LMFDB - Embedded newform 2400.4.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
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:	no (minimal twist has level 480)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.58872$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -30.3549 q^{7} +9.00000 q^{9} +20.0000 q^{11} -16.3549 q^{13} +69.0647 q^{17} +86.3549 q^{19} +91.0647 q^{21} +34.3549 q^{23} -27.0000 q^{27} -39.4196 q^{29} -217.194 q^{31} -60.0000 q^{33} -281.774 q^{37} +49.0647 q^{39} +342.710 q^{41} -373.420 q^{43} -198.614 q^{47} +578.420 q^{49} -207.194 q^{51} -91.8706 q^{53} -259.065 q^{57} -49.1608 q^{59} -309.808 q^{61} -273.194 q^{63} -651.098 q^{67} -103.065 q^{69} -850.839 q^{71} -964.388 q^{73} -607.098 q^{77} -724.484 q^{79} +81.0000 q^{81} +433.678 q^{83} +118.259 q^{87} +1264.65 q^{89} +496.451 q^{91} +651.582 q^{93} +1745.87 q^{97} +180.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 4 q^{7} + 18 q^{9} + 40 q^{11} + 24 q^{13} - 32 q^{17} + 116 q^{19} + 12 q^{21} + 12 q^{23} - 54 q^{27} + 148 q^{29} + 76 q^{31} - 120 q^{33} - 280 q^{37} - 72 q^{39} + 572 q^{41} - 520 q^{43}+ \cdots + 360 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 4 * q^7 + 18 * q^9 + 40 * q^11 + 24 * q^13 - 32 * q^17 + 116 * q^19 + 12 * q^21 + 12 * q^23 - 54 * q^27 + 148 * q^29 + 76 * q^31 - 120 * q^33 - 280 * q^37 - 72 * q^39 + 572 * q^41 - 520 * q^43 + 340 * q^47 + 930 * q^49 + 96 * q^51 - 524 * q^53 - 348 * q^57 - 552 * q^59 + 628 * q^61 - 36 * q^63 - 168 * q^67 - 36 * q^69 - 1248 * q^71 - 908 * q^73 - 80 * q^77 - 1052 * q^79 + 162 * q^81 - 40 * q^83 - 444 * q^87 + 828 * q^89 + 1560 * q^91 - 228 * q^93 + 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
$5$	0	0
$5$	0	0
$6$	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	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	0	0
$13$	−16.3549	−0.348925	−0.174463	−	0.984664i	$-0.555819\pi$
$13$	−16.3549	−0.348925	−0.174463	+	0.984664i	$0.555819\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	69.0647	0.985332	0.492666	−	0.870218i	$-0.336022\pi$
$17$	69.0647	0.985332	0.492666	+	0.870218i	$0.336022\pi$
$18$	0	0
$19$	86.3549	1.04269	0.521347	−	0.853345i	$-0.325430\pi$
$19$	86.3549	1.04269	0.521347	+	0.853345i	$0.325430\pi$
$20$	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
$25$	0	0
$26$	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	$-0.716609\pi$
$31$	−217.194	−1.25836	−0.629181	+	0.777259i	$0.716609\pi$
$32$	0	0
$33$	−60.0000	−0.316505
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−281.774	−1.25198	−0.625992	−	0.779829i	$-0.715306\pi$
$37$	−281.774	−1.25198	−0.625992	+	0.779829i	$0.715306\pi$
$38$	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
$45$	0	0
$46$	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	$-0.537985\pi$
$53$	−91.8706	−0.238102	−0.119051	+	0.992888i	$0.537985\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−259.065	−0.601999
$58$	0	0
$59$	−49.1608	−0.108478	−0.0542390	−	0.998528i	$-0.517273\pi$
$59$	−49.1608	−0.108478	−0.0542390	+	0.998528i	$0.517273\pi$
$60$	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
$65$	0	0
$66$	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	$-0.751802\pi$
$71$	−850.839	−1.42220	−0.711099	+	0.703092i	$0.751802\pi$
$72$	0	0
$73$	−964.388	−1.54621	−0.773103	−	0.634280i	$-0.781297\pi$
$73$	−964.388	−1.54621	−0.773103	+	0.634280i	$0.781297\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−607.098	−0.898509
$78$	0	0
$79$	−724.484	−1.03178	−0.515891	−	0.856654i	$-0.672539\pi$
$79$	−724.484	−1.03178	−0.515891	+	0.856654i	$0.672539\pi$
$80$	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
$85$	0	0
$86$	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
$95$	0	0
$96$	0	0
$97$	1745.87	1.82749	0.913746	−	0.406287i	$-0.133177\pi$
$97$	1745.87	1.82749	0.913746	+	0.406287i	$0.133177\pi$
$98$	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
$105$	0	0
$106$	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	$-0.571441\pi$
$113$	−534.677	−0.445116	−0.222558	+	0.974919i	$0.571441\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	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	$-0.239596\pi$
$131$	2188.58	1.45968	0.729838	+	0.683621i	$0.239596\pi$
$132$	0	0
$133$	−2621.29	−1.70898
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1896.35	1.18260	0.591301	−	0.806451i	$-0.298614\pi$
$137$	1896.35	1.18260	0.591301	+	0.806451i	$0.298614\pi$
$138$	0	0
$139$	441.128	0.269180	0.134590	−	0.990901i	$-0.457028\pi$
$139$	441.128	0.269180	0.134590	+	0.990901i	$0.457028\pi$
$140$	0	0
$141$	595.841	0.355878
$142$	0	0
$143$	−327.098	−0.191282
$144$	0	0
$145$	0	0
$146$	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	$-0.216708\pi$
$151$	2883.71	1.55413	0.777064	+	0.629421i	$0.216708\pi$
$152$	0	0
$153$	621.582	0.328444
$154$	0	0
$155$	0	0
$156$	0	0
$157$	167.579	0.0851862	0.0425931	−	0.999093i	$-0.486438\pi$
$157$	167.579	0.0851862	0.0425931	+	0.999093i	$0.486438\pi$
$158$	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
$165$	0	0
$166$	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	$-0.128362\pi$
$173$	4185.87	1.83957	0.919786	+	0.392419i	$0.128362\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	147.483	0.0626298
$178$	0	0
$179$	−1930.26	−0.806003	−0.403002	−	0.915199i	$-0.632033\pi$
$179$	−1930.26	−0.806003	−0.403002	+	0.915199i	$0.632033\pi$
$180$	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
$185$	0	0
$186$	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	$-0.842096\pi$
$191$	−4642.97	−1.75892	−0.879460	+	0.475973i	$0.842096\pi$
$192$	0	0
$193$	−4003.23	−1.49305	−0.746526	−	0.665356i	$-0.768280\pi$
$193$	−4003.23	−1.49305	−0.746526	+	0.665356i	$0.768280\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4118.97	1.48967	0.744833	−	0.667250i	$-0.232529\pi$
$197$	4118.97	1.48967	0.744833	+	0.667250i	$0.232529\pi$
$198$	0	0
$199$	−2384.68	−0.849474	−0.424737	−	0.905317i	$-0.639633\pi$
$199$	−2384.68	−0.849474	−0.424737	+	0.905317i	$0.639633\pi$
$200$	0	0
$201$	1953.29	0.685446
$202$	0	0
$203$	1196.58	0.413711
$204$	0	0
$205$	0	0
$206$	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	$-0.431431\pi$
$211$	1310.29	0.427507	0.213754	+	0.976888i	$0.431431\pi$
$212$	0	0
$213$	2552.52	0.821106
$214$	0	0
$215$	0	0
$216$	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
$225$	0	0
$226$	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	$-0.469815\pi$
$233$	673.523	0.189373	0.0946865	+	0.995507i	$0.469815\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2173.45	0.595700
$238$	0	0
$239$	−914.773	−0.247580	−0.123790	−	0.992308i	$-0.539505\pi$
$239$	−914.773	−0.247580	−0.123790	+	0.992308i	$0.539505\pi$
$240$	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
$245$	0	0
$246$	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	$-0.712509\pi$
$251$	−4923.93	−1.23823	−0.619115	+	0.785300i	$0.712509\pi$
$252$	0	0
$253$	687.098	0.170741
$254$	0	0
$255$	0	0
$256$	0	0
$257$	767.771	0.186351	0.0931756	−	0.995650i	$-0.470298\pi$
$257$	767.771	0.186351	0.0931756	+	0.995650i	$0.470298\pi$
$258$	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
$265$	0	0
$266$	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	$-0.358321\pi$
$271$	3841.51	0.861090	0.430545	+	0.902569i	$0.358321\pi$
$272$	0	0
$273$	−1489.35	−0.330182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8471.07	1.83746	0.918732	−	0.394881i	$-0.129214\pi$
$277$	8471.07	1.83746	0.918732	+	0.394881i	$0.129214\pi$
$278$	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
$285$	0	0
$286$	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	$-0.711323\pi$
$293$	−6180.79	−1.23237	−0.616187	+	0.787600i	$0.711323\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	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	$-0.751625\pi$
$311$	−7795.82	−1.42142	−0.710708	+	0.703487i	$0.751625\pi$
$312$	0	0
$313$	8138.34	1.46967	0.734834	−	0.678247i	$-0.237260\pi$
$313$	8138.34	1.46967	0.734834	+	0.678247i	$0.237260\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6911.10	1.22450	0.612249	−	0.790665i	$-0.290265\pi$
$317$	6911.10	1.22450	0.612249	+	0.790665i	$0.290265\pi$
$318$	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
$325$	0	0
$326$	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	$-0.681791\pi$
$331$	−6510.62	−1.08114	−0.540568	+	0.841300i	$0.681791\pi$
$332$	0	0
$333$	−2535.97	−0.417328
$334$	0	0
$335$	0	0
$336$	0	0
$337$	247.990	0.0400856	0.0200428	−	0.999799i	$-0.493620\pi$
$337$	247.990	0.0400856	0.0200428	+	0.999799i	$0.493620\pi$
$338$	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
$345$	0	0
$346$	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	$-0.933328\pi$
$353$	−12974.6	−1.95629	−0.978144	+	0.207929i	$0.933328\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6289.35	0.932403
$358$	0	0
$359$	5776.40	0.849211	0.424605	−	0.905379i	$-0.360413\pi$
$359$	5776.40	0.849211	0.424605	+	0.905379i	$0.360413\pi$
$360$	0	0
$361$	598.168	0.0872092
$362$	0	0
$363$	2793.00	0.403842
$364$	0	0
$365$	0	0
$366$	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	$-0.648683\pi$
$373$	−6487.77	−0.900601	−0.450300	+	0.892877i	$0.648683\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	644.703	0.0880740
$378$	0	0
$379$	1464.68	0.198510	0.0992551	−	0.995062i	$-0.468354\pi$
$379$	1464.68	0.198510	0.0992551	+	0.995062i	$0.468354\pi$
$380$	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
$385$	0	0
$386$	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
$395$	0	0
$396$	0	0
$397$	−1601.77	−0.202496	−0.101248	−	0.994861i	$-0.532283\pi$
$397$	−1601.77	−0.202496	−0.101248	+	0.994861i	$0.532283\pi$
$398$	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
$405$	0	0
$406$	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
$415$	0	0
$416$	0	0
$417$	−1323.38	−0.155411
$418$	0	0
$419$	8858.90	1.03290	0.516451	−	0.856317i	$-0.327253\pi$
$419$	8858.90	1.03290	0.516451	+	0.856317i	$0.327253\pi$
$420$	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
$425$	0	0
$426$	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	$-0.260664\pi$
$431$	12223.1	1.36605	0.683024	+	0.730396i	$0.260664\pi$
$432$	0	0
$433$	−110.192	−0.0122298	−0.00611490	−	0.999981i	$-0.501946\pi$
$433$	−110.192	−0.0122298	−0.00611490	+	0.999981i	$0.501946\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2966.71	0.324753
$438$	0	0
$439$	−10838.5	−1.17834	−0.589172	−	0.808008i	$-0.700546\pi$
$439$	−10838.5	−1.17834	−0.589172	+	0.808008i	$0.700546\pi$
$440$	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
$445$	0	0
$446$	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
$455$	0	0
$456$	0	0
$457$	1352.31	0.138421	0.0692104	−	0.997602i	$-0.477952\pi$
$457$	1352.31	0.138421	0.0692104	+	0.997602i	$0.477952\pi$
$458$	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
$465$	0	0
$466$	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
$475$	0	0
$476$	0	0
$477$	−826.836	−0.0793673
$478$	0	0
$479$	−6270.21	−0.598107	−0.299053	−	0.954236i	$-0.596671\pi$
$479$	−6270.21	−0.598107	−0.299053	+	0.954236i	$0.596671\pi$
$480$	0	0
$481$	4608.39	0.436849
$482$	0	0
$483$	3128.52	0.294726
$484$	0	0
$485$	0	0
$486$	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	$-0.578342\pi$
$491$	−5301.54	−0.487281	−0.243641	+	0.969866i	$0.578342\pi$
$492$	0	0
$493$	−2722.50	−0.248712
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25827.1	2.33100
$498$	0	0
$499$	−12167.0	−1.09152	−0.545762	−	0.837940i	$-0.683760\pi$
$499$	−12167.0	−1.09152	−0.545762	+	0.837940i	$0.683760\pi$
$500$	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
$505$	0	0
$506$	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
$515$	0	0
$516$	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
$525$	0	0
$526$	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
$535$	0	0
$536$	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
$545$	0	0
$546$	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
$555$	0	0
$556$	0	0
$557$	−11820.0	−0.899153	−0.449576	−	0.893242i	$-0.648425\pi$
$557$	−11820.0	−0.899153	−0.449576	+	0.893242i	$0.648425\pi$
$558$	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
$565$	0	0
$566$	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	$-0.466607\pi$
$571$	2857.53	0.209429	0.104714	+	0.994502i	$0.466607\pi$
$572$	0	0
$573$	13928.9	1.01551
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10000.7	0.721553	0.360776	−	0.932652i	$-0.382512\pi$
$577$	10000.7	0.721553	0.360776	+	0.932652i	$0.382512\pi$
$578$	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
$585$	0	0
$586$	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	$-0.455822\pi$
$593$	3995.50	0.276687	0.138344	+	0.990384i	$0.455822\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7154.03	0.490444
$598$	0	0
$599$	−7844.51	−0.535089	−0.267544	−	0.963546i	$-0.586212\pi$
$599$	−7844.51	−0.535089	−0.267544	+	0.963546i	$0.586212\pi$
$600$	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
$605$	0	0
$606$	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	$-0.452229\pi$
$613$	4538.37	0.299026	0.149513	+	0.988760i	$0.452229\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1715.15	−0.111911	−0.0559556	−	0.998433i	$-0.517821\pi$
$617$	−1715.15	−0.111911	−0.0559556	+	0.998433i	$0.517821\pi$
$618$	0	0
$619$	−695.376	−0.0451527	−0.0225764	−	0.999745i	$-0.507187\pi$
$619$	−695.376	−0.0451527	−0.0225764	+	0.999745i	$0.507187\pi$
$620$	0	0
$621$	−927.582	−0.0599398
$622$	0	0
$623$	−38388.2	−2.46869
$624$	0	0
$625$	0	0
$626$	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	$-0.399138\pi$
$631$	9877.82	0.623185	0.311593	+	0.950216i	$0.399138\pi$
$632$	0	0
$633$	−3930.87	−0.246821
$634$	0	0
$635$	0	0
$636$	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
$645$	0	0
$646$	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	$-0.420869\pi$
$653$	8211.29	0.492087	0.246043	+	0.969259i	$0.420869\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8679.49	−0.515402
$658$	0	0
$659$	22558.3	1.33346	0.666728	−	0.745301i	$-0.267694\pi$
$659$	22558.3	1.33346	0.666728	+	0.745301i	$0.267694\pi$
$660$	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
$665$	0	0
$666$	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	$-0.613346\pi$
$673$	−12172.8	−0.697218	−0.348609	+	0.937268i	$0.613346\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29440.0	1.67130	0.835650	−	0.549263i	$-0.185091\pi$
$677$	29440.0	1.67130	0.835650	+	0.549263i	$0.185091\pi$
$678$	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
$685$	0	0
$686$	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	$-0.519980\pi$
$691$	−2278.77	−0.125454	−0.0627268	+	0.998031i	$0.519980\pi$
$692$	0	0
$693$	−5463.88	−0.299503
$694$	0	0
$695$	0	0
$696$	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
$705$	0	0
$706$	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
$715$	0	0
$716$	0	0
$717$	2744.32	0.142941
$718$	0	0
$719$	−12797.7	−0.663804	−0.331902	−	0.943314i	$-0.607690\pi$
$719$	−12797.7	−0.663804	−0.331902	+	0.943314i	$0.607690\pi$
$720$	0	0
$721$	−7178.45	−0.370790
$722$	0	0
$723$	16549.0	0.851263
$724$	0	0
$725$	0	0
$726$	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	$-0.465263\pi$
$733$	4322.79	0.217825	0.108913	+	0.994051i	$0.465263\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13022.0	−0.650841
$738$	0	0
$739$	24280.6	1.20863	0.604315	−	0.796745i	$-0.293447\pi$
$739$	24280.6	1.20863	0.604315	+	0.796745i	$0.293447\pi$
$740$	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
$745$	0	0
$746$	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	$-0.592694\pi$
$751$	−11817.8	−0.574216	−0.287108	+	0.957898i	$0.592694\pi$
$752$	0	0
$753$	14771.8	0.714893
$754$	0	0
$755$	0	0
$756$	0	0
$757$	311.180	0.0149406	0.00747031	−	0.999972i	$-0.497622\pi$
$757$	311.180	0.0149406	0.00747031	+	0.999972i	$0.497622\pi$
$758$	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
$765$	0	0
$766$	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	$-0.301052\pi$
$773$	25149.8	1.17022	0.585108	+	0.810955i	$0.301052\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	0	0
$786$	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
$795$	0	0
$796$	0	0
$797$	−34617.1	−1.53852	−0.769260	−	0.638936i	$-0.779375\pi$
$797$	−34617.1	−1.53852	−0.769260	+	0.638936i	$0.779375\pi$
$798$	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
$805$	0	0
$806$	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	$-0.212935\pi$
$811$	36235.7	1.56894	0.784469	+	0.620168i	$0.212935\pi$
$812$	0	0
$813$	−11524.5	−0.497150
$814$	0	0
$815$	0	0
$816$	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
$825$	0	0
$826$	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
$835$	0	0
$836$	0	0
$837$	5864.24	0.242172
$838$	0	0
$839$	4074.61	0.167665	0.0838327	−	0.996480i	$-0.473284\pi$
$839$	4074.61	0.167665	0.0838327	+	0.996480i	$0.473284\pi$
$840$	0	0
$841$	−22835.1	−0.936287
$842$	0	0
$843$	−11793.3	−0.481830
$844$	0	0
$845$	0	0
$846$	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	$-0.872841\pi$
$853$	−45902.6	−1.84253	−0.921263	+	0.388941i	$0.872841\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	17316.3	0.690216	0.345108	−	0.938563i	$-0.387842\pi$
$857$	17316.3	0.690216	0.345108	+	0.938563i	$0.387842\pi$
$858$	0	0
$859$	−14329.6	−0.569171	−0.284586	−	0.958651i	$-0.591856\pi$
$859$	−14329.6	−0.569171	−0.284586	+	0.958651i	$0.591856\pi$
$860$	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
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	30048.4	1.15697	0.578485	−	0.815693i	$-0.303644\pi$
$877$	30048.4	1.15697	0.578485	+	0.815693i	$0.303644\pi$
$878$	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
$885$	0	0
$886$	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
$895$	0	0
$896$	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
$905$	0	0
$906$	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	$-0.421539\pi$
$911$	13418.4	0.488005	0.244003	+	0.969775i	$0.421539\pi$
$912$	0	0
$913$	8673.57	0.314407
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−66434.2	−2.39242
$918$	0	0
$919$	−17684.3	−0.634768	−0.317384	−	0.948297i	$-0.602804\pi$
$919$	−17684.3	−0.634768	−0.317384	+	0.948297i	$0.602804\pi$
$920$	0	0
$921$	15176.7	0.542986
$922$	0	0
$923$	13915.4	0.496241
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	−41657.0	−1.45237	−0.726187	−	0.687497i	$-0.758709\pi$
$937$	−41657.0	−1.45237	−0.726187	+	0.687497i	$0.758709\pi$
$938$	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
$945$	0	0
$946$	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	$-0.528397\pi$
$953$	−5242.25	−0.178188	−0.0890939	+	0.996023i	$0.528397\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	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	$-0.789623\pi$
$971$	−47771.8	−1.57886	−0.789428	+	0.613843i	$0.789623\pi$
$972$	0	0
$973$	−13390.4	−0.441188
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−49812.5	−1.63116	−0.815581	−	0.578644i	$-0.803582\pi$
$977$	−49812.5	−1.63116	−0.815581	+	0.578644i	$0.803582\pi$
$978$	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
$985$	0	0
$986$	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	$-0.231914\pi$
$991$	46553.2	1.49224	0.746121	+	0.665811i	$0.231914\pi$
$992$	0	0
$993$	19531.9	0.624194
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−17111.5	−0.543557	−0.271778	−	0.962360i	$-0.587612\pi$
$997$	−17111.5	−0.543557	−0.271778	+	0.962360i	$0.587612\pi$
$998$	0	0
$999$	7607.91	0.240945

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.n.1.1	✓	2	20.19	odd	2
480.4.a.p.1.2	yes	2	5.4	even	2
960.4.a.bl.1.2		2	40.29	even	2
960.4.a.bp.1.1		2	40.19	odd	2
1440.4.a.ba.1.1		2	60.59	even	2
1440.4.a.bf.1.2		2	15.14	odd	2
2400.4.a.z.1.1		2	1.1	even	1	trivial
2400.4.a.ba.1.2		2	4.3	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

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

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