LMFDB - Embedded newform 405.4.a.m.1.7

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			1.7
Root			$-5.38503$ 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)

$f(q)$	$=$	$q+5.38503 q^{2} +20.9986 q^{4} -5.00000 q^{5} +12.5702 q^{7} +69.9976 q^{8} +O(q^{10})$	\(q+5.38503 q^{2} +20.9986 q^{4} -5.00000 q^{5} +12.5702 q^{7} +69.9976 q^{8} -26.9252 q^{10} +12.8307 q^{11} +59.8645 q^{13} +67.6909 q^{14} +208.951 q^{16} -110.011 q^{17} -12.0872 q^{19} -104.993 q^{20} +69.0936 q^{22} -67.7215 q^{23} +25.0000 q^{25} +322.372 q^{26} +263.956 q^{28} +199.958 q^{29} +76.6286 q^{31} +565.226 q^{32} -592.412 q^{34} -62.8510 q^{35} -22.4815 q^{37} -65.0898 q^{38} -349.988 q^{40} +87.7615 q^{41} +119.410 q^{43} +269.426 q^{44} -364.682 q^{46} +243.155 q^{47} -184.990 q^{49} +134.626 q^{50} +1257.07 q^{52} -293.518 q^{53} -64.1534 q^{55} +879.884 q^{56} +1076.78 q^{58} -581.383 q^{59} -773.693 q^{61} +412.648 q^{62} +1372.15 q^{64} -299.322 q^{65} -231.438 q^{67} -2310.07 q^{68} -338.455 q^{70} -744.342 q^{71} -264.839 q^{73} -121.063 q^{74} -253.813 q^{76} +161.284 q^{77} +559.717 q^{79} -1044.75 q^{80} +472.598 q^{82} -1220.89 q^{83} +550.054 q^{85} +643.027 q^{86} +898.117 q^{88} -255.905 q^{89} +752.509 q^{91} -1422.05 q^{92} +1309.40 q^{94} +60.4359 q^{95} +1049.52 q^{97} -996.177 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.38503	1.90390	0.951948	−	0.306260i	$-0.0990777\pi$
$2$	5.38503	1.90390	0.951948	+	0.306260i	$0.0990777\pi$
$3$	0	0
$3$	0	0
$4$	20.9986	2.62482
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	12.5702	0.678727	0.339363	−	0.940655i	$-0.389788\pi$
$7$	12.5702	0.678727	0.339363	+	0.940655i	$0.389788\pi$
$8$	69.9976	3.09349
$9$	0	0
$10$	−26.9252	−0.851448
$11$	12.8307	0.351691	0.175845	−	0.984418i	$-0.443734\pi$
$11$	12.8307	0.351691	0.175845	+	0.984418i	$0.443734\pi$
$12$	0	0
$13$	59.8645	1.27719	0.638593	−	0.769544i	$-0.279517\pi$
$13$	59.8645	1.27719	0.638593	+	0.769544i	$0.279517\pi$
$14$	67.6909	1.29223
$15$	0	0
$16$	208.951	3.26486
$17$	−110.011	−1.56950	−0.784752	−	0.619810i	$-0.787210\pi$
$17$	−110.011	−1.56950	−0.784752	+	0.619810i	$0.787210\pi$
$18$	0	0
$19$	−12.0872	−0.145947	−0.0729733	−	0.997334i	$-0.523249\pi$
$19$	−12.0872	−0.145947	−0.0729733	+	0.997334i	$0.523249\pi$
$20$	−104.993	−1.17385
$21$	0	0
$22$	69.0936	0.669582
$23$	−67.7215	−0.613953	−0.306976	−	0.951717i	$-0.599317\pi$
$23$	−67.7215	−0.613953	−0.306976	+	0.951717i	$0.599317\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	322.372	2.43163
$27$	0	0
$28$	263.956	1.78154
$29$	199.958	1.28039	0.640194	−	0.768213i	$-0.278854\pi$
$29$	199.958	1.28039	0.640194	+	0.768213i	$0.278854\pi$
$30$	0	0
$31$	76.6286	0.443965	0.221982	−	0.975051i	$-0.428747\pi$
$31$	76.6286	0.443965	0.221982	+	0.975051i	$0.428747\pi$
$32$	565.226	3.12246
$33$	0	0
$34$	−592.412	−2.98817
$35$	−62.8510	−0.303536
$36$	0	0
$37$	−22.4815	−0.0998900	−0.0499450	−	0.998752i	$-0.515905\pi$
$37$	−22.4815	−0.0998900	−0.0499450	+	0.998752i	$0.515905\pi$
$38$	−65.0898	−0.277867
$39$	0	0
$40$	−349.988	−1.38345
$41$	87.7615	0.334294	0.167147	−	0.985932i	$-0.446545\pi$
$41$	87.7615	0.334294	0.167147	+	0.985932i	$0.446545\pi$
$42$	0	0
$43$	119.410	0.423485	0.211743	−	0.977325i	$-0.432086\pi$
$43$	119.410	0.423485	0.211743	+	0.977325i	$0.432086\pi$
$44$	269.426	0.923124
$45$	0	0
$46$	−364.682	−1.16890
$47$	243.155	0.754635	0.377317	−	0.926084i	$-0.376847\pi$
$47$	243.155	0.754635	0.377317	+	0.926084i	$0.376847\pi$
$48$	0	0
$49$	−184.990	−0.539330
$50$	134.626	0.380779
$51$	0	0
$52$	1257.07	3.35238
$53$	−293.518	−0.760712	−0.380356	−	0.924840i	$-0.624199\pi$
$53$	−293.518	−0.760712	−0.380356	+	0.924840i	$0.624199\pi$
$54$	0	0
$55$	−64.1534	−0.157281
$56$	879.884	2.09963
$57$	0	0
$58$	1076.78	2.43773
$59$	−581.383	−1.28288	−0.641438	−	0.767175i	$-0.721662\pi$
$59$	−581.383	−1.28288	−0.641438	+	0.767175i	$0.721662\pi$
$60$	0	0
$61$	−773.693	−1.62395	−0.811977	−	0.583689i	$-0.801609\pi$
$61$	−773.693	−1.62395	−0.811977	+	0.583689i	$0.801609\pi$
$62$	412.648	0.845263
$63$	0	0
$64$	1372.15	2.67998
$65$	−299.322	−0.571175
$66$	0	0
$67$	−231.438	−0.422010	−0.211005	−	0.977485i	$-0.567674\pi$
$67$	−231.438	−0.422010	−0.211005	+	0.977485i	$0.567674\pi$
$68$	−2310.07	−4.11966
$69$	0	0
$70$	−338.455	−0.577901
$71$	−744.342	−1.24418	−0.622092	−	0.782944i	$-0.713717\pi$
$71$	−744.342	−1.24418	−0.622092	+	0.782944i	$0.713717\pi$
$72$	0	0
$73$	−264.839	−0.424616	−0.212308	−	0.977203i	$-0.568098\pi$
$73$	−264.839	−0.424616	−0.212308	+	0.977203i	$0.568098\pi$
$74$	−121.063	−0.190180
$75$	0	0
$76$	−253.813	−0.383084
$77$	161.284	0.238702
$78$	0	0
$79$	559.717	0.797127	0.398564	−	0.917141i	$-0.369509\pi$
$79$	559.717	0.797127	0.398564	+	0.917141i	$0.369509\pi$
$80$	−1044.75	−1.46009
$81$	0	0
$82$	472.598	0.636460
$83$	−1220.89	−1.61458	−0.807288	−	0.590158i	$-0.799065\pi$
$83$	−1220.89	−1.61458	−0.807288	+	0.590158i	$0.799065\pi$
$84$	0	0
$85$	550.054	0.701903
$86$	643.027	0.806272
$87$	0	0
$88$	898.117	1.08795
$89$	−255.905	−0.304785	−0.152392	−	0.988320i	$-0.548698\pi$
$89$	−255.905	−0.304785	−0.152392	+	0.988320i	$0.548698\pi$
$90$	0	0
$91$	752.509	0.866861
$92$	−1422.05	−1.61151
$93$	0	0
$94$	1309.40	1.43675
$95$	60.4359	0.0652693
$96$	0	0
$97$	1049.52	1.09858	0.549291	−	0.835631i	$-0.314898\pi$
$97$	1049.52	1.09858	0.549291	+	0.835631i	$0.314898\pi$
$98$	−996.177	−1.02683
$99$	0	0
$100$	524.964	0.524964
$101$	88.7270	0.0874125	0.0437062	−	0.999044i	$-0.486083\pi$
$101$	88.7270	0.0874125	0.0437062	+	0.999044i	$0.486083\pi$
$102$	0	0
$103$	−1544.07	−1.47710	−0.738552	−	0.674196i	$-0.764490\pi$
$103$	−1544.07	−1.47710	−0.738552	+	0.674196i	$0.764490\pi$
$104$	4190.37	3.95096
$105$	0	0
$106$	−1580.60	−1.44832
$107$	−585.772	−0.529240	−0.264620	−	0.964353i	$-0.585247\pi$
$107$	−585.772	−0.529240	−0.264620	+	0.964353i	$0.585247\pi$
$108$	0	0
$109$	1367.04	1.20127	0.600634	−	0.799524i	$-0.294915\pi$
$109$	1367.04	1.20127	0.600634	+	0.799524i	$0.294915\pi$
$110$	−345.468	−0.299446
$111$	0	0
$112$	2626.55	2.21595
$113$	170.429	0.141882	0.0709408	−	0.997481i	$-0.477400\pi$
$113$	170.429	0.141882	0.0709408	+	0.997481i	$0.477400\pi$
$114$	0	0
$115$	338.608	0.274568
$116$	4198.83	3.36079
$117$	0	0
$118$	−3130.77	−2.44246
$119$	−1382.86	−1.06526
$120$	0	0
$121$	−1166.37	−0.876314
$122$	−4166.36	−3.09184
$123$	0	0
$124$	1609.09	1.16533
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1809.74	1.26447	0.632237	−	0.774775i	$-0.282137\pi$
$127$	1809.74	1.26447	0.632237	+	0.774775i	$0.282137\pi$
$128$	2867.27	1.97995
$129$	0	0
$130$	−1611.86	−1.08746
$131$	−1238.29	−0.825879	−0.412940	−	0.910758i	$-0.635498\pi$
$131$	−1238.29	−0.825879	−0.412940	+	0.910758i	$0.635498\pi$
$132$	0	0
$133$	−151.938	−0.0990580
$134$	−1246.30	−0.803462
$135$	0	0
$136$	−7700.50	−4.85524
$137$	506.183	0.315665	0.157832	−	0.987466i	$-0.449549\pi$
$137$	506.183	0.315665	0.157832	+	0.987466i	$0.449549\pi$
$138$	0	0
$139$	−213.680	−0.130389	−0.0651946	−	0.997873i	$-0.520767\pi$
$139$	−213.680	−0.130389	−0.0651946	+	0.997873i	$0.520767\pi$
$140$	−1319.78	−0.796727
$141$	0	0
$142$	−4008.30	−2.36880
$143$	768.103	0.449175
$144$	0	0
$145$	−999.790	−0.572607
$146$	−1426.16	−0.808426
$147$	0	0
$148$	−472.078	−0.262193
$149$	−905.310	−0.497758	−0.248879	−	0.968535i	$-0.580062\pi$
$149$	−905.310	−0.497758	−0.248879	+	0.968535i	$0.580062\pi$
$150$	0	0
$151$	1358.67	0.732231	0.366116	−	0.930569i	$-0.380687\pi$
$151$	1358.67	0.732231	0.366116	+	0.930569i	$0.380687\pi$
$152$	−846.073	−0.451484
$153$	0	0
$154$	868.521	0.454464
$155$	−383.143	−0.198547
$156$	0	0
$157$	1901.34	0.966519	0.483259	−	0.875477i	$-0.339453\pi$
$157$	1901.34	0.966519	0.483259	+	0.875477i	$0.339453\pi$
$158$	3014.09	1.51765
$159$	0	0
$160$	−2826.13	−1.39641
$161$	−851.273	−0.416706
$162$	0	0
$163$	−2325.15	−1.11730	−0.558649	−	0.829404i	$-0.688680\pi$
$163$	−2325.15	−1.11730	−0.558649	+	0.829404i	$0.688680\pi$
$164$	1842.86	0.877461
$165$	0	0
$166$	−6574.51	−3.07398
$167$	1965.80	0.910888	0.455444	−	0.890264i	$-0.349480\pi$
$167$	1965.80	0.910888	0.455444	+	0.890264i	$0.349480\pi$
$168$	0	0
$169$	1386.76	0.631205
$170$	2962.06	1.33635
$171$	0	0
$172$	2507.44	1.11157
$173$	−2297.90	−1.00986	−0.504932	−	0.863159i	$-0.668482\pi$
$173$	−2297.90	−1.00986	−0.504932	+	0.863159i	$0.668482\pi$
$174$	0	0
$175$	314.255	0.135745
$176$	2680.98	1.14822
$177$	0	0
$178$	−1378.05	−0.580278
$179$	−873.696	−0.364822	−0.182411	−	0.983222i	$-0.558390\pi$
$179$	−873.696	−0.364822	−0.182411	+	0.983222i	$0.558390\pi$
$180$	0	0
$181$	1494.20	0.613609	0.306805	−	0.951772i	$-0.400740\pi$
$181$	1494.20	0.613609	0.306805	+	0.951772i	$0.400740\pi$
$182$	4052.28	1.65041
$183$	0	0
$184$	−4740.34	−1.89925
$185$	112.407	0.0446722
$186$	0	0
$187$	−1411.52	−0.551980
$188$	5105.91	1.98078
$189$	0	0
$190$	325.449	0.124266
$191$	5050.26	1.91321	0.956607	−	0.291383i	$-0.0941153\pi$
$191$	5050.26	1.91321	0.956607	+	0.291383i	$0.0941153\pi$
$192$	0	0
$193$	5161.60	1.92508	0.962539	−	0.271144i	$-0.0874019\pi$
$193$	5161.60	1.92508	0.962539	+	0.271144i	$0.0874019\pi$
$194$	5651.69	2.09158
$195$	0	0
$196$	−3884.52	−1.41564
$197$	374.025	0.135270	0.0676350	−	0.997710i	$-0.478455\pi$
$197$	374.025	0.135270	0.0676350	+	0.997710i	$0.478455\pi$
$198$	0	0
$199$	603.342	0.214924	0.107462	−	0.994209i	$-0.465728\pi$
$199$	603.342	0.214924	0.107462	+	0.994209i	$0.465728\pi$
$200$	1749.94	0.618697
$201$	0	0
$202$	477.797	0.166424
$203$	2513.51	0.869034
$204$	0	0
$205$	−438.807	−0.149501
$206$	−8314.87	−2.81225
$207$	0	0
$208$	12508.7	4.16983
$209$	−155.087	−0.0513281
$210$	0	0
$211$	2411.81	0.786901	0.393450	−	0.919346i	$-0.371281\pi$
$211$	2411.81	0.786901	0.393450	+	0.919346i	$0.371281\pi$
$212$	−6163.45	−1.99673
$213$	0	0
$214$	−3154.40	−1.00762
$215$	−597.050	−0.189388
$216$	0	0
$217$	963.237	0.301331
$218$	7361.53	2.28709
$219$	0	0
$220$	−1347.13	−0.412834
$221$	−6585.75	−2.00455
$222$	0	0
$223$	5481.28	1.64598	0.822990	−	0.568056i	$-0.192304\pi$
$223$	5481.28	1.64598	0.822990	+	0.568056i	$0.192304\pi$
$224$	7105.00	2.11930
$225$	0	0
$226$	917.766	0.270128
$227$	−5472.48	−1.60009	−0.800046	−	0.599938i	$-0.795192\pi$
$227$	−5472.48	−1.60009	−0.800046	+	0.599938i	$0.795192\pi$
$228$	0	0
$229$	3275.98	0.945341	0.472670	−	0.881239i	$-0.343290\pi$
$229$	3275.98	0.945341	0.472670	+	0.881239i	$0.343290\pi$
$230$	1823.41	0.522749
$231$	0	0
$232$	13996.6	3.96086
$233$	3446.21	0.968965	0.484483	−	0.874801i	$-0.339008\pi$
$233$	3446.21	0.968965	0.484483	+	0.874801i	$0.339008\pi$
$234$	0	0
$235$	−1215.78	−0.337483
$236$	−12208.2	−3.36732
$237$	0	0
$238$	−7446.74	−2.02815
$239$	−1724.81	−0.466815	−0.233408	−	0.972379i	$-0.574988\pi$
$239$	−1724.81	−0.466815	−0.233408	+	0.972379i	$0.574988\pi$
$240$	0	0
$241$	−5575.71	−1.49030	−0.745152	−	0.666895i	$-0.767623\pi$
$241$	−5575.71	−1.49030	−0.745152	+	0.666895i	$0.767623\pi$
$242$	−6280.96	−1.66841
$243$	0	0
$244$	−16246.4	−4.26259
$245$	924.951	0.241196
$246$	0	0
$247$	−723.592	−0.186401
$248$	5363.82	1.37340
$249$	0	0
$250$	−673.129	−0.170290
$251$	−1356.38	−0.341090	−0.170545	−	0.985350i	$-0.554553\pi$
$251$	−1356.38	−0.341090	−0.170545	+	0.985350i	$0.554553\pi$
$252$	0	0
$253$	−868.913	−0.215921
$254$	9745.49	2.40743
$255$	0	0
$256$	4463.12	1.08963
$257$	−4087.82	−0.992184	−0.496092	−	0.868270i	$-0.665232\pi$
$257$	−4087.82	−0.992184	−0.496092	+	0.868270i	$0.665232\pi$
$258$	0	0
$259$	−282.596	−0.0677980
$260$	−6285.34	−1.49923
$261$	0	0
$262$	−6668.25	−1.57239
$263$	−430.260	−0.100878	−0.0504391	−	0.998727i	$-0.516062\pi$
$263$	−430.260	−0.100878	−0.0504391	+	0.998727i	$0.516062\pi$
$264$	0	0
$265$	1467.59	0.340201
$266$	−818.192	−0.188596
$267$	0	0
$268$	−4859.86	−1.10770
$269$	3467.85	0.786017	0.393008	−	0.919535i	$-0.371434\pi$
$269$	3467.85	0.786017	0.393008	+	0.919535i	$0.371434\pi$
$270$	0	0
$271$	−55.4415	−0.0124274	−0.00621371	−	0.999981i	$-0.501978\pi$
$271$	−55.4415	−0.0124274	−0.00621371	+	0.999981i	$0.501978\pi$
$272$	−22986.9	−5.12420
$273$	0	0
$274$	2725.81	0.600993
$275$	320.767	0.0703381
$276$	0	0
$277$	−1573.54	−0.341317	−0.170658	−	0.985330i	$-0.554589\pi$
$277$	−1573.54	−0.341317	−0.170658	+	0.985330i	$0.554589\pi$
$278$	−1150.67	−0.248247
$279$	0	0
$280$	−4399.42	−0.938984
$281$	8146.45	1.72945	0.864727	−	0.502242i	$-0.167491\pi$
$281$	8146.45	1.72945	0.864727	+	0.502242i	$0.167491\pi$
$282$	0	0
$283$	3098.31	0.650795	0.325398	−	0.945577i	$-0.394502\pi$
$283$	3098.31	0.650795	0.325398	+	0.945577i	$0.394502\pi$
$284$	−15630.1	−3.26576
$285$	0	0
$286$	4136.26	0.855182
$287$	1103.18	0.226894
$288$	0	0
$289$	7189.39	1.46334
$290$	−5383.90	−1.09018
$291$	0	0
$292$	−5561.23	−1.11454
$293$	122.160	0.0243572	0.0121786	−	0.999926i	$-0.496123\pi$
$293$	122.160	0.0243572	0.0121786	+	0.999926i	$0.496123\pi$
$294$	0	0
$295$	2906.92	0.573720
$296$	−1573.65	−0.309008
$297$	0	0
$298$	−4875.12	−0.947679
$299$	−4054.11	−0.784132
$300$	0	0
$301$	1501.01	0.287431
$302$	7316.48	1.39409
$303$	0	0
$304$	−2525.62	−0.476495
$305$	3868.47	0.726255
$306$	0	0
$307$	−1928.53	−0.358525	−0.179263	−	0.983801i	$-0.557371\pi$
$307$	−1928.53	−0.358525	−0.179263	+	0.983801i	$0.557371\pi$
$308$	3386.74	0.626549
$309$	0	0
$310$	−2063.24	−0.378013
$311$	7939.29	1.44758	0.723788	−	0.690023i	$-0.242399\pi$
$311$	7939.29	1.44758	0.723788	+	0.690023i	$0.242399\pi$
$312$	0	0
$313$	−6758.89	−1.22056	−0.610279	−	0.792186i	$-0.708943\pi$
$313$	−6758.89	−1.22056	−0.610279	+	0.792186i	$0.708943\pi$
$314$	10238.8	1.84015
$315$	0	0
$316$	11753.2	2.09232
$317$	8424.23	1.49259	0.746297	−	0.665614i	$-0.231830\pi$
$317$	8424.23	1.49259	0.746297	+	0.665614i	$0.231830\pi$
$318$	0	0
$319$	2565.60	0.450301
$320$	−6860.75	−1.19852
$321$	0	0
$322$	−4584.13	−0.793365
$323$	1329.72	0.229064
$324$	0	0
$325$	1496.61	0.255437
$326$	−12521.0	−2.12722
$327$	0	0
$328$	6143.09	1.03413
$329$	3056.51	0.512191
$330$	0	0
$331$	257.675	0.0427888	0.0213944	−	0.999771i	$-0.493189\pi$
$331$	257.675	0.0427888	0.0213944	+	0.999771i	$0.493189\pi$
$332$	−25636.9	−4.23797
$333$	0	0
$334$	10585.9	1.73424
$335$	1157.19	0.188728
$336$	0	0
$337$	−11099.5	−1.79415	−0.897077	−	0.441875i	$-0.854314\pi$
$337$	−11099.5	−1.79415	−0.897077	+	0.441875i	$0.854314\pi$
$338$	7467.73	1.20175
$339$	0	0
$340$	11550.3	1.84237
$341$	983.198	0.156138
$342$	0	0
$343$	−6636.94	−1.04478
$344$	8358.42	1.31005
$345$	0	0
$346$	−12374.3	−1.92267
$347$	9886.57	1.52951	0.764753	−	0.644323i	$-0.222861\pi$
$347$	9886.57	1.52951	0.764753	+	0.644323i	$0.222861\pi$
$348$	0	0
$349$	6059.20	0.929345	0.464673	−	0.885483i	$-0.346172\pi$
$349$	6059.20	0.929345	0.464673	+	0.885483i	$0.346172\pi$
$350$	1692.27	0.258445
$351$	0	0
$352$	7252.23	1.09814
$353$	−9096.69	−1.37158	−0.685790	−	0.727799i	$-0.740543\pi$
$353$	−9096.69	−1.37158	−0.685790	+	0.727799i	$0.740543\pi$
$354$	0	0
$355$	3721.71	0.556416
$356$	−5373.63	−0.800005
$357$	0	0
$358$	−4704.88	−0.694583
$359$	−8804.63	−1.29440	−0.647201	−	0.762319i	$-0.724061\pi$
$359$	−8804.63	−1.29440	−0.647201	+	0.762319i	$0.724061\pi$
$360$	0	0
$361$	−6712.90	−0.978700
$362$	8046.33	1.16825
$363$	0	0
$364$	15801.6	2.27535
$365$	1324.19	0.189894
$366$	0	0
$367$	8652.36	1.23065	0.615326	−	0.788272i	$-0.289024\pi$
$367$	8652.36	1.23065	0.615326	+	0.788272i	$0.289024\pi$
$368$	−14150.5	−2.00447
$369$	0	0
$370$	605.317	0.0850511
$371$	−3689.58	−0.516316
$372$	0	0
$373$	−11918.6	−1.65449	−0.827243	−	0.561844i	$-0.810092\pi$
$373$	−11918.6	−1.65449	−0.827243	+	0.561844i	$0.810092\pi$
$374$	−7601.05	−1.05091
$375$	0	0
$376$	17020.3	2.33445
$377$	11970.4	1.63529
$378$	0	0
$379$	5052.23	0.684738	0.342369	−	0.939566i	$-0.388771\pi$
$379$	5052.23	0.684738	0.342369	+	0.939566i	$0.388771\pi$
$380$	1269.07	0.171320
$381$	0	0
$382$	27195.8	3.64256
$383$	−5330.37	−0.711147	−0.355573	−	0.934648i	$-0.615714\pi$
$383$	−5330.37	−0.711147	−0.355573	+	0.934648i	$0.615714\pi$
$384$	0	0
$385$	−806.421	−0.106751
$386$	27795.4	3.66515
$387$	0	0
$388$	22038.4	2.88358
$389$	3339.52	0.435270	0.217635	−	0.976030i	$-0.430166\pi$
$389$	3339.52	0.435270	0.217635	+	0.976030i	$0.430166\pi$
$390$	0	0
$391$	7450.10	0.963601
$392$	−12948.9	−1.66841
$393$	0	0
$394$	2014.14	0.257540
$395$	−2798.58	−0.356486
$396$	0	0
$397$	9041.65	1.14304	0.571520	−	0.820588i	$-0.306354\pi$
$397$	9041.65	1.14304	0.571520	+	0.820588i	$0.306354\pi$
$398$	3249.02	0.409192
$399$	0	0
$400$	5223.77	0.652971
$401$	−1671.96	−0.208213	−0.104107	−	0.994566i	$-0.533198\pi$
$401$	−1671.96	−0.208213	−0.104107	+	0.994566i	$0.533198\pi$
$402$	0	0
$403$	4587.34	0.567026
$404$	1863.14	0.229442
$405$	0	0
$406$	13535.3	1.65455
$407$	−288.453	−0.0351304
$408$	0	0
$409$	472.726	0.0571511	0.0285755	−	0.999592i	$-0.490903\pi$
$409$	472.726	0.0571511	0.0285755	+	0.999592i	$0.490903\pi$
$410$	−2362.99	−0.284634
$411$	0	0
$412$	−32423.2	−3.87713
$413$	−7308.11	−0.870722
$414$	0	0
$415$	6104.43	0.722060
$416$	33836.9	3.98796
$417$	0	0
$418$	−835.147	−0.0977233
$419$	12539.3	1.46202	0.731008	−	0.682369i	$-0.239050\pi$
$419$	12539.3	1.46202	0.731008	+	0.682369i	$0.239050\pi$
$420$	0	0
$421$	−6751.76	−0.781616	−0.390808	−	0.920472i	$-0.627804\pi$
$421$	−6751.76	−0.781616	−0.390808	+	0.920472i	$0.627804\pi$
$422$	12987.7	1.49818
$423$	0	0
$424$	−20545.5	−2.35325
$425$	−2750.27	−0.313901
$426$	0	0
$427$	−9725.47	−1.10222
$428$	−12300.4	−1.38916
$429$	0	0
$430$	−3215.13	−0.360576
$431$	7535.06	0.842114	0.421057	−	0.907034i	$-0.361659\pi$
$431$	7535.06	0.842114	0.421057	+	0.907034i	$0.361659\pi$
$432$	0	0
$433$	−5135.13	−0.569928	−0.284964	−	0.958538i	$-0.591982\pi$
$433$	−5135.13	−0.569928	−0.284964	+	0.958538i	$0.591982\pi$
$434$	5187.06	0.573703
$435$	0	0
$436$	28705.8	3.15311
$437$	818.561	0.0896044
$438$	0	0
$439$	16973.2	1.84530	0.922650	−	0.385637i	$-0.126018\pi$
$439$	16973.2	1.84530	0.922650	+	0.385637i	$0.126018\pi$
$440$	−4490.59	−0.486546
$441$	0	0
$442$	−35464.4	−3.81645
$443$	11352.3	1.21753	0.608763	−	0.793352i	$-0.291666\pi$
$443$	11352.3	1.21753	0.608763	+	0.793352i	$0.291666\pi$
$444$	0	0
$445$	1279.52	0.136304
$446$	29516.9	3.13378
$447$	0	0
$448$	17248.2	1.81898
$449$	11059.8	1.16246	0.581230	−	0.813740i	$-0.302572\pi$
$449$	11059.8	1.16246	0.581230	+	0.813740i	$0.302572\pi$
$450$	0	0
$451$	1126.04	0.117568
$452$	3578.76	0.372414
$453$	0	0
$454$	−29469.5	−3.04641
$455$	−3762.54	−0.387672
$456$	0	0
$457$	356.684	0.0365098	0.0182549	−	0.999833i	$-0.494189\pi$
$457$	356.684	0.0365098	0.0182549	+	0.999833i	$0.494189\pi$
$458$	17641.3	1.79983
$459$	0	0
$460$	7110.27	0.720691
$461$	10319.8	1.04260	0.521300	−	0.853373i	$-0.325447\pi$
$461$	10319.8	1.04260	0.521300	+	0.853373i	$0.325447\pi$
$462$	0	0
$463$	−18168.5	−1.82368	−0.911839	−	0.410547i	$-0.865338\pi$
$463$	−18168.5	−1.82368	−0.911839	+	0.410547i	$0.865338\pi$
$464$	41781.4	4.18028
$465$	0	0
$466$	18558.0	1.84481
$467$	3817.15	0.378237	0.189118	−	0.981954i	$-0.439437\pi$
$467$	3817.15	0.378237	0.189118	+	0.981954i	$0.439437\pi$
$468$	0	0
$469$	−2909.22	−0.286429
$470$	−6546.99	−0.642532
$471$	0	0
$472$	−40695.4	−3.96856
$473$	1532.11	0.148936
$474$	0	0
$475$	−302.179	−0.0291893
$476$	−29038.0	−2.79613
$477$	0	0
$478$	−9288.17	−0.888768
$479$	2906.57	0.277254	0.138627	−	0.990345i	$-0.455731\pi$
$479$	2906.57	0.277254	0.138627	+	0.990345i	$0.455731\pi$
$480$	0	0
$481$	−1345.84	−0.127578
$482$	−30025.4	−2.83738
$483$	0	0
$484$	−24492.2	−2.30016
$485$	−5247.59	−0.491301
$486$	0	0
$487$	10411.1	0.968734	0.484367	−	0.874865i	$-0.339050\pi$
$487$	10411.1	0.968734	0.484367	+	0.874865i	$0.339050\pi$
$488$	−54156.7	−5.02368
$489$	0	0
$490$	4980.89	0.459211
$491$	14632.5	1.34492	0.672460	−	0.740133i	$-0.265238\pi$
$491$	14632.5	1.34492	0.672460	+	0.740133i	$0.265238\pi$
$492$	0	0
$493$	−21997.5	−2.00957
$494$	−3896.57	−0.354888
$495$	0	0
$496$	16011.6	1.44948
$497$	−9356.53	−0.844462
$498$	0	0
$499$	8166.83	0.732661	0.366330	−	0.930485i	$-0.380614\pi$
$499$	8166.83	0.732661	0.366330	+	0.930485i	$0.380614\pi$
$500$	−2624.82	−0.234771
$501$	0	0
$502$	−7304.12	−0.649400
$503$	−8080.38	−0.716275	−0.358137	−	0.933669i	$-0.616588\pi$
$503$	−8080.38	−0.716275	−0.358137	+	0.933669i	$0.616588\pi$
$504$	0	0
$505$	−443.635	−0.0390921
$506$	−4679.13	−0.411092
$507$	0	0
$508$	38001.9	3.31902
$509$	116.181	0.0101171	0.00505856	−	0.999987i	$-0.498390\pi$
$509$	116.181	0.0101171	0.00505856	+	0.999987i	$0.498390\pi$
$510$	0	0
$511$	−3329.07	−0.288199
$512$	1095.90	0.0945947
$513$	0	0
$514$	−22013.0	−1.88901
$515$	7720.35	0.660581
$516$	0	0
$517$	3119.85	0.265398
$518$	−1521.79	−0.129080
$519$	0	0
$520$	−20951.9	−1.76692
$521$	−14479.3	−1.21756	−0.608780	−	0.793339i	$-0.708341\pi$
$521$	−14479.3	−1.21756	−0.608780	+	0.793339i	$0.708341\pi$
$522$	0	0
$523$	6841.05	0.571966	0.285983	−	0.958235i	$-0.407680\pi$
$523$	6841.05	0.571966	0.285983	+	0.958235i	$0.407680\pi$
$524$	−26002.4	−2.16778
$525$	0	0
$526$	−2316.96	−0.192062
$527$	−8429.98	−0.696804
$528$	0	0
$529$	−7580.80	−0.623062
$530$	7903.01	0.647707
$531$	0	0
$532$	−3190.48	−0.260009
$533$	5253.80	0.426955
$534$	0	0
$535$	2928.86	0.236684
$536$	−16200.1	−1.30548
$537$	0	0
$538$	18674.5	1.49649
$539$	−2373.55	−0.189677
$540$	0	0
$541$	−12746.0	−1.01293	−0.506465	−	0.862261i	$-0.669048\pi$
$541$	−12746.0	−1.01293	−0.506465	+	0.862261i	$0.669048\pi$
$542$	−298.554	−0.0236605
$543$	0	0
$544$	−62181.0	−4.90071
$545$	−6835.18	−0.537224
$546$	0	0
$547$	4980.54	0.389310	0.194655	−	0.980872i	$-0.437641\pi$
$547$	4980.54	0.389310	0.194655	+	0.980872i	$0.437641\pi$
$548$	10629.1	0.828563
$549$	0	0
$550$	1727.34	0.133916
$551$	−2416.93	−0.186868
$552$	0	0
$553$	7035.75	0.541032
$554$	−8473.55	−0.649831
$555$	0	0
$556$	−4486.97	−0.342248
$557$	−635.610	−0.0483513	−0.0241756	−	0.999708i	$-0.507696\pi$
$557$	−635.610	−0.0483513	−0.0241756	+	0.999708i	$0.507696\pi$
$558$	0	0
$559$	7148.42	0.540869
$560$	−13132.8	−0.991001
$561$	0	0
$562$	43868.9	3.29270
$563$	−7537.68	−0.564254	−0.282127	−	0.959377i	$-0.591040\pi$
$563$	−7537.68	−0.564254	−0.282127	+	0.959377i	$0.591040\pi$
$564$	0	0
$565$	−852.146	−0.0634514
$566$	16684.5	1.23905
$567$	0	0
$568$	−52102.2	−3.84887
$569$	−2258.35	−0.166389	−0.0831943	−	0.996533i	$-0.526512\pi$
$569$	−2258.35	−0.166389	−0.0831943	+	0.996533i	$0.526512\pi$
$570$	0	0
$571$	11377.3	0.833847	0.416923	−	0.908942i	$-0.363108\pi$
$571$	11377.3	0.833847	0.416923	+	0.908942i	$0.363108\pi$
$572$	16129.0	1.17900
$573$	0	0
$574$	5940.66	0.431983
$575$	−1693.04	−0.122791
$576$	0	0
$577$	−25027.9	−1.80576	−0.902881	−	0.429890i	$-0.858553\pi$
$577$	−25027.9	−1.80576	−0.902881	+	0.429890i	$0.858553\pi$
$578$	38715.1	2.78605
$579$	0	0
$580$	−20994.1	−1.50299
$581$	−15346.8	−1.09586
$582$	0	0
$583$	−3766.03	−0.267535
$584$	−18538.1	−1.31355
$585$	0	0
$586$	657.835	0.0463736
$587$	148.259	0.0104247	0.00521235	−	0.999986i	$-0.498341\pi$
$587$	148.259	0.0104247	0.00521235	+	0.999986i	$0.498341\pi$
$588$	0	0
$589$	−926.224	−0.0647952
$590$	15653.8	1.09230
$591$	0	0
$592$	−4697.52	−0.326126
$593$	−27452.6	−1.90109	−0.950544	−	0.310590i	$-0.899473\pi$
$593$	−27452.6	−1.90109	−0.950544	+	0.310590i	$0.899473\pi$
$594$	0	0
$595$	6914.29	0.476401
$596$	−19010.2	−1.30652
$597$	0	0
$598$	−21831.5	−1.49291
$599$	−3812.00	−0.260024	−0.130012	−	0.991512i	$-0.541502\pi$
$599$	−3812.00	−0.260024	−0.130012	+	0.991512i	$0.541502\pi$
$600$	0	0
$601$	23168.7	1.57249	0.786247	−	0.617912i	$-0.212021\pi$
$601$	23168.7	1.57249	0.786247	+	0.617912i	$0.212021\pi$
$602$	8082.97	0.547238
$603$	0	0
$604$	28530.1	1.92197
$605$	5831.87	0.391899
$606$	0	0
$607$	7352.78	0.491664	0.245832	−	0.969312i	$-0.420939\pi$
$607$	7352.78	0.491664	0.245832	+	0.969312i	$0.420939\pi$
$608$	−6831.98	−0.455713
$609$	0	0
$610$	20831.8	1.38271
$611$	14556.4	0.963809
$612$	0	0
$613$	19902.6	1.31135	0.655676	−	0.755043i	$-0.272384\pi$
$613$	19902.6	1.31135	0.655676	+	0.755043i	$0.272384\pi$
$614$	−10385.2	−0.682595
$615$	0	0
$616$	11289.5	0.738421
$617$	25085.2	1.63678	0.818390	−	0.574664i	$-0.194867\pi$
$617$	25085.2	1.63678	0.818390	+	0.574664i	$0.194867\pi$
$618$	0	0
$619$	−3131.04	−0.203307	−0.101654	−	0.994820i	$-0.532413\pi$
$619$	−3131.04	−0.203307	−0.101654	+	0.994820i	$0.532413\pi$
$620$	−8045.45	−0.521150
$621$	0	0
$622$	42753.3	2.75603
$623$	−3216.77	−0.206866
$624$	0	0
$625$	625.000	0.0400000
$626$	−36396.8	−2.32382
$627$	0	0
$628$	39925.4	2.53694
$629$	2473.21	0.156778
$630$	0	0
$631$	−22527.2	−1.42123	−0.710614	−	0.703582i	$-0.751583\pi$
$631$	−22527.2	−1.42123	−0.710614	+	0.703582i	$0.751583\pi$
$632$	39178.8	2.46590
$633$	0	0
$634$	45364.7	2.84174
$635$	−9048.69	−0.565490
$636$	0	0
$637$	−11074.3	−0.688825
$638$	13815.8	0.857325
$639$	0	0
$640$	−14336.3	−0.885459
$641$	−6925.76	−0.426757	−0.213378	−	0.976970i	$-0.568447\pi$
$641$	−6925.76	−0.426757	−0.213378	+	0.976970i	$0.568447\pi$
$642$	0	0
$643$	−14964.5	−0.917796	−0.458898	−	0.888489i	$-0.651756\pi$
$643$	−14964.5	−0.917796	−0.458898	+	0.888489i	$0.651756\pi$
$644$	−17875.5	−1.09378
$645$	0	0
$646$	7160.58	0.436114
$647$	−2371.74	−0.144115	−0.0720577	−	0.997400i	$-0.522957\pi$
$647$	−2371.74	−0.144115	−0.0720577	+	0.997400i	$0.522957\pi$
$648$	0	0
$649$	−7459.55	−0.451176
$650$	8059.30	0.486326
$651$	0	0
$652$	−48824.7	−2.93270
$653$	−15971.7	−0.957154	−0.478577	−	0.878046i	$-0.658847\pi$
$653$	−15971.7	−0.957154	−0.478577	+	0.878046i	$0.658847\pi$
$654$	0	0
$655$	6191.47	0.369344
$656$	18337.8	1.09142
$657$	0	0
$658$	16459.4	0.975158
$659$	−6842.93	−0.404496	−0.202248	−	0.979334i	$-0.564825\pi$
$659$	−6842.93	−0.404496	−0.202248	+	0.979334i	$0.564825\pi$
$660$	0	0
$661$	−13391.0	−0.787973	−0.393986	−	0.919116i	$-0.628904\pi$
$661$	−13391.0	−0.787973	−0.393986	+	0.919116i	$0.628904\pi$
$662$	1387.59	0.0814654
$663$	0	0
$664$	−85459.1	−4.99467
$665$	759.691	0.0443001
$666$	0	0
$667$	−13541.5	−0.786098
$668$	41279.0	2.39092
$669$	0	0
$670$	6231.50	0.359319
$671$	−9927.01	−0.571130
$672$	0	0
$673$	−31902.2	−1.82725	−0.913624	−	0.406560i	$-0.866728\pi$
$673$	−31902.2	−1.82725	−0.913624	+	0.406560i	$0.866728\pi$
$674$	−59771.3	−3.41588
$675$	0	0
$676$	29119.9	1.65680
$677$	−11860.8	−0.673332	−0.336666	−	0.941624i	$-0.609299\pi$
$677$	−11860.8	−0.673332	−0.336666	+	0.941624i	$0.609299\pi$
$678$	0	0
$679$	13192.7	0.745637
$680$	38502.5	2.17133
$681$	0	0
$682$	5294.55	0.297271
$683$	7639.34	0.427981	0.213991	−	0.976836i	$-0.431354\pi$
$683$	7639.34	0.427981	0.213991	+	0.976836i	$0.431354\pi$
$684$	0	0
$685$	−2530.91	−0.141170
$686$	−35740.1	−1.98916
$687$	0	0
$688$	24950.8	1.38262
$689$	−17571.3	−0.971571
$690$	0	0
$691$	−15726.0	−0.865767	−0.432883	−	0.901450i	$-0.642504\pi$
$691$	−15726.0	−0.865767	−0.432883	+	0.901450i	$0.642504\pi$
$692$	−48252.6	−2.65071
$693$	0	0
$694$	53239.5	2.91202
$695$	1068.40	0.0583118
$696$	0	0
$697$	−9654.72	−0.524675
$698$	32629.0	1.76938
$699$	0	0
$700$	6598.90	0.356307
$701$	−6338.17	−0.341497	−0.170748	−	0.985315i	$-0.554619\pi$
$701$	−6338.17	−0.341497	−0.170748	+	0.985315i	$0.554619\pi$
$702$	0	0
$703$	271.737	0.0145786
$704$	17605.6	0.942525
$705$	0	0
$706$	−48985.9	−2.61135
$707$	1115.32	0.0593292
$708$	0	0
$709$	34383.6	1.82130	0.910651	−	0.413176i	$-0.135581\pi$
$709$	34383.6	1.82130	0.910651	+	0.413176i	$0.135581\pi$
$710$	20041.5	1.05936
$711$	0	0
$712$	−17912.7	−0.942847
$713$	−5189.41	−0.272573
$714$	0	0
$715$	−3840.51	−0.200877
$716$	−18346.4	−0.957592
$717$	0	0
$718$	−47413.2	−2.46441
$719$	−988.886	−0.0512924	−0.0256462	−	0.999671i	$-0.508164\pi$
$719$	−988.886	−0.0512924	−0.0256462	+	0.999671i	$0.508164\pi$
$720$	0	0
$721$	−19409.3	−1.00255
$722$	−36149.2	−1.86334
$723$	0	0
$724$	31376.1	1.61061
$725$	4998.95	0.256078
$726$	0	0
$727$	−13691.8	−0.698490	−0.349245	−	0.937032i	$-0.613562\pi$
$727$	−13691.8	−0.698490	−0.349245	+	0.937032i	$0.613562\pi$
$728$	52673.8	2.68162
$729$	0	0
$730$	7130.82	0.361539
$731$	−13136.4	−0.664661
$732$	0	0
$733$	−2501.94	−0.126073	−0.0630363	−	0.998011i	$-0.520078\pi$
$733$	−2501.94	−0.126073	−0.0630363	+	0.998011i	$0.520078\pi$
$734$	46593.2	2.34303
$735$	0	0
$736$	−38277.9	−1.91704
$737$	−2969.51	−0.148417
$738$	0	0
$739$	26453.1	1.31677	0.658384	−	0.752682i	$-0.271240\pi$
$739$	26453.1	1.31677	0.658384	+	0.752682i	$0.271240\pi$
$740$	2360.39	0.117256
$741$	0	0
$742$	−19868.5	−0.983012
$743$	−20882.8	−1.03111	−0.515555	−	0.856856i	$-0.672414\pi$
$743$	−20882.8	−1.03111	−0.515555	+	0.856856i	$0.672414\pi$
$744$	0	0
$745$	4526.55	0.222604
$746$	−64182.2	−3.14997
$747$	0	0
$748$	−29639.8	−1.44885
$749$	−7363.27	−0.359210
$750$	0	0
$751$	−17695.3	−0.859801	−0.429901	−	0.902876i	$-0.641451\pi$
$751$	−17695.3	−0.859801	−0.429901	+	0.902876i	$0.641451\pi$
$752$	50807.5	2.46377
$753$	0	0
$754$	64460.9	3.11343
$755$	−6793.35	−0.327464
$756$	0	0
$757$	7755.61	0.372368	0.186184	−	0.982515i	$-0.440388\pi$
$757$	7755.61	0.372368	0.186184	+	0.982515i	$0.440388\pi$
$758$	27206.4	1.30367
$759$	0	0
$760$	4230.37	0.201910
$761$	17054.3	0.812376	0.406188	−	0.913790i	$-0.366858\pi$
$761$	17054.3	0.812376	0.406188	+	0.913790i	$0.366858\pi$
$762$	0	0
$763$	17183.9	0.815333
$764$	106048.	5.02184
$765$	0	0
$766$	−28704.2	−1.35395
$767$	−34804.2	−1.63847
$768$	0	0
$769$	−10048.3	−0.471199	−0.235599	−	0.971850i	$-0.575705\pi$
$769$	−10048.3	−0.471199	−0.235599	+	0.971850i	$0.575705\pi$
$770$	−4342.60	−0.203242
$771$	0	0
$772$	108386.	5.05298
$773$	7293.24	0.339352	0.169676	−	0.985500i	$-0.445728\pi$
$773$	7293.24	0.339352	0.169676	+	0.985500i	$0.445728\pi$
$774$	0	0
$775$	1915.72	0.0887930
$776$	73463.8	3.39845
$777$	0	0
$778$	17983.4	0.828710
$779$	−1060.79	−0.0487891
$780$	0	0
$781$	−9550.42	−0.437568
$782$	40119.0	1.83460
$783$	0	0
$784$	−38653.8	−1.76083
$785$	−9506.70	−0.432240
$786$	0	0
$787$	28456.3	1.28889	0.644445	−	0.764651i	$-0.277089\pi$
$787$	28456.3	1.28889	0.644445	+	0.764651i	$0.277089\pi$
$788$	7853.99	0.355059
$789$	0	0
$790$	−15070.5	−0.678713
$791$	2142.33	0.0962989
$792$	0	0
$793$	−46316.7	−2.07409
$794$	48689.5	2.17623
$795$	0	0
$796$	12669.3	0.564136
$797$	4081.46	0.181396	0.0906981	−	0.995878i	$-0.471090\pi$
$797$	4081.46	0.181396	0.0906981	+	0.995878i	$0.471090\pi$
$798$	0	0
$799$	−26749.7	−1.18440
$800$	14130.6	0.624492
$801$	0	0
$802$	−9003.54	−0.396416
$803$	−3398.06	−0.149334
$804$	0	0
$805$	4256.36	0.186357
$806$	24702.9	1.07956
$807$	0	0
$808$	6210.67	0.270409
$809$	14209.7	0.617536	0.308768	−	0.951137i	$-0.400083\pi$
$809$	14209.7	0.617536	0.308768	+	0.951137i	$0.400083\pi$
$810$	0	0
$811$	4901.79	0.212238	0.106119	−	0.994353i	$-0.466158\pi$
$811$	4901.79	0.212238	0.106119	+	0.994353i	$0.466158\pi$
$812$	52780.1	2.28106
$813$	0	0
$814$	−1553.33	−0.0668846
$815$	11625.7	0.499671
$816$	0	0
$817$	−1443.33	−0.0618063
$818$	2545.64	0.108810
$819$	0	0
$820$	−9214.32	−0.392412
$821$	−14519.6	−0.617222	−0.308611	−	0.951188i	$-0.599864\pi$
$821$	−14519.6	−0.617222	−0.308611	+	0.951188i	$0.599864\pi$
$822$	0	0
$823$	−38406.4	−1.62669	−0.813344	−	0.581783i	$-0.802355\pi$
$823$	−38406.4	−1.62669	−0.813344	+	0.581783i	$0.802355\pi$
$824$	−108081.	−4.56940
$825$	0	0
$826$	−39354.4	−1.65776
$827$	−10446.4	−0.439248	−0.219624	−	0.975585i	$-0.570483\pi$
$827$	−10446.4	−0.439248	−0.219624	+	0.975585i	$0.570483\pi$
$828$	0	0
$829$	−6474.00	−0.271232	−0.135616	−	0.990761i	$-0.543301\pi$
$829$	−6474.00	−0.271232	−0.135616	+	0.990761i	$0.543301\pi$
$830$	32872.6	1.37473
$831$	0	0
$832$	82143.1	3.42284
$833$	20350.9	0.846480
$834$	0	0
$835$	−9829.01	−0.407362
$836$	−3256.60	−0.134727
$837$	0	0
$838$	67524.5	2.78353
$839$	15839.4	0.651773	0.325886	−	0.945409i	$-0.394337\pi$
$839$	15839.4	0.651773	0.325886	+	0.945409i	$0.394337\pi$
$840$	0	0
$841$	15594.2	0.639394
$842$	−36358.4	−1.48812
$843$	0	0
$844$	50644.6	2.06547
$845$	−6933.79	−0.282284
$846$	0	0
$847$	−14661.5	−0.594778
$848$	−61330.8	−2.48362
$849$	0	0
$850$	−14810.3	−0.597634
$851$	1522.48	0.0613277
$852$	0	0
$853$	−18840.3	−0.756249	−0.378125	−	0.925755i	$-0.623431\pi$
$853$	−18840.3	−0.756249	−0.378125	+	0.925755i	$0.623431\pi$
$854$	−52372.0	−2.09852
$855$	0	0
$856$	−41002.6	−1.63720
$857$	9056.21	0.360973	0.180487	−	0.983577i	$-0.442233\pi$
$857$	9056.21	0.360973	0.180487	+	0.983577i	$0.442233\pi$
$858$	0	0
$859$	22055.4	0.876044	0.438022	−	0.898964i	$-0.355679\pi$
$859$	22055.4	0.876044	0.438022	+	0.898964i	$0.355679\pi$
$860$	−12537.2	−0.497110
$861$	0	0
$862$	40576.5	1.60330
$863$	13105.6	0.516941	0.258471	−	0.966019i	$-0.416781\pi$
$863$	13105.6	0.516941	0.258471	+	0.966019i	$0.416781\pi$
$864$	0	0
$865$	11489.5	0.451624
$866$	−27652.8	−1.08508
$867$	0	0
$868$	20226.6	0.790939
$869$	7181.55	0.280342
$870$	0	0
$871$	−13854.9	−0.538985
$872$	95689.2	3.71611
$873$	0	0
$874$	4407.98	0.170597
$875$	−1571.27	−0.0607072
$876$	0	0
$877$	−36722.2	−1.41393	−0.706967	−	0.707246i	$-0.749937\pi$
$877$	−36722.2	−1.41393	−0.706967	+	0.707246i	$0.749937\pi$
$878$	91401.2	3.51326
$879$	0	0
$880$	−13404.9	−0.513499
$881$	36054.4	1.37878	0.689390	−	0.724390i	$-0.257879\pi$
$881$	36054.4	1.37878	0.689390	+	0.724390i	$0.257879\pi$
$882$	0	0
$883$	13524.6	0.515446	0.257723	−	0.966219i	$-0.417028\pi$
$883$	13524.6	0.515446	0.257723	+	0.966219i	$0.417028\pi$
$884$	−138291.	−5.26158
$885$	0	0
$886$	61132.5	2.31804
$887$	−39730.2	−1.50396	−0.751978	−	0.659188i	$-0.770900\pi$
$887$	−39730.2	−1.50396	−0.751978	+	0.659188i	$0.770900\pi$
$888$	0	0
$889$	22748.8	0.858233
$890$	6890.27	0.259508
$891$	0	0
$892$	115099.	4.32040
$893$	−2939.06	−0.110136
$894$	0	0
$895$	4368.48	0.163153
$896$	36042.1	1.34384
$897$	0	0
$898$	59557.3	2.21320
$899$	15322.5	0.568447
$900$	0	0
$901$	32290.1	1.19394
$902$	6063.76	0.223837
$903$	0	0
$904$	11929.6	0.438909
$905$	−7471.02	−0.274414
$906$	0	0
$907$	23022.4	0.842829	0.421415	−	0.906868i	$-0.361534\pi$
$907$	23022.4	0.842829	0.421415	+	0.906868i	$0.361534\pi$
$908$	−114914.	−4.19995
$909$	0	0
$910$	−20261.4	−0.738087
$911$	31880.2	1.15943	0.579714	−	0.814820i	$-0.303164\pi$
$911$	31880.2	1.15943	0.579714	+	0.814820i	$0.303164\pi$
$912$	0	0
$913$	−15664.8	−0.567831
$914$	1920.75	0.0695108
$915$	0	0
$916$	68790.9	2.48135
$917$	−15565.6	−0.560547
$918$	0	0
$919$	38459.6	1.38049	0.690243	−	0.723578i	$-0.257504\pi$
$919$	38459.6	1.38049	0.690243	+	0.723578i	$0.257504\pi$
$920$	23701.7	0.849372
$921$	0	0
$922$	55572.2	1.98500
$923$	−44559.7	−1.58906
$924$	0	0
$925$	−562.037	−0.0199780
$926$	−97838.1	−3.47209
$927$	0	0
$928$	113021.	3.99796
$929$	−37201.0	−1.31381	−0.656903	−	0.753975i	$-0.728134\pi$
$929$	−37201.0	−1.31381	−0.656903	+	0.753975i	$0.728134\pi$
$930$	0	0
$931$	2236.01	0.0787134
$932$	72365.5	2.54336
$933$	0	0
$934$	20555.5	0.720123
$935$	7057.58	0.246853
$936$	0	0
$937$	−23593.7	−0.822595	−0.411298	−	0.911501i	$-0.634924\pi$
$937$	−23593.7	−0.822595	−0.411298	+	0.911501i	$0.634924\pi$
$938$	−15666.2	−0.545332
$939$	0	0
$940$	−25529.5	−0.885832
$941$	36571.7	1.26695	0.633477	−	0.773762i	$-0.281627\pi$
$941$	36571.7	1.26695	0.633477	+	0.773762i	$0.281627\pi$
$942$	0	0
$943$	−5943.34	−0.205241
$944$	−121481.	−4.18841
$945$	0	0
$946$	8250.48	0.283558
$947$	32911.7	1.12934	0.564671	−	0.825316i	$-0.309003\pi$
$947$	32911.7	1.12934	0.564671	+	0.825316i	$0.309003\pi$
$948$	0	0
$949$	−15854.4	−0.542314
$950$	−1627.24	−0.0555735
$951$	0	0
$952$	−96796.8	−3.29538
$953$	−21564.5	−0.732993	−0.366497	−	0.930419i	$-0.619443\pi$
$953$	−21564.5	−0.732993	−0.366497	+	0.930419i	$0.619443\pi$
$954$	0	0
$955$	−25251.3	−0.855615
$956$	−36218.6	−1.22531
$957$	0	0
$958$	15652.0	0.527862
$959$	6362.82	0.214250
$960$	0	0
$961$	−23919.1	−0.802895
$962$	−7247.40	−0.242895
$963$	0	0
$964$	−117082.	−3.91178
$965$	−25808.0	−0.860921
$966$	0	0
$967$	29763.7	0.989801	0.494901	−	0.868950i	$-0.335204\pi$
$967$	29763.7	0.989801	0.494901	+	0.868950i	$0.335204\pi$
$968$	−81643.3	−2.71086
$969$	0	0
$970$	−28258.4	−0.935385
$971$	29812.3	0.985297	0.492648	−	0.870228i	$-0.336029\pi$
$971$	29812.3	0.985297	0.492648	+	0.870228i	$0.336029\pi$
$972$	0	0
$973$	−2686.00	−0.0884986
$974$	56064.2	1.84437
$975$	0	0
$976$	−161664.	−5.30198
$977$	−5759.73	−0.188608	−0.0943041	−	0.995543i	$-0.530063\pi$
$977$	−5759.73	−0.188608	−0.0943041	+	0.995543i	$0.530063\pi$
$978$	0	0
$979$	−3283.43	−0.107190
$980$	19422.6	0.633095
$981$	0	0
$982$	78796.5	2.56059
$983$	−17651.4	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$983$	−17651.4	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$984$	0	0
$985$	−1870.13	−0.0604946
$986$	−118457.	−3.82602
$987$	0	0
$988$	−15194.4	−0.489269
$989$	−8086.63	−0.260000
$990$	0	0
$991$	−5186.45	−0.166249	−0.0831246	−	0.996539i	$-0.526490\pi$
$991$	−5186.45	−0.166249	−0.0831246	+	0.996539i	$0.526490\pi$
$992$	43312.5	1.38626
$993$	0	0
$994$	−50385.2	−1.60777
$995$	−3016.71	−0.0961167
$996$	0	0
$997$	−16110.2	−0.511752	−0.255876	−	0.966710i	$-0.582364\pi$
$997$	−16110.2	−0.511752	−0.255876	+	0.966710i	$0.582364\pi$
$998$	43978.6	1.39491
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.m.1.7		7
3.2	odd	2		405.4.a.n.1.1		7
5.4	even	2		2025.4.a.bb.1.1		7
9.2	odd	6		135.4.e.c.91.7		14
9.4	even	3		45.4.e.c.16.1	✓	14
9.5	odd	6		135.4.e.c.46.7		14
9.7	even	3		45.4.e.c.31.1	yes	14
15.14	odd	2		2025.4.a.ba.1.7		7
45.4	even	6		225.4.e.d.151.7		14
45.7	odd	12		225.4.k.d.49.14		28
45.13	odd	12		225.4.k.d.124.14		28
45.22	odd	12		225.4.k.d.124.1		28
45.34	even	6		225.4.e.d.76.7		14
45.43	odd	12		225.4.k.d.49.1		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.1	✓	14	9.4	even	3
45.4.e.c.31.1	yes	14	9.7	even	3
135.4.e.c.46.7		14	9.5	odd	6
135.4.e.c.91.7		14	9.2	odd	6
225.4.e.d.76.7		14	45.34	even	6
225.4.e.d.151.7		14	45.4	even	6
225.4.k.d.49.1		28	45.43	odd	12
225.4.k.d.49.14		28	45.7	odd	12
225.4.k.d.124.1		28	45.22	odd	12
225.4.k.d.124.14		28	45.13	odd	12
405.4.a.m.1.7		7	1.1	even	1	trivial
405.4.a.n.1.1		7	3.2	odd	2
2025.4.a.ba.1.7		7	15.14	odd	2
2025.4.a.bb.1.1		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)

\(f(q)\)	\(=\)	\(q+5.38503 q^{2} +20.9986 q^{4} -5.00000 q^{5} +12.5702 q^{7} +69.9976 q^{8} +O(q^{10})\)	\(q+5.38503 q^{2} +20.9986 q^{4} -5.00000 q^{5} +12.5702 q^{7} +69.9976 q^{8} -26.9252 q^{10} +12.8307 q^{11} +59.8645 q^{13} +67.6909 q^{14} +208.951 q^{16} -110.011 q^{17} -12.0872 q^{19} -104.993 q^{20} +69.0936 q^{22} -67.7215 q^{23} +25.0000 q^{25} +322.372 q^{26} +263.956 q^{28} +199.958 q^{29} +76.6286 q^{31} +565.226 q^{32} -592.412 q^{34} -62.8510 q^{35} -22.4815 q^{37} -65.0898 q^{38} -349.988 q^{40} +87.7615 q^{41} +119.410 q^{43} +269.426 q^{44} -364.682 q^{46} +243.155 q^{47} -184.990 q^{49} +134.626 q^{50} +1257.07 q^{52} -293.518 q^{53} -64.1534 q^{55} +879.884 q^{56} +1076.78 q^{58} -581.383 q^{59} -773.693 q^{61} +412.648 q^{62} +1372.15 q^{64} -299.322 q^{65} -231.438 q^{67} -2310.07 q^{68} -338.455 q^{70} -744.342 q^{71} -264.839 q^{73} -121.063 q^{74} -253.813 q^{76} +161.284 q^{77} +559.717 q^{79} -1044.75 q^{80} +472.598 q^{82} -1220.89 q^{83} +550.054 q^{85} +643.027 q^{86} +898.117 q^{88} -255.905 q^{89} +752.509 q^{91} -1422.05 q^{92} +1309.40 q^{94} +60.4359 q^{95} +1049.52 q^{97} -996.177 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8}+O(q^{10}) \) 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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