LMFDB - Embedded newform 480.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(1,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("480.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.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:	$28.3209168028$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{201}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 50 $ x^2 - x - 50
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.58872$ of defining polynomial
Character	$\chi$	$=$	480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} -30.3549 q^{7} +9.00000 q^{9} -20.0000 q^{11} +16.3549 q^{13} +15.0000 q^{15} -69.0647 q^{17} -86.3549 q^{19} +91.0647 q^{21} +34.3549 q^{23} +25.0000 q^{25} -27.0000 q^{27} -39.4196 q^{29} +217.194 q^{31} +60.0000 q^{33} +151.774 q^{35} +281.774 q^{37} -49.0647 q^{39} +342.710 q^{41} -373.420 q^{43} -45.0000 q^{45} -198.614 q^{47} +578.420 q^{49} +207.194 q^{51} +91.8706 q^{53} +100.000 q^{55} +259.065 q^{57} +49.1608 q^{59} -309.808 q^{61} -273.194 q^{63} -81.7745 q^{65} -651.098 q^{67} -103.065 q^{69} +850.839 q^{71} +964.388 q^{73} -75.0000 q^{75} +607.098 q^{77} +724.484 q^{79} +81.0000 q^{81} +433.678 q^{83} +345.323 q^{85} +118.259 q^{87} +1264.65 q^{89} -496.451 q^{91} -651.582 q^{93} +431.774 q^{95} -1745.87 q^{97} -180.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 10 q^{5} - 4 q^{7} + 18 q^{9} - 40 q^{11} - 24 q^{13} + 30 q^{15} + 32 q^{17} - 116 q^{19} + 12 q^{21} + 12 q^{23} + 50 q^{25} - 54 q^{27} + 148 q^{29} - 76 q^{31} + 120 q^{33} + 20 q^{35}+ \cdots - 360 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 10 * q^5 - 4 * q^7 + 18 * q^9 - 40 * q^11 - 24 * q^13 + 30 * q^15 + 32 * q^17 - 116 * q^19 + 12 * q^21 + 12 * q^23 + 50 * q^25 - 54 * q^27 + 148 * q^29 - 76 * q^31 + 120 * q^33 + 20 * q^35 + 280 * q^37 + 72 * q^39 + 572 * q^41 - 520 * q^43 - 90 * q^45 + 340 * q^47 + 930 * q^49 - 96 * q^51 + 524 * q^53 + 200 * q^55 + 348 * q^57 + 552 * q^59 + 628 * q^61 - 36 * q^63 + 120 * q^65 - 168 * q^67 - 36 * q^69 + 1248 * q^71 + 908 * q^73 - 150 * q^75 + 80 * q^77 + 1052 * q^79 + 162 * q^81 - 40 * q^83 - 160 * q^85 - 444 * q^87 + 828 * q^89 - 1560 * q^91 + 228 * q^93 + 580 * q^95 - 316 * q^97 - 360 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−30.3549	−1.63901	−0.819505	−	0.573072i	$-0.805752\pi$
$7$	−30.3549	−1.63901	−0.819505	+	0.573072i	$0.805752\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−20.0000	−0.548202	−0.274101	−	0.961701i	$-0.588380\pi$
$11$	−20.0000	−0.548202	−0.274101	+	0.961701i	$0.588380\pi$
$12$	0	0
$13$	16.3549	0.348925	0.174463	−	0.984664i	$-0.444181\pi$
$13$	16.3549	0.348925	0.174463	+	0.984664i	$0.444181\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−69.0647	−0.985332	−0.492666	−	0.870218i	$-0.663978\pi$
$17$	−69.0647	−0.985332	−0.492666	+	0.870218i	$0.663978\pi$
$18$	0	0
$19$	−86.3549	−1.04269	−0.521347	−	0.853345i	$-0.674570\pi$
$19$	−86.3549	−1.04269	−0.521347	+	0.853345i	$0.674570\pi$
$20$	0	0
$21$	91.0647	0.946283
$22$	0	0
$23$	34.3549	0.311456	0.155728	−	0.987800i	$-0.450228\pi$
$23$	34.3549	0.311456	0.155728	+	0.987800i	$0.450228\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−39.4196	−0.252415	−0.126207	−	0.992004i	$-0.540280\pi$
$29$	−39.4196	−0.252415	−0.126207	+	0.992004i	$0.540280\pi$
$30$	0	0
$31$	217.194	1.25836	0.629181	−	0.777259i	$-0.283391\pi$
$31$	217.194	1.25836	0.629181	+	0.777259i	$0.283391\pi$
$32$	0	0
$33$	60.0000	0.316505
$34$	0	0
$35$	151.774	0.732988
$36$	0	0
$37$	281.774	1.25198	0.625992	−	0.779829i	$-0.284694\pi$
$37$	281.774	1.25198	0.625992	+	0.779829i	$0.284694\pi$
$38$	0	0
$39$	−49.0647	−0.201452
$40$	0	0
$41$	342.710	1.30542	0.652711	−	0.757607i	$-0.273632\pi$
$41$	342.710	1.30542	0.652711	+	0.757607i	$0.273632\pi$
$42$	0	0
$43$	−373.420	−1.32432	−0.662162	−	0.749361i	$-0.730361\pi$
$43$	−373.420	−1.32432	−0.662162	+	0.749361i	$0.730361\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−198.614	−0.616399	−0.308200	−	0.951322i	$-0.599727\pi$
$47$	−198.614	−0.616399	−0.308200	+	0.951322i	$0.599727\pi$
$48$	0	0
$49$	578.420	1.68635
$50$	0	0
$51$	207.194	0.568882
$52$	0	0
$53$	91.8706	0.238102	0.119051	−	0.992888i	$-0.462015\pi$
$53$	91.8706	0.238102	0.119051	+	0.992888i	$0.462015\pi$
$54$	0	0
$55$	100.000	0.245164
$56$	0	0
$57$	259.065	0.601999
$58$	0	0
$59$	49.1608	0.108478	0.0542390	−	0.998528i	$-0.482727\pi$
$59$	49.1608	0.108478	0.0542390	+	0.998528i	$0.482727\pi$
$60$	0	0
$61$	−309.808	−0.650276	−0.325138	−	0.945667i	$-0.605411\pi$
$61$	−309.808	−0.650276	−0.325138	+	0.945667i	$0.605411\pi$
$62$	0	0
$63$	−273.194	−0.546337
$64$	0	0
$65$	−81.7745	−0.156044
$66$	0	0
$67$	−651.098	−1.18723	−0.593614	−	0.804750i	$-0.702299\pi$
$67$	−651.098	−1.18723	−0.593614	+	0.804750i	$0.702299\pi$
$68$	0	0
$69$	−103.065	−0.179819
$70$	0	0
$71$	850.839	1.42220	0.711099	−	0.703092i	$-0.248198\pi$
$71$	850.839	1.42220	0.711099	+	0.703092i	$0.248198\pi$
$72$	0	0
$73$	964.388	1.54621	0.773103	−	0.634280i	$-0.218703\pi$
$73$	964.388	1.54621	0.773103	+	0.634280i	$0.218703\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	607.098	0.898509
$78$	0	0
$79$	724.484	1.03178	0.515891	−	0.856654i	$-0.327461\pi$
$79$	724.484	1.03178	0.515891	+	0.856654i	$0.327461\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	433.678	0.573523	0.286761	−	0.958002i	$-0.407421\pi$
$83$	433.678	0.573523	0.286761	+	0.958002i	$0.407421\pi$
$84$	0	0
$85$	345.323	0.440654
$86$	0	0
$87$	118.259	0.145732
$88$	0	0
$89$	1264.65	1.50621	0.753103	−	0.657903i	$-0.228556\pi$
$89$	1264.65	1.50621	0.753103	+	0.657903i	$0.228556\pi$
$90$	0	0
$91$	−496.451	−0.571892
$92$	0	0
$93$	−651.582	−0.726515
$94$	0	0
$95$	431.774	0.466307
$96$	0	0
$97$	−1745.87	−1.82749	−0.913746	−	0.406287i	$-0.866823\pi$
$97$	−1745.87	−1.82749	−0.913746	+	0.406287i	$0.866823\pi$
$98$	0	0
$99$	−180.000	−0.182734
$100$	0	0
$101$	1739.94	1.71416	0.857080	−	0.515183i	$-0.172276\pi$
$101$	1739.94	1.71416	0.857080	+	0.515183i	$0.172276\pi$
$102$	0	0
$103$	236.484	0.226228	0.113114	−	0.993582i	$-0.463917\pi$
$103$	236.484	0.226228	0.113114	+	0.993582i	$0.463917\pi$
$104$	0	0
$105$	−455.323	−0.423191
$106$	0	0
$107$	514.647	0.464979	0.232490	−	0.972599i	$-0.425313\pi$
$107$	514.647	0.464979	0.232490	+	0.972599i	$0.425313\pi$
$108$	0	0
$109$	342.902	0.301322	0.150661	−	0.988586i	$-0.451860\pi$
$109$	342.902	0.301322	0.150661	+	0.988586i	$0.451860\pi$
$110$	0	0
$111$	−845.323	−0.722834
$112$	0	0
$113$	534.677	0.445116	0.222558	−	0.974919i	$-0.428559\pi$
$113$	534.677	0.445116	0.222558	+	0.974919i	$0.428559\pi$
$114$	0	0
$115$	−171.774	−0.139287
$116$	0	0
$117$	147.194	0.116308
$118$	0	0
$119$	2096.45	1.61497
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	−1028.13	−0.753685
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	627.582	0.438495	0.219248	−	0.975669i	$-0.429640\pi$
$127$	627.582	0.438495	0.219248	+	0.975669i	$0.429640\pi$
$128$	0	0
$129$	1120.26	0.764599
$130$	0	0
$131$	−2188.58	−1.45968	−0.729838	−	0.683621i	$-0.760404\pi$
$131$	−2188.58	−1.45968	−0.729838	+	0.683621i	$0.760404\pi$
$132$	0	0
$133$	2621.29	1.70898
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−1896.35	−1.18260	−0.591301	−	0.806451i	$-0.701386\pi$
$137$	−1896.35	−1.18260	−0.591301	+	0.806451i	$0.701386\pi$
$138$	0	0
$139$	−441.128	−0.269180	−0.134590	−	0.990901i	$-0.542972\pi$
$139$	−441.128	−0.269180	−0.134590	+	0.990901i	$0.542972\pi$
$140$	0	0
$141$	595.841	0.355878
$142$	0	0
$143$	−327.098	−0.191282
$144$	0	0
$145$	197.098	0.112883
$146$	0	0
$147$	−1735.26	−0.973617
$148$	0	0
$149$	−571.294	−0.314109	−0.157054	−	0.987590i	$-0.550200\pi$
$149$	−571.294	−0.314109	−0.157054	+	0.987590i	$0.550200\pi$
$150$	0	0
$151$	−2883.71	−1.55413	−0.777064	−	0.629421i	$-0.783292\pi$
$151$	−2883.71	−1.55413	−0.777064	+	0.629421i	$0.783292\pi$
$152$	0	0
$153$	−621.582	−0.328444
$154$	0	0
$155$	−1085.97	−0.562756
$156$	0	0
$157$	−167.579	−0.0851862	−0.0425931	−	0.999093i	$-0.513562\pi$
$157$	−167.579	−0.0851862	−0.0425931	+	0.999093i	$0.513562\pi$
$158$	0	0
$159$	−275.612	−0.137468
$160$	0	0
$161$	−1042.84	−0.510480
$162$	0	0
$163$	−2257.16	−1.08463	−0.542314	−	0.840176i	$-0.682452\pi$
$163$	−2257.16	−1.08463	−0.542314	+	0.840176i	$0.682452\pi$
$164$	0	0
$165$	−300.000	−0.141545
$166$	0	0
$167$	3820.29	1.77020	0.885099	−	0.465403i	$-0.154091\pi$
$167$	3820.29	1.77020	0.885099	+	0.465403i	$0.154091\pi$
$168$	0	0
$169$	−1929.52	−0.878251
$170$	0	0
$171$	−777.194	−0.347564
$172$	0	0
$173$	−4185.87	−1.83957	−0.919786	−	0.392419i	$-0.871638\pi$
$173$	−4185.87	−1.83957	−0.919786	+	0.392419i	$0.871638\pi$
$174$	0	0
$175$	−758.872	−0.327802
$176$	0	0
$177$	−147.483	−0.0626298
$178$	0	0
$179$	1930.26	0.806003	0.403002	−	0.915199i	$-0.367967\pi$
$179$	1930.26	0.806003	0.403002	+	0.915199i	$0.367967\pi$
$180$	0	0
$181$	−2310.01	−0.948627	−0.474313	−	0.880356i	$-0.657304\pi$
$181$	−2310.01	−0.948627	−0.474313	+	0.880356i	$0.657304\pi$
$182$	0	0
$183$	929.423	0.375437
$184$	0	0
$185$	−1408.87	−0.559905
$186$	0	0
$187$	1381.29	0.540161
$188$	0	0
$189$	819.582	0.315428
$190$	0	0
$191$	4642.97	1.75892	0.879460	−	0.475973i	$-0.157904\pi$
$191$	4642.97	1.75892	0.879460	+	0.475973i	$0.157904\pi$
$192$	0	0
$193$	4003.23	1.49305	0.746526	−	0.665356i	$-0.231720\pi$
$193$	4003.23	1.49305	0.746526	+	0.665356i	$0.231720\pi$
$194$	0	0
$195$	245.323	0.0900922
$196$	0	0
$197$	−4118.97	−1.48967	−0.744833	−	0.667250i	$-0.767471\pi$
$197$	−4118.97	−1.48967	−0.744833	+	0.667250i	$0.767471\pi$
$198$	0	0
$199$	2384.68	0.849474	0.424737	−	0.905317i	$-0.360367\pi$
$199$	2384.68	0.849474	0.424737	+	0.905317i	$0.360367\pi$
$200$	0	0
$201$	1953.29	0.685446
$202$	0	0
$203$	1196.58	0.413711
$204$	0	0
$205$	−1713.55	−0.583802
$206$	0	0
$207$	309.194	0.103819
$208$	0	0
$209$	1727.10	0.571607
$210$	0	0
$211$	−1310.29	−0.427507	−0.213754	−	0.976888i	$-0.568569\pi$
$211$	−1310.29	−0.427507	−0.213754	+	0.976888i	$0.568569\pi$
$212$	0	0
$213$	−2552.52	−0.821106
$214$	0	0
$215$	1867.10	0.592256
$216$	0	0
$217$	−6592.90	−2.06247
$218$	0	0
$219$	−2893.16	−0.892703
$220$	0	0
$221$	−1129.55	−0.343807
$222$	0	0
$223$	−883.967	−0.265448	−0.132724	−	0.991153i	$-0.542372\pi$
$223$	−883.967	−0.265448	−0.132724	+	0.991153i	$0.542372\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	2407.36	0.703885	0.351942	−	0.936022i	$-0.385521\pi$
$227$	2407.36	0.703885	0.351942	+	0.936022i	$0.385521\pi$
$228$	0	0
$229$	5505.94	1.58883	0.794417	−	0.607373i	$-0.207777\pi$
$229$	5505.94	1.58883	0.794417	+	0.607373i	$0.207777\pi$
$230$	0	0
$231$	−1821.29	−0.518755
$232$	0	0
$233$	−673.523	−0.189373	−0.0946865	−	0.995507i	$-0.530185\pi$
$233$	−673.523	−0.189373	−0.0946865	+	0.995507i	$0.530185\pi$
$234$	0	0
$235$	993.068	0.275662
$236$	0	0
$237$	−2173.45	−0.595700
$238$	0	0
$239$	914.773	0.247580	0.123790	−	0.992308i	$-0.460495\pi$
$239$	914.773	0.247580	0.123790	+	0.992308i	$0.460495\pi$
$240$	0	0
$241$	−5516.33	−1.47443	−0.737216	−	0.675657i	$-0.763860\pi$
$241$	−5516.33	−1.47443	−0.737216	+	0.675657i	$0.763860\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−2892.10	−0.754161
$246$	0	0
$247$	−1412.33	−0.363822
$248$	0	0
$249$	−1301.03	−0.331123
$250$	0	0
$251$	4923.93	1.23823	0.619115	−	0.785300i	$-0.287491\pi$
$251$	4923.93	1.23823	0.619115	+	0.785300i	$0.287491\pi$
$252$	0	0
$253$	−687.098	−0.170741
$254$	0	0
$255$	−1035.97	−0.254412
$256$	0	0
$257$	−767.771	−0.186351	−0.0931756	−	0.995650i	$-0.529702\pi$
$257$	−767.771	−0.186351	−0.0931756	+	0.995650i	$0.529702\pi$
$258$	0	0
$259$	−8553.23	−2.05202
$260$	0	0
$261$	−354.776	−0.0841383
$262$	0	0
$263$	2623.97	0.615212	0.307606	−	0.951514i	$-0.400472\pi$
$263$	2623.97	0.615212	0.307606	+	0.951514i	$0.400472\pi$
$264$	0	0
$265$	−459.353	−0.106482
$266$	0	0
$267$	−3793.94	−0.869608
$268$	0	0
$269$	4539.69	1.02896	0.514479	−	0.857503i	$-0.327985\pi$
$269$	4539.69	1.02896	0.514479	+	0.857503i	$0.327985\pi$
$270$	0	0
$271$	−3841.51	−0.861090	−0.430545	−	0.902569i	$-0.641679\pi$
$271$	−3841.51	−0.861090	−0.430545	+	0.902569i	$0.641679\pi$
$272$	0	0
$273$	1489.35	0.330182
$274$	0	0
$275$	−500.000	−0.109640
$276$	0	0
$277$	−8471.07	−1.83746	−0.918732	−	0.394881i	$-0.870786\pi$
$277$	−8471.07	−1.83746	−0.918732	+	0.394881i	$0.870786\pi$
$278$	0	0
$279$	1954.75	0.419454
$280$	0	0
$281$	3931.10	0.834555	0.417277	−	0.908779i	$-0.362984\pi$
$281$	3931.10	0.834555	0.417277	+	0.908779i	$0.362984\pi$
$282$	0	0
$283$	5678.84	1.19283	0.596417	−	0.802675i	$-0.296591\pi$
$283$	5678.84	1.19283	0.596417	+	0.802675i	$0.296591\pi$
$284$	0	0
$285$	−1295.32	−0.269222
$286$	0	0
$287$	−10402.9	−2.13960
$288$	0	0
$289$	−143.070	−0.0291207
$290$	0	0
$291$	5237.62	1.05510
$292$	0	0
$293$	6180.79	1.23237	0.616187	−	0.787600i	$-0.288677\pi$
$293$	6180.79	1.23237	0.616187	+	0.787600i	$0.288677\pi$
$294$	0	0
$295$	−245.804	−0.0485128
$296$	0	0
$297$	540.000	0.105502
$298$	0	0
$299$	561.871	0.108675
$300$	0	0
$301$	11335.1	2.17058
$302$	0	0
$303$	−5219.81	−0.989671
$304$	0	0
$305$	1549.04	0.290812
$306$	0	0
$307$	−5058.91	−0.940480	−0.470240	−	0.882539i	$-0.655833\pi$
$307$	−5058.91	−0.940480	−0.470240	+	0.882539i	$0.655833\pi$
$308$	0	0
$309$	−709.453	−0.130613
$310$	0	0
$311$	7795.82	1.42142	0.710708	−	0.703487i	$-0.248375\pi$
$311$	7795.82	1.42142	0.710708	+	0.703487i	$0.248375\pi$
$312$	0	0
$313$	−8138.34	−1.46967	−0.734834	−	0.678247i	$-0.762740\pi$
$313$	−8138.34	−1.46967	−0.734834	+	0.678247i	$0.762740\pi$
$314$	0	0
$315$	1365.97	0.244329
$316$	0	0
$317$	−6911.10	−1.22450	−0.612249	−	0.790665i	$-0.709735\pi$
$317$	−6911.10	−1.22450	−0.612249	+	0.790665i	$0.709735\pi$
$318$	0	0
$319$	788.392	0.138374
$320$	0	0
$321$	−1543.94	−0.268456
$322$	0	0
$323$	5964.07	1.02740
$324$	0	0
$325$	408.872	0.0697851
$326$	0	0
$327$	−1028.71	−0.173968
$328$	0	0
$329$	6028.90	1.01028
$330$	0	0
$331$	6510.62	1.08114	0.540568	−	0.841300i	$-0.318209\pi$
$331$	6510.62	1.08114	0.540568	+	0.841300i	$0.318209\pi$
$332$	0	0
$333$	2535.97	0.417328
$334$	0	0
$335$	3255.49	0.530944
$336$	0	0
$337$	−247.990	−0.0400856	−0.0200428	−	0.999799i	$-0.506380\pi$
$337$	−247.990	−0.0400856	−0.0200428	+	0.999799i	$0.506380\pi$
$338$	0	0
$339$	−1604.03	−0.256988
$340$	0	0
$341$	−4343.88	−0.689837
$342$	0	0
$343$	−7146.14	−1.12494
$344$	0	0
$345$	515.323	0.0804176
$346$	0	0
$347$	6105.88	0.944613	0.472306	−	0.881434i	$-0.343422\pi$
$347$	6105.88	0.944613	0.472306	+	0.881434i	$0.343422\pi$
$348$	0	0
$349$	7452.90	1.14311	0.571554	−	0.820565i	$-0.306341\pi$
$349$	7452.90	1.14311	0.571554	+	0.820565i	$0.306341\pi$
$350$	0	0
$351$	−441.582	−0.0671507
$352$	0	0
$353$	12974.6	1.95629	0.978144	−	0.207929i	$-0.0666724\pi$
$353$	12974.6	1.95629	0.978144	+	0.207929i	$0.0666724\pi$
$354$	0	0
$355$	−4254.20	−0.636026
$356$	0	0
$357$	−6289.35	−0.932403
$358$	0	0
$359$	−5776.40	−0.849211	−0.424605	−	0.905379i	$-0.639587\pi$
$359$	−5776.40	−0.849211	−0.424605	+	0.905379i	$0.639587\pi$
$360$	0	0
$361$	598.168	0.0872092
$362$	0	0
$363$	2793.00	0.403842
$364$	0	0
$365$	−4821.94	−0.691485
$366$	0	0
$367$	7191.40	1.02286	0.511428	−	0.859326i	$-0.329117\pi$
$367$	7191.40	1.02286	0.511428	+	0.859326i	$0.329117\pi$
$368$	0	0
$369$	3084.39	0.435140
$370$	0	0
$371$	−2788.72	−0.390252
$372$	0	0
$373$	6487.77	0.900601	0.450300	−	0.892877i	$-0.351317\pi$
$373$	6487.77	0.900601	0.450300	+	0.892877i	$0.351317\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−644.703	−0.0880740
$378$	0	0
$379$	−1464.68	−0.198510	−0.0992551	−	0.995062i	$-0.531646\pi$
$379$	−1464.68	−0.198510	−0.0992551	+	0.995062i	$0.531646\pi$
$380$	0	0
$381$	−1882.75	−0.253165
$382$	0	0
$383$	−3568.29	−0.476060	−0.238030	−	0.971258i	$-0.576502\pi$
$383$	−3568.29	−0.476060	−0.238030	+	0.971258i	$0.576502\pi$
$384$	0	0
$385$	−3035.49	−0.401826
$386$	0	0
$387$	−3360.78	−0.441441
$388$	0	0
$389$	5562.52	0.725015	0.362507	−	0.931981i	$-0.381921\pi$
$389$	5562.52	0.725015	0.362507	+	0.931981i	$0.381921\pi$
$390$	0	0
$391$	−2372.71	−0.306888
$392$	0	0
$393$	6565.75	0.842744
$394$	0	0
$395$	−3622.42	−0.461427
$396$	0	0
$397$	1601.77	0.202496	0.101248	−	0.994861i	$-0.467717\pi$
$397$	1601.77	0.202496	0.101248	+	0.994861i	$0.467717\pi$
$398$	0	0
$399$	−7863.88	−0.986683
$400$	0	0
$401$	−5435.04	−0.676840	−0.338420	−	0.940995i	$-0.609893\pi$
$401$	−5435.04	−0.676840	−0.338420	+	0.940995i	$0.609893\pi$
$402$	0	0
$403$	3552.19	0.439074
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−5635.49	−0.686341
$408$	0	0
$409$	11535.2	1.39457	0.697283	−	0.716796i	$-0.254392\pi$
$409$	11535.2	1.39457	0.697283	+	0.716796i	$0.254392\pi$
$410$	0	0
$411$	5689.06	0.682776
$412$	0	0
$413$	−1492.27	−0.177796
$414$	0	0
$415$	−2168.39	−0.256487
$416$	0	0
$417$	1323.38	0.155411
$418$	0	0
$419$	−8858.90	−1.03290	−0.516451	−	0.856317i	$-0.672747\pi$
$419$	−8858.90	−1.03290	−0.516451	+	0.856317i	$0.672747\pi$
$420$	0	0
$421$	13361.9	1.54684	0.773422	−	0.633892i	$-0.218544\pi$
$421$	13361.9	1.54684	0.773422	+	0.633892i	$0.218544\pi$
$422$	0	0
$423$	−1787.52	−0.205466
$424$	0	0
$425$	−1726.62	−0.197066
$426$	0	0
$427$	9404.18	1.06581
$428$	0	0
$429$	981.294	0.110437
$430$	0	0
$431$	−12223.1	−1.36605	−0.683024	−	0.730396i	$-0.739336\pi$
$431$	−12223.1	−1.36605	−0.683024	+	0.730396i	$0.739336\pi$
$432$	0	0
$433$	110.192	0.0122298	0.00611490	−	0.999981i	$-0.498054\pi$
$433$	110.192	0.0122298	0.00611490	+	0.999981i	$0.498054\pi$
$434$	0	0
$435$	−591.294	−0.0651732
$436$	0	0
$437$	−2966.71	−0.324753
$438$	0	0
$439$	10838.5	1.17834	0.589172	−	0.808008i	$-0.299454\pi$
$439$	10838.5	1.17834	0.589172	+	0.808008i	$0.299454\pi$
$440$	0	0
$441$	5205.78	0.562118
$442$	0	0
$443$	14291.2	1.53272	0.766362	−	0.642409i	$-0.222065\pi$
$443$	14291.2	1.53272	0.766362	+	0.642409i	$0.222065\pi$
$444$	0	0
$445$	−6323.23	−0.673596
$446$	0	0
$447$	1713.88	0.181351
$448$	0	0
$449$	2451.92	0.257714	0.128857	−	0.991663i	$-0.458869\pi$
$449$	2451.92	0.257714	0.128857	+	0.991663i	$0.458869\pi$
$450$	0	0
$451$	−6854.20	−0.715635
$452$	0	0
$453$	8651.14	0.897276
$454$	0	0
$455$	2482.26	0.255758
$456$	0	0
$457$	−1352.31	−0.138421	−0.0692104	−	0.997602i	$-0.522048\pi$
$457$	−1352.31	−0.138421	−0.0692104	+	0.997602i	$0.522048\pi$
$458$	0	0
$459$	1864.75	0.189627
$460$	0	0
$461$	5331.29	0.538618	0.269309	−	0.963054i	$-0.413205\pi$
$461$	5331.29	0.538618	0.269309	+	0.963054i	$0.413205\pi$
$462$	0	0
$463$	−1.69784	−0.000170422 0	−8.52110e−5	−	1.00000i	$-0.500027\pi$
$463$	−1.69784	−0.000170422 0	−8.52110e−5		1.00000i	$0.500027\pi$
$464$	0	0
$465$	3257.91	0.324908
$466$	0	0
$467$	9081.89	0.899914	0.449957	−	0.893050i	$-0.351439\pi$
$467$	9081.89	0.899914	0.449957	+	0.893050i	$0.351439\pi$
$468$	0	0
$469$	19764.0	1.94588
$470$	0	0
$471$	502.736	0.0491823
$472$	0	0
$473$	7468.39	0.725998
$474$	0	0
$475$	−2158.87	−0.208539
$476$	0	0
$477$	826.836	0.0793673
$478$	0	0
$479$	6270.21	0.598107	0.299053	−	0.954236i	$-0.403329\pi$
$479$	6270.21	0.598107	0.299053	+	0.954236i	$0.403329\pi$
$480$	0	0
$481$	4608.39	0.436849
$482$	0	0
$483$	3128.52	0.294726
$484$	0	0
$485$	8729.37	0.817279
$486$	0	0
$487$	13938.7	1.29697	0.648486	−	0.761227i	$-0.275403\pi$
$487$	13938.7	1.29697	0.648486	+	0.761227i	$0.275403\pi$
$488$	0	0
$489$	6771.48	0.626211
$490$	0	0
$491$	5301.54	0.487281	0.243641	−	0.969866i	$-0.421658\pi$
$491$	5301.54	0.487281	0.243641	+	0.969866i	$0.421658\pi$
$492$	0	0
$493$	2722.50	0.248712
$494$	0	0
$495$	900.000	0.0817212
$496$	0	0
$497$	−25827.1	−2.33100
$498$	0	0
$499$	12167.0	1.09152	0.545762	−	0.837940i	$-0.316240\pi$
$499$	12167.0	1.09152	0.545762	+	0.837940i	$0.316240\pi$
$500$	0	0
$501$	−11460.9	−1.02202
$502$	0	0
$503$	−12761.8	−1.13126	−0.565628	−	0.824660i	$-0.691366\pi$
$503$	−12761.8	−1.13126	−0.565628	+	0.824660i	$0.691366\pi$
$504$	0	0
$505$	−8699.69	−0.766596
$506$	0	0
$507$	5788.55	0.507058
$508$	0	0
$509$	−1875.68	−0.163336	−0.0816680	−	0.996660i	$-0.526025\pi$
$509$	−1875.68	−0.163336	−0.0816680	+	0.996660i	$0.526025\pi$
$510$	0	0
$511$	−29273.9	−2.53425
$512$	0	0
$513$	2331.58	0.200666
$514$	0	0
$515$	−1182.42	−0.101172
$516$	0	0
$517$	3972.27	0.337912
$518$	0	0
$519$	12557.6	1.06208
$520$	0	0
$521$	12565.0	1.05659	0.528293	−	0.849062i	$-0.322832\pi$
$521$	12565.0	1.05659	0.528293	+	0.849062i	$0.322832\pi$
$522$	0	0
$523$	20006.3	1.67268	0.836341	−	0.548209i	$-0.184690\pi$
$523$	20006.3	1.67268	0.836341	+	0.548209i	$0.184690\pi$
$524$	0	0
$525$	2276.62	0.189257
$526$	0	0
$527$	−15000.4	−1.23990
$528$	0	0
$529$	−10986.7	−0.902995
$530$	0	0
$531$	442.448	0.0361593
$532$	0	0
$533$	5604.98	0.455495
$534$	0	0
$535$	−2573.23	−0.207945
$536$	0	0
$537$	−5790.79	−0.465346
$538$	0	0
$539$	−11568.4	−0.924464
$540$	0	0
$541$	3930.27	0.312339	0.156170	−	0.987730i	$-0.450085\pi$
$541$	3930.27	0.312339	0.156170	+	0.987730i	$0.450085\pi$
$542$	0	0
$543$	6930.02	0.547690
$544$	0	0
$545$	−1714.51	−0.134755
$546$	0	0
$547$	19908.5	1.55618	0.778088	−	0.628156i	$-0.216190\pi$
$547$	19908.5	1.55618	0.778088	+	0.628156i	$0.216190\pi$
$548$	0	0
$549$	−2788.27	−0.216759
$550$	0	0
$551$	3404.07	0.263191
$552$	0	0
$553$	−21991.6	−1.69110
$554$	0	0
$555$	4226.62	0.323261
$556$	0	0
$557$	11820.0	0.899153	0.449576	−	0.893242i	$-0.351575\pi$
$557$	11820.0	0.899153	0.449576	+	0.893242i	$0.351575\pi$
$558$	0	0
$559$	−6107.24	−0.462091
$560$	0	0
$561$	−4143.88	−0.311862
$562$	0	0
$563$	−21578.2	−1.61530	−0.807648	−	0.589665i	$-0.799260\pi$
$563$	−21578.2	−1.61530	−0.807648	+	0.589665i	$0.799260\pi$
$564$	0	0
$565$	−2673.38	−0.199062
$566$	0	0
$567$	−2458.75	−0.182112
$568$	0	0
$569$	22874.5	1.68533	0.842663	−	0.538441i	$-0.180986\pi$
$569$	22874.5	1.68533	0.842663	+	0.538441i	$0.180986\pi$
$570$	0	0
$571$	−2857.53	−0.209429	−0.104714	−	0.994502i	$-0.533393\pi$
$571$	−2857.53	−0.209429	−0.104714	+	0.994502i	$0.533393\pi$
$572$	0	0
$573$	−13928.9	−1.01551
$574$	0	0
$575$	858.872	0.0622912
$576$	0	0
$577$	−10000.7	−0.721553	−0.360776	−	0.932652i	$-0.617488\pi$
$577$	−10000.7	−0.721553	−0.360776	+	0.932652i	$0.617488\pi$
$578$	0	0
$579$	−12009.7	−0.862014
$580$	0	0
$581$	−13164.3	−0.940009
$582$	0	0
$583$	−1837.41	−0.130528
$584$	0	0
$585$	−735.970	−0.0520147
$586$	0	0
$587$	−10215.0	−0.718262	−0.359131	−	0.933287i	$-0.616927\pi$
$587$	−10215.0	−0.718262	−0.359131	+	0.933287i	$0.616927\pi$
$588$	0	0
$589$	−18755.8	−1.31208
$590$	0	0
$591$	12356.9	0.860060
$592$	0	0
$593$	−3995.50	−0.276687	−0.138344	−	0.990384i	$-0.544178\pi$
$593$	−3995.50	−0.276687	−0.138344	+	0.990384i	$0.544178\pi$
$594$	0	0
$595$	−10482.3	−0.722236
$596$	0	0
$597$	−7154.03	−0.490444
$598$	0	0
$599$	7844.51	0.535089	0.267544	−	0.963546i	$-0.413788\pi$
$599$	7844.51	0.535089	0.267544	+	0.963546i	$0.413788\pi$
$600$	0	0
$601$	5265.24	0.357361	0.178680	−	0.983907i	$-0.442817\pi$
$601$	5265.24	0.357361	0.178680	+	0.983907i	$0.442817\pi$
$602$	0	0
$603$	−5859.88	−0.395743
$604$	0	0
$605$	4655.00	0.312814
$606$	0	0
$607$	−16213.6	−1.08416	−0.542082	−	0.840325i	$-0.682364\pi$
$607$	−16213.6	−1.08416	−0.542082	+	0.840325i	$0.682364\pi$
$608$	0	0
$609$	−3589.73	−0.238856
$610$	0	0
$611$	−3248.30	−0.215077
$612$	0	0
$613$	−4538.37	−0.299026	−0.149513	−	0.988760i	$-0.547771\pi$
$613$	−4538.37	−0.299026	−0.149513	+	0.988760i	$0.547771\pi$
$614$	0	0
$615$	5140.65	0.337058
$616$	0	0
$617$	1715.15	0.111911	0.0559556	−	0.998433i	$-0.482179\pi$
$617$	1715.15	0.111911	0.0559556	+	0.998433i	$0.482179\pi$
$618$	0	0
$619$	695.376	0.0451527	0.0225764	−	0.999745i	$-0.492813\pi$
$619$	695.376	0.0451527	0.0225764	+	0.999745i	$0.492813\pi$
$620$	0	0
$621$	−927.582	−0.0599398
$622$	0	0
$623$	−38388.2	−2.46869
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−5181.29	−0.330017
$628$	0	0
$629$	−19460.7	−1.23362
$630$	0	0
$631$	−9877.82	−0.623185	−0.311593	−	0.950216i	$-0.600862\pi$
$631$	−9877.82	−0.623185	−0.311593	+	0.950216i	$0.600862\pi$
$632$	0	0
$633$	3930.87	0.246821
$634$	0	0
$635$	−3137.91	−0.196101
$636$	0	0
$637$	9459.99	0.588412
$638$	0	0
$639$	7657.55	0.474066
$640$	0	0
$641$	2441.03	0.150413	0.0752067	−	0.997168i	$-0.476038\pi$
$641$	2441.03	0.150413	0.0752067	+	0.997168i	$0.476038\pi$
$642$	0	0
$643$	8080.27	0.495575	0.247788	−	0.968814i	$-0.420296\pi$
$643$	8080.27	0.495575	0.247788	+	0.968814i	$0.420296\pi$
$644$	0	0
$645$	−5601.29	−0.341939
$646$	0	0
$647$	1852.94	0.112591	0.0562957	−	0.998414i	$-0.482071\pi$
$647$	1852.94	0.112591	0.0562957	+	0.998414i	$0.482071\pi$
$648$	0	0
$649$	−983.217	−0.0594679
$650$	0	0
$651$	19778.7	1.19077
$652$	0	0
$653$	−8211.29	−0.492087	−0.246043	−	0.969259i	$-0.579131\pi$
$653$	−8211.29	−0.492087	−0.246043	+	0.969259i	$0.579131\pi$
$654$	0	0
$655$	10942.9	0.652787
$656$	0	0
$657$	8679.49	0.515402
$658$	0	0
$659$	−22558.3	−1.33346	−0.666728	−	0.745301i	$-0.732306\pi$
$659$	−22558.3	−1.33346	−0.666728	+	0.745301i	$0.732306\pi$
$660$	0	0
$661$	−20058.9	−1.18033	−0.590166	−	0.807282i	$-0.700938\pi$
$661$	−20058.9	−1.18033	−0.590166	+	0.807282i	$0.700938\pi$
$662$	0	0
$663$	3388.64	0.198497
$664$	0	0
$665$	−13106.5	−0.764281
$666$	0	0
$667$	−1354.26	−0.0786162
$668$	0	0
$669$	2651.90	0.153256
$670$	0	0
$671$	6196.15	0.356483
$672$	0	0
$673$	12172.8	0.697218	0.348609	−	0.937268i	$-0.386654\pi$
$673$	12172.8	0.697218	0.348609	+	0.937268i	$0.386654\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−29440.0	−1.67130	−0.835650	−	0.549263i	$-0.814909\pi$
$677$	−29440.0	−1.67130	−0.835650	+	0.549263i	$0.814909\pi$
$678$	0	0
$679$	52995.8	2.99528
$680$	0	0
$681$	−7222.07	−0.406388
$682$	0	0
$683$	21223.7	1.18902	0.594512	−	0.804087i	$-0.297345\pi$
$683$	21223.7	1.18902	0.594512	+	0.804087i	$0.297345\pi$
$684$	0	0
$685$	9481.77	0.528876
$686$	0	0
$687$	−16517.8	−0.917313
$688$	0	0
$689$	1502.53	0.0830798
$690$	0	0
$691$	2278.77	0.125454	0.0627268	−	0.998031i	$-0.480020\pi$
$691$	2278.77	0.125454	0.0627268	+	0.998031i	$0.480020\pi$
$692$	0	0
$693$	5463.88	0.299503
$694$	0	0
$695$	2205.64	0.120381
$696$	0	0
$697$	−23669.1	−1.28627
$698$	0	0
$699$	2020.57	0.109335
$700$	0	0
$701$	−8985.49	−0.484133	−0.242067	−	0.970260i	$-0.577825\pi$
$701$	−8985.49	−0.484133	−0.242067	+	0.970260i	$0.577825\pi$
$702$	0	0
$703$	−24332.6	−1.30544
$704$	0	0
$705$	−2979.20	−0.159154
$706$	0	0
$707$	−52815.6	−2.80953
$708$	0	0
$709$	−530.836	−0.0281184	−0.0140592	−	0.999901i	$-0.504475\pi$
$709$	−530.836	−0.0281184	−0.0140592	+	0.999901i	$0.504475\pi$
$710$	0	0
$711$	6520.36	0.343928
$712$	0	0
$713$	7461.68	0.391924
$714$	0	0
$715$	1635.49	0.0855438
$716$	0	0
$717$	−2744.32	−0.142941
$718$	0	0
$719$	12797.7	0.663804	0.331902	−	0.943314i	$-0.392310\pi$
$719$	12797.7	0.663804	0.331902	+	0.943314i	$0.392310\pi$
$720$	0	0
$721$	−7178.45	−0.370790
$722$	0	0
$723$	16549.0	0.851263
$724$	0	0
$725$	−985.489	−0.0504830
$726$	0	0
$727$	30477.7	1.55482	0.777411	−	0.628993i	$-0.216533\pi$
$727$	30477.7	1.55482	0.777411	+	0.628993i	$0.216533\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	25790.1	1.30490
$732$	0	0
$733$	−4322.79	−0.217825	−0.108913	−	0.994051i	$-0.534737\pi$
$733$	−4322.79	−0.217825	−0.108913	+	0.994051i	$0.534737\pi$
$734$	0	0
$735$	8676.29	0.435415
$736$	0	0
$737$	13022.0	0.650841
$738$	0	0
$739$	−24280.6	−1.20863	−0.604315	−	0.796745i	$-0.706553\pi$
$739$	−24280.6	−1.20863	−0.604315	+	0.796745i	$0.706553\pi$
$740$	0	0
$741$	4236.98	0.210053
$742$	0	0
$743$	11638.4	0.574658	0.287329	−	0.957832i	$-0.407233\pi$
$743$	11638.4	0.574658	0.287329	+	0.957832i	$0.407233\pi$
$744$	0	0
$745$	2856.47	0.140474
$746$	0	0
$747$	3903.10	0.191174
$748$	0	0
$749$	−15622.0	−0.762106
$750$	0	0
$751$	11817.8	0.574216	0.287108	−	0.957898i	$-0.407306\pi$
$751$	11817.8	0.574216	0.287108	+	0.957898i	$0.407306\pi$
$752$	0	0
$753$	−14771.8	−0.714893
$754$	0	0
$755$	14418.6	0.695027
$756$	0	0
$757$	−311.180	−0.0149406	−0.00747031	−	0.999972i	$-0.502378\pi$
$757$	−311.180	−0.0149406	−0.00747031	+	0.999972i	$0.502378\pi$
$758$	0	0
$759$	2061.29	0.0985774
$760$	0	0
$761$	−7381.11	−0.351597	−0.175798	−	0.984426i	$-0.556251\pi$
$761$	−7381.11	−0.351597	−0.175798	+	0.984426i	$0.556251\pi$
$762$	0	0
$763$	−10408.8	−0.493869
$764$	0	0
$765$	3107.91	0.146885
$766$	0	0
$767$	804.020	0.0378507
$768$	0	0
$769$	−6695.05	−0.313953	−0.156976	−	0.987602i	$-0.550175\pi$
$769$	−6695.05	−0.313953	−0.156976	+	0.987602i	$0.550175\pi$
$770$	0	0
$771$	2303.31	0.107590
$772$	0	0
$773$	−25149.8	−1.17022	−0.585108	−	0.810955i	$-0.698948\pi$
$773$	−25149.8	−1.17022	−0.585108	+	0.810955i	$0.698948\pi$
$774$	0	0
$775$	5429.85	0.251672
$776$	0	0
$777$	25659.7	1.18473
$778$	0	0
$779$	−29594.7	−1.36115
$780$	0	0
$781$	−17016.8	−0.779652
$782$	0	0
$783$	1064.33	0.0485773
$784$	0	0
$785$	837.894	0.0380964
$786$	0	0
$787$	25047.8	1.13451	0.567254	−	0.823543i	$-0.308006\pi$
$787$	25047.8	1.13451	0.567254	+	0.823543i	$0.308006\pi$
$788$	0	0
$789$	−7871.90	−0.355193
$790$	0	0
$791$	−16230.1	−0.729550
$792$	0	0
$793$	−5066.87	−0.226898
$794$	0	0
$795$	1378.06	0.0614777
$796$	0	0
$797$	34617.1	1.53852	0.769260	−	0.638936i	$-0.220625\pi$
$797$	34617.1	1.53852	0.769260	+	0.638936i	$0.220625\pi$
$798$	0	0
$799$	13717.2	0.607358
$800$	0	0
$801$	11381.8	0.502069
$802$	0	0
$803$	−19287.8	−0.847634
$804$	0	0
$805$	5214.20	0.228293
$806$	0	0
$807$	−13619.1	−0.594069
$808$	0	0
$809$	−39309.0	−1.70832	−0.854160	−	0.520011i	$-0.825928\pi$
$809$	−39309.0	−1.70832	−0.854160	+	0.520011i	$0.825928\pi$
$810$	0	0
$811$	−36235.7	−1.56894	−0.784469	−	0.620168i	$-0.787065\pi$
$811$	−36235.7	−1.56894	−0.784469	+	0.620168i	$0.787065\pi$
$812$	0	0
$813$	11524.5	0.497150
$814$	0	0
$815$	11285.8	0.485061
$816$	0	0
$817$	32246.6	1.38086
$818$	0	0
$819$	−4468.06	−0.190631
$820$	0	0
$821$	4570.56	0.194292	0.0971459	−	0.995270i	$-0.469029\pi$
$821$	4570.56	0.194292	0.0971459	+	0.995270i	$0.469029\pi$
$822$	0	0
$823$	18993.8	0.804473	0.402237	−	0.915536i	$-0.368233\pi$
$823$	18993.8	0.804473	0.402237	+	0.915536i	$0.368233\pi$
$824$	0	0
$825$	1500.00	0.0633010
$826$	0	0
$827$	−32809.1	−1.37954	−0.689772	−	0.724027i	$-0.742289\pi$
$827$	−32809.1	−1.37954	−0.689772	+	0.724027i	$0.742289\pi$
$828$	0	0
$829$	5188.36	0.217369	0.108685	−	0.994076i	$-0.465336\pi$
$829$	5188.36	0.217369	0.108685	+	0.994076i	$0.465336\pi$
$830$	0	0
$831$	25413.2	1.06086
$832$	0	0
$833$	−39948.4	−1.66162
$834$	0	0
$835$	−19101.5	−0.791657
$836$	0	0
$837$	−5864.24	−0.242172
$838$	0	0
$839$	−4074.61	−0.167665	−0.0838327	−	0.996480i	$-0.526716\pi$
$839$	−4074.61	−0.167665	−0.0838327	+	0.996480i	$0.526716\pi$
$840$	0	0
$841$	−22835.1	−0.936287
$842$	0	0
$843$	−11793.3	−0.481830
$844$	0	0
$845$	9647.59	0.392766
$846$	0	0
$847$	28260.4	1.14645
$848$	0	0
$849$	−17036.5	−0.688683
$850$	0	0
$851$	9680.33	0.389938
$852$	0	0
$853$	45902.6	1.84253	0.921263	−	0.388941i	$-0.127159\pi$
$853$	45902.6	1.84253	0.921263	+	0.388941i	$0.127159\pi$
$854$	0	0
$855$	3885.97	0.155436
$856$	0	0
$857$	−17316.3	−0.690216	−0.345108	−	0.938563i	$-0.612158\pi$
$857$	−17316.3	−0.690216	−0.345108	+	0.938563i	$0.612158\pi$
$858$	0	0
$859$	14329.6	0.569171	0.284586	−	0.958651i	$-0.408144\pi$
$859$	14329.6	0.569171	0.284586	+	0.958651i	$0.408144\pi$
$860$	0	0
$861$	31208.8	1.23530
$862$	0	0
$863$	−28679.6	−1.13125	−0.565623	−	0.824664i	$-0.691364\pi$
$863$	−28679.6	−1.13125	−0.565623	+	0.824664i	$0.691364\pi$
$864$	0	0
$865$	20929.4	0.822682
$866$	0	0
$867$	429.209	0.0168128
$868$	0	0
$869$	−14489.7	−0.565626
$870$	0	0
$871$	−10648.6	−0.414254
$872$	0	0
$873$	−15712.9	−0.609164
$874$	0	0
$875$	3794.36	0.146598
$876$	0	0
$877$	−30048.4	−1.15697	−0.578485	−	0.815693i	$-0.696356\pi$
$877$	−30048.4	−1.15697	−0.578485	+	0.815693i	$0.696356\pi$
$878$	0	0
$879$	−18542.4	−0.711511
$880$	0	0
$881$	12576.6	0.480948	0.240474	−	0.970656i	$-0.422697\pi$
$881$	12576.6	0.480948	0.240474	+	0.970656i	$0.422697\pi$
$882$	0	0
$883$	18031.5	0.687213	0.343607	−	0.939114i	$-0.388351\pi$
$883$	18031.5	0.687213	0.343607	+	0.939114i	$0.388351\pi$
$884$	0	0
$885$	737.413	0.0280089
$886$	0	0
$887$	7089.07	0.268351	0.134176	−	0.990958i	$-0.457161\pi$
$887$	7089.07	0.268351	0.134176	+	0.990958i	$0.457161\pi$
$888$	0	0
$889$	−19050.2	−0.718698
$890$	0	0
$891$	−1620.00	−0.0609114
$892$	0	0
$893$	17151.3	0.642716
$894$	0	0
$895$	−9651.31	−0.360456
$896$	0	0
$897$	−1685.61	−0.0627435
$898$	0	0
$899$	−8561.70	−0.317629
$900$	0	0
$901$	−6345.02	−0.234609
$902$	0	0
$903$	−34005.3	−1.25319
$904$	0	0
$905$	11550.0	0.424239
$906$	0	0
$907$	18949.2	0.693714	0.346857	−	0.937918i	$-0.387249\pi$
$907$	18949.2	0.693714	0.346857	+	0.937918i	$0.387249\pi$
$908$	0	0
$909$	15659.4	0.571387
$910$	0	0
$911$	−13418.4	−0.488005	−0.244003	−	0.969775i	$-0.578461\pi$
$911$	−13418.4	−0.488005	−0.244003	+	0.969775i	$0.578461\pi$
$912$	0	0
$913$	−8673.57	−0.314407
$914$	0	0
$915$	−4647.11	−0.167900
$916$	0	0
$917$	66434.2	2.39242
$918$	0	0
$919$	17684.3	0.634768	0.317384	−	0.948297i	$-0.397196\pi$
$919$	17684.3	0.634768	0.317384	+	0.948297i	$0.397196\pi$
$920$	0	0
$921$	15176.7	0.542986
$922$	0	0
$923$	13915.4	0.496241
$924$	0	0
$925$	7044.36	0.250397
$926$	0	0
$927$	2128.36	0.0754093
$928$	0	0
$929$	34376.8	1.21406	0.607032	−	0.794677i	$-0.292360\pi$
$929$	34376.8	1.21406	0.607032	+	0.794677i	$0.292360\pi$
$930$	0	0
$931$	−49949.4	−1.75835
$932$	0	0
$933$	−23387.5	−0.820655
$934$	0	0
$935$	−6906.47	−0.241568
$936$	0	0
$937$	41657.0	1.45237	0.726187	−	0.687497i	$-0.241291\pi$
$937$	41657.0	1.45237	0.726187	+	0.687497i	$0.241291\pi$
$938$	0	0
$939$	24415.0	0.848513
$940$	0	0
$941$	13683.4	0.474035	0.237017	−	0.971505i	$-0.423830\pi$
$941$	13683.4	0.474035	0.237017	+	0.971505i	$0.423830\pi$
$942$	0	0
$943$	11773.8	0.406581
$944$	0	0
$945$	−4097.91	−0.141064
$946$	0	0
$947$	13500.9	0.463273	0.231637	−	0.972802i	$-0.425592\pi$
$947$	13500.9	0.463273	0.231637	+	0.972802i	$0.425592\pi$
$948$	0	0
$949$	15772.5	0.539511
$950$	0	0
$951$	20733.3	0.706965
$952$	0	0
$953$	5242.25	0.178188	0.0890939	−	0.996023i	$-0.471603\pi$
$953$	5242.25	0.178188	0.0890939	+	0.996023i	$0.471603\pi$
$954$	0	0
$955$	−23214.9	−0.786613
$956$	0	0
$957$	−2365.17	−0.0798905
$958$	0	0
$959$	57563.7	1.93830
$960$	0	0
$961$	17382.3	0.583473
$962$	0	0
$963$	4631.82	0.154993
$964$	0	0
$965$	−20016.2	−0.667713
$966$	0	0
$967$	45761.1	1.52180	0.760898	−	0.648871i	$-0.224759\pi$
$967$	45761.1	1.52180	0.760898	+	0.648871i	$0.224759\pi$
$968$	0	0
$969$	−17892.2	−0.593169
$970$	0	0
$971$	47771.8	1.57886	0.789428	−	0.613843i	$-0.210377\pi$
$971$	47771.8	1.57886	0.789428	+	0.613843i	$0.210377\pi$
$972$	0	0
$973$	13390.4	0.441188
$974$	0	0
$975$	−1226.62	−0.0402904
$976$	0	0
$977$	49812.5	1.63116	0.815581	−	0.578644i	$-0.196418\pi$
$977$	49812.5	1.63116	0.815581	+	0.578644i	$0.196418\pi$
$978$	0	0
$979$	−25292.9	−0.825706
$980$	0	0
$981$	3086.12	0.100441
$982$	0	0
$983$	49131.6	1.59416	0.797078	−	0.603877i	$-0.206378\pi$
$983$	49131.6	1.59416	0.797078	+	0.603877i	$0.206378\pi$
$984$	0	0
$985$	20594.8	0.666199
$986$	0	0
$987$	−18086.7	−0.583288
$988$	0	0
$989$	−12828.8	−0.412469
$990$	0	0
$991$	−46553.2	−1.49224	−0.746121	−	0.665811i	$-0.768086\pi$
$991$	−46553.2	−1.49224	−0.746121	+	0.665811i	$0.768086\pi$
$992$	0	0
$993$	−19531.9	−0.624194
$994$	0	0
$995$	−11923.4	−0.379896
$996$	0	0
$997$	17111.5	0.543557	0.271778	−	0.962360i	$-0.412388\pi$
$997$	17111.5	0.543557	0.271778	+	0.962360i	$0.412388\pi$
$998$	0	0
$999$	−7607.91	−0.240945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.a.n.1.1	✓	2
3.2	odd	2		1440.4.a.ba.1.1		2
4.3	odd	2		480.4.a.p.1.2	yes	2
5.4	even	2		2400.4.a.ba.1.2		2
8.3	odd	2		960.4.a.bl.1.2		2
8.5	even	2		960.4.a.bp.1.1		2
12.11	even	2		1440.4.a.bf.1.2		2
20.19	odd	2		2400.4.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.n.1.1	✓	2	1.1	even	1	trivial
480.4.a.p.1.2	yes	2	4.3	odd	2
960.4.a.bl.1.2		2	8.3	odd	2
960.4.a.bp.1.1		2	8.5	even	2
1440.4.a.ba.1.1		2	3.2	odd	2
1440.4.a.bf.1.2		2	12.11	even	2
2400.4.a.z.1.1		2	20.19	odd	2
2400.4.a.ba.1.2		2	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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} -30.3549 q^{7} +9.00000 q^{9} -20.0000 q^{11} +16.3549 q^{13} +15.0000 q^{15} -69.0647 q^{17} -86.3549 q^{19} +91.0647 q^{21} +34.3549 q^{23} +25.0000 q^{25} -27.0000 q^{27} -39.4196 q^{29} +217.194 q^{31} +60.0000 q^{33} +151.774 q^{35} +281.774 q^{37} -49.0647 q^{39} +342.710 q^{41} -373.420 q^{43} -45.0000 q^{45} -198.614 q^{47} +578.420 q^{49} +207.194 q^{51} +91.8706 q^{53} +100.000 q^{55} +259.065 q^{57} +49.1608 q^{59} -309.808 q^{61} -273.194 q^{63} -81.7745 q^{65} -651.098 q^{67} -103.065 q^{69} +850.839 q^{71} +964.388 q^{73} -75.0000 q^{75} +607.098 q^{77} +724.484 q^{79} +81.0000 q^{81} +433.678 q^{83} +345.323 q^{85} +118.259 q^{87} +1264.65 q^{89} -496.451 q^{91} -651.582 q^{93} +431.774 q^{95} -1745.87 q^{97} -180.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 10 q^{5} - 4 q^{7} + 18 q^{9} - 40 q^{11} - 24 q^{13} + 30 q^{15} + 32 q^{17} - 116 q^{19} + 12 q^{21} + 12 q^{23} + 50 q^{25} - 54 q^{27} + 148 q^{29} - 76 q^{31} + 120 q^{33} + 20 q^{35}+ \cdots - 360 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 10 * q^5 - 4 * q^7 + 18 * q^9 - 40 * q^11 - 24 * q^13 + 30 * q^15 + 32 * q^17 - 116 * q^19 + 12 * q^21 + 12 * q^23 + 50 * q^25 - 54 * q^27 + 148 * q^29 - 76 * q^31 + 120 * q^33 + 20 * q^35 + 280 * q^37 + 72 * q^39 + 572 * q^41 - 520 * q^43 - 90 * q^45 + 340 * q^47 + 930 * q^49 - 96 * q^51 + 524 * q^53 + 200 * q^55 + 348 * q^57 + 552 * q^59 + 628 * q^61 - 36 * q^63 + 120 * q^65 - 168 * q^67 - 36 * q^69 + 1248 * q^71 + 908 * q^73 - 150 * q^75 + 80 * q^77 + 1052 * q^79 + 162 * q^81 - 40 * q^83 - 160 * q^85 - 444 * q^87 + 828 * q^89 - 1560 * q^91 + 228 * q^93 + 580 * q^95 - 316 * q^97 - 360 * q^99

Introduction

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