LMFDB - Embedded newform 405.4.a.m.1.2

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			1.2
Root			$4.26178$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.26178 q^{2} +10.1628 q^{4} -5.00000 q^{5} +30.7639 q^{7} -9.21718 q^{8} +O(q^{10})$	\(q-4.26178 q^{2} +10.1628 q^{4} -5.00000 q^{5} +30.7639 q^{7} -9.21718 q^{8} +21.3089 q^{10} +40.7146 q^{11} +63.2178 q^{13} -131.109 q^{14} -42.0204 q^{16} -6.58990 q^{17} +75.3803 q^{19} -50.8138 q^{20} -173.517 q^{22} -62.3628 q^{23} +25.0000 q^{25} -269.420 q^{26} +312.646 q^{28} +49.6084 q^{29} +103.004 q^{31} +252.819 q^{32} +28.0847 q^{34} -153.820 q^{35} -282.029 q^{37} -321.254 q^{38} +46.0859 q^{40} -157.540 q^{41} -337.814 q^{43} +413.773 q^{44} +265.776 q^{46} +44.5158 q^{47} +603.420 q^{49} -106.544 q^{50} +642.467 q^{52} +26.2752 q^{53} -203.573 q^{55} -283.557 q^{56} -211.420 q^{58} -425.926 q^{59} +850.595 q^{61} -438.981 q^{62} -741.296 q^{64} -316.089 q^{65} +96.3076 q^{67} -66.9715 q^{68} +655.545 q^{70} +952.164 q^{71} -50.8558 q^{73} +1201.95 q^{74} +766.071 q^{76} +1252.54 q^{77} -197.279 q^{79} +210.102 q^{80} +671.400 q^{82} -197.739 q^{83} +32.9495 q^{85} +1439.69 q^{86} -375.274 q^{88} +1364.54 q^{89} +1944.83 q^{91} -633.778 q^{92} -189.717 q^{94} -376.901 q^{95} -1431.16 q^{97} -2571.64 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.26178	−1.50677	−0.753383	−	0.657582i	$-0.771579\pi$
$2$	−4.26178	−1.50677	−0.753383	+	0.657582i	$0.771579\pi$
$3$	0	0
$3$	0	0
$4$	10.1628	1.27034
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	30.7639	1.66110	0.830548	−	0.556947i	$-0.188027\pi$
$7$	30.7639	1.66110	0.830548	+	0.556947i	$0.188027\pi$
$8$	−9.21718	−0.407346
$9$	0	0
$10$	21.3089	0.673846
$11$	40.7146	1.11599	0.557996	−	0.829843i	$-0.311570\pi$
$11$	40.7146	1.11599	0.557996	+	0.829843i	$0.311570\pi$
$12$	0	0
$13$	63.2178	1.34873	0.674364	−	0.738399i	$-0.264418\pi$
$13$	63.2178	1.34873	0.674364	+	0.738399i	$0.264418\pi$
$14$	−131.109	−2.50288
$15$	0	0
$16$	−42.0204	−0.656569
$17$	−6.58990	−0.0940168	−0.0470084	−	0.998894i	$-0.514969\pi$
$17$	−6.58990	−0.0940168	−0.0470084	+	0.998894i	$0.514969\pi$
$18$	0	0
$19$	75.3803	0.910180	0.455090	−	0.890445i	$-0.349607\pi$
$19$	75.3803	0.910180	0.455090	+	0.890445i	$0.349607\pi$
$20$	−50.8138	−0.568115
$21$	0	0
$22$	−173.517	−1.68154
$23$	−62.3628	−0.565371	−0.282686	−	0.959213i	$-0.591225\pi$
$23$	−62.3628	−0.565371	−0.282686	+	0.959213i	$0.591225\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−269.420	−2.03222
$27$	0	0
$28$	312.646	2.11016
$29$	49.6084	0.317657	0.158828	−	0.987306i	$-0.449228\pi$
$29$	49.6084	0.317657	0.158828	+	0.987306i	$0.449228\pi$
$30$	0	0
$31$	103.004	0.596777	0.298389	−	0.954444i	$-0.403551\pi$
$31$	103.004	0.596777	0.298389	+	0.954444i	$0.403551\pi$
$32$	252.819	1.39664
$33$	0	0
$34$	28.0847	0.141661
$35$	−153.820	−0.742865
$36$	0	0
$37$	−282.029	−1.25312	−0.626559	−	0.779374i	$-0.715537\pi$
$37$	−282.029	−1.25312	−0.626559	+	0.779374i	$0.715537\pi$
$38$	−321.254	−1.37143
$39$	0	0
$40$	46.0859	0.182171
$41$	−157.540	−0.600088	−0.300044	−	0.953925i	$-0.597001\pi$
$41$	−157.540	−0.600088	−0.300044	+	0.953925i	$0.597001\pi$
$42$	0	0
$43$	−337.814	−1.19805	−0.599025	−	0.800730i	$-0.704445\pi$
$43$	−337.814	−1.19805	−0.599025	+	0.800730i	$0.704445\pi$
$44$	413.773	1.41770
$45$	0	0
$46$	265.776	0.851882
$47$	44.5158	0.138155	0.0690777	−	0.997611i	$-0.477994\pi$
$47$	44.5158	0.138155	0.0690777	+	0.997611i	$0.477994\pi$
$48$	0	0
$49$	603.420	1.75924
$50$	−106.544	−0.301353
$51$	0	0
$52$	642.467	1.71335
$53$	26.2752	0.0680978	0.0340489	−	0.999420i	$-0.489160\pi$
$53$	26.2752	0.0680978	0.0340489	+	0.999420i	$0.489160\pi$
$54$	0	0
$55$	−203.573	−0.499087
$56$	−283.557	−0.676641
$57$	0	0
$58$	−211.420	−0.478634
$59$	−425.926	−0.939845	−0.469923	−	0.882708i	$-0.655718\pi$
$59$	−425.926	−0.939845	−0.469923	+	0.882708i	$0.655718\pi$
$60$	0	0
$61$	850.595	1.78537	0.892684	−	0.450682i	$-0.148819\pi$
$61$	850.595	1.78537	0.892684	+	0.450682i	$0.148819\pi$
$62$	−438.981	−0.899204
$63$	0	0
$64$	−741.296	−1.44784
$65$	−316.089	−0.603170
$66$	0	0
$67$	96.3076	0.175610	0.0878048	−	0.996138i	$-0.472015\pi$
$67$	96.3076	0.175610	0.0878048	+	0.996138i	$0.472015\pi$
$68$	−66.9715	−0.119434
$69$	0	0
$70$	655.545	1.11932
$71$	952.164	1.59156	0.795782	−	0.605583i	$-0.207060\pi$
$71$	952.164	1.59156	0.795782	+	0.605583i	$0.207060\pi$
$72$	0	0
$73$	−50.8558	−0.0815373	−0.0407686	−	0.999169i	$-0.512981\pi$
$73$	−50.8558	−0.0815373	−0.0407686	+	0.999169i	$0.512981\pi$
$74$	1201.95	1.88815
$75$	0	0
$76$	766.071	1.15624
$77$	1252.54	1.85377
$78$	0	0
$79$	−197.279	−0.280957	−0.140479	−	0.990084i	$-0.544864\pi$
$79$	−197.279	−0.280957	−0.140479	+	0.990084i	$0.544864\pi$
$80$	210.102	0.293627
$81$	0	0
$82$	671.400	0.904192
$83$	−197.739	−0.261502	−0.130751	−	0.991415i	$-0.541739\pi$
$83$	−197.739	−0.261502	−0.130751	+	0.991415i	$0.541739\pi$
$84$	0	0
$85$	32.9495	0.0420456
$86$	1439.69	1.80518
$87$	0	0
$88$	−375.274	−0.454595
$89$	1364.54	1.62519	0.812593	−	0.582832i	$-0.198056\pi$
$89$	1364.54	1.62519	0.812593	+	0.582832i	$0.198056\pi$
$90$	0	0
$91$	1944.83	2.24037
$92$	−633.778	−0.718216
$93$	0	0
$94$	−189.717	−0.208168
$95$	−376.901	−0.407045
$96$	0	0
$97$	−1431.16	−1.49806	−0.749031	−	0.662535i	$-0.769481\pi$
$97$	−1431.16	−1.49806	−0.749031	+	0.662535i	$0.769481\pi$
$98$	−2571.64	−2.65076
$99$	0	0
$100$	254.069	0.254069
$101$	1107.62	1.09121	0.545604	−	0.838043i	$-0.316300\pi$
$101$	1107.62	1.09121	0.545604	+	0.838043i	$0.316300\pi$
$102$	0	0
$103$	528.874	0.505937	0.252969	−	0.967475i	$-0.418593\pi$
$103$	528.874	0.505937	0.252969	+	0.967475i	$0.418593\pi$
$104$	−582.690	−0.549399
$105$	0	0
$106$	−111.979	−0.102607
$107$	490.910	0.443533	0.221766	−	0.975100i	$-0.428818\pi$
$107$	490.910	0.443533	0.221766	+	0.975100i	$0.428818\pi$
$108$	0	0
$109$	−351.634	−0.308994	−0.154497	−	0.987993i	$-0.549376\pi$
$109$	−351.634	−0.308994	−0.154497	+	0.987993i	$0.549376\pi$
$110$	867.584	0.752008
$111$	0	0
$112$	−1292.71	−1.09063
$113$	1762.45	1.46724	0.733618	−	0.679563i	$-0.237830\pi$
$113$	1762.45	1.46724	0.733618	+	0.679563i	$0.237830\pi$
$114$	0	0
$115$	311.814	0.252842
$116$	504.158	0.403533
$117$	0	0
$118$	1815.20	1.41613
$119$	−202.731	−0.156171
$120$	0	0
$121$	326.681	0.245440
$122$	−3625.05	−2.69013
$123$	0	0
$124$	1046.81	0.758113
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1506.12	−1.05234	−0.526169	−	0.850380i	$-0.676372\pi$
$127$	−1506.12	−1.05234	−0.526169	+	0.850380i	$0.676372\pi$
$128$	1136.69	0.784920
$129$	0	0
$130$	1347.10	0.908835
$131$	1275.12	0.850444	0.425222	−	0.905089i	$-0.360196\pi$
$131$	1275.12	0.850444	0.425222	+	0.905089i	$0.360196\pi$
$132$	0	0
$133$	2318.99	1.51190
$134$	−410.441	−0.264603
$135$	0	0
$136$	60.7403	0.0382973
$137$	1439.42	0.897652	0.448826	−	0.893619i	$-0.351842\pi$
$137$	1439.42	0.897652	0.448826	+	0.893619i	$0.351842\pi$
$138$	0	0
$139$	−2539.50	−1.54962	−0.774811	−	0.632193i	$-0.782155\pi$
$139$	−2539.50	−1.54962	−0.774811	+	0.632193i	$0.782155\pi$
$140$	−1563.23	−0.943694
$141$	0	0
$142$	−4057.91	−2.39812
$143$	2573.89	1.50517
$144$	0	0
$145$	−248.042	−0.142060
$146$	216.736	0.122858
$147$	0	0
$148$	−2866.20	−1.59189
$149$	93.7592	0.0515507	0.0257754	−	0.999668i	$-0.491795\pi$
$149$	93.7592	0.0515507	0.0257754	+	0.999668i	$0.491795\pi$
$150$	0	0
$151$	−1068.13	−0.575650	−0.287825	−	0.957683i	$-0.592932\pi$
$151$	−1068.13	−0.575650	−0.287825	+	0.957683i	$0.592932\pi$
$152$	−694.794	−0.370758
$153$	0	0
$154$	−5338.06	−2.79320
$155$	−515.021	−0.266887
$156$	0	0
$157$	181.715	0.0923721	0.0461861	−	0.998933i	$-0.485293\pi$
$157$	181.715	0.0923721	0.0461861	+	0.998933i	$0.485293\pi$
$158$	840.760	0.423337
$159$	0	0
$160$	−1264.10	−0.624598
$161$	−1918.52	−0.939136
$162$	0	0
$163$	−1103.07	−0.530056	−0.265028	−	0.964241i	$-0.585381\pi$
$163$	−1103.07	−0.530056	−0.265028	+	0.964241i	$0.585381\pi$
$164$	−1601.04	−0.762319
$165$	0	0
$166$	842.718	0.394022
$167$	−4041.70	−1.87279	−0.936396	−	0.350945i	$-0.885860\pi$
$167$	−4041.70	−1.87279	−0.936396	+	0.350945i	$0.885860\pi$
$168$	0	0
$169$	1799.49	0.819067
$170$	−140.423	−0.0633529
$171$	0	0
$172$	−3433.12	−1.52194
$173$	−2747.76	−1.20756	−0.603780	−	0.797151i	$-0.706340\pi$
$173$	−2747.76	−1.20756	−0.603780	+	0.797151i	$0.706340\pi$
$174$	0	0
$175$	769.098	0.332219
$176$	−1710.85	−0.732727
$177$	0	0
$178$	−5815.39	−2.44877
$179$	−2838.32	−1.18517	−0.592587	−	0.805506i	$-0.701893\pi$
$179$	−2838.32	−1.18517	−0.592587	+	0.805506i	$0.701893\pi$
$180$	0	0
$181$	3442.25	1.41359	0.706796	−	0.707417i	$-0.250140\pi$
$181$	3442.25	1.41359	0.706796	+	0.707417i	$0.250140\pi$
$182$	−8288.43	−3.37571
$183$	0	0
$184$	574.809	0.230302
$185$	1410.15	0.560411
$186$	0	0
$187$	−268.305	−0.104922
$188$	452.403	0.175505
$189$	0	0
$190$	1606.27	0.613322
$191$	−746.295	−0.282723	−0.141361	−	0.989958i	$-0.545148\pi$
$191$	−746.295	−0.282723	−0.141361	+	0.989958i	$0.545148\pi$
$192$	0	0
$193$	107.908	0.0402454	0.0201227	−	0.999798i	$-0.493594\pi$
$193$	107.908	0.0402454	0.0201227	+	0.999798i	$0.493594\pi$
$194$	6099.28	2.25723
$195$	0	0
$196$	6132.41	2.23484
$197$	−3361.02	−1.21555	−0.607774	−	0.794110i	$-0.707937\pi$
$197$	−3361.02	−1.21555	−0.607774	+	0.794110i	$0.707937\pi$
$198$	0	0
$199$	1368.99	0.487663	0.243831	−	0.969818i	$-0.421596\pi$
$199$	1368.99	0.487663	0.243831	+	0.969818i	$0.421596\pi$
$200$	−230.430	−0.0814692
$201$	0	0
$202$	−4720.42	−1.64419
$203$	1526.15	0.527658
$204$	0	0
$205$	787.700	0.268368
$206$	−2253.94	−0.762329
$207$	0	0
$208$	−2656.44	−0.885534
$209$	3069.08	1.01575
$210$	0	0
$211$	2502.55	0.816504	0.408252	−	0.912869i	$-0.366138\pi$
$211$	2502.55	0.816504	0.408252	+	0.912869i	$0.366138\pi$
$212$	267.029	0.0865077
$213$	0	0
$214$	−2092.15	−0.668300
$215$	1689.07	0.535784
$216$	0	0
$217$	3168.81	0.991305
$218$	1498.58	0.465582
$219$	0	0
$220$	−2068.86	−0.634013
$221$	−416.599	−0.126803
$222$	0	0
$223$	1874.53	0.562904	0.281452	−	0.959575i	$-0.409184\pi$
$223$	1874.53	0.562904	0.281452	+	0.959575i	$0.409184\pi$
$224$	7777.72	2.31996
$225$	0	0
$226$	−7511.18	−2.21078
$227$	5078.74	1.48497	0.742484	−	0.669864i	$-0.233647\pi$
$227$	5078.74	1.48497	0.742484	+	0.669864i	$0.233647\pi$
$228$	0	0
$229$	865.865	0.249860	0.124930	−	0.992166i	$-0.460129\pi$
$229$	865.865	0.249860	0.124930	+	0.992166i	$0.460129\pi$
$230$	−1328.88	−0.380973
$231$	0	0
$232$	−457.249	−0.129396
$233$	1142.10	0.321122	0.160561	−	0.987026i	$-0.448670\pi$
$233$	1142.10	0.321122	0.160561	+	0.987026i	$0.448670\pi$
$234$	0	0
$235$	−222.579	−0.0617849
$236$	−4328.58	−1.19393
$237$	0	0
$238$	863.995	0.235313
$239$	−3149.37	−0.852366	−0.426183	−	0.904637i	$-0.640142\pi$
$239$	−3149.37	−0.852366	−0.426183	+	0.904637i	$0.640142\pi$
$240$	0	0
$241$	4504.37	1.20395	0.601975	−	0.798515i	$-0.294381\pi$
$241$	4504.37	1.20395	0.601975	+	0.798515i	$0.294381\pi$
$242$	−1392.24	−0.369821
$243$	0	0
$244$	8644.39	2.26803
$245$	−3017.10	−0.786756
$246$	0	0
$247$	4765.38	1.22759
$248$	−949.409	−0.243095
$249$	0	0
$250$	532.722	0.134769
$251$	886.861	0.223021	0.111510	−	0.993763i	$-0.464431\pi$
$251$	886.861	0.223021	0.111510	+	0.993763i	$0.464431\pi$
$252$	0	0
$253$	−2539.08	−0.630950
$254$	6418.77	1.58563
$255$	0	0
$256$	1086.07	0.265153
$257$	2262.04	0.549035	0.274517	−	0.961582i	$-0.411482\pi$
$257$	2262.04	0.549035	0.274517	+	0.961582i	$0.411482\pi$
$258$	0	0
$259$	−8676.33	−2.08155
$260$	−3212.34	−0.766233
$261$	0	0
$262$	−5434.30	−1.28142
$263$	7280.62	1.70701	0.853503	−	0.521088i	$-0.174474\pi$
$263$	7280.62	1.70701	0.853503	+	0.521088i	$0.174474\pi$
$264$	0	0
$265$	−131.376	−0.0304543
$266$	−9883.04	−2.27807
$267$	0	0
$268$	978.750	0.223085
$269$	−106.781	−0.0242028	−0.0121014	−	0.999927i	$-0.503852\pi$
$269$	−106.781	−0.0242028	−0.0121014	+	0.999927i	$0.503852\pi$
$270$	0	0
$271$	5908.85	1.32449	0.662246	−	0.749287i	$-0.269603\pi$
$271$	5908.85	1.32449	0.662246	+	0.749287i	$0.269603\pi$
$272$	276.910	0.0617285
$273$	0	0
$274$	−6134.51	−1.35255
$275$	1017.87	0.223199
$276$	0	0
$277$	3290.08	0.713653	0.356827	−	0.934171i	$-0.383859\pi$
$277$	3290.08	0.713653	0.356827	+	0.934171i	$0.383859\pi$
$278$	10822.8	2.33492
$279$	0	0
$280$	1417.78	0.302603
$281$	339.723	0.0721216	0.0360608	−	0.999350i	$-0.488519\pi$
$281$	339.723	0.0721216	0.0360608	+	0.999350i	$0.488519\pi$
$282$	0	0
$283$	2599.66	0.546056	0.273028	−	0.962006i	$-0.411975\pi$
$283$	2599.66	0.546056	0.273028	+	0.962006i	$0.411975\pi$
$284$	9676.61	2.02184
$285$	0	0
$286$	−10969.3	−2.26794
$287$	−4846.55	−0.996804
$288$	0	0
$289$	−4869.57	−0.991161
$290$	1057.10	0.214052
$291$	0	0
$292$	−516.835	−0.103580
$293$	178.085	0.0355081	0.0177540	−	0.999842i	$-0.494348\pi$
$293$	178.085	0.0355081	0.0177540	+	0.999842i	$0.494348\pi$
$294$	0	0
$295$	2129.63	0.420312
$296$	2599.52	0.510452
$297$	0	0
$298$	−399.581	−0.0776749
$299$	−3942.44	−0.762532
$300$	0	0
$301$	−10392.5	−1.99008
$302$	4552.13	0.867370
$303$	0	0
$304$	−3167.51	−0.597596
$305$	−4252.97	−0.798441
$306$	0	0
$307$	1537.60	0.285848	0.142924	−	0.989734i	$-0.454349\pi$
$307$	1537.60	0.285848	0.142924	+	0.989734i	$0.454349\pi$
$308$	12729.3	2.35493
$309$	0	0
$310$	2194.91	0.402136
$311$	945.323	0.172361	0.0861807	−	0.996280i	$-0.472534\pi$
$311$	945.323	0.172361	0.0861807	+	0.996280i	$0.472534\pi$
$312$	0	0
$313$	1985.53	0.358559	0.179280	−	0.983798i	$-0.442623\pi$
$313$	1985.53	0.358559	0.179280	+	0.983798i	$0.442623\pi$
$314$	−774.428	−0.139183
$315$	0	0
$316$	−2004.90	−0.356913
$317$	−5183.24	−0.918358	−0.459179	−	0.888344i	$-0.651856\pi$
$317$	−5183.24	−0.918358	−0.459179	+	0.888344i	$0.651856\pi$
$318$	0	0
$319$	2019.79	0.354503
$320$	3706.48	0.647496
$321$	0	0
$322$	8176.33	1.41506
$323$	−496.748	−0.0855722
$324$	0	0
$325$	1580.45	0.269746
$326$	4701.05	0.798671
$327$	0	0
$328$	1452.07	0.244443
$329$	1369.48	0.229489
$330$	0	0
$331$	−2173.36	−0.360903	−0.180451	−	0.983584i	$-0.557756\pi$
$331$	−2173.36	−0.360903	−0.180451	+	0.983584i	$0.557756\pi$
$332$	−2009.57	−0.332197
$333$	0	0
$334$	17224.8	2.82186
$335$	−481.538	−0.0785350
$336$	0	0
$337$	−7850.31	−1.26894	−0.634472	−	0.772946i	$-0.718782\pi$
$337$	−7850.31	−1.26894	−0.634472	+	0.772946i	$0.718782\pi$
$338$	−7669.03	−1.23414
$339$	0	0
$340$	334.858	0.0534124
$341$	4193.78	0.665999
$342$	0	0
$343$	8011.53	1.26117
$344$	3113.69	0.488021
$345$	0	0
$346$	11710.3	1.81951
$347$	−9372.24	−1.44994	−0.724969	−	0.688782i	$-0.758146\pi$
$347$	−9372.24	−1.44994	−0.724969	+	0.688782i	$0.758146\pi$
$348$	0	0
$349$	−1177.90	−0.180664	−0.0903321	−	0.995912i	$-0.528793\pi$
$349$	−1177.90	−0.180664	−0.0903321	+	0.995912i	$0.528793\pi$
$350$	−3277.73	−0.500577
$351$	0	0
$352$	10293.4	1.55864
$353$	−7262.13	−1.09497	−0.547484	−	0.836816i	$-0.684414\pi$
$353$	−7262.13	−1.09497	−0.547484	+	0.836816i	$0.684414\pi$
$354$	0	0
$355$	−4760.82	−0.711769
$356$	13867.5	2.06454
$357$	0	0
$358$	12096.3	1.78578
$359$	−1939.95	−0.285199	−0.142600	−	0.989780i	$-0.545546\pi$
$359$	−1939.95	−0.285199	−0.142600	+	0.989780i	$0.545546\pi$
$360$	0	0
$361$	−1176.81	−0.171572
$362$	−14670.1	−2.12995
$363$	0	0
$364$	19764.8	2.84604
$365$	254.279	0.0364646
$366$	0	0
$367$	6501.94	0.924792	0.462396	−	0.886674i	$-0.346990\pi$
$367$	6501.94	0.924792	0.462396	+	0.886674i	$0.346990\pi$
$368$	2620.51	0.371205
$369$	0	0
$370$	−6009.73	−0.844409
$371$	808.330	0.113117
$372$	0	0
$373$	13318.0	1.84874	0.924372	−	0.381492i	$-0.124589\pi$
$373$	13318.0	1.84874	0.924372	+	0.381492i	$0.124589\pi$
$374$	1143.46	0.158093
$375$	0	0
$376$	−410.311	−0.0562770
$377$	3136.13	0.428432
$378$	0	0
$379$	−3198.42	−0.433488	−0.216744	−	0.976228i	$-0.569544\pi$
$379$	−3198.42	−0.433488	−0.216744	+	0.976228i	$0.569544\pi$
$380$	−3830.36	−0.517087
$381$	0	0
$382$	3180.55	0.425997
$383$	−2340.50	−0.312256	−0.156128	−	0.987737i	$-0.549901\pi$
$383$	−2340.50	−0.312256	−0.156128	+	0.987737i	$0.549901\pi$
$384$	0	0
$385$	−6262.71	−0.829032
$386$	−459.878	−0.0606403
$387$	0	0
$388$	−14544.5	−1.90305
$389$	3991.94	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$389$	3991.94	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$390$	0	0
$391$	410.964	0.0531544
$392$	−5561.83	−0.716619
$393$	0	0
$394$	14323.9	1.83155
$395$	986.395	0.125648
$396$	0	0
$397$	−4960.90	−0.627155	−0.313577	−	0.949563i	$-0.601528\pi$
$397$	−4960.90	−0.627155	−0.313577	+	0.949563i	$0.601528\pi$
$398$	−5834.31	−0.734793
$399$	0	0
$400$	−1050.51	−0.131314
$401$	827.622	0.103066	0.0515330	−	0.998671i	$-0.483589\pi$
$401$	827.622	0.103066	0.0515330	+	0.998671i	$0.483589\pi$
$402$	0	0
$403$	6511.70	0.804890
$404$	11256.4	1.38621
$405$	0	0
$406$	−6504.11	−0.795058
$407$	−11482.7	−1.39847
$408$	0	0
$409$	−8367.64	−1.01162	−0.505811	−	0.862645i	$-0.668807\pi$
$409$	−8367.64	−1.01162	−0.505811	+	0.862645i	$0.668807\pi$
$410$	−3357.00	−0.404367
$411$	0	0
$412$	5374.82	0.642714
$413$	−13103.2	−1.56117
$414$	0	0
$415$	988.693	0.116947
$416$	15982.7	1.88369
$417$	0	0
$418$	−13079.7	−1.53050
$419$	4200.72	0.489782	0.244891	−	0.969551i	$-0.421248\pi$
$419$	4200.72	0.489782	0.244891	+	0.969551i	$0.421248\pi$
$420$	0	0
$421$	6938.76	0.803265	0.401632	−	0.915801i	$-0.368443\pi$
$421$	6938.76	0.803265	0.401632	+	0.915801i	$0.368443\pi$
$422$	−10665.3	−1.23028
$423$	0	0
$424$	−242.184	−0.0277394
$425$	−164.747	−0.0188034
$426$	0	0
$427$	26167.6	2.96567
$428$	4988.99	0.563439
$429$	0	0
$430$	−7198.44	−0.807301
$431$	6827.09	0.762991	0.381496	−	0.924371i	$-0.375409\pi$
$431$	6827.09	0.762991	0.381496	+	0.924371i	$0.375409\pi$
$432$	0	0
$433$	−2199.59	−0.244124	−0.122062	−	0.992522i	$-0.538951\pi$
$433$	−2199.59	−0.244124	−0.122062	+	0.992522i	$0.538951\pi$
$434$	−13504.8	−1.49366
$435$	0	0
$436$	−3573.57	−0.392529
$437$	−4700.92	−0.514590
$438$	0	0
$439$	8324.32	0.905007	0.452504	−	0.891763i	$-0.350531\pi$
$439$	8324.32	0.905007	0.452504	+	0.891763i	$0.350531\pi$
$440$	1876.37	0.203301
$441$	0	0
$442$	1775.45	0.191063
$443$	−10866.8	−1.16546	−0.582729	−	0.812667i	$-0.698015\pi$
$443$	−10866.8	−1.16546	−0.582729	+	0.812667i	$0.698015\pi$
$444$	0	0
$445$	−6822.72	−0.726805
$446$	−7988.81	−0.848164
$447$	0	0
$448$	−22805.2	−2.40501
$449$	−4947.73	−0.520040	−0.260020	−	0.965603i	$-0.583729\pi$
$449$	−4947.73	−0.520040	−0.260020	+	0.965603i	$0.583729\pi$
$450$	0	0
$451$	−6414.18	−0.669694
$452$	17911.4	1.86389
$453$	0	0
$454$	−21644.5	−2.23750
$455$	−9724.14	−1.00192
$456$	0	0
$457$	−14197.7	−1.45326	−0.726630	−	0.687029i	$-0.758915\pi$
$457$	−14197.7	−1.45326	−0.726630	+	0.687029i	$0.758915\pi$
$458$	−3690.13	−0.376481
$459$	0	0
$460$	3168.89	0.321196
$461$	−18228.0	−1.84157	−0.920786	−	0.390069i	$-0.872451\pi$
$461$	−18228.0	−1.84157	−0.920786	+	0.390069i	$0.872451\pi$
$462$	0	0
$463$	4341.19	0.435750	0.217875	−	0.975977i	$-0.430088\pi$
$463$	4341.19	0.435750	0.217875	+	0.975977i	$0.430088\pi$
$464$	−2084.57	−0.208564
$465$	0	0
$466$	−4867.38	−0.483856
$467$	4919.63	0.487481	0.243740	−	0.969841i	$-0.421626\pi$
$467$	4919.63	0.487481	0.243740	+	0.969841i	$0.421626\pi$
$468$	0	0
$469$	2962.80	0.291704
$470$	948.583	0.0930954
$471$	0	0
$472$	3925.84	0.382842
$473$	−13754.0	−1.33702
$474$	0	0
$475$	1884.51	0.182036
$476$	−2060.31	−0.198391
$477$	0	0
$478$	13421.9	1.28432
$479$	4572.83	0.436196	0.218098	−	0.975927i	$-0.430015\pi$
$479$	4572.83	0.436196	0.218098	+	0.975927i	$0.430015\pi$
$480$	0	0
$481$	−17829.3	−1.69011
$482$	−19196.6	−1.81407
$483$	0	0
$484$	3319.98	0.311794
$485$	7155.79	0.669954
$486$	0	0
$487$	15751.5	1.46564	0.732822	−	0.680421i	$-0.238203\pi$
$487$	15751.5	1.46564	0.732822	+	0.680421i	$0.238203\pi$
$488$	−7840.09	−0.727263
$489$	0	0
$490$	12858.2	1.18546
$491$	15308.4	1.40704	0.703520	−	0.710676i	$-0.251611\pi$
$491$	15308.4	1.40704	0.703520	+	0.710676i	$0.251611\pi$
$492$	0	0
$493$	−326.914	−0.0298650
$494$	−20309.0	−1.84968
$495$	0	0
$496$	−4328.28	−0.391826
$497$	29292.3	2.64374
$498$	0	0
$499$	17732.1	1.59078	0.795389	−	0.606099i	$-0.207267\pi$
$499$	17732.1	1.59078	0.795389	+	0.606099i	$0.207267\pi$
$500$	−1270.34	−0.113623
$501$	0	0
$502$	−3779.61	−0.336040
$503$	−10511.5	−0.931775	−0.465887	−	0.884844i	$-0.654265\pi$
$503$	−10511.5	−0.931775	−0.465887	+	0.884844i	$0.654265\pi$
$504$	0	0
$505$	−5538.08	−0.488003
$506$	10821.0	0.950695
$507$	0	0
$508$	−15306.4	−1.33683
$509$	19630.8	1.70947	0.854737	−	0.519062i	$-0.173719\pi$
$509$	19630.8	1.70947	0.854737	+	0.519062i	$0.173719\pi$
$510$	0	0
$511$	−1564.52	−0.135441
$512$	−13722.1	−1.18444
$513$	0	0
$514$	−9640.30	−0.827267
$515$	−2644.37	−0.226262
$516$	0	0
$517$	1812.45	0.154180
$518$	36976.6	3.13641
$519$	0	0
$520$	2913.45	0.245699
$521$	88.4336	0.00743636	0.00371818	−	0.999993i	$-0.498816\pi$
$521$	88.4336	0.00743636	0.00371818	+	0.999993i	$0.498816\pi$
$522$	0	0
$523$	21346.4	1.78473	0.892363	−	0.451317i	$-0.149046\pi$
$523$	21346.4	1.78473	0.892363	+	0.451317i	$0.149046\pi$
$524$	12958.8	1.08036
$525$	0	0
$526$	−31028.4	−2.57206
$527$	−678.787	−0.0561071
$528$	0	0
$529$	−8277.88	−0.680355
$530$	559.896	0.0458874
$531$	0	0
$532$	23567.4	1.92063
$533$	−9959.33	−0.809356
$534$	0	0
$535$	−2454.55	−0.198354
$536$	−887.684	−0.0715338
$537$	0	0
$538$	455.077	0.0364679
$539$	24568.0	1.96330
$540$	0	0
$541$	−14432.6	−1.14696	−0.573480	−	0.819220i	$-0.694407\pi$
$541$	−14432.6	−1.14696	−0.573480	+	0.819220i	$0.694407\pi$
$542$	−25182.2	−1.99570
$543$	0	0
$544$	−1666.05	−0.131308
$545$	1758.17	0.138187
$546$	0	0
$547$	−16569.6	−1.29518	−0.647592	−	0.761988i	$-0.724224\pi$
$547$	−16569.6	−1.29518	−0.647592	+	0.761988i	$0.724224\pi$
$548$	14628.5	1.14033
$549$	0	0
$550$	−4337.92	−0.336308
$551$	3739.49	0.289125
$552$	0	0
$553$	−6069.08	−0.466697
$554$	−14021.6	−1.07531
$555$	0	0
$556$	−25808.3	−1.96855
$557$	11597.0	0.882191	0.441096	−	0.897460i	$-0.354590\pi$
$557$	11597.0	0.882191	0.441096	+	0.897460i	$0.354590\pi$
$558$	0	0
$559$	−21355.9	−1.61584
$560$	6463.57	0.487742
$561$	0	0
$562$	−1447.82	−0.108670
$563$	−23049.3	−1.72542	−0.862709	−	0.505700i	$-0.831234\pi$
$563$	−23049.3	−1.72542	−0.862709	+	0.505700i	$0.831234\pi$
$564$	0	0
$565$	−8812.26	−0.656168
$566$	−11079.2	−0.822778
$567$	0	0
$568$	−8776.27	−0.648317
$569$	14733.9	1.08555	0.542775	−	0.839878i	$-0.317374\pi$
$569$	14733.9	1.08555	0.542775	+	0.839878i	$0.317374\pi$
$570$	0	0
$571$	15111.9	1.10755	0.553776	−	0.832666i	$-0.313186\pi$
$571$	15111.9	1.10755	0.553776	+	0.832666i	$0.313186\pi$
$572$	26157.8	1.91209
$573$	0	0
$574$	20654.9	1.50195
$575$	−1559.07	−0.113074
$576$	0	0
$577$	−26150.5	−1.88676	−0.943380	−	0.331715i	$-0.892373\pi$
$577$	−26150.5	−1.88676	−0.943380	+	0.331715i	$0.892373\pi$
$578$	20753.0	1.49345
$579$	0	0
$580$	−2520.79	−0.180466
$581$	−6083.22	−0.434379
$582$	0	0
$583$	1069.79	0.0759967
$584$	468.747	0.0332139
$585$	0	0
$586$	−758.961	−0.0535023
$587$	2091.54	0.147065	0.0735326	−	0.997293i	$-0.476573\pi$
$587$	2091.54	0.147065	0.0735326	+	0.997293i	$0.476573\pi$
$588$	0	0
$589$	7764.49	0.543175
$590$	−9076.02	−0.633311
$591$	0	0
$592$	11851.0	0.822759
$593$	3260.34	0.225778	0.112889	−	0.993608i	$-0.463990\pi$
$593$	3260.34	0.225778	0.112889	+	0.993608i	$0.463990\pi$
$594$	0	0
$595$	1013.66	0.0698417
$596$	952.852	0.0654872
$597$	0	0
$598$	16801.8	1.14896
$599$	−22999.2	−1.56882	−0.784410	−	0.620243i	$-0.787034\pi$
$599$	−22999.2	−1.56882	−0.784410	+	0.620243i	$0.787034\pi$
$600$	0	0
$601$	−14293.9	−0.970148	−0.485074	−	0.874473i	$-0.661207\pi$
$601$	−14293.9	−0.970148	−0.485074	+	0.874473i	$0.661207\pi$
$602$	44290.5	2.99858
$603$	0	0
$604$	−10855.1	−0.731274
$605$	−1633.40	−0.109764
$606$	0	0
$607$	−16434.1	−1.09891	−0.549455	−	0.835523i	$-0.685165\pi$
$607$	−16434.1	−1.09891	−0.549455	+	0.835523i	$0.685165\pi$
$608$	19057.6	1.27120
$609$	0	0
$610$	18125.2	1.20306
$611$	2814.19	0.186334
$612$	0	0
$613$	3674.45	0.242104	0.121052	−	0.992646i	$-0.461373\pi$
$613$	3674.45	0.242104	0.121052	+	0.992646i	$0.461373\pi$
$614$	−6552.91	−0.430707
$615$	0	0
$616$	−11544.9	−0.755126
$617$	8627.57	0.562938	0.281469	−	0.959570i	$-0.409178\pi$
$617$	8627.57	0.562938	0.281469	+	0.959570i	$0.409178\pi$
$618$	0	0
$619$	−1309.81	−0.0850496	−0.0425248	−	0.999095i	$-0.513540\pi$
$619$	−1309.81	−0.0850496	−0.0425248	+	0.999095i	$0.513540\pi$
$620$	−5234.03	−0.339038
$621$	0	0
$622$	−4028.76	−0.259708
$623$	41978.8	2.69959
$624$	0	0
$625$	625.000	0.0400000
$626$	−8461.91	−0.540265
$627$	0	0
$628$	1846.72	0.117344
$629$	1858.54	0.117814
$630$	0	0
$631$	14447.2	0.911463	0.455731	−	0.890117i	$-0.349378\pi$
$631$	14447.2	0.911463	0.455731	+	0.890117i	$0.349378\pi$
$632$	1818.36	0.114447
$633$	0	0
$634$	22089.8	1.38375
$635$	7530.62	0.470620
$636$	0	0
$637$	38146.9	2.37274
$638$	−8607.88	−0.534152
$639$	0	0
$640$	−5683.43	−0.351027
$641$	−20073.1	−1.23688	−0.618440	−	0.785832i	$-0.712235\pi$
$641$	−20073.1	−1.23688	−0.618440	+	0.785832i	$0.712235\pi$
$642$	0	0
$643$	−17735.9	−1.08777	−0.543885	−	0.839160i	$-0.683047\pi$
$643$	−17735.9	−1.08777	−0.543885	+	0.839160i	$0.683047\pi$
$644$	−19497.5	−1.19303
$645$	0	0
$646$	2117.03	0.128937
$647$	−11456.8	−0.696158	−0.348079	−	0.937465i	$-0.613166\pi$
$647$	−11456.8	−0.696158	−0.348079	+	0.937465i	$0.613166\pi$
$648$	0	0
$649$	−17341.4	−1.04886
$650$	−6735.51	−0.406444
$651$	0	0
$652$	−11210.2	−0.673354
$653$	29071.9	1.74223	0.871113	−	0.491083i	$-0.163399\pi$
$653$	29071.9	1.74223	0.871113	+	0.491083i	$0.163399\pi$
$654$	0	0
$655$	−6375.62	−0.380330
$656$	6619.90	0.393999
$657$	0	0
$658$	−5836.43	−0.345787
$659$	−4355.09	−0.257436	−0.128718	−	0.991681i	$-0.541086\pi$
$659$	−4355.09	−0.257436	−0.128718	+	0.991681i	$0.541086\pi$
$660$	0	0
$661$	27295.3	1.60615	0.803075	−	0.595878i	$-0.203196\pi$
$661$	27295.3	1.60615	0.803075	+	0.595878i	$0.203196\pi$
$662$	9262.38	0.543796
$663$	0	0
$664$	1822.59	0.106522
$665$	−11595.0	−0.676141
$666$	0	0
$667$	−3093.72	−0.179594
$668$	−41074.8	−2.37909
$669$	0	0
$670$	2052.21	0.118334
$671$	34631.7	1.99246
$672$	0	0
$673$	−17760.0	−1.01723	−0.508617	−	0.860993i	$-0.669843\pi$
$673$	−17760.0	−1.01723	−0.508617	+	0.860993i	$0.669843\pi$
$674$	33456.3	1.91200
$675$	0	0
$676$	18287.8	1.04050
$677$	1937.33	0.109982	0.0549910	−	0.998487i	$-0.482487\pi$
$677$	1937.33	0.109982	0.0549910	+	0.998487i	$0.482487\pi$
$678$	0	0
$679$	−44028.0	−2.48843
$680$	−303.702	−0.0171271
$681$	0	0
$682$	−17873.0	−1.00351
$683$	−2125.71	−0.119090	−0.0595448	−	0.998226i	$-0.518965\pi$
$683$	−2125.71	−0.119090	−0.0595448	+	0.998226i	$0.518965\pi$
$684$	0	0
$685$	−7197.12	−0.401442
$686$	−34143.4	−1.90029
$687$	0	0
$688$	14195.1	0.786603
$689$	1661.06	0.0918454
$690$	0	0
$691$	13826.3	0.761180	0.380590	−	0.924744i	$-0.375721\pi$
$691$	13826.3	0.761180	0.380590	+	0.924744i	$0.375721\pi$
$692$	−27924.8	−1.53402
$693$	0	0
$694$	39942.4	2.18472
$695$	12697.5	0.693012
$696$	0	0
$697$	1038.17	0.0564183
$698$	5019.97	0.272219
$699$	0	0
$700$	7816.16	0.422033
$701$	−24464.0	−1.31810	−0.659052	−	0.752097i	$-0.729042\pi$
$701$	−24464.0	−1.31810	−0.659052	+	0.752097i	$0.729042\pi$
$702$	0	0
$703$	−21259.5	−1.14056
$704$	−30181.6	−1.61578
$705$	0	0
$706$	30949.6	1.64986
$707$	34074.6	1.81260
$708$	0	0
$709$	−29498.3	−1.56253	−0.781263	−	0.624202i	$-0.785424\pi$
$709$	−29498.3	−1.56253	−0.781263	+	0.624202i	$0.785424\pi$
$710$	20289.6	1.07247
$711$	0	0
$712$	−12577.3	−0.662012
$713$	−6423.63	−0.337401
$714$	0	0
$715$	−12869.4	−0.673133
$716$	−28845.2	−1.50558
$717$	0	0
$718$	8267.63	0.429729
$719$	−3857.66	−0.200093	−0.100046	−	0.994983i	$-0.531899\pi$
$719$	−3857.66	−0.200093	−0.100046	+	0.994983i	$0.531899\pi$
$720$	0	0
$721$	16270.2	0.840410
$722$	5015.32	0.258519
$723$	0	0
$724$	34982.7	1.79575
$725$	1240.21	0.0635313
$726$	0	0
$727$	8745.10	0.446132	0.223066	−	0.974803i	$-0.428394\pi$
$727$	8745.10	0.446132	0.223066	+	0.974803i	$0.428394\pi$
$728$	−17925.8	−0.912604
$729$	0	0
$730$	−1083.68	−0.0549436
$731$	2226.16	0.112637
$732$	0	0
$733$	−9418.40	−0.474593	−0.237296	−	0.971437i	$-0.576261\pi$
$733$	−9418.40	−0.474593	−0.237296	+	0.971437i	$0.576261\pi$
$734$	−27709.8	−1.39345
$735$	0	0
$736$	−15766.5	−0.789621
$737$	3921.13	0.195979
$738$	0	0
$739$	6219.42	0.309587	0.154794	−	0.987947i	$-0.450529\pi$
$739$	6219.42	0.309587	0.154794	+	0.987947i	$0.450529\pi$
$740$	14331.0	0.711915
$741$	0	0
$742$	−3444.92	−0.170441
$743$	−29387.4	−1.45104	−0.725518	−	0.688203i	$-0.758400\pi$
$743$	−29387.4	−1.45104	−0.725518	+	0.688203i	$0.758400\pi$
$744$	0	0
$745$	−468.796	−0.0230542
$746$	−56758.5	−2.78563
$747$	0	0
$748$	−2726.72	−0.133287
$749$	15102.3	0.736751
$750$	0	0
$751$	−21647.1	−1.05182	−0.525909	−	0.850541i	$-0.676275\pi$
$751$	−21647.1	−1.05182	−0.525909	+	0.850541i	$0.676275\pi$
$752$	−1870.57	−0.0907086
$753$	0	0
$754$	−13365.5	−0.645547
$755$	5340.65	0.257438
$756$	0	0
$757$	−13907.2	−0.667722	−0.333861	−	0.942622i	$-0.608352\pi$
$757$	−13907.2	−0.667722	−0.333861	+	0.942622i	$0.608352\pi$
$758$	13631.0	0.653165
$759$	0	0
$760$	3473.97	0.165808
$761$	−23906.1	−1.13876	−0.569379	−	0.822075i	$-0.692816\pi$
$761$	−23906.1	−1.13876	−0.569379	+	0.822075i	$0.692816\pi$
$762$	0	0
$763$	−10817.6	−0.513270
$764$	−7584.42	−0.359155
$765$	0	0
$766$	9974.70	0.470497
$767$	−26926.1	−1.26760
$768$	0	0
$769$	−25111.6	−1.17757	−0.588783	−	0.808291i	$-0.700393\pi$
$769$	−25111.6	−1.17757	−0.588783	+	0.808291i	$0.700393\pi$
$770$	26690.3	1.24916
$771$	0	0
$772$	1096.64	0.0511255
$773$	14909.4	0.693729	0.346865	−	0.937915i	$-0.387246\pi$
$773$	14909.4	0.693729	0.346865	+	0.937915i	$0.387246\pi$
$774$	0	0
$775$	2575.10	0.119355
$776$	13191.2	0.610229
$777$	0	0
$778$	−17012.7	−0.783980
$779$	−11875.4	−0.546188
$780$	0	0
$781$	38767.0	1.77617
$782$	−1751.44	−0.0800912
$783$	0	0
$784$	−25356.0	−1.15506
$785$	−908.574	−0.0413101
$786$	0	0
$787$	−2658.77	−0.120426	−0.0602128	−	0.998186i	$-0.519178\pi$
$787$	−2658.77	−0.120426	−0.0602128	+	0.998186i	$0.519178\pi$
$788$	−34157.2	−1.54416
$789$	0	0
$790$	−4203.80	−0.189322
$791$	54220.0	2.43722
$792$	0	0
$793$	53772.7	2.40798
$794$	21142.3	0.944976
$795$	0	0
$796$	13912.7	0.619499
$797$	−14172.2	−0.629870	−0.314935	−	0.949113i	$-0.601983\pi$
$797$	−14172.2	−0.629870	−0.314935	+	0.949113i	$0.601983\pi$
$798$	0	0
$799$	−293.355	−0.0129889
$800$	6320.48	0.279329
$801$	0	0
$802$	−3527.14	−0.155296
$803$	−2070.57	−0.0909950
$804$	0	0
$805$	9592.62	0.419994
$806$	−27751.4	−1.21278
$807$	0	0
$808$	−10209.1	−0.444499
$809$	−21077.7	−0.916012	−0.458006	−	0.888949i	$-0.651436\pi$
$809$	−21077.7	−0.916012	−0.458006	+	0.888949i	$0.651436\pi$
$810$	0	0
$811$	−11937.4	−0.516868	−0.258434	−	0.966029i	$-0.583206\pi$
$811$	−11937.4	−0.516868	−0.258434	+	0.966029i	$0.583206\pi$
$812$	15509.9	0.670308
$813$	0	0
$814$	48936.8	2.10717
$815$	5515.36	0.237048
$816$	0	0
$817$	−25464.5	−1.09044
$818$	35661.0	1.52428
$819$	0	0
$820$	8005.20	0.340919
$821$	14800.5	0.629161	0.314580	−	0.949231i	$-0.398136\pi$
$821$	14800.5	0.629161	0.314580	+	0.949231i	$0.398136\pi$
$822$	0	0
$823$	33176.6	1.40518	0.702591	−	0.711594i	$-0.252026\pi$
$823$	33176.6	1.40518	0.702591	+	0.711594i	$0.252026\pi$
$824$	−4874.73	−0.206091
$825$	0	0
$826$	55842.8	2.35232
$827$	−29001.3	−1.21944	−0.609718	−	0.792619i	$-0.708717\pi$
$827$	−29001.3	−1.21944	−0.609718	+	0.792619i	$0.708717\pi$
$828$	0	0
$829$	13221.1	0.553904	0.276952	−	0.960884i	$-0.410676\pi$
$829$	13221.1	0.553904	0.276952	+	0.960884i	$0.410676\pi$
$830$	−4213.59	−0.176212
$831$	0	0
$832$	−46863.1	−1.95275
$833$	−3976.47	−0.165398
$834$	0	0
$835$	20208.5	0.837538
$836$	31190.3	1.29036
$837$	0	0
$838$	−17902.5	−0.737987
$839$	−42123.9	−1.73335	−0.866675	−	0.498874i	$-0.833747\pi$
$839$	−42123.9	−1.73335	−0.866675	+	0.498874i	$0.833747\pi$
$840$	0	0
$841$	−21928.0	−0.899094
$842$	−29571.5	−1.21033
$843$	0	0
$844$	25432.8	1.03724
$845$	−8997.45	−0.366298
$846$	0	0
$847$	10050.0	0.407700
$848$	−1104.10	−0.0447109
$849$	0	0
$850$	702.117	0.0283323
$851$	17588.1	0.708476
$852$	0	0
$853$	14431.7	0.579287	0.289643	−	0.957135i	$-0.406463\pi$
$853$	14431.7	0.579287	0.289643	+	0.957135i	$0.406463\pi$
$854$	−111521.	−4.46857
$855$	0	0
$856$	−4524.80	−0.180671
$857$	16113.4	0.642268	0.321134	−	0.947034i	$-0.395936\pi$
$857$	16113.4	0.642268	0.321134	+	0.947034i	$0.395936\pi$
$858$	0	0
$859$	−22500.4	−0.893717	−0.446858	−	0.894605i	$-0.647457\pi$
$859$	−22500.4	−0.893717	−0.446858	+	0.894605i	$0.647457\pi$
$860$	17165.6	0.680630
$861$	0	0
$862$	−29095.5	−1.14965
$863$	−43396.8	−1.71175	−0.855877	−	0.517180i	$-0.826982\pi$
$863$	−43396.8	−1.71175	−0.855877	+	0.517180i	$0.826982\pi$
$864$	0	0
$865$	13738.8	0.540038
$866$	9374.18	0.367838
$867$	0	0
$868$	32203.9	1.25930
$869$	−8032.14	−0.313546
$870$	0	0
$871$	6088.35	0.236850
$872$	3241.07	0.125868
$873$	0	0
$874$	20034.3	0.775366
$875$	−3845.49	−0.148573
$876$	0	0
$877$	−2465.53	−0.0949317	−0.0474659	−	0.998873i	$-0.515115\pi$
$877$	−2465.53	−0.0949317	−0.0474659	+	0.998873i	$0.515115\pi$
$878$	−35476.4	−1.36363
$879$	0	0
$880$	8554.23	0.327685
$881$	24512.9	0.937412	0.468706	−	0.883354i	$-0.344720\pi$
$881$	24512.9	0.937412	0.468706	+	0.883354i	$0.344720\pi$
$882$	0	0
$883$	24236.1	0.923679	0.461840	−	0.886963i	$-0.347190\pi$
$883$	24236.1	0.923679	0.461840	+	0.886963i	$0.347190\pi$
$884$	−4233.79	−0.161084
$885$	0	0
$886$	46311.9	1.75607
$887$	25198.7	0.953877	0.476939	−	0.878937i	$-0.341746\pi$
$887$	25198.7	0.953877	0.476939	+	0.878937i	$0.341746\pi$
$888$	0	0
$889$	−46334.3	−1.74803
$890$	29076.9	1.09512
$891$	0	0
$892$	19050.3	0.715081
$893$	3355.61	0.125746
$894$	0	0
$895$	14191.6	0.530026
$896$	34968.9	1.30383
$897$	0	0
$898$	21086.1	0.783579
$899$	5109.87	0.189570
$900$	0	0
$901$	−173.151	−0.00640233
$902$	27335.8	1.00907
$903$	0	0
$904$	−16244.8	−0.597672
$905$	−17211.2	−0.632178
$906$	0	0
$907$	26608.1	0.974098	0.487049	−	0.873375i	$-0.338073\pi$
$907$	26608.1	0.974098	0.487049	+	0.873375i	$0.338073\pi$
$908$	51614.0	1.88642
$909$	0	0
$910$	41442.1	1.50966
$911$	24832.1	0.903101	0.451550	−	0.892246i	$-0.350871\pi$
$911$	24832.1	0.903101	0.451550	+	0.892246i	$0.350871\pi$
$912$	0	0
$913$	−8050.86	−0.291834
$914$	60507.4	2.18972
$915$	0	0
$916$	8799.58	0.317409
$917$	39227.8	1.41267
$918$	0	0
$919$	−26107.2	−0.937102	−0.468551	−	0.883436i	$-0.655224\pi$
$919$	−26107.2	−0.937102	−0.468551	+	0.883436i	$0.655224\pi$
$920$	−2874.05	−0.102994
$921$	0	0
$922$	77683.9	2.77482
$923$	60193.7	2.14659
$924$	0	0
$925$	−7050.73	−0.250623
$926$	−18501.2	−0.656573
$927$	0	0
$928$	12542.0	0.443653
$929$	−23361.5	−0.825046	−0.412523	−	0.910947i	$-0.635352\pi$
$929$	−23361.5	−0.825046	−0.412523	+	0.910947i	$0.635352\pi$
$930$	0	0
$931$	45485.9	1.60123
$932$	11606.9	0.407936
$933$	0	0
$934$	−20966.4	−0.734519
$935$	1341.53	0.0469226
$936$	0	0
$937$	10548.5	0.367775	0.183888	−	0.982947i	$-0.441132\pi$
$937$	10548.5	0.367775	0.183888	+	0.982947i	$0.441132\pi$
$938$	−12626.8	−0.439530
$939$	0	0
$940$	−2262.02	−0.0784881
$941$	−6490.14	−0.224838	−0.112419	−	0.993661i	$-0.535860\pi$
$941$	−6490.14	−0.224838	−0.112419	+	0.993661i	$0.535860\pi$
$942$	0	0
$943$	9824.63	0.339273
$944$	17897.6	0.617074
$945$	0	0
$946$	58616.4	2.01457
$947$	23506.0	0.806590	0.403295	−	0.915070i	$-0.367865\pi$
$947$	23506.0	0.806590	0.403295	+	0.915070i	$0.367865\pi$
$948$	0	0
$949$	−3214.99	−0.109972
$950$	−8031.35	−0.274286
$951$	0	0
$952$	1868.61	0.0636156
$953$	6740.26	0.229106	0.114553	−	0.993417i	$-0.463456\pi$
$953$	6740.26	0.229106	0.114553	+	0.993417i	$0.463456\pi$
$954$	0	0
$955$	3731.48	0.126437
$956$	−32006.2	−1.08280
$957$	0	0
$958$	−19488.4	−0.657245
$959$	44282.3	1.49109
$960$	0	0
$961$	−19181.1	−0.643857
$962$	75984.4	2.54661
$963$	0	0
$964$	45776.8	1.52943
$965$	−539.538	−0.0179983
$966$	0	0
$967$	16782.5	0.558108	0.279054	−	0.960275i	$-0.409979\pi$
$967$	16782.5	0.558108	0.279054	+	0.960275i	$0.409979\pi$
$968$	−3011.08	−0.0999791
$969$	0	0
$970$	−30496.4	−1.00946
$971$	−46282.7	−1.52964	−0.764822	−	0.644242i	$-0.777173\pi$
$971$	−46282.7	−1.52964	−0.764822	+	0.644242i	$0.777173\pi$
$972$	0	0
$973$	−78124.9	−2.57407
$974$	−67129.4	−2.20838
$975$	0	0
$976$	−35742.4	−1.17222
$977$	6767.95	0.221623	0.110812	−	0.993841i	$-0.464655\pi$
$977$	6767.95	0.221623	0.110812	+	0.993841i	$0.464655\pi$
$978$	0	0
$979$	55556.9	1.81370
$980$	−30662.0	−0.999452
$981$	0	0
$982$	−65240.8	−2.12008
$983$	−50074.9	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$983$	−50074.9	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$984$	0	0
$985$	16805.1	0.543609
$986$	1393.24	0.0449996
$987$	0	0
$988$	48429.3	1.55946
$989$	21067.0	0.677343
$990$	0	0
$991$	−2658.60	−0.0852200	−0.0426100	−	0.999092i	$-0.513567\pi$
$991$	−2658.60	−0.0852200	−0.0426100	+	0.999092i	$0.513567\pi$
$992$	26041.4	0.833485
$993$	0	0
$994$	−124837.	−3.98350
$995$	−6844.93	−0.218089
$996$	0	0
$997$	−60620.1	−1.92563	−0.962817	−	0.270153i	$-0.912926\pi$
$997$	−60620.1	−1.92563	−0.962817	+	0.270153i	$0.912926\pi$
$998$	−75570.3	−2.39693
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.m.1.2		7
3.2	odd	2		405.4.a.n.1.6		7
5.4	even	2		2025.4.a.bb.1.6		7
9.2	odd	6		135.4.e.c.91.2		14
9.4	even	3		45.4.e.c.16.6	✓	14
9.5	odd	6		135.4.e.c.46.2		14
9.7	even	3		45.4.e.c.31.6	yes	14
15.14	odd	2		2025.4.a.ba.1.2		7
45.4	even	6		225.4.e.d.151.2		14
45.7	odd	12		225.4.k.d.49.3		28
45.13	odd	12		225.4.k.d.124.3		28
45.22	odd	12		225.4.k.d.124.12		28
45.34	even	6		225.4.e.d.76.2		14
45.43	odd	12		225.4.k.d.49.12		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.6	✓	14	9.4	even	3
45.4.e.c.31.6	yes	14	9.7	even	3
135.4.e.c.46.2		14	9.5	odd	6
135.4.e.c.91.2		14	9.2	odd	6
225.4.e.d.76.2		14	45.34	even	6
225.4.e.d.151.2		14	45.4	even	6
225.4.k.d.49.3		28	45.7	odd	12
225.4.k.d.49.12		28	45.43	odd	12
225.4.k.d.124.3		28	45.13	odd	12
225.4.k.d.124.12		28	45.22	odd	12
405.4.a.m.1.2		7	1.1	even	1	trivial
405.4.a.n.1.6		7	3.2	odd	2
2025.4.a.ba.1.2		7	15.14	odd	2
2025.4.a.bb.1.6		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)

\(f(q)\)	\(=\)	\(q-4.26178 q^{2} +10.1628 q^{4} -5.00000 q^{5} +30.7639 q^{7} -9.21718 q^{8} +O(q^{10})\)	\(q-4.26178 q^{2} +10.1628 q^{4} -5.00000 q^{5} +30.7639 q^{7} -9.21718 q^{8} +21.3089 q^{10} +40.7146 q^{11} +63.2178 q^{13} -131.109 q^{14} -42.0204 q^{16} -6.58990 q^{17} +75.3803 q^{19} -50.8138 q^{20} -173.517 q^{22} -62.3628 q^{23} +25.0000 q^{25} -269.420 q^{26} +312.646 q^{28} +49.6084 q^{29} +103.004 q^{31} +252.819 q^{32} +28.0847 q^{34} -153.820 q^{35} -282.029 q^{37} -321.254 q^{38} +46.0859 q^{40} -157.540 q^{41} -337.814 q^{43} +413.773 q^{44} +265.776 q^{46} +44.5158 q^{47} +603.420 q^{49} -106.544 q^{50} +642.467 q^{52} +26.2752 q^{53} -203.573 q^{55} -283.557 q^{56} -211.420 q^{58} -425.926 q^{59} +850.595 q^{61} -438.981 q^{62} -741.296 q^{64} -316.089 q^{65} +96.3076 q^{67} -66.9715 q^{68} +655.545 q^{70} +952.164 q^{71} -50.8558 q^{73} +1201.95 q^{74} +766.071 q^{76} +1252.54 q^{77} -197.279 q^{79} +210.102 q^{80} +671.400 q^{82} -197.739 q^{83} +32.9495 q^{85} +1439.69 q^{86} -375.274 q^{88} +1364.54 q^{89} +1944.83 q^{91} -633.778 q^{92} -189.717 q^{94} -376.901 q^{95} -1431.16 q^{97} -2571.64 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8}+O(q^{10}) \) 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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