LMFDB - Embedded newform 2205.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.41421 q^{2} +21.3137 q^{4} -5.00000 q^{5} -72.0833 q^{8} +O(q^{10})$	$q-5.41421 q^{2} +21.3137 q^{4} -5.00000 q^{5} -72.0833 q^{8} +27.0711 q^{10} +52.2548 q^{11} -30.6569 q^{13} +219.765 q^{16} +37.2254 q^{17} -80.2254 q^{19} -106.569 q^{20} -282.919 q^{22} -25.8335 q^{23} +25.0000 q^{25} +165.983 q^{26} -20.9411 q^{29} +314.558 q^{31} -613.186 q^{32} -201.546 q^{34} +197.147 q^{37} +434.357 q^{38} +360.416 q^{40} +11.3625 q^{41} -33.8335 q^{43} +1113.74 q^{44} +139.868 q^{46} -361.676 q^{47} -135.355 q^{50} -653.411 q^{52} -153.019 q^{53} -261.274 q^{55} +113.380 q^{58} -616.000 q^{59} -15.2649 q^{61} -1703.09 q^{62} +1561.80 q^{64} +153.284 q^{65} -166.510 q^{67} +793.411 q^{68} +952.000 q^{71} +148.489 q^{73} -1067.40 q^{74} -1709.90 q^{76} +857.725 q^{79} -1098.82 q^{80} -61.5189 q^{82} +660.528 q^{83} -186.127 q^{85} +183.182 q^{86} -3766.70 q^{88} -45.7746 q^{89} -550.607 q^{92} +1958.19 q^{94} +401.127 q^{95} -1682.13 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8}+O(q^{10}) $ 2 * q - 8 * q^2 + 20 * q^4 - 10 * q^5 - 48 * q^8	$ 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8} + 40 q^{10} + 14 q^{11} - 50 q^{13} + 168 q^{16} - 50 q^{17} - 36 q^{19} - 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} + 216 q^{26} + 26 q^{29} + 120 q^{31} - 672 q^{32} + 24 q^{34} + 564 q^{37} + 320 q^{38} + 240 q^{40} - 328 q^{41} - 260 q^{43} + 1164 q^{44} + 704 q^{46} - 350 q^{47} - 200 q^{50} - 628 q^{52} + 56 q^{53} - 70 q^{55} - 8 q^{58} - 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} + 250 q^{65} - 152 q^{67} + 908 q^{68} + 1904 q^{71} - 676 q^{73} - 2016 q^{74} - 1768 q^{76} + 1014 q^{79} - 840 q^{80} + 816 q^{82} - 376 q^{83} + 250 q^{85} + 768 q^{86} - 4688 q^{88} - 216 q^{89} - 264 q^{92} + 1928 q^{94} + 180 q^{95} - 2742 q^{97}+O(q^{100}) $ 2 * q - 8 * q^2 + 20 * q^4 - 10 * q^5 - 48 * q^8 + 40 * q^10 + 14 * q^11 - 50 * q^13 + 168 * q^16 - 50 * q^17 - 36 * q^19 - 100 * q^20 - 184 * q^22 - 244 * q^23 + 50 * q^25 + 216 * q^26 + 26 * q^29 + 120 * q^31 - 672 * q^32 + 24 * q^34 + 564 * q^37 + 320 * q^38 + 240 * q^40 - 328 * q^41 - 260 * q^43 + 1164 * q^44 + 704 * q^46 - 350 * q^47 - 200 * q^50 - 628 * q^52 + 56 * q^53 - 70 * q^55 - 8 * q^58 - 1232 * q^59 - 336 * q^61 - 1200 * q^62 + 2128 * q^64 + 250 * q^65 - 152 * q^67 + 908 * q^68 + 1904 * q^71 - 676 * q^73 - 2016 * q^74 - 1768 * q^76 + 1014 * q^79 - 840 * q^80 + 816 * q^82 - 376 * q^83 + 250 * q^85 + 768 * q^86 - 4688 * q^88 - 216 * q^89 - 264 * q^92 + 1928 * q^94 + 180 * q^95 - 2742 * q^97

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 $

$2$	−5.41421	−1.91421	−0.957107	−	0.289735i	$-0.906433\pi$
$2$	−5.41421	−1.91421	−0.957107	+	0.289735i	$0.906433\pi$
$3$	0	0
$3$	0	0
$4$	21.3137	2.66421
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−72.0833	−3.18566
$9$	0	0
$10$	27.0711	0.856062
$11$	52.2548	1.43231	0.716156	−	0.697941i	$-0.245900\pi$
$11$	52.2548	1.43231	0.716156	+	0.697941i	$0.245900\pi$
$12$	0	0
$13$	−30.6569	−0.654052	−0.327026	−	0.945015i	$-0.606047\pi$
$13$	−30.6569	−0.654052	−0.327026	+	0.945015i	$0.606047\pi$
$14$	0	0
$15$	0	0
$16$	219.765	3.43382
$17$	37.2254	0.531087	0.265544	−	0.964099i	$-0.414449\pi$
$17$	37.2254	0.531087	0.265544	+	0.964099i	$0.414449\pi$
$18$	0	0
$19$	−80.2254	−0.968683	−0.484341	−	0.874879i	$-0.660941\pi$
$19$	−80.2254	−0.968683	−0.484341	+	0.874879i	$0.660941\pi$
$20$	−106.569	−1.19147
$21$	0	0
$22$	−282.919	−2.74175
$23$	−25.8335	−0.234202	−0.117101	−	0.993120i	$-0.537360\pi$
$23$	−25.8335	−0.234202	−0.117101	+	0.993120i	$0.537360\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	165.983	1.25200
$27$	0	0
$28$	0	0
$29$	−20.9411	−0.134092	−0.0670460	−	0.997750i	$-0.521357\pi$
$29$	−20.9411	−0.134092	−0.0670460	+	0.997750i	$0.521357\pi$
$30$	0	0
$31$	314.558	1.82246	0.911232	−	0.411894i	$-0.135133\pi$
$31$	314.558	1.82246	0.911232	+	0.411894i	$0.135133\pi$
$32$	−613.186	−3.38741
$33$	0	0
$34$	−201.546	−1.01661
$35$	0	0
$36$	0	0
$37$	197.147	0.875968	0.437984	−	0.898983i	$-0.355693\pi$
$37$	197.147	0.875968	0.437984	+	0.898983i	$0.355693\pi$
$38$	434.357	1.85427
$39$	0	0
$40$	360.416	1.42467
$41$	11.3625	0.0432810	0.0216405	−	0.999766i	$-0.493111\pi$
$41$	11.3625	0.0432810	0.0216405	+	0.999766i	$0.493111\pi$
$42$	0	0
$43$	−33.8335	−0.119990	−0.0599948	−	0.998199i	$-0.519108\pi$
$43$	−33.8335	−0.119990	−0.0599948	+	0.998199i	$0.519108\pi$
$44$	1113.74	3.81598
$45$	0	0
$46$	139.868	0.448313
$47$	−361.676	−1.12247	−0.561233	−	0.827658i	$-0.689673\pi$
$47$	−361.676	−1.12247	−0.561233	+	0.827658i	$0.689673\pi$
$48$	0	0
$49$	0	0
$50$	−135.355	−0.382843
$51$	0	0
$52$	−653.411	−1.74254
$53$	−153.019	−0.396582	−0.198291	−	0.980143i	$-0.563539\pi$
$53$	−153.019	−0.396582	−0.198291	+	0.980143i	$0.563539\pi$
$54$	0	0
$55$	−261.274	−0.640549
$56$	0	0
$57$	0	0
$58$	113.380	0.256681
$59$	−616.000	−1.35926	−0.679630	−	0.733555i	$-0.737860\pi$
$59$	−616.000	−1.35926	−0.679630	+	0.733555i	$0.737860\pi$
$60$	0	0
$61$	−15.2649	−0.0320406	−0.0160203	−	0.999872i	$-0.505100\pi$
$61$	−15.2649	−0.0320406	−0.0160203	+	0.999872i	$0.505100\pi$
$62$	−1703.09	−3.48858
$63$	0	0
$64$	1561.80	3.05040
$65$	153.284	0.292501
$66$	0	0
$67$	−166.510	−0.303618	−0.151809	−	0.988410i	$-0.548510\pi$
$67$	−166.510	−0.303618	−0.151809	+	0.988410i	$0.548510\pi$
$68$	793.411	1.41493
$69$	0	0
$70$	0	0
$71$	952.000	1.59129	0.795645	−	0.605763i	$-0.207132\pi$
$71$	952.000	1.59129	0.795645	+	0.605763i	$0.207132\pi$
$72$	0	0
$73$	148.489	0.238074	0.119037	−	0.992890i	$-0.462019\pi$
$73$	148.489	0.238074	0.119037	+	0.992890i	$0.462019\pi$
$74$	−1067.40	−1.67679
$75$	0	0
$76$	−1709.90	−2.58078
$77$	0	0
$78$	0	0
$79$	857.725	1.22154	0.610770	−	0.791808i	$-0.290860\pi$
$79$	857.725	1.22154	0.610770	+	0.791808i	$0.290860\pi$
$80$	−1098.82	−1.53565
$81$	0	0
$82$	−61.5189	−0.0828491
$83$	660.528	0.873523	0.436761	−	0.899577i	$-0.356125\pi$
$83$	660.528	0.873523	0.436761	+	0.899577i	$0.356125\pi$
$84$	0	0
$85$	−186.127	−0.237509
$86$	183.182	0.229686
$87$	0	0
$88$	−3766.70	−4.56286
$89$	−45.7746	−0.0545180	−0.0272590	−	0.999628i	$-0.508678\pi$
$89$	−45.7746	−0.0545180	−0.0272590	+	0.999628i	$0.508678\pi$
$90$	0	0
$91$	0	0
$92$	−550.607	−0.623965
$93$	0	0
$94$	1958.19	2.14864
$95$	401.127	0.433208
$96$	0	0
$97$	−1682.13	−1.76076	−0.880382	−	0.474265i	$-0.842714\pi$
$97$	−1682.13	−1.76076	−0.880382	+	0.474265i	$0.842714\pi$
$98$	0	0
$99$	0	0
$100$	532.843	0.532843
$101$	−434.167	−0.427734	−0.213867	−	0.976863i	$-0.568606\pi$
$101$	−434.167	−0.427734	−0.213867	+	0.976863i	$0.568606\pi$
$102$	0	0
$103$	−345.577	−0.330589	−0.165295	−	0.986244i	$-0.552858\pi$
$103$	−345.577	−0.330589	−0.165295	+	0.986244i	$0.552858\pi$
$104$	2209.85	2.08359
$105$	0	0
$106$	828.479	0.759142
$107$	−217.119	−0.196165	−0.0980825	−	0.995178i	$-0.531271\pi$
$107$	−217.119	−0.196165	−0.0980825	+	0.995178i	$0.531271\pi$
$108$	0	0
$109$	1734.41	1.52409	0.762047	−	0.647521i	$-0.224194\pi$
$109$	1734.41	1.52409	0.762047	+	0.647521i	$0.224194\pi$
$110$	1414.59	1.22615
$111$	0	0
$112$	0	0
$113$	1854.20	1.54362	0.771809	−	0.635855i	$-0.219352\pi$
$113$	1854.20	1.54362	0.771809	+	0.635855i	$0.219352\pi$
$114$	0	0
$115$	129.167	0.104738
$116$	−446.333	−0.357250
$117$	0	0
$118$	3335.16	2.60191
$119$	0	0
$120$	0	0
$121$	1399.57	1.05152
$122$	82.6476	0.0613325
$123$	0	0
$124$	6704.41	4.85543
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1394.51	0.974352	0.487176	−	0.873304i	$-0.338027\pi$
$127$	1394.51	0.974352	0.487176	+	0.873304i	$0.338027\pi$
$128$	−3550.45	−2.45171
$129$	0	0
$130$	−829.914	−0.559910
$131$	1762.42	1.17544	0.587722	−	0.809063i	$-0.300025\pi$
$131$	1762.42	1.17544	0.587722	+	0.809063i	$0.300025\pi$
$132$	0	0
$133$	0	0
$134$	901.519	0.581189
$135$	0	0
$136$	−2683.33	−1.69186
$137$	922.949	0.575568	0.287784	−	0.957695i	$-0.407081\pi$
$137$	922.949	0.575568	0.287784	+	0.957695i	$0.407081\pi$
$138$	0	0
$139$	196.039	0.119624	0.0598122	−	0.998210i	$-0.480950\pi$
$139$	196.039	0.119624	0.0598122	+	0.998210i	$0.480950\pi$
$140$	0	0
$141$	0	0
$142$	−5154.33	−3.04607
$143$	−1601.97	−0.936807
$144$	0	0
$145$	104.706	0.0599678
$146$	−803.954	−0.455724
$147$	0	0
$148$	4201.94	2.33376
$149$	−780.372	−0.429064	−0.214532	−	0.976717i	$-0.568823\pi$
$149$	−780.372	−0.429064	−0.214532	+	0.976717i	$0.568823\pi$
$150$	0	0
$151$	−2319.43	−1.25002	−0.625008	−	0.780618i	$-0.714904\pi$
$151$	−2319.43	−1.25002	−0.625008	+	0.780618i	$0.714904\pi$
$152$	5782.91	3.08589
$153$	0	0
$154$	0	0
$155$	−1572.79	−0.815030
$156$	0	0
$157$	−1022.90	−0.519977	−0.259989	−	0.965612i	$-0.583719\pi$
$157$	−1022.90	−0.519977	−0.259989	+	0.965612i	$0.583719\pi$
$158$	−4643.91	−2.33829
$159$	0	0
$160$	3065.93	1.51489
$161$	0	0
$162$	0	0
$163$	−1350.63	−0.649013	−0.324507	−	0.945883i	$-0.605198\pi$
$163$	−1350.63	−0.649013	−0.324507	+	0.945883i	$0.605198\pi$
$164$	242.177	0.115310
$165$	0	0
$166$	−3576.24	−1.67211
$167$	−1230.58	−0.570209	−0.285105	−	0.958496i	$-0.592028\pi$
$167$	−1230.58	−0.570209	−0.285105	+	0.958496i	$0.592028\pi$
$168$	0	0
$169$	−1257.16	−0.572215
$170$	1007.73	0.454644
$171$	0	0
$172$	−721.117	−0.319678
$173$	−2487.65	−1.09325	−0.546626	−	0.837377i	$-0.684088\pi$
$173$	−2487.65	−1.09325	−0.546626	+	0.837377i	$0.684088\pi$
$174$	0	0
$175$	0	0
$176$	11483.8	4.91830
$177$	0	0
$178$	247.833	0.104359
$179$	−1621.18	−0.676941	−0.338471	−	0.940977i	$-0.609910\pi$
$179$	−1621.18	−0.676941	−0.338471	+	0.940977i	$0.609910\pi$
$180$	0	0
$181$	−2593.69	−1.06512	−0.532561	−	0.846392i	$-0.678770\pi$
$181$	−2593.69	−1.06512	−0.532561	+	0.846392i	$0.678770\pi$
$182$	0	0
$183$	0	0
$184$	1862.16	0.746089
$185$	−985.736	−0.391745
$186$	0	0
$187$	1945.21	0.760682
$188$	−7708.66	−2.99049
$189$	0	0
$190$	−2171.79	−0.829253
$191$	1823.08	0.690645	0.345323	−	0.938484i	$-0.387769\pi$
$191$	1823.08	0.690645	0.345323	+	0.938484i	$0.387769\pi$
$192$	0	0
$193$	−1541.03	−0.574744	−0.287372	−	0.957819i	$-0.592782\pi$
$193$	−1541.03	−0.574744	−0.287372	+	0.957819i	$0.592782\pi$
$194$	9107.39	3.37048
$195$	0	0
$196$	0	0
$197$	−701.243	−0.253612	−0.126806	−	0.991928i	$-0.540473\pi$
$197$	−701.243	−0.253612	−0.126806	+	0.991928i	$0.540473\pi$
$198$	0	0
$199$	−3294.96	−1.17374	−0.586868	−	0.809682i	$-0.699639\pi$
$199$	−3294.96	−1.17374	−0.586868	+	0.809682i	$0.699639\pi$
$200$	−1802.08	−0.637132
$201$	0	0
$202$	2350.67	0.818775
$203$	0	0
$204$	0	0
$205$	−56.8124	−0.0193559
$206$	1871.03	0.632819
$207$	0	0
$208$	−6737.29	−2.24590
$209$	−4192.16	−1.38746
$210$	0	0
$211$	4082.35	1.33195	0.665974	−	0.745975i	$-0.268016\pi$
$211$	4082.35	1.33195	0.665974	+	0.745975i	$0.268016\pi$
$212$	−3261.41	−1.05658
$213$	0	0
$214$	1175.53	0.375502
$215$	169.167	0.0536610
$216$	0	0
$217$	0	0
$218$	−9390.46	−2.91744
$219$	0	0
$220$	−5568.72	−1.70656
$221$	−1141.21	−0.347359
$222$	0	0
$223$	−747.161	−0.224366	−0.112183	−	0.993688i	$-0.535784\pi$
$223$	−747.161	−0.224366	−0.112183	+	0.993688i	$0.535784\pi$
$224$	0	0
$225$	0	0
$226$	−10039.1	−2.95481
$227$	1665.67	0.487025	0.243513	−	0.969898i	$-0.421700\pi$
$227$	1665.67	0.487025	0.243513	+	0.969898i	$0.421700\pi$
$228$	0	0
$229$	6628.35	1.91272	0.956362	−	0.292183i	$-0.0943816\pi$
$229$	6628.35	1.91272	0.956362	+	0.292183i	$0.0943816\pi$
$230$	−699.340	−0.200492
$231$	0	0
$232$	1509.50	0.427172
$233$	432.431	0.121586	0.0607929	−	0.998150i	$-0.480637\pi$
$233$	432.431	0.121586	0.0607929	+	0.998150i	$0.480637\pi$
$234$	0	0
$235$	1808.38	0.501982
$236$	−13129.2	−3.62136
$237$	0	0
$238$	0	0
$239$	−5580.44	−1.51033	−0.755165	−	0.655535i	$-0.772443\pi$
$239$	−5580.44	−1.51033	−0.755165	+	0.655535i	$0.772443\pi$
$240$	0	0
$241$	6296.87	1.68306	0.841529	−	0.540212i	$-0.181656\pi$
$241$	6296.87	1.68306	0.841529	+	0.540212i	$0.181656\pi$
$242$	−7577.56	−2.01283
$243$	0	0
$244$	−325.352	−0.0853629
$245$	0	0
$246$	0	0
$247$	2459.46	0.633569
$248$	−22674.4	−5.80575
$249$	0	0
$250$	676.777	0.171212
$251$	311.921	0.0784393	0.0392197	−	0.999231i	$-0.487513\pi$
$251$	311.921	0.0784393	0.0392197	+	0.999231i	$0.487513\pi$
$252$	0	0
$253$	−1349.92	−0.335451
$254$	−7550.17	−1.86512
$255$	0	0
$256$	6728.46	1.64269
$257$	−7861.39	−1.90809	−0.954046	−	0.299659i	$-0.903127\pi$
$257$	−7861.39	−1.90809	−0.954046	+	0.299659i	$0.903127\pi$
$258$	0	0
$259$	0	0
$260$	3267.06	0.779285
$261$	0	0
$262$	−9542.11	−2.25005
$263$	−5227.09	−1.22554	−0.612769	−	0.790262i	$-0.709944\pi$
$263$	−5227.09	−1.22554	−0.612769	+	0.790262i	$0.709944\pi$
$264$	0	0
$265$	765.097	0.177357
$266$	0	0
$267$	0	0
$268$	−3548.94	−0.808903
$269$	1281.71	0.290510	0.145255	−	0.989394i	$-0.453600\pi$
$269$	1281.71	0.290510	0.145255	+	0.989394i	$0.453600\pi$
$270$	0	0
$271$	−4704.14	−1.05445	−0.527226	−	0.849725i	$-0.676768\pi$
$271$	−4704.14	−1.05445	−0.527226	+	0.849725i	$0.676768\pi$
$272$	8180.82	1.82366
$273$	0	0
$274$	−4997.04	−1.10176
$275$	1306.37	0.286462
$276$	0	0
$277$	8958.56	1.94321	0.971603	−	0.236619i	$-0.0760393\pi$
$277$	8958.56	1.94321	0.971603	+	0.236619i	$0.0760393\pi$
$278$	−1061.40	−0.228987
$279$	0	0
$280$	0	0
$281$	370.904	0.0787412	0.0393706	−	0.999225i	$-0.487465\pi$
$281$	370.904	0.0787412	0.0393706	+	0.999225i	$0.487465\pi$
$282$	0	0
$283$	5822.26	1.22296	0.611479	−	0.791261i	$-0.290575\pi$
$283$	5822.26	1.22296	0.611479	+	0.791261i	$0.290575\pi$
$284$	20290.7	4.23954
$285$	0	0
$286$	8673.40	1.79325
$287$	0	0
$288$	0	0
$289$	−3527.27	−0.717946
$290$	−566.899	−0.114791
$291$	0	0
$292$	3164.86	0.634279
$293$	7443.79	1.48420	0.742100	−	0.670289i	$-0.233830\pi$
$293$	7443.79	1.48420	0.742100	+	0.670289i	$0.233830\pi$
$294$	0	0
$295$	3080.00	0.607880
$296$	−14211.0	−2.79053
$297$	0	0
$298$	4225.10	0.821320
$299$	791.973	0.153181
$300$	0	0
$301$	0	0
$302$	12557.9	2.39280
$303$	0	0
$304$	−17630.7	−3.32628
$305$	76.3247	0.0143290
$306$	0	0
$307$	761.674	0.141600	0.0707998	−	0.997491i	$-0.477445\pi$
$307$	761.674	0.141600	0.0707998	+	0.997491i	$0.477445\pi$
$308$	0	0
$309$	0	0
$310$	8515.43	1.56014
$311$	7718.69	1.40735	0.703677	−	0.710520i	$-0.251540\pi$
$311$	7718.69	1.40735	0.703677	+	0.710520i	$0.251540\pi$
$312$	0	0
$313$	−8556.00	−1.54509	−0.772546	−	0.634959i	$-0.781017\pi$
$313$	−8556.00	−1.54509	−0.772546	+	0.634959i	$0.781017\pi$
$314$	5538.21	0.995348
$315$	0	0
$316$	18281.3	3.25444
$317$	7780.95	1.37862	0.689309	−	0.724468i	$-0.257914\pi$
$317$	7780.95	1.37862	0.689309	+	0.724468i	$0.257914\pi$
$318$	0	0
$319$	−1094.28	−0.192062
$320$	−7809.02	−1.36418
$321$	0	0
$322$	0	0
$323$	−2986.42	−0.514455
$324$	0	0
$325$	−766.421	−0.130810
$326$	7312.58	1.24235
$327$	0	0
$328$	−819.045	−0.137879
$329$	0	0
$330$	0	0
$331$	−4932.12	−0.819015	−0.409507	−	0.912307i	$-0.634299\pi$
$331$	−4932.12	−0.819015	−0.409507	+	0.912307i	$0.634299\pi$
$332$	14078.3	2.32725
$333$	0	0
$334$	6662.61	1.09150
$335$	832.548	0.135782
$336$	0	0
$337$	−7121.13	−1.15108	−0.575538	−	0.817775i	$-0.695207\pi$
$337$	−7121.13	−1.15108	−0.575538	+	0.817775i	$0.695207\pi$
$338$	6806.52	1.09534
$339$	0	0
$340$	−3967.06	−0.632776
$341$	16437.2	2.61034
$342$	0	0
$343$	0	0
$344$	2438.83	0.382246
$345$	0	0
$346$	13468.7	2.09272
$347$	−9540.58	−1.47598	−0.737991	−	0.674811i	$-0.764225\pi$
$347$	−9540.58	−1.47598	−0.737991	+	0.674811i	$0.764225\pi$
$348$	0	0
$349$	−1281.65	−0.196576	−0.0982880	−	0.995158i	$-0.531337\pi$
$349$	−1281.65	−0.196576	−0.0982880	+	0.995158i	$0.531337\pi$
$350$	0	0
$351$	0	0
$352$	−32041.9	−4.85182
$353$	5798.07	0.874221	0.437110	−	0.899408i	$-0.356002\pi$
$353$	5798.07	0.874221	0.437110	+	0.899408i	$0.356002\pi$
$354$	0	0
$355$	−4760.00	−0.711647
$356$	−975.627	−0.145247
$357$	0	0
$358$	8777.40	1.29581
$359$	−2267.29	−0.333323	−0.166662	−	0.986014i	$-0.553299\pi$
$359$	−2267.29	−0.333323	−0.166662	+	0.986014i	$0.553299\pi$
$360$	0	0
$361$	−422.886	−0.0616541
$362$	14042.8	2.03887
$363$	0	0
$364$	0	0
$365$	−742.447	−0.106470
$366$	0	0
$367$	7372.85	1.04866	0.524332	−	0.851514i	$-0.324315\pi$
$367$	7372.85	1.04866	0.524332	+	0.851514i	$0.324315\pi$
$368$	−5677.28	−0.804209
$369$	0	0
$370$	5336.98	0.749883
$371$	0	0
$372$	0	0
$373$	6447.14	0.894961	0.447480	−	0.894294i	$-0.352321\pi$
$373$	6447.14	0.894961	0.447480	+	0.894294i	$0.352321\pi$
$374$	−10531.8	−1.45611
$375$	0	0
$376$	26070.8	3.57579
$377$	641.989	0.0877032
$378$	0	0
$379$	−4247.57	−0.575680	−0.287840	−	0.957678i	$-0.592937\pi$
$379$	−4247.57	−0.575680	−0.287840	+	0.957678i	$0.592937\pi$
$380$	8549.50	1.15416
$381$	0	0
$382$	−9870.53	−1.32204
$383$	−6681.86	−0.891454	−0.445727	−	0.895169i	$-0.647055\pi$
$383$	−6681.86	−0.891454	−0.445727	+	0.895169i	$0.647055\pi$
$384$	0	0
$385$	0	0
$386$	8343.45	1.10018
$387$	0	0
$388$	−35852.4	−4.69105
$389$	6371.78	0.830494	0.415247	−	0.909709i	$-0.363695\pi$
$389$	6371.78	0.830494	0.415247	+	0.909709i	$0.363695\pi$
$390$	0	0
$391$	−961.661	−0.124382
$392$	0	0
$393$	0	0
$394$	3796.68	0.485467
$395$	−4288.62	−0.546289
$396$	0	0
$397$	−4247.93	−0.537021	−0.268510	−	0.963277i	$-0.586531\pi$
$397$	−4247.93	−0.537021	−0.268510	+	0.963277i	$0.586531\pi$
$398$	17839.6	2.24678
$399$	0	0
$400$	5494.11	0.686764
$401$	8833.62	1.10008	0.550038	−	0.835140i	$-0.314613\pi$
$401$	8833.62	1.10008	0.550038	+	0.835140i	$0.314613\pi$
$402$	0	0
$403$	−9643.37	−1.19199
$404$	−9253.70	−1.13958
$405$	0	0
$406$	0	0
$407$	10301.9	1.25466
$408$	0	0
$409$	319.205	0.0385908	0.0192954	−	0.999814i	$-0.493858\pi$
$409$	319.205	0.0385908	0.0192954	+	0.999814i	$0.493858\pi$
$410$	307.595	0.0370512
$411$	0	0
$412$	−7365.53	−0.880761
$413$	0	0
$414$	0	0
$415$	−3302.64	−0.390651
$416$	18798.3	2.21554
$417$	0	0
$418$	22697.3	2.65589
$419$	−12789.2	−1.49115	−0.745577	−	0.666420i	$-0.767826\pi$
$419$	−12789.2	−1.49115	−0.745577	+	0.666420i	$0.767826\pi$
$420$	0	0
$421$	−6747.40	−0.781112	−0.390556	−	0.920579i	$-0.627717\pi$
$421$	−6747.40	−0.781112	−0.390556	+	0.920579i	$0.627717\pi$
$422$	−22102.7	−2.54963
$423$	0	0
$424$	11030.1	1.26337
$425$	930.635	0.106217
$426$	0	0
$427$	0	0
$428$	−4627.60	−0.522625
$429$	0	0
$430$	−915.908	−0.102719
$431$	5184.75	0.579444	0.289722	−	0.957111i	$-0.406437\pi$
$431$	5184.75	0.579444	0.289722	+	0.957111i	$0.406437\pi$
$432$	0	0
$433$	4242.03	0.470806	0.235403	−	0.971898i	$-0.424359\pi$
$433$	4242.03	0.470806	0.235403	+	0.971898i	$0.424359\pi$
$434$	0	0
$435$	0	0
$436$	36966.7	4.06051
$437$	2072.50	0.226868
$438$	0	0
$439$	5434.12	0.590789	0.295394	−	0.955375i	$-0.404549\pi$
$439$	5434.12	0.590789	0.295394	+	0.955375i	$0.404549\pi$
$440$	18833.5	2.04057
$441$	0	0
$442$	6178.77	0.664919
$443$	11493.8	1.23270	0.616350	−	0.787472i	$-0.288611\pi$
$443$	11493.8	1.23270	0.616350	+	0.787472i	$0.288611\pi$
$444$	0	0
$445$	228.873	0.0243812
$446$	4045.29	0.429484
$447$	0	0
$448$	0	0
$449$	16849.3	1.77098	0.885489	−	0.464661i	$-0.153824\pi$
$449$	16849.3	1.77098	0.885489	+	0.464661i	$0.153824\pi$
$450$	0	0
$451$	593.745	0.0619919
$452$	39520.0	4.11253
$453$	0	0
$454$	−9018.32	−0.932270
$455$	0	0
$456$	0	0
$457$	15348.5	1.57106	0.785528	−	0.618826i	$-0.212391\pi$
$457$	15348.5	1.57106	0.785528	+	0.618826i	$0.212391\pi$
$458$	−35887.3	−3.66136
$459$	0	0
$460$	2753.04	0.279046
$461$	14038.4	1.41830	0.709148	−	0.705059i	$-0.249080\pi$
$461$	14038.4	1.41830	0.709148	+	0.705059i	$0.249080\pi$
$462$	0	0
$463$	−8661.23	−0.869377	−0.434689	−	0.900581i	$-0.643142\pi$
$463$	−8661.23	−0.869377	−0.434689	+	0.900581i	$0.643142\pi$
$464$	−4602.12	−0.460448
$465$	0	0
$466$	−2341.27	−0.232741
$467$	7014.71	0.695079	0.347539	−	0.937665i	$-0.387017\pi$
$467$	7014.71	0.695079	0.347539	+	0.937665i	$0.387017\pi$
$468$	0	0
$469$	0	0
$470$	−9790.96	−0.960901
$471$	0	0
$472$	44403.3	4.33014
$473$	−1767.96	−0.171863
$474$	0	0
$475$	−2005.63	−0.193737
$476$	0	0
$477$	0	0
$478$	30213.7	2.89109
$479$	18134.7	1.72984	0.864922	−	0.501907i	$-0.167368\pi$
$479$	18134.7	1.72984	0.864922	+	0.501907i	$0.167368\pi$
$480$	0	0
$481$	−6043.91	−0.572929
$482$	−34092.6	−3.22173
$483$	0	0
$484$	29830.0	2.80146
$485$	8410.63	0.787438
$486$	0	0
$487$	16537.8	1.53881	0.769405	−	0.638761i	$-0.220553\pi$
$487$	16537.8	1.53881	0.769405	+	0.638761i	$0.220553\pi$
$488$	1100.35	0.102070
$489$	0	0
$490$	0	0
$491$	−220.608	−0.0202768	−0.0101384	−	0.999949i	$-0.503227\pi$
$491$	−220.608	−0.0202768	−0.0101384	+	0.999949i	$0.503227\pi$
$492$	0	0
$493$	−779.542	−0.0712146
$494$	−13316.0	−1.21279
$495$	0	0
$496$	69128.8	6.25801
$497$	0	0
$498$	0	0
$499$	5939.04	0.532801	0.266401	−	0.963862i	$-0.414166\pi$
$499$	5939.04	0.532801	0.266401	+	0.963862i	$0.414166\pi$
$500$	−2664.21	−0.238295
$501$	0	0
$502$	−1688.81	−0.150150
$503$	−11604.8	−1.02869	−0.514345	−	0.857584i	$-0.671965\pi$
$503$	−11604.8	−1.02869	−0.514345	+	0.857584i	$0.671965\pi$
$504$	0	0
$505$	2170.83	0.191289
$506$	7308.78	0.642124
$507$	0	0
$508$	29722.2	2.59588
$509$	−1867.67	−0.162639	−0.0813193	−	0.996688i	$-0.525913\pi$
$509$	−1867.67	−0.162639	−0.0813193	+	0.996688i	$0.525913\pi$
$510$	0	0
$511$	0	0
$512$	−8025.75	−0.692757
$513$	0	0
$514$	42563.2	3.65250
$515$	1727.88	0.147844
$516$	0	0
$517$	−18899.3	−1.60772
$518$	0	0
$519$	0	0
$520$	−11049.2	−0.931809
$521$	6117.21	0.514395	0.257197	−	0.966359i	$-0.417201\pi$
$521$	6117.21	0.514395	0.257197	+	0.966359i	$0.417201\pi$
$522$	0	0
$523$	16685.6	1.39505	0.697524	−	0.716561i	$-0.254285\pi$
$523$	16685.6	1.39505	0.697524	+	0.716561i	$0.254285\pi$
$524$	37563.7	3.13164
$525$	0	0
$526$	28300.6	2.34594
$527$	11709.6	0.967887
$528$	0	0
$529$	−11499.6	−0.945149
$530$	−4142.40	−0.339499
$531$	0	0
$532$	0	0
$533$	−348.338	−0.0283081
$534$	0	0
$535$	1085.59	0.0877276
$536$	12002.6	0.967223
$537$	0	0
$538$	−6939.44	−0.556097
$539$	0	0
$540$	0	0
$541$	9309.03	0.739790	0.369895	−	0.929074i	$-0.379394\pi$
$541$	9309.03	0.739790	0.369895	+	0.929074i	$0.379394\pi$
$542$	25469.2	2.01845
$543$	0	0
$544$	−22826.1	−1.79901
$545$	−8672.05	−0.681596
$546$	0	0
$547$	10894.7	0.851598	0.425799	−	0.904818i	$-0.359993\pi$
$547$	10894.7	0.851598	0.425799	+	0.904818i	$0.359993\pi$
$548$	19671.5	1.53344
$549$	0	0
$550$	−7072.97	−0.548350
$551$	1680.01	0.129893
$552$	0	0
$553$	0	0
$554$	−48503.6	−3.71971
$555$	0	0
$556$	4178.31	0.318705
$557$	7873.90	0.598973	0.299486	−	0.954101i	$-0.403185\pi$
$557$	7873.90	0.598973	0.299486	+	0.954101i	$0.403185\pi$
$558$	0	0
$559$	1037.23	0.0784796
$560$	0	0
$561$	0	0
$562$	−2008.15	−0.150728
$563$	21770.7	1.62971	0.814854	−	0.579666i	$-0.196817\pi$
$563$	21770.7	1.62971	0.814854	+	0.579666i	$0.196817\pi$
$564$	0	0
$565$	−9271.02	−0.690327
$566$	−31522.9	−2.34100
$567$	0	0
$568$	−68623.3	−5.06931
$569$	12381.3	0.912213	0.456106	−	0.889925i	$-0.349244\pi$
$569$	12381.3	0.912213	0.456106	+	0.889925i	$0.349244\pi$
$570$	0	0
$571$	−5768.38	−0.422765	−0.211383	−	0.977403i	$-0.567797\pi$
$571$	−5768.38	−0.422765	−0.211383	+	0.977403i	$0.567797\pi$
$572$	−34143.9	−2.49585
$573$	0	0
$574$	0	0
$575$	−645.837	−0.0468405
$576$	0	0
$577$	−4733.38	−0.341513	−0.170757	−	0.985313i	$-0.554621\pi$
$577$	−4733.38	−0.341513	−0.170757	+	0.985313i	$0.554621\pi$
$578$	19097.4	1.37430
$579$	0	0
$580$	2231.67	0.159767
$581$	0	0
$582$	0	0
$583$	−7996.00	−0.568028
$584$	−10703.6	−0.758422
$585$	0	0
$586$	−40302.3	−2.84108
$587$	−8441.67	−0.593569	−0.296785	−	0.954944i	$-0.595914\pi$
$587$	−8441.67	−0.593569	−0.296785	+	0.954944i	$0.595914\pi$
$588$	0	0
$589$	−25235.6	−1.76539
$590$	−16675.8	−1.16361
$591$	0	0
$592$	43326.0	3.00792
$593$	18939.9	1.31158	0.655791	−	0.754943i	$-0.272335\pi$
$593$	18939.9	1.31158	0.655791	+	0.754943i	$0.272335\pi$
$594$	0	0
$595$	0	0
$596$	−16632.6	−1.14312
$597$	0	0
$598$	−4287.91	−0.293220
$599$	−22655.3	−1.54536	−0.772681	−	0.634794i	$-0.781085\pi$
$599$	−22655.3	−1.54536	−0.772681	+	0.634794i	$0.781085\pi$
$600$	0	0
$601$	15947.4	1.08237	0.541187	−	0.840902i	$-0.317975\pi$
$601$	15947.4	1.08237	0.541187	+	0.840902i	$0.317975\pi$
$602$	0	0
$603$	0	0
$604$	−49435.6	−3.33031
$605$	−6997.84	−0.470252
$606$	0	0
$607$	25993.2	1.73811	0.869053	−	0.494719i	$-0.164729\pi$
$607$	25993.2	1.73811	0.869053	+	0.494719i	$0.164729\pi$
$608$	49193.1	3.28132
$609$	0	0
$610$	−413.238	−0.0274287
$611$	11087.9	0.734152
$612$	0	0
$613$	665.408	0.0438427	0.0219213	−	0.999760i	$-0.493022\pi$
$613$	665.408	0.0438427	0.0219213	+	0.999760i	$0.493022\pi$
$614$	−4123.87	−0.271052
$615$	0	0
$616$	0	0
$617$	−18401.3	−1.20066	−0.600330	−	0.799752i	$-0.704964\pi$
$617$	−18401.3	−1.20066	−0.600330	+	0.799752i	$0.704964\pi$
$618$	0	0
$619$	11150.6	0.724040	0.362020	−	0.932170i	$-0.382087\pi$
$619$	11150.6	0.724040	0.362020	+	0.932170i	$0.382087\pi$
$620$	−33522.0	−2.17142
$621$	0	0
$622$	−41790.6	−2.69397
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	46324.0	2.95764
$627$	0	0
$628$	−21801.8	−1.38533
$629$	7338.88	0.465215
$630$	0	0
$631$	5381.79	0.339534	0.169767	−	0.985484i	$-0.445699\pi$
$631$	5381.79	0.339534	0.169767	+	0.985484i	$0.445699\pi$
$632$	−61827.6	−3.89141
$633$	0	0
$634$	−42127.7	−2.63897
$635$	−6972.55	−0.435744
$636$	0	0
$637$	0	0
$638$	5924.64	0.367647
$639$	0	0
$640$	17752.2	1.09644
$641$	19455.1	1.19880	0.599398	−	0.800451i	$-0.295407\pi$
$641$	19455.1	1.19880	0.599398	+	0.800451i	$0.295407\pi$
$642$	0	0
$643$	14695.8	0.901317	0.450658	−	0.892696i	$-0.351189\pi$
$643$	14695.8	0.901317	0.450658	+	0.892696i	$0.351189\pi$
$644$	0	0
$645$	0	0
$646$	16169.1	0.984777
$647$	−12694.8	−0.771383	−0.385691	−	0.922628i	$-0.626037\pi$
$647$	−12694.8	−0.771383	−0.385691	+	0.922628i	$0.626037\pi$
$648$	0	0
$649$	−32189.0	−1.94688
$650$	4149.57	0.250399
$651$	0	0
$652$	−28786.8	−1.72911
$653$	12385.6	0.742247	0.371124	−	0.928583i	$-0.378973\pi$
$653$	12385.6	0.742247	0.371124	+	0.928583i	$0.378973\pi$
$654$	0	0
$655$	−8812.09	−0.525675
$656$	2497.07	0.148619
$657$	0	0
$658$	0	0
$659$	2072.18	0.122489	0.0612447	−	0.998123i	$-0.480493\pi$
$659$	2072.18	0.122489	0.0612447	+	0.998123i	$0.480493\pi$
$660$	0	0
$661$	−1074.36	−0.0632193	−0.0316096	−	0.999500i	$-0.510063\pi$
$661$	−1074.36	−0.0632193	−0.0316096	+	0.999500i	$0.510063\pi$
$662$	26703.6	1.56777
$663$	0	0
$664$	−47613.0	−2.78275
$665$	0	0
$666$	0	0
$667$	540.982	0.0314047
$668$	−26228.2	−1.51916
$669$	0	0
$670$	−4507.59	−0.259916
$671$	−797.667	−0.0458921
$672$	0	0
$673$	26195.2	1.50037	0.750186	−	0.661226i	$-0.229964\pi$
$673$	26195.2	1.50037	0.750186	+	0.661226i	$0.229964\pi$
$674$	38555.3	2.20341
$675$	0	0
$676$	−26794.7	−1.52450
$677$	−4228.44	−0.240047	−0.120024	−	0.992771i	$-0.538297\pi$
$677$	−4228.44	−0.240047	−0.120024	+	0.992771i	$0.538297\pi$
$678$	0	0
$679$	0	0
$680$	13416.6	0.756624
$681$	0	0
$682$	−88994.5	−4.99674
$683$	−27525.5	−1.54207	−0.771036	−	0.636792i	$-0.780261\pi$
$683$	−27525.5	−1.54207	−0.771036	+	0.636792i	$0.780261\pi$
$684$	0	0
$685$	−4614.74	−0.257402
$686$	0	0
$687$	0	0
$688$	−7435.40	−0.412023
$689$	4691.09	0.259385
$690$	0	0
$691$	33324.4	1.83462	0.917309	−	0.398177i	$-0.130357\pi$
$691$	33324.4	1.83462	0.917309	+	0.398177i	$0.130357\pi$
$692$	−53021.1	−2.91266
$693$	0	0
$694$	51654.8	2.82534
$695$	−980.193	−0.0534976
$696$	0	0
$697$	422.973	0.0229860
$698$	6939.12	0.376289
$699$	0	0
$700$	0	0
$701$	33262.9	1.79219	0.896094	−	0.443864i	$-0.146393\pi$
$701$	33262.9	1.79219	0.896094	+	0.443864i	$0.146393\pi$
$702$	0	0
$703$	−15816.2	−0.848534
$704$	81611.8	4.36912
$705$	0	0
$706$	−31392.0	−1.67345
$707$	0	0
$708$	0	0
$709$	13703.0	0.725851	0.362926	−	0.931818i	$-0.381778\pi$
$709$	13703.0	0.725851	0.362926	+	0.931818i	$0.381778\pi$
$710$	25771.7	1.36224
$711$	0	0
$712$	3299.58	0.173676
$713$	−8126.14	−0.426825
$714$	0	0
$715$	8009.84	0.418953
$716$	−34553.3	−1.80352
$717$	0	0
$718$	12275.6	0.638052
$719$	−8074.93	−0.418838	−0.209419	−	0.977826i	$-0.567157\pi$
$719$	−8074.93	−0.418838	−0.209419	+	0.977826i	$0.567157\pi$
$720$	0	0
$721$	0	0
$722$	2289.59	0.118019
$723$	0	0
$724$	−55281.1	−2.83771
$725$	−523.528	−0.0268184
$726$	0	0
$727$	3668.70	0.187159	0.0935794	−	0.995612i	$-0.470169\pi$
$727$	3668.70	0.187159	0.0935794	+	0.995612i	$0.470169\pi$
$728$	0	0
$729$	0	0
$730$	4019.77	0.203806
$731$	−1259.46	−0.0637250
$732$	0	0
$733$	14980.3	0.754857	0.377428	−	0.926039i	$-0.376808\pi$
$733$	14980.3	0.754857	0.377428	+	0.926039i	$0.376808\pi$
$734$	−39918.2	−2.00737
$735$	0	0
$736$	15840.7	0.793338
$737$	−8700.94	−0.434875
$738$	0	0
$739$	6530.59	0.325077	0.162538	−	0.986702i	$-0.448032\pi$
$739$	6530.59	0.325077	0.162538	+	0.986702i	$0.448032\pi$
$740$	−21009.7	−1.04369
$741$	0	0
$742$	0	0
$743$	−25952.0	−1.28141	−0.640704	−	0.767788i	$-0.721357\pi$
$743$	−25952.0	−1.28141	−0.640704	+	0.767788i	$0.721357\pi$
$744$	0	0
$745$	3901.86	0.191883
$746$	−34906.2	−1.71315
$747$	0	0
$748$	41459.6	2.02662
$749$	0	0
$750$	0	0
$751$	−14093.9	−0.684813	−0.342407	−	0.939552i	$-0.611242\pi$
$751$	−14093.9	−0.684813	−0.342407	+	0.939552i	$0.611242\pi$
$752$	−79483.6	−3.85435
$753$	0	0
$754$	−3475.87	−0.167883
$755$	11597.1	0.559024
$756$	0	0
$757$	−2554.41	−0.122644	−0.0613220	−	0.998118i	$-0.519532\pi$
$757$	−2554.41	−0.122644	−0.0613220	+	0.998118i	$0.519532\pi$
$758$	22997.2	1.10197
$759$	0	0
$760$	−28914.5	−1.38005
$761$	2219.08	0.105705	0.0528527	−	0.998602i	$-0.483169\pi$
$761$	2219.08	0.105705	0.0528527	+	0.998602i	$0.483169\pi$
$762$	0	0
$763$	0	0
$764$	38856.5	1.84003
$765$	0	0
$766$	36177.0	1.70643
$767$	18884.6	0.889028
$768$	0	0
$769$	22466.2	1.05352	0.526758	−	0.850015i	$-0.323408\pi$
$769$	22466.2	1.05352	0.526758	+	0.850015i	$0.323408\pi$
$770$	0	0
$771$	0	0
$772$	−32845.0	−1.53124
$773$	9674.79	0.450165	0.225083	−	0.974340i	$-0.427735\pi$
$773$	9674.79	0.450165	0.225083	+	0.974340i	$0.427735\pi$
$774$	0	0
$775$	7863.96	0.364493
$776$	121253.	5.60920
$777$	0	0
$778$	−34498.2	−1.58974
$779$	−911.560	−0.0419256
$780$	0	0
$781$	49746.6	2.27922
$782$	5206.64	0.238093
$783$	0	0
$784$	0	0
$785$	5114.51	0.232541
$786$	0	0
$787$	20942.8	0.948577	0.474288	−	0.880370i	$-0.342705\pi$
$787$	20942.8	0.948577	0.474288	+	0.880370i	$0.342705\pi$
$788$	−14946.1	−0.675676
$789$	0	0
$790$	23219.5	1.04571
$791$	0	0
$792$	0	0
$793$	467.975	0.0209562
$794$	22999.2	1.02797
$795$	0	0
$796$	−70227.8	−3.12708
$797$	23526.6	1.04561	0.522807	−	0.852451i	$-0.324885\pi$
$797$	23526.6	1.04561	0.522807	+	0.852451i	$0.324885\pi$
$798$	0	0
$799$	−13463.5	−0.596127
$800$	−15329.6	−0.677481
$801$	0	0
$802$	−47827.1	−2.10578
$803$	7759.29	0.340996
$804$	0	0
$805$	0	0
$806$	52211.3	2.28172
$807$	0	0
$808$	31296.1	1.36262
$809$	18202.2	0.791047	0.395523	−	0.918456i	$-0.370563\pi$
$809$	18202.2	0.791047	0.395523	+	0.918456i	$0.370563\pi$
$810$	0	0
$811$	2510.24	0.108689	0.0543443	−	0.998522i	$-0.482693\pi$
$811$	2510.24	0.108689	0.0543443	+	0.998522i	$0.482693\pi$
$812$	0	0
$813$	0	0
$814$	−55776.7	−2.40168
$815$	6753.13	0.290248
$816$	0	0
$817$	2714.30	0.116232
$818$	−1728.24	−0.0738711
$819$	0	0
$820$	−1210.88	−0.0515681
$821$	−17899.6	−0.760903	−0.380451	−	0.924801i	$-0.624231\pi$
$821$	−17899.6	−0.760903	−0.380451	+	0.924801i	$0.624231\pi$
$822$	0	0
$823$	14039.5	0.594637	0.297318	−	0.954778i	$-0.403908\pi$
$823$	14039.5	0.594637	0.297318	+	0.954778i	$0.403908\pi$
$824$	24910.3	1.05315
$825$	0	0
$826$	0	0
$827$	−15127.4	−0.636073	−0.318036	−	0.948079i	$-0.603023\pi$
$827$	−15127.4	−0.636073	−0.318036	+	0.948079i	$0.603023\pi$
$828$	0	0
$829$	−21986.5	−0.921136	−0.460568	−	0.887624i	$-0.652354\pi$
$829$	−21986.5	−0.921136	−0.460568	+	0.887624i	$0.652354\pi$
$830$	17881.2	0.747790
$831$	0	0
$832$	−47880.0	−1.99512
$833$	0	0
$834$	0	0
$835$	6152.89	0.255005
$836$	−89350.6	−3.69648
$837$	0	0
$838$	69243.5	2.85439
$839$	−2276.89	−0.0936914	−0.0468457	−	0.998902i	$-0.514917\pi$
$839$	−2276.89	−0.0936914	−0.0468457	+	0.998902i	$0.514917\pi$
$840$	0	0
$841$	−23950.5	−0.982019
$842$	36531.8	1.49521
$843$	0	0
$844$	87010.1	3.54859
$845$	6285.79	0.255903
$846$	0	0
$847$	0	0
$848$	−33628.2	−1.36179
$849$	0	0
$850$	−5038.66	−0.203323
$851$	−5093.00	−0.205154
$852$	0	0
$853$	−13342.6	−0.535570	−0.267785	−	0.963479i	$-0.586292\pi$
$853$	−13342.6	−0.535570	−0.267785	+	0.963479i	$0.586292\pi$
$854$	0	0
$855$	0	0
$856$	15650.6	0.624915
$857$	18690.9	0.745003	0.372502	−	0.928032i	$-0.378500\pi$
$857$	18690.9	0.745003	0.372502	+	0.928032i	$0.378500\pi$
$858$	0	0
$859$	−18318.9	−0.727628	−0.363814	−	0.931472i	$-0.618526\pi$
$859$	−18318.9	−0.727628	−0.363814	+	0.931472i	$0.618526\pi$
$860$	3605.58	0.142964
$861$	0	0
$862$	−28071.3	−1.10918
$863$	−38133.1	−1.50413	−0.752067	−	0.659087i	$-0.770943\pi$
$863$	−38133.1	−1.50413	−0.752067	+	0.659087i	$0.770943\pi$
$864$	0	0
$865$	12438.3	0.488917
$866$	−22967.3	−0.901223
$867$	0	0
$868$	0	0
$869$	44820.3	1.74962
$870$	0	0
$871$	5104.66	0.198582
$872$	−125022.	−4.85525
$873$	0	0
$874$	−11221.0	−0.434273
$875$	0	0
$876$	0	0
$877$	−19707.5	−0.758807	−0.379404	−	0.925231i	$-0.623871\pi$
$877$	−19707.5	−0.758807	−0.379404	+	0.925231i	$0.623871\pi$
$878$	−29421.5	−1.13090
$879$	0	0
$880$	−57418.8	−2.19953
$881$	−14091.5	−0.538883	−0.269441	−	0.963017i	$-0.586839\pi$
$881$	−14091.5	−0.538883	−0.269441	+	0.963017i	$0.586839\pi$
$882$	0	0
$883$	3115.87	0.118751	0.0593757	−	0.998236i	$-0.481089\pi$
$883$	3115.87	0.118751	0.0593757	+	0.998236i	$0.481089\pi$
$884$	−24323.5	−0.925438
$885$	0	0
$886$	−62229.8	−2.35965
$887$	38734.6	1.46627	0.733134	−	0.680084i	$-0.238057\pi$
$887$	38734.6	1.46627	0.733134	+	0.680084i	$0.238057\pi$
$888$	0	0
$889$	0	0
$890$	−1239.17	−0.0466708
$891$	0	0
$892$	−15924.8	−0.597759
$893$	29015.6	1.08731
$894$	0	0
$895$	8105.89	0.302737
$896$	0	0
$897$	0	0
$898$	−91225.8	−3.39003
$899$	−6587.21	−0.244378
$900$	0	0
$901$	−5696.21	−0.210619
$902$	−3214.66	−0.118666
$903$	0	0
$904$	−133657.	−4.91744
$905$	12968.4	0.476337
$906$	0	0
$907$	19242.9	0.704464	0.352232	−	0.935913i	$-0.385423\pi$
$907$	19242.9	0.704464	0.352232	+	0.935913i	$0.385423\pi$
$908$	35501.7	1.29754
$909$	0	0
$910$	0	0
$911$	−34613.3	−1.25882	−0.629412	−	0.777072i	$-0.716704\pi$
$911$	−34613.3	−1.25882	−0.629412	+	0.777072i	$0.716704\pi$
$912$	0	0
$913$	34515.8	1.25116
$914$	−83100.1	−3.00734
$915$	0	0
$916$	141275.	5.09591
$917$	0	0
$918$	0	0
$919$	25826.4	0.927022	0.463511	−	0.886091i	$-0.346589\pi$
$919$	25826.4	0.927022	0.463511	+	0.886091i	$0.346589\pi$
$920$	−9310.81	−0.333661
$921$	0	0
$922$	−76007.0	−2.71492
$923$	−29185.3	−1.04079
$924$	0	0
$925$	4928.68	0.175194
$926$	46893.8	1.66417
$927$	0	0
$928$	12840.8	0.454224
$929$	19451.6	0.686960	0.343480	−	0.939160i	$-0.388394\pi$
$929$	19451.6	0.686960	0.343480	+	0.939160i	$0.388394\pi$
$930$	0	0
$931$	0	0
$932$	9216.70	0.323930
$933$	0	0
$934$	−37979.1	−1.33053
$935$	−9726.03	−0.340188
$936$	0	0
$937$	−34469.1	−1.20177	−0.600884	−	0.799336i	$-0.705185\pi$
$937$	−34469.1	−1.20177	−0.600884	+	0.799336i	$0.705185\pi$
$938$	0	0
$939$	0	0
$940$	38543.3	1.33739
$941$	14156.4	0.490419	0.245209	−	0.969470i	$-0.421143\pi$
$941$	14156.4	0.490419	0.245209	+	0.969470i	$0.421143\pi$
$942$	0	0
$943$	−293.532	−0.0101365
$944$	−135375.	−4.66746
$945$	0	0
$946$	9572.13	0.328982
$947$	38092.4	1.30711	0.653557	−	0.756877i	$-0.273276\pi$
$947$	38092.4	1.30711	0.653557	+	0.756877i	$0.273276\pi$
$948$	0	0
$949$	−4552.22	−0.155713
$950$	10858.9	0.370853
$951$	0	0
$952$	0	0
$953$	−5037.40	−0.171225	−0.0856126	−	0.996329i	$-0.527285\pi$
$953$	−5037.40	−0.171225	−0.0856126	+	0.996329i	$0.527285\pi$
$954$	0	0
$955$	−9115.39	−0.308866
$956$	−118940.	−4.02384
$957$	0	0
$958$	−98185.1	−3.31129
$959$	0	0
$960$	0	0
$961$	69156.0	2.32137
$962$	32723.0	1.09671
$963$	0	0
$964$	134210.	4.48403
$965$	7705.14	0.257033
$966$	0	0
$967$	11495.3	0.382278	0.191139	−	0.981563i	$-0.438782\pi$
$967$	11495.3	0.382278	0.191139	+	0.981563i	$0.438782\pi$
$968$	−100885.	−3.34977
$969$	0	0
$970$	−45537.0	−1.50732
$971$	22352.7	0.738757	0.369379	−	0.929279i	$-0.379571\pi$
$971$	22352.7	0.738757	0.369379	+	0.929279i	$0.379571\pi$
$972$	0	0
$973$	0	0
$974$	−89539.4	−2.94561
$975$	0	0
$976$	−3354.69	−0.110022
$977$	−14345.7	−0.469765	−0.234882	−	0.972024i	$-0.575470\pi$
$977$	−14345.7	−0.469765	−0.234882	+	0.972024i	$0.575470\pi$
$978$	0	0
$979$	−2391.94	−0.0780867
$980$	0	0
$981$	0	0
$982$	1194.42	0.0388141
$983$	34460.9	1.11814	0.559070	−	0.829120i	$-0.311158\pi$
$983$	34460.9	1.11814	0.559070	+	0.829120i	$0.311158\pi$
$984$	0	0
$985$	3506.21	0.113419
$986$	4220.61	0.136320
$987$	0	0
$988$	52420.2	1.68796
$989$	874.036	0.0281019
$990$	0	0
$991$	−35189.6	−1.12799	−0.563993	−	0.825780i	$-0.690735\pi$
$991$	−35189.6	−1.12799	−0.563993	+	0.825780i	$0.690735\pi$
$992$	−192883.	−6.17342
$993$	0	0
$994$	0	0
$995$	16474.8	0.524911
$996$	0	0
$997$	50730.0	1.61147	0.805734	−	0.592277i	$-0.201771\pi$
$997$	50730.0	1.61147	0.805734	+	0.592277i	$0.201771\pi$
$998$	−32155.2	−1.01990
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.u.1.1		2
3.2	odd	2		245.4.a.k.1.2		2
7.6	odd	2		315.4.a.f.1.1		2
15.14	odd	2		1225.4.a.m.1.1		2
21.2	odd	6		245.4.e.i.116.1		4
21.5	even	6		245.4.e.h.116.1		4
21.11	odd	6		245.4.e.i.226.1		4
21.17	even	6		245.4.e.h.226.1		4
21.20	even	2		35.4.a.b.1.2	✓	2
35.34	odd	2		1575.4.a.z.1.2		2
84.83	odd	2		560.4.a.r.1.2		2
105.62	odd	4		175.4.b.c.99.4		4
105.83	odd	4		175.4.b.c.99.1		4
105.104	even	2		175.4.a.c.1.1		2
168.83	odd	2		2240.4.a.bo.1.1		2
168.125	even	2		2240.4.a.bn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.b.1.2	✓	2	21.20	even	2
175.4.a.c.1.1		2	105.104	even	2
175.4.b.c.99.1		4	105.83	odd	4
175.4.b.c.99.4		4	105.62	odd	4
245.4.a.k.1.2		2	3.2	odd	2
245.4.e.h.116.1		4	21.5	even	6
245.4.e.h.226.1		4	21.17	even	6
245.4.e.i.116.1		4	21.2	odd	6
245.4.e.i.226.1		4	21.11	odd	6
315.4.a.f.1.1		2	7.6	odd	2
560.4.a.r.1.2		2	84.83	odd	2
1225.4.a.m.1.1		2	15.14	odd	2
1575.4.a.z.1.2		2	35.34	odd	2
2205.4.a.u.1.1		2	1.1	even	1	trivial
2240.4.a.bn.1.2		2	168.125	even	2
2240.4.a.bo.1.1		2	168.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-5.41421 q^{2} +21.3137 q^{4} -5.00000 q^{5} -72.0833 q^{8} +O(q^{10})\)	\(q-5.41421 q^{2} +21.3137 q^{4} -5.00000 q^{5} -72.0833 q^{8} +27.0711 q^{10} +52.2548 q^{11} -30.6569 q^{13} +219.765 q^{16} +37.2254 q^{17} -80.2254 q^{19} -106.569 q^{20} -282.919 q^{22} -25.8335 q^{23} +25.0000 q^{25} +165.983 q^{26} -20.9411 q^{29} +314.558 q^{31} -613.186 q^{32} -201.546 q^{34} +197.147 q^{37} +434.357 q^{38} +360.416 q^{40} +11.3625 q^{41} -33.8335 q^{43} +1113.74 q^{44} +139.868 q^{46} -361.676 q^{47} -135.355 q^{50} -653.411 q^{52} -153.019 q^{53} -261.274 q^{55} +113.380 q^{58} -616.000 q^{59} -15.2649 q^{61} -1703.09 q^{62} +1561.80 q^{64} +153.284 q^{65} -166.510 q^{67} +793.411 q^{68} +952.000 q^{71} +148.489 q^{73} -1067.40 q^{74} -1709.90 q^{76} +857.725 q^{79} -1098.82 q^{80} -61.5189 q^{82} +660.528 q^{83} -186.127 q^{85} +183.182 q^{86} -3766.70 q^{88} -45.7746 q^{89} -550.607 q^{92} +1958.19 q^{94} +401.127 q^{95} -1682.13 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8}+O(q^{10}) \) 2 * q - 8 * q^2 + 20 * q^4 - 10 * q^5 - 48 * q^8	\( 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8} + 40 q^{10} + 14 q^{11} - 50 q^{13} + 168 q^{16} - 50 q^{17} - 36 q^{19} - 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} + 216 q^{26} + 26 q^{29} + 120 q^{31} - 672 q^{32} + 24 q^{34} + 564 q^{37} + 320 q^{38} + 240 q^{40} - 328 q^{41} - 260 q^{43} + 1164 q^{44} + 704 q^{46} - 350 q^{47} - 200 q^{50} - 628 q^{52} + 56 q^{53} - 70 q^{55} - 8 q^{58} - 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} + 250 q^{65} - 152 q^{67} + 908 q^{68} + 1904 q^{71} - 676 q^{73} - 2016 q^{74} - 1768 q^{76} + 1014 q^{79} - 840 q^{80} + 816 q^{82} - 376 q^{83} + 250 q^{85} + 768 q^{86} - 4688 q^{88} - 216 q^{89} - 264 q^{92} + 1928 q^{94} + 180 q^{95} - 2742 q^{97}+O(q^{100}) \) 2 * q - 8 * q^2 + 20 * q^4 - 10 * q^5 - 48 * q^8 + 40 * q^10 + 14 * q^11 - 50 * q^13 + 168 * q^16 - 50 * q^17 - 36 * q^19 - 100 * q^20 - 184 * q^22 - 244 * q^23 + 50 * q^25 + 216 * q^26 + 26 * q^29 + 120 * q^31 - 672 * q^32 + 24 * q^34 + 564 * q^37 + 320 * q^38 + 240 * q^40 - 328 * q^41 - 260 * q^43 + 1164 * q^44 + 704 * q^46 - 350 * q^47 - 200 * q^50 - 628 * q^52 + 56 * q^53 - 70 * q^55 - 8 * q^58 - 1232 * q^59 - 336 * q^61 - 1200 * q^62 + 2128 * q^64 + 250 * q^65 - 152 * q^67 + 908 * q^68 + 1904 * q^71 - 676 * q^73 - 2016 * q^74 - 1768 * q^76 + 1014 * q^79 - 840 * q^80 + 816 * q^82 - 376 * q^83 + 250 * q^85 + 768 * q^86 - 4688 * q^88 - 216 * q^89 - 264 * q^92 + 1928 * q^94 + 180 * q^95 - 2742 * q^97

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