i	$0.646349\pi$
$948$	0	0
$949$	−1043.66	−0.0356992
$950$	−5248.59	−0.179249
$951$	0	0
$952$	248.530	0.00846102
$953$	−1060.03	−0.0360312	−0.0180156	−	0.999838i	$-0.505735\pi$
$953$	−1060.03	−0.0360312	−0.0180156	+	0.999838i	$0.505735\pi$
$954$	0	0
$955$	1952.18	0.0661476
$956$	2791.61	0.0944426
$957$	0	0
$958$	7614.56	0.256801
$959$	8223.16	0.276892
$960$	0	0
$961$	48127.3	1.61550
$962$	12285.9	0.411759
$963$	0	0
$964$	1521.71	0.0508412
$965$	−19575.2	−0.653004
$966$	0	0
$967$	−5185.01	−0.172429	−0.0862145	−	0.996277i	$-0.527477\pi$
$967$	−5185.01	−0.172429	−0.0862145	+	0.996277i	$0.527477\pi$
$968$	−35391.0	−1.17511
$969$	0	0
$970$	−7958.54	−0.263436
$971$	−28314.9	−0.935805	−0.467903	−	0.883780i	$-0.654990\pi$
$971$	−28314.9	−0.935805	−0.467903	+	0.883780i	$0.654990\pi$
$972$	0	0
$973$	−5793.85	−0.190897
$974$	−30811.3	−1.01361
$975$	0	0
$976$	−13864.8	−0.454713
$977$	−44896.7	−1.47019	−0.735093	−	0.677967i	$-0.762861\pi$
$977$	−44896.7	−1.47019	−0.735093	+	0.677967i	$0.762861\pi$
$978$	0	0
$979$	−34946.3	−1.14084
$980$	5339.45	0.174043
$981$	0	0
$982$	−21539.4	−0.699950
$983$	14572.8	0.472838	0.236419	−	0.971651i	$-0.424026\pi$
$983$	14572.8	0.472838	0.236419	+	0.971651i	$0.424026\pi$
$984$	0	0
$985$	4463.40	0.144382
$986$	−1829.88	−0.0591026
$987$	0	0
$988$	6231.24	0.200650
$989$	37622.0	1.20962
$990$	0	0
$991$	10602.1	0.339844	0.169922	−	0.985457i	$-0.445648\pi$
$991$	10602.1	0.339844	0.169922	+	0.985457i	$0.445648\pi$
$992$	37421.6	1.19772
$993$	0	0
$994$	1883.76	0.0601100
$995$	−13852.5	−0.441361
$996$	0	0
$997$	27897.0	0.886164	0.443082	−	0.896481i	$-0.353885\pi$
$997$	27897.0	0.886164	0.443082	+	0.896481i	$0.353885\pi$
$998$	−26447.4	−0.838857
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.n.1.5		7
3.2	odd	2		405.4.a.m.1.3		7
5.4	even	2		2025.4.a.ba.1.3		7
9.2	odd	6		45.4.e.c.31.5	yes	14
9.4	even	3		135.4.e.c.46.3		14
9.5	odd	6		45.4.e.c.16.5	✓	14
9.7	even	3		135.4.e.c.91.3		14
15.14	odd	2		2025.4.a.bb.1.5		7
45.2	even	12		225.4.k.d.49.5		28
45.14	odd	6		225.4.e.d.151.3		14
45.23	even	12		225.4.k.d.124.5		28
45.29	odd	6		225.4.e.d.76.3		14
45.32	even	12		225.4.k.d.124.10		28
45.38	even	12		225.4.k.d.49.10		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.5	✓	14	9.5	odd	6
45.4.e.c.31.5	yes	14	9.2	odd	6
135.4.e.c.46.3		14	9.4	even	3
135.4.e.c.91.3		14	9.7	even	3
225.4.e.d.76.3		14	45.29	odd	6
225.4.e.d.151.3		14	45.14	odd	6
225.4.k.d.49.5		28	45.2	even	12
225.4.k.d.49.10		28	45.38	even	12
225.4.k.d.124.5		28	45.23	even	12
225.4.k.d.124.10		28	45.32	even	12
405.4.a.m.1.3		7	3.2	odd	2
405.4.a.n.1.5		7	1.1	even	1	trivial
2025.4.a.ba.1.3		7	5.4	even	2
2025.4.a.bb.1.5		7	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)

\(f(q)\)	\(=\)	\(q+2.19444 q^{2} -3.18442 q^{4} +5.00000 q^{5} +2.76605 q^{7} -24.5436 q^{8} +O(q^{10})\)	\(q+2.19444 q^{2} -3.18442 q^{4} +5.00000 q^{5} +2.76605 q^{7} -24.5436 q^{8} +10.9722 q^{10} +52.6590 q^{11} +20.4535 q^{13} +6.06994 q^{14} -28.3842 q^{16} -3.66084 q^{17} -95.6705 q^{19} -15.9221 q^{20} +115.557 q^{22} +89.8411 q^{23} +25.0000 q^{25} +44.8840 q^{26} -8.80825 q^{28} +227.780 q^{29} +279.139 q^{31} +134.061 q^{32} -8.03351 q^{34} +13.8302 q^{35} +273.725 q^{37} -209.944 q^{38} -122.718 q^{40} -64.8647 q^{41} +418.762 q^{43} -167.688 q^{44} +197.151 q^{46} -138.709 q^{47} -335.349 q^{49} +54.8611 q^{50} -65.1323 q^{52} +197.063 q^{53} +263.295 q^{55} -67.8887 q^{56} +499.852 q^{58} -741.103 q^{59} +488.468 q^{61} +612.554 q^{62} +521.263 q^{64} +102.267 q^{65} -411.468 q^{67} +11.6576 q^{68} +30.3497 q^{70} +310.343 q^{71} -51.0260 q^{73} +600.675 q^{74} +304.655 q^{76} +145.657 q^{77} +1208.00 q^{79} -141.921 q^{80} -142.342 q^{82} -905.221 q^{83} -18.3042 q^{85} +918.949 q^{86} -1292.44 q^{88} -663.633 q^{89} +56.5752 q^{91} -286.092 q^{92} -304.389 q^{94} -478.353 q^{95} -725.336 q^{97} -735.905 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 36 * q^4 + 35 * q^5 + 22 * q^7 + 18 * q^8	\( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + 10 q^{10} + 23 q^{11} + 96 q^{13} - 21 q^{14} + 324 q^{16} + 161 q^{17} + 279 q^{19} + 180 q^{20} + 311 q^{22} + 96 q^{23} + 175 q^{25} - 358 q^{26} + 337 q^{28} - 296 q^{29} + 244 q^{31} - 314 q^{32} + 125 q^{34} + 110 q^{35} + 404 q^{37} + 305 q^{38} + 90 q^{40} - 47 q^{41} + 525 q^{43} + 55 q^{44} + 717 q^{46} + 164 q^{47} + 1225 q^{49} + 50 q^{50} + 1682 q^{52} + 506 q^{53} + 115 q^{55} - 981 q^{56} + 1183 q^{58} - 85 q^{59} + 828 q^{61} - 786 q^{62} + 2236 q^{64} + 480 q^{65} + 1093 q^{67} + 2473 q^{68} - 105 q^{70} + 328 q^{71} + 2085 q^{73} - 1316 q^{74} + 2789 q^{76} + 24 q^{77} + 2110 q^{79} + 1620 q^{80} - 62 q^{82} + 1290 q^{83} + 805 q^{85} - 2569 q^{86} + 2271 q^{88} - 3048 q^{89} + 3338 q^{91} + 2763 q^{92} - 517 q^{94} + 1395 q^{95} + 1787 q^{97} + 1279 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 36 * q^4 + 35 * q^5 + 22 * q^7 + 18 * q^8 + 10 * q^10 + 23 * q^11 + 96 * q^13 - 21 * q^14 + 324 * q^16 + 161 * q^17 + 279 * q^19 + 180 * q^20 + 311 * q^22 + 96 * q^23 + 175 * q^25 - 358 * q^26 + 337 * q^28 - 296 * q^29 + 244 * q^31 - 314 * q^32 + 125 * q^34 + 110 * q^35 + 404 * q^37 + 305 * q^38 + 90 * q^40 - 47 * q^41 + 525 * q^43 + 55 * q^44 + 717 * q^46 + 164 * q^47 + 1225 * q^49 + 50 * q^50 + 1682 * q^52 + 506 * q^53 + 115 * q^55 - 981 * q^56 + 1183 * q^58 - 85 * q^59 + 828 * q^61 - 786 * q^62 + 2236 * q^64 + 480 * q^65 + 1093 * q^67 + 2473 * q^68 - 105 * q^70 + 328 * q^71 + 2085 * q^73 - 1316 * q^74 + 2789 * q^76 + 24 * q^77 + 2110 * q^79 + 1620 * q^80 - 62 * q^82 + 1290 * q^83 + 805 * q^85 - 2569 * q^86 + 2271 * q^88 - 3048 * q^89 + 3338 * q^91 + 2763 * q^92 - 517 * q^94 + 1395 * q^95 + 1787 * q^97 + 1279 * q^98

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