LMFDB - Embedded newform 405.4.a.m.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			$-1.57021$ 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)

$f(q)$	$=$	$q+1.57021 q^{2} -5.53444 q^{4} -5.00000 q^{5} +34.2398 q^{7} -21.2519 q^{8} +O(q^{10})$	\(q+1.57021 q^{2} -5.53444 q^{4} -5.00000 q^{5} +34.2398 q^{7} -21.2519 q^{8} -7.85104 q^{10} -26.8960 q^{11} -19.4880 q^{13} +53.7636 q^{14} +10.9056 q^{16} +29.1232 q^{17} +47.1006 q^{19} +27.6722 q^{20} -42.2323 q^{22} +113.217 q^{23} +25.0000 q^{25} -30.6003 q^{26} -189.498 q^{28} +81.2439 q^{29} -10.8362 q^{31} +187.139 q^{32} +45.7295 q^{34} -171.199 q^{35} +410.002 q^{37} +73.9578 q^{38} +106.260 q^{40} -443.510 q^{41} +339.496 q^{43} +148.854 q^{44} +177.774 q^{46} +236.241 q^{47} +829.363 q^{49} +39.2552 q^{50} +107.855 q^{52} +609.634 q^{53} +134.480 q^{55} -727.661 q^{56} +127.570 q^{58} +15.5454 q^{59} +61.0509 q^{61} -17.0151 q^{62} +206.603 q^{64} +97.4401 q^{65} +216.889 q^{67} -161.181 q^{68} -268.818 q^{70} +65.4315 q^{71} +711.811 q^{73} +643.789 q^{74} -260.676 q^{76} -920.913 q^{77} -957.752 q^{79} -54.5281 q^{80} -696.404 q^{82} -522.440 q^{83} -145.616 q^{85} +533.079 q^{86} +571.591 q^{88} -1602.24 q^{89} -667.266 q^{91} -626.591 q^{92} +370.948 q^{94} -235.503 q^{95} +801.938 q^{97} +1302.27 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.57021	0.555153	0.277576	−	0.960704i	$-0.410469\pi$
$2$	1.57021	0.555153	0.277576	+	0.960704i	$0.410469\pi$
$3$	0	0
$3$	0	0
$4$	−5.53444	−0.691805
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	34.2398	1.84877	0.924387	−	0.381455i	$-0.124577\pi$
$7$	34.2398	1.84877	0.924387	+	0.381455i	$0.124577\pi$
$8$	−21.2519	−0.939210
$9$	0	0
$10$	−7.85104	−0.248272
$11$	−26.8960	−0.737223	−0.368611	−	0.929584i	$-0.620167\pi$
$11$	−26.8960	−0.737223	−0.368611	+	0.929584i	$0.620167\pi$
$12$	0	0
$13$	−19.4880	−0.415770	−0.207885	−	0.978153i	$-0.566658\pi$
$13$	−19.4880	−0.415770	−0.207885	+	0.978153i	$0.566658\pi$
$14$	53.7636	1.02635
$15$	0	0
$16$	10.9056	0.170400
$17$	29.1232	0.415495	0.207747	−	0.978182i	$-0.433387\pi$
$17$	29.1232	0.415495	0.207747	+	0.978182i	$0.433387\pi$
$18$	0	0
$19$	47.1006	0.568717	0.284358	−	0.958718i	$-0.408219\pi$
$19$	47.1006	0.568717	0.284358	+	0.958718i	$0.408219\pi$
$20$	27.6722	0.309385
$21$	0	0
$22$	−42.2323	−0.409271
$23$	113.217	1.02640	0.513202	−	0.858268i	$-0.328459\pi$
$23$	113.217	1.02640	0.513202	+	0.858268i	$0.328459\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−30.6003	−0.230816
$27$	0	0
$28$	−189.498	−1.27899
$29$	81.2439	0.520228	0.260114	−	0.965578i	$-0.416240\pi$
$29$	81.2439	0.520228	0.260114	+	0.965578i	$0.416240\pi$
$30$	0	0
$31$	−10.8362	−0.0627820	−0.0313910	−	0.999507i	$-0.509994\pi$
$31$	−10.8362	−0.0627820	−0.0313910	+	0.999507i	$0.509994\pi$
$32$	187.139	1.03381
$33$	0	0
$34$	45.7295	0.230663
$35$	−171.199	−0.826797
$36$	0	0
$37$	410.002	1.82173	0.910864	−	0.412706i	$-0.135416\pi$
$37$	410.002	1.82173	0.910864	+	0.412706i	$0.135416\pi$
$38$	73.9578	0.315725
$39$	0	0
$40$	106.260	0.420028
$41$	−443.510	−1.68938	−0.844691	−	0.535254i	$-0.820216\pi$
$41$	−443.510	−1.68938	−0.844691	+	0.535254i	$0.820216\pi$
$42$	0	0
$43$	339.496	1.20401	0.602007	−	0.798491i	$-0.294368\pi$
$43$	339.496	1.20401	0.602007	+	0.798491i	$0.294368\pi$
$44$	148.854	0.510015
$45$	0	0
$46$	177.774	0.569811
$47$	236.241	0.733178	0.366589	−	0.930383i	$-0.380526\pi$
$47$	236.241	0.733178	0.366589	+	0.930383i	$0.380526\pi$
$48$	0	0
$49$	829.363	2.41797
$50$	39.2552	0.111031
$51$	0	0
$52$	107.855	0.287632
$53$	609.634	1.57999	0.789997	−	0.613111i	$-0.210082\pi$
$53$	609.634	1.57999	0.789997	+	0.613111i	$0.210082\pi$
$54$	0	0
$55$	134.480	0.329696
$56$	−727.661	−1.73639
$57$	0	0
$58$	127.570	0.288806
$59$	15.5454	0.0343024	0.0171512	−	0.999853i	$-0.494540\pi$
$59$	15.5454	0.0343024	0.0171512	+	0.999853i	$0.494540\pi$
$60$	0	0
$61$	61.0509	0.128144	0.0640719	−	0.997945i	$-0.479591\pi$
$61$	61.0509	0.128144	0.0640719	+	0.997945i	$0.479591\pi$
$62$	−17.0151	−0.0348536
$63$	0	0
$64$	206.603	0.403521
$65$	97.4401	0.185938
$66$	0	0
$67$	216.889	0.395481	0.197741	−	0.980254i	$-0.436640\pi$
$67$	216.889	0.395481	0.197741	+	0.980254i	$0.436640\pi$
$68$	−161.181	−0.287442
$69$	0	0
$70$	−268.818	−0.458999
$71$	65.4315	0.109370	0.0546852	−	0.998504i	$-0.482584\pi$
$71$	65.4315	0.109370	0.0546852	+	0.998504i	$0.482584\pi$
$72$	0	0
$73$	711.811	1.14125	0.570624	−	0.821211i	$-0.306701\pi$
$73$	711.811	1.14125	0.570624	+	0.821211i	$0.306701\pi$
$74$	643.789	1.01134
$75$	0	0
$76$	−260.676	−0.393441
$77$	−920.913	−1.36296
$78$	0	0
$79$	−957.752	−1.36399	−0.681997	−	0.731355i	$-0.738888\pi$
$79$	−957.752	−1.36399	−0.681997	+	0.731355i	$0.738888\pi$
$80$	−54.5281	−0.0762053
$81$	0	0
$82$	−696.404	−0.937865
$83$	−522.440	−0.690907	−0.345453	−	0.938436i	$-0.612275\pi$
$83$	−522.440	−0.690907	−0.345453	+	0.938436i	$0.612275\pi$
$84$	0	0
$85$	−145.616	−0.185815
$86$	533.079	0.668412
$87$	0	0
$88$	571.591	0.692407
$89$	−1602.24	−1.90828	−0.954141	−	0.299359i	$-0.903227\pi$
$89$	−1602.24	−1.90828	−0.954141	+	0.299359i	$0.903227\pi$
$90$	0	0
$91$	−667.266	−0.768664
$92$	−626.591	−0.710072
$93$	0	0
$94$	370.948	0.407026
$95$	−235.503	−0.254338
$96$	0	0
$97$	801.938	0.839427	0.419714	−	0.907657i	$-0.362130\pi$
$97$	801.938	0.839427	0.419714	+	0.907657i	$0.362130\pi$
$98$	1302.27	1.34234
$99$	0	0
$100$	−138.361	−0.138361
$101$	586.130	0.577447	0.288724	−	0.957413i	$-0.406769\pi$
$101$	586.130	0.577447	0.288724	+	0.957413i	$0.406769\pi$
$102$	0	0
$103$	−820.675	−0.785082	−0.392541	−	0.919734i	$-0.628404\pi$
$103$	−820.675	−0.785082	−0.392541	+	0.919734i	$0.628404\pi$
$104$	414.158	0.390495
$105$	0	0
$106$	957.252	0.877138
$107$	−1358.33	−1.22724	−0.613621	−	0.789600i	$-0.710288\pi$
$107$	−1358.33	−1.22724	−0.613621	+	0.789600i	$0.710288\pi$
$108$	0	0
$109$	489.495	0.430139	0.215069	−	0.976599i	$-0.431002\pi$
$109$	489.495	0.430139	0.215069	+	0.976599i	$0.431002\pi$
$110$	211.162	0.183032
$111$	0	0
$112$	373.406	0.315032
$113$	797.663	0.664052	0.332026	−	0.943270i	$-0.392268\pi$
$113$	797.663	0.664052	0.332026	+	0.943270i	$0.392268\pi$
$114$	0	0
$115$	−566.083	−0.459022
$116$	−449.640	−0.359897
$117$	0	0
$118$	24.4096	0.0190431
$119$	997.172	0.768156
$120$	0	0
$121$	−607.605	−0.456503
$122$	95.8627	0.0711394
$123$	0	0
$124$	59.9725	0.0434330
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1728.22	1.20752	0.603759	−	0.797167i	$-0.293669\pi$
$127$	1728.22	1.20752	0.603759	+	0.797167i	$0.293669\pi$
$128$	−1172.70	−0.809793
$129$	0	0
$130$	153.001	0.103224
$131$	639.101	0.426248	0.213124	−	0.977025i	$-0.431636\pi$
$131$	639.101	0.426248	0.213124	+	0.977025i	$0.431636\pi$
$132$	0	0
$133$	1612.71	1.05143
$134$	340.562	0.219553
$135$	0	0
$136$	−618.923	−0.390237
$137$	−2835.94	−1.76855	−0.884274	−	0.466969i	$-0.845346\pi$
$137$	−2835.94	−1.76855	−0.884274	+	0.466969i	$0.845346\pi$
$138$	0	0
$139$	174.996	0.106784	0.0533920	−	0.998574i	$-0.482997\pi$
$139$	174.996	0.106784	0.0533920	+	0.998574i	$0.482997\pi$
$140$	947.491	0.571983
$141$	0	0
$142$	102.741	0.0607172
$143$	524.150	0.306515
$144$	0	0
$145$	−406.219	−0.232653
$146$	1117.69	0.633567
$147$	0	0
$148$	−2269.13	−1.26028
$149$	1023.53	0.562758	0.281379	−	0.959597i	$-0.409208\pi$
$149$	1023.53	0.562758	0.281379	+	0.959597i	$0.409208\pi$
$150$	0	0
$151$	−873.996	−0.471025	−0.235512	−	0.971871i	$-0.575677\pi$
$151$	−873.996	−0.471025	−0.235512	+	0.971871i	$0.575677\pi$
$152$	−1000.98	−0.534145
$153$	0	0
$154$	−1446.03	−0.756650
$155$	54.1811	0.0280770
$156$	0	0
$157$	123.552	0.0628061	0.0314031	−	0.999507i	$-0.490002\pi$
$157$	123.552	0.0628061	0.0314031	+	0.999507i	$0.490002\pi$
$158$	−1503.87	−0.757225
$159$	0	0
$160$	−935.697	−0.462333
$161$	3876.51	1.89759
$162$	0	0
$163$	−1410.89	−0.677972	−0.338986	−	0.940791i	$-0.610084\pi$
$163$	−1410.89	−0.677972	−0.338986	+	0.940791i	$0.610084\pi$
$164$	2454.58	1.16872
$165$	0	0
$166$	−820.340	−0.383559
$167$	−2027.70	−0.939571	−0.469786	−	0.882781i	$-0.655669\pi$
$167$	−2027.70	−0.939571	−0.469786	+	0.882781i	$0.655669\pi$
$168$	0	0
$169$	−1817.22	−0.827136
$170$	−228.648	−0.103156
$171$	0	0
$172$	−1878.92	−0.832944
$173$	2703.98	1.18832	0.594160	−	0.804347i	$-0.297485\pi$
$173$	2703.98	1.18832	0.594160	+	0.804347i	$0.297485\pi$
$174$	0	0
$175$	855.995	0.369755
$176$	−293.317	−0.125623
$177$	0	0
$178$	−2515.85	−1.05939
$179$	−750.694	−0.313461	−0.156730	−	0.987641i	$-0.550095\pi$
$179$	−750.694	−0.313461	−0.156730	+	0.987641i	$0.550095\pi$
$180$	0	0
$181$	−1134.08	−0.465722	−0.232861	−	0.972510i	$-0.574809\pi$
$181$	−1134.08	−0.465722	−0.232861	+	0.972510i	$0.574809\pi$
$182$	−1047.75	−0.426726
$183$	0	0
$184$	−2406.07	−0.964010
$185$	−2050.01	−0.814702
$186$	0	0
$187$	−783.297	−0.306312
$188$	−1307.46	−0.507216
$189$	0	0
$190$	−369.789	−0.141196
$191$	−535.255	−0.202773	−0.101387	−	0.994847i	$-0.532328\pi$
$191$	−535.255	−0.202773	−0.101387	+	0.994847i	$0.532328\pi$
$192$	0	0
$193$	2623.55	0.978483	0.489241	−	0.872148i	$-0.337274\pi$
$193$	2623.55	0.978483	0.489241	+	0.872148i	$0.337274\pi$
$194$	1259.21	0.466010
$195$	0	0
$196$	−4590.06	−1.67276
$197$	2954.08	1.06837	0.534186	−	0.845367i	$-0.320618\pi$
$197$	2954.08	1.06837	0.534186	+	0.845367i	$0.320618\pi$
$198$	0	0
$199$	2639.35	0.940192	0.470096	−	0.882615i	$-0.344219\pi$
$199$	2639.35	0.940192	0.470096	+	0.882615i	$0.344219\pi$
$200$	−531.298	−0.187842
$201$	0	0
$202$	920.347	0.320571
$203$	2781.77	0.961784
$204$	0	0
$205$	2217.55	0.755515
$206$	−1288.63	−0.435841
$207$	0	0
$208$	−212.529	−0.0708473
$209$	−1266.82	−0.419271
$210$	0	0
$211$	−1990.65	−0.649489	−0.324745	−	0.945802i	$-0.605278\pi$
$211$	−1990.65	−0.649489	−0.324745	+	0.945802i	$0.605278\pi$
$212$	−3373.98	−1.09305
$213$	0	0
$214$	−2132.87	−0.681307
$215$	−1697.48	−0.538452
$216$	0	0
$217$	−371.030	−0.116070
$218$	768.609	0.238793
$219$	0	0
$220$	−744.272	−0.228085
$221$	−567.554	−0.172750
$222$	0	0
$223$	−674.900	−0.202667	−0.101333	−	0.994853i	$-0.532311\pi$
$223$	−674.900	−0.202667	−0.101333	+	0.994853i	$0.532311\pi$
$224$	6407.61	1.91128
$225$	0	0
$226$	1252.50	0.368650
$227$	1891.85	0.553157	0.276579	−	0.960991i	$-0.410799\pi$
$227$	1891.85	0.553157	0.276579	+	0.960991i	$0.410799\pi$
$228$	0	0
$229$	3408.51	0.983585	0.491792	−	0.870712i	$-0.336342\pi$
$229$	3408.51	0.983585	0.491792	+	0.870712i	$0.336342\pi$
$230$	−888.869	−0.254827
$231$	0	0
$232$	−1726.59	−0.488603
$233$	−2841.86	−0.799041	−0.399521	−	0.916724i	$-0.630823\pi$
$233$	−2841.86	−0.799041	−0.399521	+	0.916724i	$0.630823\pi$
$234$	0	0
$235$	−1181.21	−0.327887
$236$	−86.0353	−0.0237306
$237$	0	0
$238$	1565.77	0.426444
$239$	1619.61	0.438341	0.219171	−	0.975687i	$-0.429665\pi$
$239$	1619.61	0.438341	0.219171	+	0.975687i	$0.429665\pi$
$240$	0	0
$241$	26.7008	0.00713671	0.00356835	−	0.999994i	$-0.498864\pi$
$241$	26.7008	0.00713671	0.00356835	+	0.999994i	$0.498864\pi$
$242$	−954.068	−0.253429
$243$	0	0
$244$	−337.883	−0.0886506
$245$	−4146.81	−1.08135
$246$	0	0
$247$	−917.898	−0.236455
$248$	230.290	0.0589656
$249$	0	0
$250$	−196.276	−0.0496544
$251$	−427.354	−0.107467	−0.0537337	−	0.998555i	$-0.517112\pi$
$251$	−427.354	−0.107467	−0.0537337	+	0.998555i	$0.517112\pi$
$252$	0	0
$253$	−3045.07	−0.756689
$254$	2713.67	0.670357
$255$	0	0
$256$	−3494.22	−0.853080
$257$	3841.55	0.932409	0.466204	−	0.884677i	$-0.345621\pi$
$257$	3841.55	0.932409	0.466204	+	0.884677i	$0.345621\pi$
$258$	0	0
$259$	14038.4	3.36797
$260$	−539.277	−0.128633
$261$	0	0
$262$	1003.52	0.236633
$263$	−798.874	−0.187303	−0.0936514	−	0.995605i	$-0.529854\pi$
$263$	−798.874	−0.187303	−0.0936514	+	0.995605i	$0.529854\pi$
$264$	0	0
$265$	−3048.17	−0.706594
$266$	2532.30	0.583704
$267$	0	0
$268$	−1200.36	−0.273596
$269$	7934.42	1.79840	0.899201	−	0.437536i	$-0.144149\pi$
$269$	7934.42	1.79840	0.899201	+	0.437536i	$0.144149\pi$
$270$	0	0
$271$	−2309.09	−0.517592	−0.258796	−	0.965932i	$-0.583326\pi$
$271$	−2309.09	−0.517592	−0.258796	+	0.965932i	$0.583326\pi$
$272$	317.606	0.0708004
$273$	0	0
$274$	−4453.02	−0.981814
$275$	−672.400	−0.147445
$276$	0	0
$277$	761.303	0.165134	0.0825672	−	0.996585i	$-0.473688\pi$
$277$	761.303	0.165134	0.0825672	+	0.996585i	$0.473688\pi$
$278$	274.781	0.0592815
$279$	0	0
$280$	3638.30	0.776536
$281$	−1287.28	−0.273283	−0.136641	−	0.990621i	$-0.543631\pi$
$281$	−1287.28	−0.273283	−0.136641	+	0.990621i	$0.543631\pi$
$282$	0	0
$283$	−2726.51	−0.572699	−0.286350	−	0.958125i	$-0.592442\pi$
$283$	−2726.51	−0.572699	−0.286350	+	0.958125i	$0.592442\pi$
$284$	−362.127	−0.0756630
$285$	0	0
$286$	823.025	0.170163
$287$	−15185.7	−3.12329
$288$	0	0
$289$	−4064.84	−0.827364
$290$	−637.849	−0.129158
$291$	0	0
$292$	−3939.48	−0.789522
$293$	−96.3739	−0.0192158	−0.00960789	−	0.999954i	$-0.503058\pi$
$293$	−96.3739	−0.0192158	−0.00960789	+	0.999954i	$0.503058\pi$
$294$	0	0
$295$	−77.7272	−0.0153405
$296$	−8713.33	−1.71099
$297$	0	0
$298$	1607.16	0.312417
$299$	−2206.37	−0.426748
$300$	0	0
$301$	11624.3	2.22595
$302$	−1372.36	−0.261491
$303$	0	0
$304$	513.661	0.0969095
$305$	−305.255	−0.0573076
$306$	0	0
$307$	5258.02	0.977495	0.488747	−	0.872425i	$-0.337454\pi$
$307$	5258.02	0.977495	0.488747	+	0.872425i	$0.337454\pi$
$308$	5096.74	0.942902
$309$	0	0
$310$	85.0757	0.0155870
$311$	−5231.31	−0.953828	−0.476914	−	0.878950i	$-0.658245\pi$
$311$	−5231.31	−0.953828	−0.476914	+	0.878950i	$0.658245\pi$
$312$	0	0
$313$	−3992.66	−0.721017	−0.360509	−	0.932756i	$-0.617397\pi$
$313$	−3992.66	−0.721017	−0.360509	+	0.932756i	$0.617397\pi$
$314$	194.003	0.0348670
$315$	0	0
$316$	5300.63	0.943619
$317$	−4733.58	−0.838688	−0.419344	−	0.907827i	$-0.637740\pi$
$317$	−4733.58	−0.838688	−0.419344	+	0.907827i	$0.637740\pi$
$318$	0	0
$319$	−2185.13	−0.383524
$320$	−1033.01	−0.180460
$321$	0	0
$322$	6086.94	1.05345
$323$	1371.72	0.236299
$324$	0	0
$325$	−487.201	−0.0831539
$326$	−2215.39	−0.376378
$327$	0	0
$328$	9425.44	1.58669
$329$	8088.85	1.35548
$330$	0	0
$331$	9369.84	1.55593	0.777965	−	0.628308i	$-0.216252\pi$
$331$	9369.84	1.55593	0.777965	+	0.628308i	$0.216252\pi$
$332$	2891.42	0.477973
$333$	0	0
$334$	−3183.92	−0.521605
$335$	−1084.45	−0.176865
$336$	0	0
$337$	5448.83	0.880762	0.440381	−	0.897811i	$-0.354843\pi$
$337$	5448.83	0.880762	0.440381	+	0.897811i	$0.354843\pi$
$338$	−2853.41	−0.459187
$339$	0	0
$340$	805.903	0.128548
$341$	291.451	0.0462843
$342$	0	0
$343$	16653.0	2.62150
$344$	−7214.93	−1.13082
$345$	0	0
$346$	4245.81	0.659699
$347$	−3897.55	−0.602972	−0.301486	−	0.953471i	$-0.597483\pi$
$347$	−3897.55	−0.602972	−0.301486	+	0.953471i	$0.597483\pi$
$348$	0	0
$349$	−6259.06	−0.960000	−0.480000	−	0.877269i	$-0.659363\pi$
$349$	−6259.06	−0.960000	−0.480000	+	0.877269i	$0.659363\pi$
$350$	1344.09	0.205270
$351$	0	0
$352$	−5033.30	−0.762147
$353$	342.866	0.0516967	0.0258483	−	0.999666i	$-0.491771\pi$
$353$	342.866	0.0516967	0.0258483	+	0.999666i	$0.491771\pi$
$354$	0	0
$355$	−327.158	−0.0489119
$356$	8867.50	1.32016
$357$	0	0
$358$	−1178.75	−0.174019
$359$	10448.9	1.53613	0.768066	−	0.640371i	$-0.221219\pi$
$359$	10448.9	1.53613	0.768066	+	0.640371i	$0.221219\pi$
$360$	0	0
$361$	−4640.53	−0.676561
$362$	−1780.75	−0.258547
$363$	0	0
$364$	3692.95	0.531766
$365$	−3559.05	−0.510382
$366$	0	0
$367$	−5173.73	−0.735876	−0.367938	−	0.929850i	$-0.619936\pi$
$367$	−5173.73	−0.735876	−0.367938	+	0.929850i	$0.619936\pi$
$368$	1234.70	0.174900
$369$	0	0
$370$	−3218.95	−0.452284
$371$	20873.7	2.92105
$372$	0	0
$373$	2101.07	0.291660	0.145830	−	0.989310i	$-0.453415\pi$
$373$	2101.07	0.291660	0.145830	+	0.989310i	$0.453415\pi$
$374$	−1229.94	−0.170050
$375$	0	0
$376$	−5020.58	−0.688608
$377$	−1583.28	−0.216295
$378$	0	0
$379$	11242.6	1.52374	0.761868	−	0.647732i	$-0.224282\pi$
$379$	11242.6	1.52374	0.761868	+	0.647732i	$0.224282\pi$
$380$	1303.38	0.175952
$381$	0	0
$382$	−840.462	−0.112570
$383$	6794.87	0.906532	0.453266	−	0.891375i	$-0.350259\pi$
$383$	6794.87	0.906532	0.453266	+	0.891375i	$0.350259\pi$
$384$	0	0
$385$	4604.57	0.609533
$386$	4119.52	0.543207
$387$	0	0
$388$	−4438.28	−0.580721
$389$	7049.40	0.918815	0.459407	−	0.888226i	$-0.348062\pi$
$389$	7049.40	0.918815	0.459407	+	0.888226i	$0.348062\pi$
$390$	0	0
$391$	3297.23	0.426466
$392$	−17625.5	−2.27098
$393$	0	0
$394$	4638.52	0.593110
$395$	4788.76	0.609997
$396$	0	0
$397$	3011.36	0.380694	0.190347	−	0.981717i	$-0.439039\pi$
$397$	3011.36	0.380694	0.190347	+	0.981717i	$0.439039\pi$
$398$	4144.32	0.521950
$399$	0	0
$400$	272.640	0.0340801
$401$	−4168.15	−0.519071	−0.259535	−	0.965734i	$-0.583569\pi$
$401$	−4168.15	−0.519071	−0.259535	+	0.965734i	$0.583569\pi$
$402$	0	0
$403$	211.177	0.0261029
$404$	−3243.91	−0.399481
$405$	0	0
$406$	4367.96	0.533937
$407$	−11027.4	−1.34302
$408$	0	0
$409$	−12605.4	−1.52395	−0.761974	−	0.647608i	$-0.775769\pi$
$409$	−12605.4	−1.52395	−0.761974	+	0.647608i	$0.775769\pi$
$410$	3482.02	0.419426
$411$	0	0
$412$	4541.98	0.543124
$413$	532.272	0.0634175
$414$	0	0
$415$	2612.20	0.308983
$416$	−3646.98	−0.429826
$417$	0	0
$418$	−1989.17	−0.232759
$419$	−5144.47	−0.599819	−0.299909	−	0.953968i	$-0.596956\pi$
$419$	−5144.47	−0.599819	−0.299909	+	0.953968i	$0.596956\pi$
$420$	0	0
$421$	−3370.07	−0.390136	−0.195068	−	0.980790i	$-0.562493\pi$
$421$	−3370.07	−0.390136	−0.195068	+	0.980790i	$0.562493\pi$
$422$	−3125.74	−0.360566
$423$	0	0
$424$	−12955.9	−1.48395
$425$	728.080	0.0830990
$426$	0	0
$427$	2090.37	0.236909
$428$	7517.61	0.849013
$429$	0	0
$430$	−2665.40	−0.298923
$431$	14948.3	1.67061	0.835306	−	0.549785i	$-0.185290\pi$
$431$	14948.3	1.67061	0.835306	+	0.549785i	$0.185290\pi$
$432$	0	0
$433$	−14011.1	−1.55504	−0.777520	−	0.628858i	$-0.783523\pi$
$433$	−14011.1	−1.55504	−0.777520	+	0.628858i	$0.783523\pi$
$434$	−582.595	−0.0644365
$435$	0	0
$436$	−2709.08	−0.297572
$437$	5332.57	0.583734
$438$	0	0
$439$	−4782.19	−0.519913	−0.259956	−	0.965620i	$-0.583708\pi$
$439$	−4782.19	−0.519913	−0.259956	+	0.965620i	$0.583708\pi$
$440$	−2857.96	−0.309654
$441$	0	0
$442$	−891.178	−0.0959027
$443$	4774.01	0.512009	0.256005	−	0.966676i	$-0.417594\pi$
$443$	4774.01	0.512009	0.256005	+	0.966676i	$0.417594\pi$
$444$	0	0
$445$	8011.20	0.853409
$446$	−1059.73	−0.112511
$447$	0	0
$448$	7074.04	0.746020
$449$	−855.812	−0.0899516	−0.0449758	−	0.998988i	$-0.514321\pi$
$449$	−855.812	−0.0899516	−0.0449758	+	0.998988i	$0.514321\pi$
$450$	0	0
$451$	11928.7	1.24545
$452$	−4414.62	−0.459395
$453$	0	0
$454$	2970.60	0.307087
$455$	3336.33	0.343757
$456$	0	0
$457$	−13418.0	−1.37345	−0.686725	−	0.726917i	$-0.740952\pi$
$457$	−13418.0	−1.37345	−0.686725	+	0.726917i	$0.740952\pi$
$458$	5352.08	0.546040
$459$	0	0
$460$	3132.96	0.317554
$461$	−15886.0	−1.60495	−0.802477	−	0.596682i	$-0.796485\pi$
$461$	−15886.0	−1.60495	−0.802477	+	0.596682i	$0.796485\pi$
$462$	0	0
$463$	6561.38	0.658603	0.329302	−	0.944225i	$-0.393187\pi$
$463$	6561.38	0.658603	0.329302	+	0.944225i	$0.393187\pi$
$464$	886.015	0.0886470
$465$	0	0
$466$	−4462.32	−0.443590
$467$	−6792.37	−0.673048	−0.336524	−	0.941675i	$-0.609251\pi$
$467$	−6792.37	−0.673048	−0.336524	+	0.941675i	$0.609251\pi$
$468$	0	0
$469$	7426.25	0.731156
$470$	−1854.74	−0.182027
$471$	0	0
$472$	−330.370	−0.0322172
$473$	−9131.08	−0.887626
$474$	0	0
$475$	1177.51	0.113743
$476$	−5518.79	−0.531415
$477$	0	0
$478$	2543.12	0.243346
$479$	17652.5	1.68385	0.841923	−	0.539598i	$-0.181424\pi$
$479$	17652.5	1.68385	0.841923	+	0.539598i	$0.181424\pi$
$480$	0	0
$481$	−7990.14	−0.757420
$482$	41.9258	0.00396196
$483$	0	0
$484$	3362.76	0.315811
$485$	−4009.69	−0.375403
$486$	0	0
$487$	−5318.16	−0.494844	−0.247422	−	0.968908i	$-0.579583\pi$
$487$	−5318.16	−0.494844	−0.247422	+	0.968908i	$0.579583\pi$
$488$	−1297.45	−0.120354
$489$	0	0
$490$	−6511.36	−0.600313
$491$	10870.0	0.999092	0.499546	−	0.866287i	$-0.333500\pi$
$491$	10870.0	0.999092	0.499546	+	0.866287i	$0.333500\pi$
$492$	0	0
$493$	2366.08	0.216152
$494$	−1441.29	−0.131269
$495$	0	0
$496$	−118.176	−0.0106981
$497$	2240.36	0.202201
$498$	0	0
$499$	10370.5	0.930355	0.465178	−	0.885217i	$-0.345990\pi$
$499$	10370.5	0.930355	0.465178	+	0.885217i	$0.345990\pi$
$500$	691.805	0.0618770
$501$	0	0
$502$	−671.035	−0.0596608
$503$	−12396.4	−1.09886	−0.549432	−	0.835538i	$-0.685156\pi$
$503$	−12396.4	−1.09886	−0.549432	+	0.835538i	$0.685156\pi$
$504$	0	0
$505$	−2930.65	−0.258242
$506$	−4781.40	−0.420078
$507$	0	0
$508$	−9564.73	−0.835367
$509$	5561.04	0.484261	0.242130	−	0.970244i	$-0.422154\pi$
$509$	5561.04	0.484261	0.242130	+	0.970244i	$0.422154\pi$
$510$	0	0
$511$	24372.2	2.10991
$512$	3894.99	0.336203
$513$	0	0
$514$	6032.03	0.517629
$515$	4103.37	0.351100
$516$	0	0
$517$	−6353.95	−0.540515
$518$	22043.2	1.86974
$519$	0	0
$520$	−2070.79	−0.174635
$521$	−2613.39	−0.219760	−0.109880	−	0.993945i	$-0.535047\pi$
$521$	−2613.39	−0.219760	−0.109880	+	0.993945i	$0.535047\pi$
$522$	0	0
$523$	−2927.73	−0.244781	−0.122391	−	0.992482i	$-0.539056\pi$
$523$	−2927.73	−0.244781	−0.122391	+	0.992482i	$0.539056\pi$
$524$	−3537.07	−0.294881
$525$	0	0
$526$	−1254.40	−0.103982
$527$	−315.586	−0.0260856
$528$	0	0
$529$	651.012	0.0535064
$530$	−4786.26	−0.392268
$531$	0	0
$532$	−8925.48	−0.727384
$533$	8643.14	0.702394
$534$	0	0
$535$	6791.66	0.548840
$536$	−4609.31	−0.371440
$537$	0	0
$538$	12458.7	0.998388
$539$	−22306.5	−1.78258
$540$	0	0
$541$	−4023.02	−0.319710	−0.159855	−	0.987140i	$-0.551103\pi$
$541$	−4023.02	−0.319710	−0.159855	+	0.987140i	$0.551103\pi$
$542$	−3625.76	−0.287343
$543$	0	0
$544$	5450.10	0.429542
$545$	−2447.47	−0.192364
$546$	0	0
$547$	3305.54	0.258382	0.129191	−	0.991620i	$-0.458762\pi$
$547$	3305.54	0.258382	0.129191	+	0.991620i	$0.458762\pi$
$548$	15695.4	1.22349
$549$	0	0
$550$	−1055.81	−0.0818542
$551$	3826.64	0.295862
$552$	0	0
$553$	−32793.2	−2.52172
$554$	1195.40	0.0916748
$555$	0	0
$556$	−968.507	−0.0738738
$557$	6472.87	0.492396	0.246198	−	0.969220i	$-0.420819\pi$
$557$	6472.87	0.492396	0.246198	+	0.969220i	$0.420819\pi$
$558$	0	0
$559$	−6616.10	−0.500593
$560$	−1867.03	−0.140886
$561$	0	0
$562$	−2021.29	−0.151714
$563$	−20555.3	−1.53872	−0.769362	−	0.638813i	$-0.779426\pi$
$563$	−20555.3	−1.53872	−0.769362	+	0.638813i	$0.779426\pi$
$564$	0	0
$565$	−3988.32	−0.296973
$566$	−4281.18	−0.317936
$567$	0	0
$568$	−1390.54	−0.102722
$569$	1270.62	0.0936156	0.0468078	−	0.998904i	$-0.485095\pi$
$569$	1270.62	0.0936156	0.0468078	+	0.998904i	$0.485095\pi$
$570$	0	0
$571$	−16753.3	−1.22785	−0.613927	−	0.789363i	$-0.710411\pi$
$571$	−16753.3	−1.22785	−0.613927	+	0.789363i	$0.710411\pi$
$572$	−2900.88	−0.212049
$573$	0	0
$574$	−23844.7	−1.73390
$575$	2830.42	0.205281
$576$	0	0
$577$	−7800.01	−0.562770	−0.281385	−	0.959595i	$-0.590794\pi$
$577$	−7800.01	−0.562770	−0.281385	+	0.959595i	$0.590794\pi$
$578$	−6382.65	−0.459313
$579$	0	0
$580$	2248.20	0.160951
$581$	−17888.2	−1.27733
$582$	0	0
$583$	−16396.7	−1.16481
$584$	−15127.3	−1.07187
$585$	0	0
$586$	−151.327	−0.0106677
$587$	−2369.25	−0.166592	−0.0832958	−	0.996525i	$-0.526545\pi$
$587$	−2369.25	−0.166592	−0.0832958	+	0.996525i	$0.526545\pi$
$588$	0	0
$589$	−510.393	−0.0357052
$590$	−122.048	−0.00851633
$591$	0	0
$592$	4471.33	0.310423
$593$	11974.3	0.829218	0.414609	−	0.910000i	$-0.363918\pi$
$593$	11974.3	0.829218	0.414609	+	0.910000i	$0.363918\pi$
$594$	0	0
$595$	−4985.86	−0.343530
$596$	−5664.68	−0.389319
$597$	0	0
$598$	−3464.46	−0.236910
$599$	22199.5	1.51427	0.757134	−	0.653260i	$-0.226599\pi$
$599$	22199.5	1.51427	0.757134	+	0.653260i	$0.226599\pi$
$600$	0	0
$601$	−11498.9	−0.780452	−0.390226	−	0.920719i	$-0.627603\pi$
$601$	−11498.9	−0.780452	−0.390226	+	0.920719i	$0.627603\pi$
$602$	18252.5	1.23574
$603$	0	0
$604$	4837.08	0.325858
$605$	3038.03	0.204154
$606$	0	0
$607$	−15273.0	−1.02127	−0.510636	−	0.859797i	$-0.670590\pi$
$607$	−15273.0	−1.02127	−0.510636	+	0.859797i	$0.670590\pi$
$608$	8814.38	0.587944
$609$	0	0
$610$	−479.314	−0.0318145
$611$	−4603.88	−0.304833
$612$	0	0
$613$	−16440.8	−1.08326	−0.541630	−	0.840617i	$-0.682192\pi$
$613$	−16440.8	−1.08326	−0.541630	+	0.840617i	$0.682192\pi$
$614$	8256.19	0.542659
$615$	0	0
$616$	19571.2	1.28010
$617$	9359.13	0.610672	0.305336	−	0.952245i	$-0.401231\pi$
$617$	9359.13	0.610672	0.305336	+	0.952245i	$0.401231\pi$
$618$	0	0
$619$	−25484.6	−1.65478	−0.827392	−	0.561625i	$-0.810176\pi$
$619$	−25484.6	−1.65478	−0.827392	+	0.561625i	$0.810176\pi$
$620$	−299.862	−0.0194238
$621$	0	0
$622$	−8214.25	−0.529520
$623$	−54860.3	−3.52798
$624$	0	0
$625$	625.000	0.0400000
$626$	−6269.31	−0.400275
$627$	0	0
$628$	−683.794	−0.0434496
$629$	11940.6	0.756919
$630$	0	0
$631$	−21733.8	−1.37117	−0.685585	−	0.727993i	$-0.740454\pi$
$631$	−21733.8	−1.37117	−0.685585	+	0.727993i	$0.740454\pi$
$632$	20354.1	1.28108
$633$	0	0
$634$	−7432.71	−0.465600
$635$	−8641.10	−0.540018
$636$	0	0
$637$	−16162.6	−1.00532
$638$	−3431.12	−0.212914
$639$	0	0
$640$	5863.52	0.362150
$641$	−20324.8	−1.25239	−0.626195	−	0.779666i	$-0.715389\pi$
$641$	−20324.8	−1.25239	−0.626195	+	0.779666i	$0.715389\pi$
$642$	0	0
$643$	6727.44	0.412604	0.206302	−	0.978488i	$-0.433857\pi$
$643$	6727.44	0.412604	0.206302	+	0.978488i	$0.433857\pi$
$644$	−21454.3	−1.31276
$645$	0	0
$646$	2153.89	0.131182
$647$	6724.69	0.408616	0.204308	−	0.978907i	$-0.434506\pi$
$647$	6724.69	0.408616	0.204308	+	0.978907i	$0.434506\pi$
$648$	0	0
$649$	−418.110	−0.0252885
$650$	−765.007	−0.0461631
$651$	0	0
$652$	7808.49	0.469025
$653$	−30796.3	−1.84556	−0.922782	−	0.385321i	$-0.874091\pi$
$653$	−30796.3	−1.84556	−0.922782	+	0.385321i	$0.874091\pi$
$654$	0	0
$655$	−3195.51	−0.190624
$656$	−4836.75	−0.287871
$657$	0	0
$658$	12701.2	0.752498
$659$	16534.3	0.977368	0.488684	−	0.872461i	$-0.337477\pi$
$659$	16534.3	0.977368	0.488684	+	0.872461i	$0.337477\pi$
$660$	0	0
$661$	1937.05	0.113983	0.0569914	−	0.998375i	$-0.481849\pi$
$661$	1937.05	0.113983	0.0569914	+	0.998375i	$0.481849\pi$
$662$	14712.6	0.863779
$663$	0	0
$664$	11102.8	0.648907
$665$	−8063.57	−0.470213
$666$	0	0
$667$	9198.16	0.533964
$668$	11222.2	0.650000
$669$	0	0
$670$	−1702.81	−0.0981869
$671$	−1642.03	−0.0944705
$672$	0	0
$673$	−460.588	−0.0263809	−0.0131905	−	0.999913i	$-0.504199\pi$
$673$	−460.588	−0.0263809	−0.0131905	+	0.999913i	$0.504199\pi$
$674$	8555.80	0.488957
$675$	0	0
$676$	10057.3	0.572217
$677$	29059.6	1.64971	0.824854	−	0.565346i	$-0.191257\pi$
$677$	29059.6	1.64971	0.824854	+	0.565346i	$0.191257\pi$
$678$	0	0
$679$	27458.2	1.55191
$680$	3094.62	0.174519
$681$	0	0
$682$	457.639	0.0256949
$683$	14366.9	0.804881	0.402440	−	0.915446i	$-0.368162\pi$
$683$	14366.9	0.804881	0.402440	+	0.915446i	$0.368162\pi$
$684$	0	0
$685$	14179.7	0.790918
$686$	26148.6	1.45533
$687$	0	0
$688$	3702.41	0.205164
$689$	−11880.6	−0.656913
$690$	0	0
$691$	−12707.9	−0.699613	−0.349807	−	0.936822i	$-0.613753\pi$
$691$	−12707.9	−0.699613	−0.349807	+	0.936822i	$0.613753\pi$
$692$	−14965.0	−0.822087
$693$	0	0
$694$	−6119.97	−0.334742
$695$	−874.981	−0.0477553
$696$	0	0
$697$	−12916.4	−0.701930
$698$	−9828.04	−0.532946
$699$	0	0
$700$	−4737.45	−0.255798
$701$	5959.06	0.321071	0.160535	−	0.987030i	$-0.448678\pi$
$701$	5959.06	0.321071	0.160535	+	0.987030i	$0.448678\pi$
$702$	0	0
$703$	19311.4	1.03605
$704$	−5556.79	−0.297485
$705$	0	0
$706$	538.372	0.0286995
$707$	20069.0	1.06757
$708$	0	0
$709$	451.825	0.0239332	0.0119666	−	0.999928i	$-0.496191\pi$
$709$	451.825	0.0239332	0.0119666	+	0.999928i	$0.496191\pi$
$710$	−513.706	−0.0271536
$711$	0	0
$712$	34050.6	1.79228
$713$	−1226.84	−0.0644398
$714$	0	0
$715$	−2620.75	−0.137078
$716$	4154.67	0.216854
$717$	0	0
$718$	16406.9	0.852788
$719$	−13488.4	−0.699627	−0.349814	−	0.936819i	$-0.613755\pi$
$719$	−13488.4	−0.699627	−0.349814	+	0.936819i	$0.613755\pi$
$720$	0	0
$721$	−28099.7	−1.45144
$722$	−7286.61	−0.375595
$723$	0	0
$724$	6276.51	0.322189
$725$	2031.10	0.104046
$726$	0	0
$727$	29183.1	1.48878	0.744389	−	0.667746i	$-0.232741\pi$
$727$	29183.1	1.48878	0.744389	+	0.667746i	$0.232741\pi$
$728$	14180.7	0.721938
$729$	0	0
$730$	−5588.46	−0.283340
$731$	9887.20	0.500262
$732$	0	0
$733$	12512.5	0.630504	0.315252	−	0.949008i	$-0.397911\pi$
$733$	12512.5	0.630504	0.315252	+	0.949008i	$0.397911\pi$
$734$	−8123.84	−0.408524
$735$	0	0
$736$	21187.3	1.06111
$737$	−5833.46	−0.291558
$738$	0	0
$739$	−32248.2	−1.60524	−0.802619	−	0.596492i	$-0.796561\pi$
$739$	−32248.2	−1.60524	−0.802619	+	0.596492i	$0.796561\pi$
$740$	11345.7	0.563615
$741$	0	0
$742$	32776.1	1.62163
$743$	18489.0	0.912914	0.456457	−	0.889745i	$-0.349118\pi$
$743$	18489.0	0.912914	0.456457	+	0.889745i	$0.349118\pi$
$744$	0	0
$745$	−5117.66	−0.251673
$746$	3299.12	0.161916
$747$	0	0
$748$	4335.12	0.211908
$749$	−46509.0	−2.26889
$750$	0	0
$751$	−2052.20	−0.0997151	−0.0498575	−	0.998756i	$-0.515877\pi$
$751$	−2052.20	−0.0997151	−0.0498575	+	0.998756i	$0.515877\pi$
$752$	2576.36	0.124934
$753$	0	0
$754$	−2486.08	−0.120077
$755$	4369.98	0.210649
$756$	0	0
$757$	10118.9	0.485834	0.242917	−	0.970047i	$-0.421896\pi$
$757$	10118.9	0.485834	0.242917	+	0.970047i	$0.421896\pi$
$758$	17653.3	0.845906
$759$	0	0
$760$	5004.89	0.238877
$761$	27304.0	1.30062	0.650309	−	0.759670i	$-0.274640\pi$
$761$	27304.0	1.30062	0.650309	+	0.759670i	$0.274640\pi$
$762$	0	0
$763$	16760.2	0.795229
$764$	2962.34	0.140280
$765$	0	0
$766$	10669.4	0.503264
$767$	−302.950	−0.0142619
$768$	0	0
$769$	−22473.8	−1.05387	−0.526935	−	0.849905i	$-0.676659\pi$
$769$	−22473.8	−1.05387	−0.526935	+	0.849905i	$0.676659\pi$
$770$	7230.13	0.338384
$771$	0	0
$772$	−14519.9	−0.676920
$773$	−35405.6	−1.64741	−0.823707	−	0.567015i	$-0.808098\pi$
$773$	−35405.6	−1.64741	−0.823707	+	0.567015i	$0.808098\pi$
$774$	0	0
$775$	−270.906	−0.0125564
$776$	−17042.7	−0.788399
$777$	0	0
$778$	11069.0	0.510083
$779$	−20889.6	−0.960780
$780$	0	0
$781$	−1759.85	−0.0806303
$782$	5177.34	0.236754
$783$	0	0
$784$	9044.71	0.412022
$785$	−617.762	−0.0280877
$786$	0	0
$787$	−12180.0	−0.551680	−0.275840	−	0.961204i	$-0.588956\pi$
$787$	−12180.0	−0.551680	−0.275840	+	0.961204i	$0.588956\pi$
$788$	−16349.2	−0.739106
$789$	0	0
$790$	7519.36	0.338641
$791$	27311.8	1.22768
$792$	0	0
$793$	−1189.76	−0.0532783
$794$	4728.46	0.211344
$795$	0	0
$796$	−14607.3	−0.650430
$797$	2293.08	0.101913	0.0509567	−	0.998701i	$-0.483773\pi$
$797$	2293.08	0.101913	0.0509567	+	0.998701i	$0.483773\pi$
$798$	0	0
$799$	6880.10	0.304632
$800$	4678.48	0.206762
$801$	0	0
$802$	−6544.86	−0.288164
$803$	−19144.9	−0.841354
$804$	0	0
$805$	−19382.6	−0.848628
$806$	331.591	0.0144911
$807$	0	0
$808$	−12456.4	−0.542344
$809$	−25264.0	−1.09794	−0.548972	−	0.835841i	$-0.684981\pi$
$809$	−25264.0	−1.09794	−0.548972	+	0.835841i	$0.684981\pi$
$810$	0	0
$811$	18781.2	0.813190	0.406595	−	0.913608i	$-0.366716\pi$
$811$	18781.2	0.813190	0.406595	+	0.913608i	$0.366716\pi$
$812$	−15395.6	−0.665368
$813$	0	0
$814$	−17315.4	−0.745581
$815$	7054.45	0.303198
$816$	0	0
$817$	15990.5	0.684743
$818$	−19793.0	−0.846023
$819$	0	0
$820$	−12272.9	−0.522669
$821$	31574.8	1.34223	0.671114	−	0.741354i	$-0.265816\pi$
$821$	31574.8	1.34223	0.671114	+	0.741354i	$0.265816\pi$
$822$	0	0
$823$	−946.258	−0.0400784	−0.0200392	−	0.999799i	$-0.506379\pi$
$823$	−946.258	−0.0400784	−0.0200392	+	0.999799i	$0.506379\pi$
$824$	17440.9	0.737358
$825$	0	0
$826$	835.779	0.0352064
$827$	35044.7	1.47355	0.736774	−	0.676139i	$-0.236348\pi$
$827$	35044.7	1.47355	0.736774	+	0.676139i	$0.236348\pi$
$828$	0	0
$829$	31029.2	1.29999	0.649993	−	0.759940i	$-0.274772\pi$
$829$	31029.2	1.29999	0.649993	+	0.759940i	$0.274772\pi$
$830$	4101.70	0.171533
$831$	0	0
$832$	−4026.28	−0.167772
$833$	24153.7	1.00465
$834$	0	0
$835$	10138.5	0.420189
$836$	7011.13	0.290054
$837$	0	0
$838$	−8077.90	−0.332991
$839$	−21199.9	−0.872350	−0.436175	−	0.899862i	$-0.643667\pi$
$839$	−21199.9	−0.872350	−0.436175	+	0.899862i	$0.643667\pi$
$840$	0	0
$841$	−17788.4	−0.729363
$842$	−5291.71	−0.216585
$843$	0	0
$844$	11017.2	0.449320
$845$	9086.08	0.369906
$846$	0	0
$847$	−20804.3	−0.843971
$848$	6648.43	0.269231
$849$	0	0
$850$	1143.24	0.0461326
$851$	46419.1	1.86983
$852$	0	0
$853$	10979.9	0.440731	0.220365	−	0.975417i	$-0.429275\pi$
$853$	10979.9	0.440731	0.220365	+	0.975417i	$0.429275\pi$
$854$	3282.32	0.131521
$855$	0	0
$856$	28867.2	1.15264
$857$	44814.4	1.78627	0.893133	−	0.449793i	$-0.148502\pi$
$857$	44814.4	1.78627	0.893133	+	0.449793i	$0.148502\pi$
$858$	0	0
$859$	37386.2	1.48498	0.742492	−	0.669855i	$-0.233644\pi$
$859$	37386.2	1.48498	0.742492	+	0.669855i	$0.233644\pi$
$860$	9394.60	0.372504
$861$	0	0
$862$	23471.9	0.927445
$863$	−30536.3	−1.20448	−0.602241	−	0.798314i	$-0.705725\pi$
$863$	−30536.3	−1.20448	−0.602241	+	0.798314i	$0.705725\pi$
$864$	0	0
$865$	−13519.9	−0.531433
$866$	−22000.4	−0.863285
$867$	0	0
$868$	2053.44	0.0802978
$869$	25759.7	1.00557
$870$	0	0
$871$	−4226.75	−0.164429
$872$	−10402.7	−0.403991
$873$	0	0
$874$	8373.25	0.324061
$875$	−4279.97	−0.165359
$876$	0	0
$877$	34327.5	1.32173	0.660864	−	0.750506i	$-0.270190\pi$
$877$	34327.5	1.32173	0.660864	+	0.750506i	$0.270190\pi$
$878$	−7509.04	−0.288631
$879$	0	0
$880$	1466.59	0.0561803
$881$	−18009.6	−0.688718	−0.344359	−	0.938838i	$-0.611904\pi$
$881$	−18009.6	−0.688718	−0.344359	+	0.938838i	$0.611904\pi$
$882$	0	0
$883$	−29074.4	−1.10808	−0.554039	−	0.832491i	$-0.686914\pi$
$883$	−29074.4	−1.10808	−0.554039	+	0.832491i	$0.686914\pi$
$884$	3141.09	0.119510
$885$	0	0
$886$	7496.20	0.284243
$887$	−32902.6	−1.24550	−0.622752	−	0.782420i	$-0.713985\pi$
$887$	−32902.6	−1.24550	−0.622752	+	0.782420i	$0.713985\pi$
$888$	0	0
$889$	59173.9	2.23243
$890$	12579.3	0.473772
$891$	0	0
$892$	3735.20	0.140206
$893$	11127.1	0.416970
$894$	0	0
$895$	3753.47	0.140184
$896$	−40153.2	−1.49712
$897$	0	0
$898$	−1343.80	−0.0499369
$899$	−880.377	−0.0326610
$900$	0	0
$901$	17754.5	0.656479
$902$	18730.5	0.691415
$903$	0	0
$904$	−16951.9	−0.623684
$905$	5670.41	0.208277
$906$	0	0
$907$	−815.815	−0.0298663	−0.0149331	−	0.999888i	$-0.504754\pi$
$907$	−815.815	−0.0298663	−0.0149331	+	0.999888i	$0.504754\pi$
$908$	−10470.4	−0.382677
$909$	0	0
$910$	5238.73	0.190838
$911$	15842.8	0.576176	0.288088	−	0.957604i	$-0.406980\pi$
$911$	15842.8	0.576176	0.288088	+	0.957604i	$0.406980\pi$
$912$	0	0
$913$	14051.5	0.509352
$914$	−21069.0	−0.762474
$915$	0	0
$916$	−18864.2	−0.680449
$917$	21882.7	0.788037
$918$	0	0
$919$	23848.4	0.856025	0.428013	−	0.903773i	$-0.359214\pi$
$919$	23848.4	0.856025	0.428013	+	0.903773i	$0.359214\pi$
$920$	12030.3	0.431118
$921$	0	0
$922$	−24944.3	−0.890995
$923$	−1275.13	−0.0454729
$924$	0	0
$925$	10250.1	0.364346
$926$	10302.7	0.365625
$927$	0	0
$928$	15203.9	0.537816
$929$	−39592.0	−1.39825	−0.699123	−	0.715001i	$-0.746426\pi$
$929$	−39592.0	−1.39825	−0.699123	+	0.715001i	$0.746426\pi$
$930$	0	0
$931$	39063.5	1.37514
$932$	15728.1	0.552781
$933$	0	0
$934$	−10665.4	−0.373644
$935$	3916.49	0.136987
$936$	0	0
$937$	10255.0	0.357540	0.178770	−	0.983891i	$-0.442788\pi$
$937$	10255.0	0.357540	0.178770	+	0.983891i	$0.442788\pi$
$938$	11660.8	0.405903
$939$	0	0
$940$	6537.32	0.226834
$941$	−23894.3	−0.827771	−0.413886	−	0.910329i	$-0.635829\pi$
$941$	−23894.3	−0.827771	−0.413886	+	0.910329i	$0.635829\pi$
$942$	0	0
$943$	−50212.8	−1.73399
$944$	169.533	0.00584514
$945$	0	0
$946$	−14337.7	−0.492768
$947$	−36257.3	−1.24414	−0.622072	−	0.782960i	$-0.713709\pi$
$947$	−36257.3	−1.24414	−0.622072	+	0.782960i	$0.713709\pi$
$948$	0	0
$949$	−13871.8	−0.474497
$950$	1848.94	0.0631449
$951$	0	0
$952$	−21191.8	−0.721460
$953$	−46736.7	−1.58861	−0.794307	−	0.607516i	$-0.792166\pi$
$953$	−46736.7	−1.58861	−0.794307	+	0.607516i	$0.792166\pi$
$954$	0	0
$955$	2676.27	0.0906830
$956$	−8963.62	−0.303247
$957$	0	0
$958$	27718.1	0.934792
$959$	−97102.1	−3.26965
$960$	0	0
$961$	−29673.6	−0.996058
$962$	−12546.2	−0.420484
$963$	0	0
$964$	−147.774	−0.00493721
$965$	−13117.7	−0.437591
$966$	0	0
$967$	44357.0	1.47510	0.737552	−	0.675290i	$-0.235982\pi$
$967$	44357.0	1.47510	0.737552	+	0.675290i	$0.235982\pi$
$968$	12912.8	0.428752
$969$	0	0
$970$	−6296.05	−0.208406
$971$	−650.540	−0.0215003	−0.0107502	−	0.999942i	$-0.503422\pi$
$971$	−650.540	−0.0215003	−0.0107502	+	0.999942i	$0.503422\pi$
$972$	0	0
$973$	5991.83	0.197420
$974$	−8350.62	−0.274714
$975$	0	0
$976$	665.798	0.0218357
$977$	51675.0	1.69215	0.846075	−	0.533064i	$-0.178960\pi$
$977$	51675.0	1.69215	0.846075	+	0.533064i	$0.178960\pi$
$978$	0	0
$979$	43093.8	1.40683
$980$	22950.3	0.748082
$981$	0	0
$982$	17068.1	0.554649
$983$	23519.0	0.763111	0.381556	−	0.924346i	$-0.375388\pi$
$983$	23519.0	0.763111	0.381556	+	0.924346i	$0.375388\pi$
$984$	0	0
$985$	−14770.4	−0.477791
$986$	3715.24	0.119997
$987$	0	0
$988$	5080.05	0.163581
$989$	38436.6	1.23581
$990$	0	0
$991$	−27732.9	−0.888966	−0.444483	−	0.895787i	$-0.646613\pi$
$991$	−27732.9	−0.888966	−0.444483	+	0.895787i	$0.646613\pi$
$992$	−2027.88	−0.0649046
$993$	0	0
$994$	3517.83	0.112252
$995$	−13196.7	−0.420467
$996$	0	0
$997$	−34327.8	−1.09044	−0.545221	−	0.838292i	$-0.683554\pi$
$997$	−34327.8	−1.09044	−0.545221	+	0.838292i	$0.683554\pi$
$998$	16283.9	0.516489
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.m.1.5		7
3.2	odd	2		405.4.a.n.1.3		7
5.4	even	2		2025.4.a.bb.1.3		7
9.2	odd	6		135.4.e.c.91.5		14
9.4	even	3		45.4.e.c.16.3	✓	14
9.5	odd	6		135.4.e.c.46.5		14
9.7	even	3		45.4.e.c.31.3	yes	14
15.14	odd	2		2025.4.a.ba.1.5		7
45.4	even	6		225.4.e.d.151.5		14
45.7	odd	12		225.4.k.d.49.9		28
45.13	odd	12		225.4.k.d.124.9		28
45.22	odd	12		225.4.k.d.124.6		28
45.34	even	6		225.4.e.d.76.5		14
45.43	odd	12		225.4.k.d.49.6		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.3	✓	14	9.4	even	3
45.4.e.c.31.3	yes	14	9.7	even	3
135.4.e.c.46.5		14	9.5	odd	6
135.4.e.c.91.5		14	9.2	odd	6
225.4.e.d.76.5		14	45.34	even	6
225.4.e.d.151.5		14	45.4	even	6
225.4.k.d.49.6		28	45.43	odd	12
225.4.k.d.49.9		28	45.7	odd	12
225.4.k.d.124.6		28	45.22	odd	12
225.4.k.d.124.9		28	45.13	odd	12
405.4.a.m.1.5		7	1.1	even	1	trivial
405.4.a.n.1.3		7	3.2	odd	2
2025.4.a.ba.1.5		7	15.14	odd	2
2025.4.a.bb.1.3		7	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+1.57021 q^{2} -5.53444 q^{4} -5.00000 q^{5} +34.2398 q^{7} -21.2519 q^{8} +O(q^{10})\)	\(q+1.57021 q^{2} -5.53444 q^{4} -5.00000 q^{5} +34.2398 q^{7} -21.2519 q^{8} -7.85104 q^{10} -26.8960 q^{11} -19.4880 q^{13} +53.7636 q^{14} +10.9056 q^{16} +29.1232 q^{17} +47.1006 q^{19} +27.6722 q^{20} -42.2323 q^{22} +113.217 q^{23} +25.0000 q^{25} -30.6003 q^{26} -189.498 q^{28} +81.2439 q^{29} -10.8362 q^{31} +187.139 q^{32} +45.7295 q^{34} -171.199 q^{35} +410.002 q^{37} +73.9578 q^{38} +106.260 q^{40} -443.510 q^{41} +339.496 q^{43} +148.854 q^{44} +177.774 q^{46} +236.241 q^{47} +829.363 q^{49} +39.2552 q^{50} +107.855 q^{52} +609.634 q^{53} +134.480 q^{55} -727.661 q^{56} +127.570 q^{58} +15.5454 q^{59} +61.0509 q^{61} -17.0151 q^{62} +206.603 q^{64} +97.4401 q^{65} +216.889 q^{67} -161.181 q^{68} -268.818 q^{70} +65.4315 q^{71} +711.811 q^{73} +643.789 q^{74} -260.676 q^{76} -920.913 q^{77} -957.752 q^{79} -54.5281 q^{80} -696.404 q^{82} -522.440 q^{83} -145.616 q^{85} +533.079 q^{86} +571.591 q^{88} -1602.24 q^{89} -667.266 q^{91} -626.591 q^{92} +370.948 q^{94} -235.503 q^{95} +801.938 q^{97} +1302.27 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more