LMFDB - Embedded newform 1440.4.a.bk.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,4,Mod(1,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,15,0,14,0,0,0,22,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$84.9627504083$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.16773.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 17x + 18 $ x^3 - x^2 - 17*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.06301$ of defining polynomial
Character	$\chi$	$=$	1440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +1.74795 q^{7} -28.9759 q^{11} -48.2362 q^{13} -16.2198 q^{17} -130.692 q^{19} +182.188 q^{23} +25.0000 q^{25} -291.384 q^{29} +219.636 q^{31} +8.73976 q^{35} +436.044 q^{37} +339.840 q^{41} +316.505 q^{43} -335.605 q^{47} -339.945 q^{49} +520.736 q^{53} -144.880 q^{55} +589.386 q^{59} -566.087 q^{61} -241.181 q^{65} +407.400 q^{67} +486.328 q^{71} +143.417 q^{73} -50.6485 q^{77} +968.012 q^{79} -532.718 q^{83} -81.0989 q^{85} -67.8074 q^{89} -84.3145 q^{91} -653.461 q^{95} +218.087 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} + 14 q^{7} + 22 q^{11} + 8 q^{13} + 34 q^{17} - 4 q^{19} + 176 q^{23} + 75 q^{25} - 98 q^{29} + 88 q^{31} + 70 q^{35} + 284 q^{37} + 8 q^{41} + 504 q^{43} + 280 q^{47} - 409 q^{49} + 150 q^{53}+ \cdots - 1394 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 + 14 * q^7 + 22 * q^11 + 8 * q^13 + 34 * q^17 - 4 * q^19 + 176 * q^23 + 75 * q^25 - 98 * q^29 + 88 * q^31 + 70 * q^35 + 284 * q^37 + 8 * q^41 + 504 * q^43 + 280 * q^47 - 409 * q^49 + 150 * q^53 + 110 * q^55 + 350 * q^59 + 350 * q^61 + 40 * q^65 + 804 * q^67 + 500 * q^71 - 486 * q^73 - 788 * q^77 + 1592 * q^79 - 684 * q^83 + 170 * q^85 + 668 * q^89 + 1920 * q^91 - 20 * q^95 - 1394 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	1.74795	0.0943805	0.0471903	−	0.998886i	$-0.484973\pi$
$7$	1.74795	0.0943805	0.0471903	+	0.998886i	$0.484973\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−28.9759	−0.794234	−0.397117	−	0.917768i	$-0.629989\pi$
$11$	−28.9759	−0.794234	−0.397117	+	0.917768i	$0.629989\pi$
$12$	0	0
$13$	−48.2362	−1.02910	−0.514550	−	0.857460i	$-0.672041\pi$
$13$	−48.2362	−1.02910	−0.514550	+	0.857460i	$0.672041\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−16.2198	−0.231404	−0.115702	−	0.993284i	$-0.536912\pi$
$17$	−16.2198	−0.231404	−0.115702	+	0.993284i	$0.536912\pi$
$18$	0	0
$19$	−130.692	−1.57804	−0.789022	−	0.614365i	$-0.789412\pi$
$19$	−130.692	−1.57804	−0.789022	+	0.614365i	$0.789412\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	182.188	1.65169	0.825844	−	0.563899i	$-0.190699\pi$
$23$	182.188	1.65169	0.825844	+	0.563899i	$0.190699\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−291.384	−1.86582	−0.932909	−	0.360113i	$-0.882738\pi$
$29$	−291.384	−1.86582	−0.932909	+	0.360113i	$0.882738\pi$
$30$	0	0
$31$	219.636	1.27251	0.636254	−	0.771480i	$-0.280483\pi$
$31$	219.636	1.27251	0.636254	+	0.771480i	$0.280483\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.73976	0.0422083
$36$	0	0
$37$	436.044	1.93744	0.968718	−	0.248165i	$-0.0798274\pi$
$37$	436.044	1.93744	0.968718	+	0.248165i	$0.0798274\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	339.840	1.29449	0.647245	−	0.762282i	$-0.275921\pi$
$41$	339.840	1.29449	0.647245	+	0.762282i	$0.275921\pi$
$42$	0	0
$43$	316.505	1.12248	0.561239	−	0.827654i	$-0.310325\pi$
$43$	316.505	1.12248	0.561239	+	0.827654i	$0.310325\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−335.605	−1.04155	−0.520777	−	0.853693i	$-0.674358\pi$
$47$	−335.605	−1.04155	−0.520777	+	0.853693i	$0.674358\pi$
$48$	0	0
$49$	−339.945	−0.991092
$50$	0	0
$51$	0	0
$52$	0	0
$53$	520.736	1.34960	0.674798	−	0.738003i	$-0.264231\pi$
$53$	520.736	1.34960	0.674798	+	0.738003i	$0.264231\pi$
$54$	0	0
$55$	−144.880	−0.355192
$56$	0	0
$57$	0	0
$58$	0	0
$59$	589.386	1.30053	0.650267	−	0.759706i	$-0.274657\pi$
$59$	589.386	1.30053	0.650267	+	0.759706i	$0.274657\pi$
$60$	0	0
$61$	−566.087	−1.18820	−0.594099	−	0.804392i	$-0.702491\pi$
$61$	−566.087	−1.18820	−0.594099	+	0.804392i	$0.702491\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−241.181	−0.460228
$66$	0	0
$67$	407.400	0.742862	0.371431	−	0.928460i	$-0.378867\pi$
$67$	407.400	0.742862	0.371431	+	0.928460i	$0.378867\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	486.328	0.812908	0.406454	−	0.913671i	$-0.366765\pi$
$71$	486.328	0.812908	0.406454	+	0.913671i	$0.366765\pi$
$72$	0	0
$73$	143.417	0.229941	0.114970	−	0.993369i	$-0.463323\pi$
$73$	143.417	0.229941	0.114970	+	0.993369i	$0.463323\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−50.6485	−0.0749602
$78$	0	0
$79$	968.012	1.37861	0.689303	−	0.724473i	$-0.257917\pi$
$79$	968.012	1.37861	0.689303	+	0.724473i	$0.257917\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−532.718	−0.704499	−0.352250	−	0.935906i	$-0.614583\pi$
$83$	−532.718	−0.704499	−0.352250	+	0.935906i	$0.614583\pi$
$84$	0	0
$85$	−81.0989	−0.103487
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−67.8074	−0.0807592	−0.0403796	−	0.999184i	$-0.512857\pi$
$89$	−67.8074	−0.0807592	−0.0403796	+	0.999184i	$0.512857\pi$
$90$	0	0
$91$	−84.3145	−0.0971271
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−653.461	−0.705723
$96$	0	0
$97$	218.087	0.228282	0.114141	−	0.993465i	$-0.463588\pi$
$97$	218.087	0.228282	0.114141	+	0.993465i	$0.463588\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1191.31	−1.17366	−0.586829	−	0.809711i	$-0.699624\pi$
$101$	−1191.31	−1.17366	−0.586829	+	0.809711i	$0.699624\pi$
$102$	0	0
$103$	−530.052	−0.507064	−0.253532	−	0.967327i	$-0.581592\pi$
$103$	−530.052	−0.507064	−0.253532	+	0.967327i	$0.581592\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−658.684	−0.595116	−0.297558	−	0.954704i	$-0.596172\pi$
$107$	−658.684	−0.595116	−0.297558	+	0.954704i	$0.596172\pi$
$108$	0	0
$109$	1686.34	1.48185	0.740926	−	0.671587i	$-0.234387\pi$
$109$	1686.34	1.48185	0.740926	+	0.671587i	$0.234387\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−99.1521	−0.0825437	−0.0412719	−	0.999148i	$-0.513141\pi$
$113$	−99.1521	−0.0825437	−0.0412719	+	0.999148i	$0.513141\pi$
$114$	0	0
$115$	910.940	0.738657
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−28.3514	−0.0218401
$120$	0	0
$121$	−491.396	−0.369193
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1588.17	1.10966	0.554832	−	0.831962i	$-0.312782\pi$
$127$	1588.17	1.10966	0.554832	+	0.831962i	$0.312782\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−194.359	−0.129628	−0.0648139	−	0.997897i	$-0.520645\pi$
$131$	−194.359	−0.129628	−0.0648139	+	0.997897i	$0.520645\pi$
$132$	0	0
$133$	−228.444	−0.148937
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1100.45	0.686261	0.343130	−	0.939288i	$-0.388513\pi$
$137$	1100.45	0.686261	0.343130	+	0.939288i	$0.388513\pi$
$138$	0	0
$139$	2353.62	1.43620	0.718099	−	0.695941i	$-0.245012\pi$
$139$	2353.62	1.43620	0.718099	+	0.695941i	$0.245012\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1397.69	0.817346
$144$	0	0
$145$	−1456.92	−0.834419
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1186.36	0.652284	0.326142	−	0.945321i	$-0.394251\pi$
$149$	1186.36	0.652284	0.326142	+	0.945321i	$0.394251\pi$
$150$	0	0
$151$	3235.58	1.74376	0.871880	−	0.489720i	$-0.162901\pi$
$151$	3235.58	1.74376	0.871880	+	0.489720i	$0.162901\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1098.18	0.569083
$156$	0	0
$157$	−1651.82	−0.839680	−0.419840	−	0.907598i	$-0.637914\pi$
$157$	−1651.82	−0.839680	−0.419840	+	0.907598i	$0.637914\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	318.456	0.155887
$162$	0	0
$163$	3838.72	1.84461	0.922305	−	0.386462i	$-0.126303\pi$
$163$	3838.72	1.84461	0.922305	+	0.386462i	$0.126303\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2100.49	−0.973299	−0.486650	−	0.873597i	$-0.661781\pi$
$167$	−2100.49	−0.973299	−0.486650	+	0.873597i	$0.661781\pi$
$168$	0	0
$169$	129.728	0.0590477
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3582.79	−1.57453	−0.787266	−	0.616613i	$-0.788504\pi$
$173$	−3582.79	−1.57453	−0.787266	+	0.616613i	$0.788504\pi$
$174$	0	0
$175$	43.6988	0.0188761
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2433.21	−1.01601	−0.508007	−	0.861353i	$-0.669618\pi$
$179$	−2433.21	−1.01601	−0.508007	+	0.861353i	$0.669618\pi$
$180$	0	0
$181$	572.506	0.235105	0.117553	−	0.993067i	$-0.462495\pi$
$181$	572.506	0.235105	0.117553	+	0.993067i	$0.462495\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2180.22	0.866448
$186$	0	0
$187$	469.983	0.183789
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−491.456	−0.186181	−0.0930903	−	0.995658i	$-0.529675\pi$
$191$	−491.456	−0.186181	−0.0930903	+	0.995658i	$0.529675\pi$
$192$	0	0
$193$	2798.12	1.04359	0.521795	−	0.853071i	$-0.325263\pi$
$193$	2798.12	1.04359	0.521795	+	0.853071i	$0.325263\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3111.06	−1.12515	−0.562574	−	0.826747i	$-0.690189\pi$
$197$	−3111.06	−1.12515	−0.562574	+	0.826747i	$0.690189\pi$
$198$	0	0
$199$	33.7597	0.0120259	0.00601297	−	0.999982i	$-0.498086\pi$
$199$	33.7597	0.0120259	0.00601297	+	0.999982i	$0.498086\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−509.326	−0.176097
$204$	0	0
$205$	1699.20	0.578914
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3786.93	1.25334
$210$	0	0
$211$	2956.21	0.964521	0.482260	−	0.876028i	$-0.339816\pi$
$211$	2956.21	0.964521	0.482260	+	0.876028i	$0.339816\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1582.53	0.501988
$216$	0	0
$217$	383.913	0.120100
$218$	0	0
$219$	0	0
$220$	0	0
$221$	782.380	0.238138
$222$	0	0
$223$	6111.29	1.83517	0.917583	−	0.397544i	$-0.130138\pi$
$223$	6111.29	1.83517	0.917583	+	0.397544i	$0.130138\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5529.84	1.61687	0.808433	−	0.588588i	$-0.200316\pi$
$227$	5529.84	1.61687	0.808433	+	0.588588i	$0.200316\pi$
$228$	0	0
$229$	−2033.37	−0.586765	−0.293382	−	0.955995i	$-0.594781\pi$
$229$	−2033.37	−0.586765	−0.293382	+	0.955995i	$0.594781\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4598.67	1.29300	0.646500	−	0.762914i	$-0.276232\pi$
$233$	4598.67	1.29300	0.646500	+	0.762914i	$0.276232\pi$
$234$	0	0
$235$	−1678.03	−0.465797
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1866.96	0.505286	0.252643	−	0.967560i	$-0.418700\pi$
$239$	1866.96	0.505286	0.252643	+	0.967560i	$0.418700\pi$
$240$	0	0
$241$	1313.63	0.351112	0.175556	−	0.984469i	$-0.443828\pi$
$241$	1313.63	0.351112	0.175556	+	0.984469i	$0.443828\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1699.72	−0.443230
$246$	0	0
$247$	6304.09	1.62397
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4412.72	1.10967	0.554837	−	0.831959i	$-0.312781\pi$
$251$	4412.72	1.10967	0.554837	+	0.831959i	$0.312781\pi$
$252$	0	0
$253$	−5279.07	−1.31183
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−858.193	−0.208298	−0.104149	−	0.994562i	$-0.533212\pi$
$257$	−858.193	−0.208298	−0.104149	+	0.994562i	$0.533212\pi$
$258$	0	0
$259$	762.183	0.182856
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5550.09	−1.30127	−0.650634	−	0.759392i	$-0.725497\pi$
$263$	−5550.09	−1.30127	−0.650634	+	0.759392i	$0.725497\pi$
$264$	0	0
$265$	2603.68	0.603557
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7497.76	−1.69943	−0.849715	−	0.527242i	$-0.823226\pi$
$269$	−7497.76	−1.69943	−0.849715	+	0.527242i	$0.823226\pi$
$270$	0	0
$271$	−2085.27	−0.467421	−0.233711	−	0.972306i	$-0.575087\pi$
$271$	−2085.27	−0.467421	−0.233711	+	0.972306i	$0.575087\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−724.398	−0.158847
$276$	0	0
$277$	3545.01	0.768949	0.384474	−	0.923136i	$-0.374383\pi$
$277$	3545.01	0.768949	0.384474	+	0.923136i	$0.374383\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	791.597	0.168052	0.0840262	−	0.996464i	$-0.473222\pi$
$281$	791.597	0.168052	0.0840262	+	0.996464i	$0.473222\pi$
$282$	0	0
$283$	−807.921	−0.169703	−0.0848514	−	0.996394i	$-0.527042\pi$
$283$	−807.921	−0.169703	−0.0848514	+	0.996394i	$0.527042\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	594.024	0.122175
$288$	0	0
$289$	−4649.92	−0.946452
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2084.86	−0.415696	−0.207848	−	0.978161i	$-0.566646\pi$
$293$	−2084.86	−0.415696	−0.207848	+	0.978161i	$0.566646\pi$
$294$	0	0
$295$	2946.93	0.581616
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8788.05	−1.69975
$300$	0	0
$301$	553.236	0.105940
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2830.44	−0.531378
$306$	0	0
$307$	5952.64	1.10663	0.553315	−	0.832972i	$-0.313363\pi$
$307$	5952.64	1.10663	0.553315	+	0.832972i	$0.313363\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8347.40	1.52199	0.760993	−	0.648760i	$-0.224712\pi$
$311$	8347.40	1.52199	0.760993	+	0.648760i	$0.224712\pi$
$312$	0	0
$313$	−2717.74	−0.490786	−0.245393	−	0.969424i	$-0.578917\pi$
$313$	−2717.74	−0.490786	−0.245393	+	0.969424i	$0.578917\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4175.01	0.739722	0.369861	−	0.929087i	$-0.379405\pi$
$317$	4175.01	0.739722	0.369861	+	0.929087i	$0.379405\pi$
$318$	0	0
$319$	8443.13	1.48189
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2119.80	0.365166
$324$	0	0
$325$	−1205.90	−0.205820
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−586.622	−0.0983024
$330$	0	0
$331$	9455.96	1.57023	0.785115	−	0.619350i	$-0.212604\pi$
$331$	9455.96	1.57023	0.785115	+	0.619350i	$0.212604\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2037.00	0.332218
$336$	0	0
$337$	−3821.38	−0.617698	−0.308849	−	0.951111i	$-0.599944\pi$
$337$	−3821.38	−0.617698	−0.308849	+	0.951111i	$0.599944\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6364.15	−1.01067
$342$	0	0
$343$	−1193.75	−0.187920
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8406.55	−1.30054	−0.650270	−	0.759704i	$-0.725344\pi$
$347$	−8406.55	−1.30054	−0.650270	+	0.759704i	$0.725344\pi$
$348$	0	0
$349$	−2414.18	−0.370282	−0.185141	−	0.982712i	$-0.559274\pi$
$349$	−2414.18	−0.370282	−0.185141	+	0.982712i	$0.559274\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7565.70	1.14074	0.570370	−	0.821388i	$-0.306800\pi$
$353$	7565.70	1.14074	0.570370	+	0.821388i	$0.306800\pi$
$354$	0	0
$355$	2431.64	0.363544
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−763.942	−0.112310	−0.0561550	−	0.998422i	$-0.517884\pi$
$359$	−763.942	−0.112310	−0.0561550	+	0.998422i	$0.517884\pi$
$360$	0	0
$361$	10221.4	1.49022
$362$	0	0
$363$	0	0
$364$	0	0
$365$	717.085	0.102833
$366$	0	0
$367$	−12441.4	−1.76959	−0.884793	−	0.465984i	$-0.845701\pi$
$367$	−12441.4	−1.76959	−0.884793	+	0.465984i	$0.845701\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	910.221	0.127376
$372$	0	0
$373$	8266.81	1.14756	0.573779	−	0.819010i	$-0.305477\pi$
$373$	8266.81	1.14756	0.573779	+	0.819010i	$0.305477\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14055.3	1.92011
$378$	0	0
$379$	5511.15	0.746936	0.373468	−	0.927643i	$-0.378169\pi$
$379$	5511.15	0.746936	0.373468	+	0.927643i	$0.378169\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2056.33	0.274344	0.137172	−	0.990547i	$-0.456199\pi$
$383$	2056.33	0.274344	0.137172	+	0.990547i	$0.456199\pi$
$384$	0	0
$385$	−253.243	−0.0335232
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5799.26	−0.755872	−0.377936	−	0.925832i	$-0.623366\pi$
$389$	−5799.26	−0.755872	−0.377936	+	0.925832i	$0.623366\pi$
$390$	0	0
$391$	−2955.05	−0.382208
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4840.06	0.616531
$396$	0	0
$397$	−2929.02	−0.370285	−0.185143	−	0.982712i	$-0.559275\pi$
$397$	−2929.02	−0.370285	−0.185143	+	0.982712i	$0.559275\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13545.0	1.68680	0.843399	−	0.537288i	$-0.180551\pi$
$401$	13545.0	1.68680	0.843399	+	0.537288i	$0.180551\pi$
$402$	0	0
$403$	−10594.4	−1.30954
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12634.8	−1.53878
$408$	0	0
$409$	10860.3	1.31298	0.656488	−	0.754337i	$-0.272041\pi$
$409$	10860.3	1.31298	0.656488	+	0.754337i	$0.272041\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1030.22	0.122745
$414$	0	0
$415$	−2663.59	−0.315062
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7253.42	0.845711	0.422856	−	0.906197i	$-0.361028\pi$
$419$	7253.42	0.845711	0.422856	+	0.906197i	$0.361028\pi$
$420$	0	0
$421$	−4768.55	−0.552030	−0.276015	−	0.961153i	$-0.589014\pi$
$421$	−4768.55	−0.552030	−0.276015	+	0.961153i	$0.589014\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−405.495	−0.0462809
$426$	0	0
$427$	−989.493	−0.112143
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6035.15	−0.674485	−0.337242	−	0.941418i	$-0.609494\pi$
$431$	−6035.15	−0.674485	−0.337242	+	0.941418i	$0.609494\pi$
$432$	0	0
$433$	−2347.84	−0.260577	−0.130289	−	0.991476i	$-0.541590\pi$
$433$	−2347.84	−0.260577	−0.130289	+	0.991476i	$0.541590\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−23810.5	−2.60644
$438$	0	0
$439$	−8024.30	−0.872389	−0.436195	−	0.899852i	$-0.643674\pi$
$439$	−8024.30	−0.872389	−0.436195	+	0.899852i	$0.643674\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1463.73	0.156984	0.0784922	−	0.996915i	$-0.474989\pi$
$443$	1463.73	0.156984	0.0784922	+	0.996915i	$0.474989\pi$
$444$	0	0
$445$	−339.037	−0.0361166
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7194.97	0.756240	0.378120	−	0.925757i	$-0.376571\pi$
$449$	7194.97	0.756240	0.378120	+	0.925757i	$0.376571\pi$
$450$	0	0
$451$	−9847.18	−1.02813
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−421.573	−0.0434365
$456$	0	0
$457$	−13383.9	−1.36996	−0.684979	−	0.728563i	$-0.740189\pi$
$457$	−13383.9	−1.36996	−0.684979	+	0.728563i	$0.740189\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10270.0	−1.03757	−0.518786	−	0.854904i	$-0.673616\pi$
$461$	−10270.0	−1.03757	−0.518786	+	0.854904i	$0.673616\pi$
$462$	0	0
$463$	7562.55	0.759096	0.379548	−	0.925172i	$-0.376080\pi$
$463$	7562.55	0.759096	0.379548	+	0.925172i	$0.376080\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18059.3	−1.78947	−0.894736	−	0.446596i	$-0.852636\pi$
$467$	−18059.3	−1.78947	−0.894736	+	0.446596i	$0.852636\pi$
$468$	0	0
$469$	712.115	0.0701118
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9171.03	−0.891510
$474$	0	0
$475$	−3267.30	−0.315609
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7023.57	−0.669969	−0.334985	−	0.942224i	$-0.608731\pi$
$479$	−7023.57	−0.669969	−0.334985	+	0.942224i	$0.608731\pi$
$480$	0	0
$481$	−21033.1	−1.99382
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1090.44	0.102091
$486$	0	0
$487$	6354.87	0.591307	0.295653	−	0.955295i	$-0.404463\pi$
$487$	6354.87	0.591307	0.295653	+	0.955295i	$0.404463\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19124.1	1.75775	0.878877	−	0.477049i	$-0.158294\pi$
$491$	19124.1	1.75775	0.878877	+	0.477049i	$0.158294\pi$
$492$	0	0
$493$	4726.19	0.431758
$494$	0	0
$495$	0	0
$496$	0	0
$497$	850.078	0.0767227
$498$	0	0
$499$	−19093.0	−1.71287	−0.856433	−	0.516258i	$-0.827325\pi$
$499$	−19093.0	−1.71287	−0.856433	+	0.516258i	$0.827325\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5689.49	0.504338	0.252169	−	0.967683i	$-0.418856\pi$
$503$	5689.49	0.504338	0.252169	+	0.967683i	$0.418856\pi$
$504$	0	0
$505$	−5956.54	−0.524876
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3732.80	0.325056	0.162528	−	0.986704i	$-0.448035\pi$
$509$	3732.80	0.325056	0.162528	+	0.986704i	$0.448035\pi$
$510$	0	0
$511$	250.686	0.0217020
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2650.26	−0.226766
$516$	0	0
$517$	9724.47	0.827237
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9445.51	0.794271	0.397136	−	0.917760i	$-0.370004\pi$
$521$	9445.51	0.794271	0.397136	+	0.917760i	$0.370004\pi$
$522$	0	0
$523$	3021.83	0.252649	0.126325	−	0.991989i	$-0.459682\pi$
$523$	3021.83	0.252649	0.126325	+	0.991989i	$0.459682\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3562.44	−0.294464
$528$	0	0
$529$	21025.5	1.72807
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−16392.6	−1.33216
$534$	0	0
$535$	−3293.42	−0.266144
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9850.21	0.787159
$540$	0	0
$541$	16850.9	1.33915	0.669574	−	0.742745i	$-0.266477\pi$
$541$	16850.9	1.33915	0.669574	+	0.742745i	$0.266477\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8431.69	0.662704
$546$	0	0
$547$	−3424.29	−0.267664	−0.133832	−	0.991004i	$-0.542728\pi$
$547$	−3424.29	−0.267664	−0.133832	+	0.991004i	$0.542728\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	38081.6	2.94434
$552$	0	0
$553$	1692.04	0.130114
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2100.17	−0.159761	−0.0798807	−	0.996804i	$-0.525454\pi$
$557$	−2100.17	−0.159761	−0.0798807	+	0.996804i	$0.525454\pi$
$558$	0	0
$559$	−15267.0	−1.15514
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20424.7	−1.52895	−0.764475	−	0.644654i	$-0.777002\pi$
$563$	−20424.7	−1.52895	−0.764475	+	0.644654i	$0.777002\pi$
$564$	0	0
$565$	−495.760	−0.0369147
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24001.0	1.76832	0.884161	−	0.467182i	$-0.154731\pi$
$569$	24001.0	1.76832	0.884161	+	0.467182i	$0.154731\pi$
$570$	0	0
$571$	−11915.7	−0.873305	−0.436653	−	0.899630i	$-0.643836\pi$
$571$	−11915.7	−0.873305	−0.436653	+	0.899630i	$0.643836\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4554.70	0.330338
$576$	0	0
$577$	18995.8	1.37055	0.685273	−	0.728286i	$-0.259683\pi$
$577$	18995.8	1.37055	0.685273	+	0.728286i	$0.259683\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−931.166	−0.0664910
$582$	0	0
$583$	−15088.8	−1.07189
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21167.0	−1.48834	−0.744170	−	0.667990i	$-0.767155\pi$
$587$	−21167.0	−1.48834	−0.744170	+	0.667990i	$0.767155\pi$
$588$	0	0
$589$	−28704.7	−2.00807
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−820.766	−0.0568378	−0.0284189	−	0.999596i	$-0.509047\pi$
$593$	−820.766	−0.0568378	−0.0284189	+	0.999596i	$0.509047\pi$
$594$	0	0
$595$	−141.757	−0.00976718
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23985.1	−1.63607	−0.818035	−	0.575168i	$-0.804937\pi$
$599$	−23985.1	−1.63607	−0.818035	+	0.575168i	$0.804937\pi$
$600$	0	0
$601$	−6991.04	−0.474493	−0.237247	−	0.971449i	$-0.576245\pi$
$601$	−6991.04	−0.474493	−0.237247	+	0.971449i	$0.576245\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2456.98	−0.165108
$606$	0	0
$607$	−9572.17	−0.640070	−0.320035	−	0.947406i	$-0.603695\pi$
$607$	−9572.17	−0.640070	−0.320035	+	0.947406i	$0.603695\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16188.3	1.07186
$612$	0	0
$613$	−11648.4	−0.767496	−0.383748	−	0.923438i	$-0.625367\pi$
$613$	−11648.4	−0.767496	−0.383748	+	0.923438i	$0.625367\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8519.93	0.555915	0.277957	−	0.960593i	$-0.410343\pi$
$617$	8519.93	0.555915	0.277957	+	0.960593i	$0.410343\pi$
$618$	0	0
$619$	−152.812	−0.00992251	−0.00496125	−	0.999988i	$-0.501579\pi$
$619$	−152.812	−0.00992251	−0.00496125	+	0.999988i	$0.501579\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−118.524	−0.00762210
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7072.53	−0.448331
$630$	0	0
$631$	−10877.8	−0.686274	−0.343137	−	0.939285i	$-0.611489\pi$
$631$	−10877.8	−0.686274	−0.343137	+	0.939285i	$0.611489\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7940.86	0.496257
$636$	0	0
$637$	16397.6	1.01993
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−25590.5	−1.57685	−0.788427	−	0.615129i	$-0.789104\pi$
$641$	−25590.5	−1.57685	−0.788427	+	0.615129i	$0.789104\pi$
$642$	0	0
$643$	4144.13	0.254166	0.127083	−	0.991892i	$-0.459439\pi$
$643$	4144.13	0.254166	0.127083	+	0.991892i	$0.459439\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3745.27	−0.227576	−0.113788	−	0.993505i	$-0.536299\pi$
$647$	−3745.27	−0.227576	−0.113788	+	0.993505i	$0.536299\pi$
$648$	0	0
$649$	−17078.0	−1.03293
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1982.21	0.118790	0.0593951	−	0.998235i	$-0.481083\pi$
$653$	1982.21	0.118790	0.0593951	+	0.998235i	$0.481083\pi$
$654$	0	0
$655$	−971.796	−0.0579713
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6452.60	0.381423	0.190711	−	0.981646i	$-0.438921\pi$
$659$	6452.60	0.381423	0.190711	+	0.981646i	$0.438921\pi$
$660$	0	0
$661$	−15677.5	−0.922518	−0.461259	−	0.887265i	$-0.652602\pi$
$661$	−15677.5	−0.922518	−0.461259	+	0.887265i	$0.652602\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1142.22	−0.0666065
$666$	0	0
$667$	−53086.7	−3.08175
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16402.9	0.943706
$672$	0	0
$673$	−7672.17	−0.439436	−0.219718	−	0.975563i	$-0.570514\pi$
$673$	−7672.17	−0.439436	−0.219718	+	0.975563i	$0.570514\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27895.3	1.58361	0.791806	−	0.610773i	$-0.209141\pi$
$677$	27895.3	1.58361	0.791806	+	0.610773i	$0.209141\pi$
$678$	0	0
$679$	381.206	0.0215454
$680$	0	0
$681$	0	0
$682$	0	0
$683$	23138.6	1.29630	0.648150	−	0.761512i	$-0.275543\pi$
$683$	23138.6	1.29630	0.648150	+	0.761512i	$0.275543\pi$
$684$	0	0
$685$	5502.25	0.306905
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−25118.3	−1.38887
$690$	0	0
$691$	−16370.6	−0.901257	−0.450629	−	0.892711i	$-0.648800\pi$
$691$	−16370.6	−0.901257	−0.450629	+	0.892711i	$0.648800\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11768.1	0.642288
$696$	0	0
$697$	−5512.13	−0.299551
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3963.47	0.213550	0.106775	−	0.994283i	$-0.465948\pi$
$701$	3963.47	0.213550	0.106775	+	0.994283i	$0.465948\pi$
$702$	0	0
$703$	−56987.5	−3.05736
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2082.35	−0.110771
$708$	0	0
$709$	13370.4	0.708229	0.354115	−	0.935202i	$-0.384782\pi$
$709$	13370.4	0.708229	0.354115	+	0.935202i	$0.384782\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	40015.0	2.10179
$714$	0	0
$715$	6988.44	0.365528
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12373.7	−0.641809	−0.320904	−	0.947112i	$-0.603987\pi$
$719$	−12373.7	−0.641809	−0.320904	+	0.947112i	$0.603987\pi$
$720$	0	0
$721$	−926.506	−0.0478570
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7284.61	−0.373163
$726$	0	0
$727$	−1756.74	−0.0896203	−0.0448102	−	0.998996i	$-0.514268\pi$
$727$	−1756.74	−0.0896203	−0.0448102	+	0.998996i	$0.514268\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5133.64	−0.259746
$732$	0	0
$733$	4564.43	0.230001	0.115001	−	0.993365i	$-0.463313\pi$
$733$	4564.43	0.230001	0.115001	+	0.993365i	$0.463313\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11804.8	−0.590006
$738$	0	0
$739$	−73.8682	−0.00367698	−0.00183849	−	0.999998i	$-0.500585\pi$
$739$	−73.8682	−0.00367698	−0.00183849	+	0.999998i	$0.500585\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24005.9	−1.18532	−0.592660	−	0.805453i	$-0.701922\pi$
$743$	−24005.9	−1.18532	−0.592660	+	0.805453i	$0.701922\pi$
$744$	0	0
$745$	5931.80	0.291710
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1151.35	−0.0561674
$750$	0	0
$751$	7708.95	0.374572	0.187286	−	0.982305i	$-0.440031\pi$
$751$	7708.95	0.374572	0.187286	+	0.982305i	$0.440031\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16177.9	0.779833
$756$	0	0
$757$	18327.3	0.879941	0.439971	−	0.898012i	$-0.354989\pi$
$757$	18327.3	0.879941	0.439971	+	0.898012i	$0.354989\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21820.5	1.03941	0.519705	−	0.854346i	$-0.326042\pi$
$761$	21820.5	1.03941	0.519705	+	0.854346i	$0.326042\pi$
$762$	0	0
$763$	2947.64	0.139858
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28429.7	−1.33838
$768$	0	0
$769$	19246.4	0.902527	0.451263	−	0.892391i	$-0.350973\pi$
$769$	19246.4	0.902527	0.451263	+	0.892391i	$0.350973\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25012.3	−1.16382	−0.581909	−	0.813254i	$-0.697694\pi$
$773$	−25012.3	−1.16382	−0.581909	+	0.813254i	$0.697694\pi$
$774$	0	0
$775$	5490.89	0.254502
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−44414.4	−2.04276
$780$	0	0
$781$	−14091.8	−0.645639
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8259.11	−0.375516
$786$	0	0
$787$	28460.6	1.28909	0.644544	−	0.764568i	$-0.277047\pi$
$787$	28460.6	1.28909	0.644544	+	0.764568i	$0.277047\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−173.313	−0.00779052
$792$	0	0
$793$	27305.9	1.22277
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16315.9	−0.725145	−0.362573	−	0.931956i	$-0.618101\pi$
$797$	−16315.9	−0.725145	−0.362573	+	0.931956i	$0.618101\pi$
$798$	0	0
$799$	5443.44	0.241020
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4155.64	−0.182627
$804$	0	0
$805$	1592.28	0.0697149
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5461.82	0.237364	0.118682	−	0.992932i	$-0.462133\pi$
$809$	5461.82	0.237364	0.118682	+	0.992932i	$0.462133\pi$
$810$	0	0
$811$	−23278.4	−1.00791	−0.503956	−	0.863729i	$-0.668123\pi$
$811$	−23278.4	−1.00791	−0.503956	+	0.863729i	$0.668123\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	19193.6	0.824935
$816$	0	0
$817$	−41364.7	−1.77132
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−879.972	−0.0374071	−0.0187035	−	0.999825i	$-0.505954\pi$
$821$	−879.972	−0.0374071	−0.0187035	+	0.999825i	$0.505954\pi$
$822$	0	0
$823$	12811.2	0.542614	0.271307	−	0.962493i	$-0.412544\pi$
$823$	12811.2	0.542614	0.271307	+	0.962493i	$0.412544\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29331.8	−1.23333	−0.616666	−	0.787225i	$-0.711517\pi$
$827$	−29331.8	−1.23333	−0.616666	+	0.787225i	$0.711517\pi$
$828$	0	0
$829$	−13073.7	−0.547732	−0.273866	−	0.961768i	$-0.588303\pi$
$829$	−13073.7	−0.547732	−0.273866	+	0.961768i	$0.588303\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5513.83	0.229343
$834$	0	0
$835$	−10502.5	−0.435273
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2423.58	0.0997276	0.0498638	−	0.998756i	$-0.484121\pi$
$839$	2423.58	0.0997276	0.0498638	+	0.998756i	$0.484121\pi$
$840$	0	0
$841$	60515.8	2.48127
$842$	0	0
$843$	0	0
$844$	0	0
$845$	648.638	0.0264069
$846$	0	0
$847$	−858.936	−0.0348446
$848$	0	0
$849$	0	0
$850$	0	0
$851$	79441.9	3.20004
$852$	0	0
$853$	−28668.5	−1.15075	−0.575375	−	0.817890i	$-0.695144\pi$
$853$	−28668.5	−1.15075	−0.575375	+	0.817890i	$0.695144\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25991.2	−1.03599	−0.517994	−	0.855384i	$-0.673321\pi$
$857$	−25991.2	−1.03599	−0.517994	+	0.855384i	$0.673321\pi$
$858$	0	0
$859$	−12131.6	−0.481866	−0.240933	−	0.970542i	$-0.577453\pi$
$859$	−12131.6	−0.481866	−0.240933	+	0.970542i	$0.577453\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1174.43	0.0463244	0.0231622	−	0.999732i	$-0.492627\pi$
$863$	1174.43	0.0463244	0.0231622	+	0.999732i	$0.492627\pi$
$864$	0	0
$865$	−17913.9	−0.704152
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−28049.0	−1.09493
$870$	0	0
$871$	−19651.4	−0.764480
$872$	0	0
$873$	0	0
$874$	0	0
$875$	218.494	0.00844165
$876$	0	0
$877$	−15509.8	−0.597181	−0.298591	−	0.954381i	$-0.596517\pi$
$877$	−15509.8	−0.597181	−0.298591	+	0.954381i	$0.596517\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1050.07	0.0401565	0.0200783	−	0.999798i	$-0.493608\pi$
$881$	1050.07	0.0401565	0.0200783	+	0.999798i	$0.493608\pi$
$882$	0	0
$883$	−22517.4	−0.858176	−0.429088	−	0.903263i	$-0.641165\pi$
$883$	−22517.4	−0.858176	−0.429088	+	0.903263i	$0.641165\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2762.26	0.104563	0.0522816	−	0.998632i	$-0.483351\pi$
$887$	2762.26	0.104563	0.0522816	+	0.998632i	$0.483351\pi$
$888$	0	0
$889$	2776.05	0.104731
$890$	0	0
$891$	0	0
$892$	0	0
$893$	43860.9	1.64362
$894$	0	0
$895$	−12166.0	−0.454375
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−63998.4	−2.37427
$900$	0	0
$901$	−8446.22	−0.312302
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2862.53	0.105142
$906$	0	0
$907$	36714.4	1.34408	0.672040	−	0.740515i	$-0.265418\pi$
$907$	36714.4	1.34408	0.672040	+	0.740515i	$0.265418\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51426.7	1.87030	0.935150	−	0.354252i	$-0.115264\pi$
$911$	51426.7	1.87030	0.935150	+	0.354252i	$0.115264\pi$
$912$	0	0
$913$	15436.0	0.559537
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−339.731	−0.0122343
$918$	0	0
$919$	−33475.8	−1.20159	−0.600797	−	0.799402i	$-0.705150\pi$
$919$	−33475.8	−1.20159	−0.600797	+	0.799402i	$0.705150\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−23458.6	−0.836564
$924$	0	0
$925$	10901.1	0.387487
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35514.9	−1.25426	−0.627128	−	0.778916i	$-0.715770\pi$
$929$	−35514.9	−1.25426	−0.627128	+	0.778916i	$0.715770\pi$
$930$	0	0
$931$	44428.1	1.56399
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2349.92	0.0821930
$936$	0	0
$937$	−53917.6	−1.87984	−0.939921	−	0.341392i	$-0.889102\pi$
$937$	−53917.6	−1.87984	−0.939921	+	0.341392i	$0.889102\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31034.7	1.07513	0.537567	−	0.843221i	$-0.319343\pi$
$941$	31034.7	1.07513	0.537567	+	0.843221i	$0.319343\pi$
$942$	0	0
$943$	61914.8	2.13810
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19423.7	−0.666510	−0.333255	−	0.942837i	$-0.608147\pi$
$947$	−19423.7	−0.666510	−0.333255	+	0.942837i	$0.608147\pi$
$948$	0	0
$949$	−6917.89	−0.236632
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−23270.7	−0.790987	−0.395493	−	0.918469i	$-0.629426\pi$
$953$	−23270.7	−0.790987	−0.395493	+	0.918469i	$0.629426\pi$
$954$	0	0
$955$	−2457.28	−0.0832625
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1923.53	0.0647697
$960$	0	0
$961$	18448.9	0.619277
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13990.6	0.466708
$966$	0	0
$967$	−33988.2	−1.13028	−0.565142	−	0.824993i	$-0.691179\pi$
$967$	−33988.2	−1.13028	−0.565142	+	0.824993i	$0.691179\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31963.3	1.05639	0.528194	−	0.849124i	$-0.322870\pi$
$971$	31963.3	1.05639	0.528194	+	0.849124i	$0.322870\pi$
$972$	0	0
$973$	4114.02	0.135549
$974$	0	0
$975$	0	0
$976$	0	0
$977$	35977.7	1.17813	0.589063	−	0.808087i	$-0.299497\pi$
$977$	35977.7	1.17813	0.589063	+	0.808087i	$0.299497\pi$
$978$	0	0
$979$	1964.78	0.0641417
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−34020.8	−1.10386	−0.551931	−	0.833890i	$-0.686109\pi$
$983$	−34020.8	−1.10386	−0.551931	+	0.833890i	$0.686109\pi$
$984$	0	0
$985$	−15555.3	−0.503182
$986$	0	0
$987$	0	0
$988$	0	0
$989$	57663.4	1.85398
$990$	0	0
$991$	20226.2	0.648343	0.324172	−	0.945998i	$-0.394914\pi$
$991$	20226.2	0.648343	0.324172	+	0.945998i	$0.394914\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	168.799	0.00537817
$996$	0	0
$997$	19657.5	0.624431	0.312216	−	0.950011i	$-0.398929\pi$
$997$	19657.5	0.624431	0.312216	+	0.950011i	$0.398929\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.4.a.bk.1.2	yes	3
3.2	odd	2		1440.4.a.bi.1.2	yes	3
4.3	odd	2		1440.4.a.bj.1.2	yes	3
12.11	even	2		1440.4.a.bh.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.4.a.bh.1.2	✓	3	12.11	even	2
1440.4.a.bi.1.2	yes	3	3.2	odd	2
1440.4.a.bj.1.2	yes	3	4.3	odd	2
1440.4.a.bk.1.2	yes	3	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +1.74795 q^{7} -28.9759 q^{11} -48.2362 q^{13} -16.2198 q^{17} -130.692 q^{19} +182.188 q^{23} +25.0000 q^{25} -291.384 q^{29} +219.636 q^{31} +8.73976 q^{35} +436.044 q^{37} +339.840 q^{41} +316.505 q^{43} -335.605 q^{47} -339.945 q^{49} +520.736 q^{53} -144.880 q^{55} +589.386 q^{59} -566.087 q^{61} -241.181 q^{65} +407.400 q^{67} +486.328 q^{71} +143.417 q^{73} -50.6485 q^{77} +968.012 q^{79} -532.718 q^{83} -81.0989 q^{85} -67.8074 q^{89} -84.3145 q^{91} -653.461 q^{95} +218.087 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 14 q^{7} + 22 q^{11} + 8 q^{13} + 34 q^{17} - 4 q^{19} + 176 q^{23} + 75 q^{25} - 98 q^{29} + 88 q^{31} + 70 q^{35} + 284 q^{37} + 8 q^{41} + 504 q^{43} + 280 q^{47} - 409 q^{49} + 150 q^{53}+ \cdots - 1394 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 + 14 * q^7 + 22 * q^11 + 8 * q^13 + 34 * q^17 - 4 * q^19 + 176 * q^23 + 75 * q^25 - 98 * q^29 + 88 * q^31 + 70 * q^35 + 284 * q^37 + 8 * q^41 + 504 * q^43 + 280 * q^47 - 409 * q^49 + 150 * q^53 + 110 * q^55 + 350 * q^59 + 350 * q^61 + 40 * q^65 + 804 * q^67 + 500 * q^71 - 486 * q^73 - 788 * q^77 + 1592 * q^79 - 684 * q^83 + 170 * q^85 + 668 * q^89 + 1920 * q^91 - 20 * q^95 - 1394 * q^97

Introduction

L-functions

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