LMFDB - Embedded newform 1872.4.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.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:	$110.451575531$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.73549$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.90776 q^{5} -36.4129 q^{7} +O(q^{10})$	$q+3.90776 q^{5} -36.4129 q^{7} +19.1943 q^{11} +13.0000 q^{13} +83.8839 q^{17} -46.8492 q^{19} +103.905 q^{23} -109.729 q^{25} -108.341 q^{29} +147.532 q^{31} -142.293 q^{35} -160.012 q^{37} -231.490 q^{41} +340.314 q^{43} +119.653 q^{47} +982.902 q^{49} +732.879 q^{53} +75.0067 q^{55} -229.782 q^{59} +108.943 q^{61} +50.8009 q^{65} -10.3955 q^{67} -869.201 q^{71} -1099.07 q^{73} -698.920 q^{77} -140.410 q^{79} -159.474 q^{83} +327.799 q^{85} -1067.93 q^{89} -473.368 q^{91} -183.075 q^{95} +858.881 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{5} - 30 q^{7}+O(q^{10}) $ 3 * q - 4 * q^5 - 30 * q^7	$ 3 q - 4 q^{5} - 30 q^{7} - 16 q^{11} + 39 q^{13} + 146 q^{17} - 94 q^{19} - 48 q^{23} + 145 q^{25} + 2 q^{29} - 302 q^{31} + 80 q^{35} + 374 q^{37} - 480 q^{41} + 260 q^{43} - 24 q^{47} + 447 q^{49} + 678 q^{53} + 1552 q^{55} - 1788 q^{59} + 230 q^{61} - 52 q^{65} - 74 q^{67} - 948 q^{71} - 222 q^{73} - 112 q^{77} + 24 q^{79} - 796 q^{83} - 248 q^{85} - 1436 q^{89} - 390 q^{91} - 4032 q^{95} + 3242 q^{97}+O(q^{100}) $ 3 * q - 4 * q^5 - 30 * q^7 - 16 * q^11 + 39 * q^13 + 146 * q^17 - 94 * q^19 - 48 * q^23 + 145 * q^25 + 2 * q^29 - 302 * q^31 + 80 * q^35 + 374 * q^37 - 480 * q^41 + 260 * q^43 - 24 * q^47 + 447 * q^49 + 678 * q^53 + 1552 * q^55 - 1788 * q^59 + 230 * q^61 - 52 * q^65 - 74 * q^67 - 948 * q^71 - 222 * q^73 - 112 * q^77 + 24 * q^79 - 796 * q^83 - 248 * q^85 - 1436 * q^89 - 390 * q^91 - 4032 * q^95 + 3242 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.90776	0.349521	0.174761	−	0.984611i	$-0.444085\pi$
$5$	3.90776	0.349521	0.174761	+	0.984611i	$0.444085\pi$
$6$	0	0
$7$	−36.4129	−1.96611	−0.983057	−	0.183301i	$-0.941322\pi$
$7$	−36.4129	−1.96611	−0.983057	+	0.183301i	$0.941322\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	19.1943	0.526117	0.263059	−	0.964780i	$-0.415269\pi$
$11$	19.1943	0.526117	0.263059	+	0.964780i	$0.415269\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	83.8839	1.19676	0.598378	−	0.801214i	$-0.295812\pi$
$17$	83.8839	1.19676	0.598378	+	0.801214i	$0.295812\pi$
$18$	0	0
$19$	−46.8492	−0.565681	−0.282840	−	0.959167i	$-0.591277\pi$
$19$	−46.8492	−0.565681	−0.282840	+	0.959167i	$0.591277\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	103.905	0.941983	0.470991	−	0.882138i	$-0.343896\pi$
$23$	103.905	0.941983	0.470991	+	0.882138i	$0.343896\pi$
$24$	0	0
$25$	−109.729	−0.877835
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−108.341	−0.693738	−0.346869	−	0.937914i	$-0.612755\pi$
$29$	−108.341	−0.693738	−0.346869	+	0.937914i	$0.612755\pi$
$30$	0	0
$31$	147.532	0.854759	0.427379	−	0.904072i	$-0.359437\pi$
$31$	147.532	0.854759	0.427379	+	0.904072i	$0.359437\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−142.293	−0.687198
$36$	0	0
$37$	−160.012	−0.710969	−0.355484	−	0.934682i	$-0.615684\pi$
$37$	−160.012	−0.710969	−0.355484	+	0.934682i	$0.615684\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−231.490	−0.881772	−0.440886	−	0.897563i	$-0.645336\pi$
$41$	−231.490	−0.881772	−0.440886	+	0.897563i	$0.645336\pi$
$42$	0	0
$43$	340.314	1.20692	0.603458	−	0.797395i	$-0.293789\pi$
$43$	340.314	1.20692	0.603458	+	0.797395i	$0.293789\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	119.653	0.371346	0.185673	−	0.982612i	$-0.440554\pi$
$47$	119.653	0.371346	0.185673	+	0.982612i	$0.440554\pi$
$48$	0	0
$49$	982.902	2.86560
$50$	0	0
$51$	0	0
$52$	0	0
$53$	732.879	1.89941	0.949705	−	0.313146i	$-0.101383\pi$
$53$	732.879	1.89941	0.949705	+	0.313146i	$0.101383\pi$
$54$	0	0
$55$	75.0067	0.183889
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−229.782	−0.507035	−0.253518	−	0.967331i	$-0.581588\pi$
$59$	−229.782	−0.507035	−0.253518	+	0.967331i	$0.581588\pi$
$60$	0	0
$61$	108.943	0.228668	0.114334	−	0.993442i	$-0.463527\pi$
$61$	108.943	0.228668	0.114334	+	0.993442i	$0.463527\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	50.8009	0.0969397
$66$	0	0
$67$	−10.3955	−0.0189555	−0.00947774	−	0.999955i	$-0.503017\pi$
$67$	−10.3955	−0.0189555	−0.00947774	+	0.999955i	$0.503017\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−869.201	−1.45289	−0.726445	−	0.687224i	$-0.758829\pi$
$71$	−869.201	−1.45289	−0.726445	+	0.687224i	$0.758829\pi$
$72$	0	0
$73$	−1099.07	−1.76214	−0.881072	−	0.472982i	$-0.843178\pi$
$73$	−1099.07	−1.76214	−0.881072	+	0.472982i	$0.843178\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−698.920	−1.03441
$78$	0	0
$79$	−140.410	−0.199967	−0.0999835	−	0.994989i	$-0.531879\pi$
$79$	−140.410	−0.199967	−0.0999835	+	0.994989i	$0.531879\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−159.474	−0.210898	−0.105449	−	0.994425i	$-0.533628\pi$
$83$	−159.474	−0.210898	−0.105449	+	0.994425i	$0.533628\pi$
$84$	0	0
$85$	327.799	0.418291
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1067.93	−1.27192	−0.635959	−	0.771723i	$-0.719395\pi$
$89$	−1067.93	−1.27192	−0.635959	+	0.771723i	$0.719395\pi$
$90$	0	0
$91$	−473.368	−0.545302
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−183.075	−0.197717
$96$	0	0
$97$	858.881	0.899032	0.449516	−	0.893272i	$-0.351596\pi$
$97$	858.881	0.899032	0.449516	+	0.893272i	$0.351596\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1574.16	1.55084	0.775421	−	0.631444i	$-0.217538\pi$
$101$	1574.16	1.55084	0.775421	+	0.631444i	$0.217538\pi$
$102$	0	0
$103$	129.724	0.124098	0.0620489	−	0.998073i	$-0.480237\pi$
$103$	129.724	0.124098	0.0620489	+	0.998073i	$0.480237\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1957.43	−1.76853	−0.884263	−	0.466990i	$-0.845339\pi$
$107$	−1957.43	−1.76853	−0.884263	+	0.466990i	$0.845339\pi$
$108$	0	0
$109$	1228.77	1.07977	0.539886	−	0.841738i	$-0.318467\pi$
$109$	1228.77	1.07977	0.539886	+	0.841738i	$0.318467\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1629.50	−1.35655	−0.678275	−	0.734808i	$-0.737272\pi$
$113$	−1629.50	−1.35655	−0.678275	+	0.734808i	$0.737272\pi$
$114$	0	0
$115$	406.035	0.329243
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3054.46	−2.35296
$120$	0	0
$121$	−962.580	−0.723201
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−917.267	−0.656343
$126$	0	0
$127$	−276.112	−0.192921	−0.0964607	−	0.995337i	$-0.530752\pi$
$127$	−276.112	−0.192921	−0.0964607	+	0.995337i	$0.530752\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−96.2240	−0.0641765	−0.0320883	−	0.999485i	$-0.510216\pi$
$131$	−96.2240	−0.0641765	−0.0320883	+	0.999485i	$0.510216\pi$
$132$	0	0
$133$	1705.92	1.11219
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2618.38	−1.63287	−0.816435	−	0.577438i	$-0.804053\pi$
$137$	−2618.38	−1.63287	−0.816435	+	0.577438i	$0.804053\pi$
$138$	0	0
$139$	−1963.34	−1.19805	−0.599023	−	0.800732i	$-0.704444\pi$
$139$	−1963.34	−1.19805	−0.599023	+	0.800732i	$0.704444\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	249.526	0.145919
$144$	0	0
$145$	−423.370	−0.242476
$146$	0	0
$147$	0	0
$148$	0	0
$149$	301.111	0.165557	0.0827784	−	0.996568i	$-0.473621\pi$
$149$	301.111	0.165557	0.0827784	+	0.996568i	$0.473621\pi$
$150$	0	0
$151$	−342.973	−0.184839	−0.0924197	−	0.995720i	$-0.529460\pi$
$151$	−342.973	−0.184839	−0.0924197	+	0.995720i	$0.529460\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	576.520	0.298756
$156$	0	0
$157$	−1286.97	−0.654211	−0.327106	−	0.944988i	$-0.606073\pi$
$157$	−1286.97	−0.654211	−0.327106	+	0.944988i	$0.606073\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3783.47	−1.85205
$162$	0	0
$163$	−532.561	−0.255910	−0.127955	−	0.991780i	$-0.540841\pi$
$163$	−532.561	−0.255910	−0.127955	+	0.991780i	$0.540841\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	41.9542	0.0194402	0.00972011	−	0.999953i	$-0.496906\pi$
$167$	41.9542	0.0194402	0.00972011	+	0.999953i	$0.496906\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1066.50	0.468694	0.234347	−	0.972153i	$-0.424705\pi$
$173$	1066.50	0.468694	0.234347	+	0.972153i	$0.424705\pi$
$174$	0	0
$175$	3995.57	1.72592
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3174.61	1.32559	0.662797	−	0.748799i	$-0.269369\pi$
$179$	3174.61	1.32559	0.662797	+	0.748799i	$0.269369\pi$
$180$	0	0
$181$	−2725.43	−1.11923	−0.559613	−	0.828754i	$-0.689050\pi$
$181$	−2725.43	−1.11923	−0.559613	+	0.828754i	$0.689050\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−625.290	−0.248498
$186$	0	0
$187$	1610.09	0.629634
$188$	0	0
$189$	0	0
$190$	0	0
$191$	784.888	0.297343	0.148672	−	0.988887i	$-0.452500\pi$
$191$	784.888	0.297343	0.148672	+	0.988887i	$0.452500\pi$
$192$	0	0
$193$	−1255.87	−0.468391	−0.234195	−	0.972190i	$-0.575246\pi$
$193$	−1255.87	−0.468391	−0.234195	+	0.972190i	$0.575246\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2777.35	1.00446	0.502229	−	0.864734i	$-0.332513\pi$
$197$	2777.35	1.00446	0.502229	+	0.864734i	$0.332513\pi$
$198$	0	0
$199$	−1490.43	−0.530924	−0.265462	−	0.964121i	$-0.585524\pi$
$199$	−1490.43	−0.530924	−0.265462	+	0.964121i	$0.585524\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3945.01	1.36397
$204$	0	0
$205$	−904.608	−0.308198
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−899.236	−0.297614
$210$	0	0
$211$	−2305.63	−0.752255	−0.376127	−	0.926568i	$-0.622745\pi$
$211$	−2305.63	−0.752255	−0.376127	+	0.926568i	$0.622745\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1329.87	0.421842
$216$	0	0
$217$	−5372.07	−1.68055
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1090.49	0.331920
$222$	0	0
$223$	−1241.98	−0.372956	−0.186478	−	0.982459i	$-0.559707\pi$
$223$	−1241.98	−0.372956	−0.186478	+	0.982459i	$0.559707\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1724.76	−0.504300	−0.252150	−	0.967688i	$-0.581138\pi$
$227$	−1724.76	−0.504300	−0.252150	+	0.967688i	$0.581138\pi$
$228$	0	0
$229$	−3273.72	−0.944688	−0.472344	−	0.881414i	$-0.656592\pi$
$229$	−3273.72	−0.944688	−0.472344	+	0.881414i	$0.656592\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2129.52	0.598752	0.299376	−	0.954135i	$-0.403222\pi$
$233$	2129.52	0.598752	0.299376	+	0.954135i	$0.403222\pi$
$234$	0	0
$235$	467.577	0.129793
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5082.38	−1.37553	−0.687765	−	0.725933i	$-0.741408\pi$
$239$	−5082.38	−1.37553	−0.687765	+	0.725933i	$0.741408\pi$
$240$	0	0
$241$	4765.65	1.27379	0.636893	−	0.770953i	$-0.280219\pi$
$241$	4765.65	1.27379	0.636893	+	0.770953i	$0.280219\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3840.95	1.00159
$246$	0	0
$247$	−609.039	−0.156892
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4339.96	−1.09138	−0.545689	−	0.837988i	$-0.683732\pi$
$251$	−4339.96	−1.09138	−0.545689	+	0.837988i	$0.683732\pi$
$252$	0	0
$253$	1994.37	0.495594
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4359.49	−1.05812	−0.529062	−	0.848583i	$-0.677456\pi$
$257$	−4359.49	−1.05812	−0.529062	+	0.848583i	$0.677456\pi$
$258$	0	0
$259$	5826.51	1.39785
$260$	0	0
$261$	0	0
$262$	0	0
$263$	608.077	0.142569	0.0712844	−	0.997456i	$-0.477290\pi$
$263$	608.077	0.142569	0.0712844	+	0.997456i	$0.477290\pi$
$264$	0	0
$265$	2863.92	0.663884
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3454.29	−0.782942	−0.391471	−	0.920190i	$-0.628034\pi$
$269$	−3454.29	−0.782942	−0.391471	+	0.920190i	$0.628034\pi$
$270$	0	0
$271$	−3703.72	−0.830204	−0.415102	−	0.909775i	$-0.636254\pi$
$271$	−3703.72	−0.830204	−0.415102	+	0.909775i	$0.636254\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2106.18	−0.461844
$276$	0	0
$277$	−3566.89	−0.773696	−0.386848	−	0.922144i	$-0.626436\pi$
$277$	−3566.89	−0.773696	−0.386848	+	0.922144i	$0.626436\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	117.474	0.0249392	0.0124696	−	0.999922i	$-0.496031\pi$
$281$	117.474	0.0249392	0.0124696	+	0.999922i	$0.496031\pi$
$282$	0	0
$283$	1737.62	0.364984	0.182492	−	0.983207i	$-0.441584\pi$
$283$	1737.62	0.364984	0.182492	+	0.983207i	$0.441584\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8429.23	1.73366
$288$	0	0
$289$	2123.51	0.432223
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1904.05	−0.379643	−0.189822	−	0.981819i	$-0.560791\pi$
$293$	−1904.05	−0.379643	−0.189822	+	0.981819i	$0.560791\pi$
$294$	0	0
$295$	−897.934	−0.177219
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1350.76	0.261259
$300$	0	0
$301$	−12391.8	−2.37293
$302$	0	0
$303$	0	0
$304$	0	0
$305$	425.724	0.0799242
$306$	0	0
$307$	−2862.39	−0.532134	−0.266067	−	0.963955i	$-0.585724\pi$
$307$	−2862.39	−0.532134	−0.266067	+	0.963955i	$0.585724\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4201.55	0.766071	0.383036	−	0.923734i	$-0.374879\pi$
$311$	4201.55	0.766071	0.383036	+	0.923734i	$0.374879\pi$
$312$	0	0
$313$	3427.74	0.619002	0.309501	−	0.950899i	$-0.399838\pi$
$313$	3427.74	0.619002	0.309501	+	0.950899i	$0.399838\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1676.09	−0.296966	−0.148483	−	0.988915i	$-0.547439\pi$
$317$	−1676.09	−0.296966	−0.148483	+	0.988915i	$0.547439\pi$
$318$	0	0
$319$	−2079.52	−0.364987
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3929.89	−0.676982
$324$	0	0
$325$	−1426.48	−0.243468
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4356.93	−0.730108
$330$	0	0
$331$	−11156.6	−1.85264	−0.926319	−	0.376740i	$-0.877045\pi$
$331$	−11156.6	−1.85264	−0.926319	+	0.376740i	$0.877045\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−40.6233	−0.00662534
$336$	0	0
$337$	1636.44	0.264517	0.132259	−	0.991215i	$-0.457777\pi$
$337$	1636.44	0.264517	0.132259	+	0.991215i	$0.457777\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2831.77	0.449703
$342$	0	0
$343$	−23300.7	−3.66799
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2977.87	−0.460693	−0.230347	−	0.973109i	$-0.573986\pi$
$347$	−2977.87	−0.460693	−0.230347	+	0.973109i	$0.573986\pi$
$348$	0	0
$349$	9847.29	1.51035	0.755177	−	0.655521i	$-0.227551\pi$
$349$	9847.29	1.51035	0.755177	+	0.655521i	$0.227551\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4687.34	−0.706747	−0.353374	−	0.935482i	$-0.614966\pi$
$353$	−4687.34	−0.706747	−0.353374	+	0.935482i	$0.614966\pi$
$354$	0	0
$355$	−3396.63	−0.507816
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2069.88	−0.304301	−0.152151	−	0.988357i	$-0.548620\pi$
$359$	−2069.88	−0.304301	−0.152151	+	0.988357i	$0.548620\pi$
$360$	0	0
$361$	−4664.16	−0.680005
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4294.91	−0.615906
$366$	0	0
$367$	7299.16	1.03818	0.519092	−	0.854719i	$-0.326270\pi$
$367$	7299.16	1.03818	0.519092	+	0.854719i	$0.326270\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−26686.3	−3.73446
$372$	0	0
$373$	−8964.32	−1.24438	−0.622192	−	0.782865i	$-0.713758\pi$
$373$	−8964.32	−1.24438	−0.622192	+	0.782865i	$0.713758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1408.43	−0.192408
$378$	0	0
$379$	4399.26	0.596239	0.298120	−	0.954529i	$-0.403641\pi$
$379$	4399.26	0.596239	0.298120	+	0.954529i	$0.403641\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3529.74	0.470917	0.235459	−	0.971884i	$-0.424341\pi$
$383$	3529.74	0.470917	0.235459	+	0.971884i	$0.424341\pi$
$384$	0	0
$385$	−2731.21	−0.361547
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3034.77	0.395549	0.197775	−	0.980248i	$-0.436629\pi$
$389$	3034.77	0.395549	0.197775	+	0.980248i	$0.436629\pi$
$390$	0	0
$391$	8715.93	1.12732
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−548.690	−0.0698927
$396$	0	0
$397$	−3997.36	−0.505344	−0.252672	−	0.967552i	$-0.581309\pi$
$397$	−3997.36	−0.505344	−0.252672	+	0.967552i	$0.581309\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9092.88	1.13236	0.566181	−	0.824281i	$-0.308420\pi$
$401$	9092.88	1.13236	0.566181	+	0.824281i	$0.308420\pi$
$402$	0	0
$403$	1917.92	0.237067
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3071.32	−0.374053
$408$	0	0
$409$	7143.54	0.863631	0.431816	−	0.901962i	$-0.357873\pi$
$409$	7143.54	0.863631	0.431816	+	0.901962i	$0.357873\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8367.04	0.996889
$414$	0	0
$415$	−623.187	−0.0737134
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8213.84	0.957691	0.478845	−	0.877899i	$-0.341055\pi$
$419$	8213.84	0.957691	0.478845	+	0.877899i	$0.341055\pi$
$420$	0	0
$421$	−7997.40	−0.925818	−0.462909	−	0.886406i	$-0.653194\pi$
$421$	−7997.40	−0.925818	−0.462909	+	0.886406i	$0.653194\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−9204.53	−1.05055
$426$	0	0
$427$	−3966.94	−0.449587
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13694.8	−1.53053	−0.765263	−	0.643718i	$-0.777391\pi$
$431$	−13694.8	−1.53053	−0.765263	+	0.643718i	$0.777391\pi$
$432$	0	0
$433$	6716.57	0.745445	0.372722	−	0.927943i	$-0.378424\pi$
$433$	6716.57	0.745445	0.372722	+	0.927943i	$0.378424\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4867.84	−0.532862
$438$	0	0
$439$	5933.32	0.645061	0.322531	−	0.946559i	$-0.395466\pi$
$439$	5933.32	0.645061	0.322531	+	0.946559i	$0.395466\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6923.40	−0.742530	−0.371265	−	0.928527i	$-0.621076\pi$
$443$	−6923.40	−0.742530	−0.371265	+	0.928527i	$0.621076\pi$
$444$	0	0
$445$	−4173.23	−0.444562
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8886.78	−0.934061	−0.467030	−	0.884241i	$-0.654676\pi$
$449$	−8886.78	−0.934061	−0.467030	+	0.884241i	$0.654676\pi$
$450$	0	0
$451$	−4443.28	−0.463916
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1849.81	−0.190594
$456$	0	0
$457$	10965.0	1.12237	0.561184	−	0.827691i	$-0.310346\pi$
$457$	10965.0	1.12237	0.561184	+	0.827691i	$0.310346\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10069.2	−1.01729	−0.508644	−	0.860977i	$-0.669853\pi$
$461$	−10069.2	−1.01729	−0.508644	+	0.860977i	$0.669853\pi$
$462$	0	0
$463$	−5599.72	−0.562076	−0.281038	−	0.959697i	$-0.590679\pi$
$463$	−5599.72	−0.562076	−0.281038	+	0.959697i	$0.590679\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13247.8	−1.31271	−0.656355	−	0.754452i	$-0.727903\pi$
$467$	−13247.8	−1.31271	−0.656355	+	0.754452i	$0.727903\pi$
$468$	0	0
$469$	378.532	0.0372686
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6532.08	0.634979
$474$	0	0
$475$	5140.73	0.496575
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16725.4	1.59541	0.797707	−	0.603045i	$-0.206046\pi$
$479$	16725.4	1.59541	0.797707	+	0.603045i	$0.206046\pi$
$480$	0	0
$481$	−2080.16	−0.197187
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3356.30	0.314231
$486$	0	0
$487$	5305.86	0.493699	0.246850	−	0.969054i	$-0.420605\pi$
$487$	5305.86	0.493699	0.246850	+	0.969054i	$0.420605\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16200.2	1.48901	0.744506	−	0.667616i	$-0.232685\pi$
$491$	16200.2	1.48901	0.744506	+	0.667616i	$0.232685\pi$
$492$	0	0
$493$	−9088.05	−0.830234
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31650.2	2.85655
$498$	0	0
$499$	−4392.70	−0.394076	−0.197038	−	0.980396i	$-0.563132\pi$
$499$	−4392.70	−0.394076	−0.197038	+	0.980396i	$0.563132\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14955.2	−1.32568	−0.662841	−	0.748760i	$-0.730650\pi$
$503$	−14955.2	−1.32568	−0.662841	+	0.748760i	$0.730650\pi$
$504$	0	0
$505$	6151.46	0.542052
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13403.4	−1.16719	−0.583593	−	0.812047i	$-0.698353\pi$
$509$	−13403.4	−1.16719	−0.583593	+	0.812047i	$0.698353\pi$
$510$	0	0
$511$	40020.4	3.46458
$512$	0	0
$513$	0	0
$514$	0	0
$515$	506.930	0.0433748
$516$	0	0
$517$	2296.66	0.195371
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19643.0	−1.65178	−0.825888	−	0.563834i	$-0.809326\pi$
$521$	−19643.0	−1.65178	−0.825888	+	0.563834i	$0.809326\pi$
$522$	0	0
$523$	−14657.4	−1.22548	−0.612738	−	0.790286i	$-0.709932\pi$
$523$	−14657.4	−1.22548	−0.612738	+	0.790286i	$0.709932\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12375.6	1.02294
$528$	0	0
$529$	−1370.83	−0.112668
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3009.37	−0.244560
$534$	0	0
$535$	−7649.19	−0.618137
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18866.1	1.50764
$540$	0	0
$541$	13921.3	1.10633	0.553164	−	0.833072i	$-0.313420\pi$
$541$	13921.3	1.10633	0.553164	+	0.833072i	$0.313420\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4801.75	0.377403
$546$	0	0
$547$	2324.11	0.181667	0.0908335	−	0.995866i	$-0.471047\pi$
$547$	2324.11	0.181667	0.0908335	+	0.995866i	$0.471047\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5075.68	0.392434
$552$	0	0
$553$	5112.75	0.393158
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16962.8	1.29037	0.645185	−	0.764027i	$-0.276780\pi$
$557$	16962.8	1.29037	0.645185	+	0.764027i	$0.276780\pi$
$558$	0	0
$559$	4424.08	0.334738
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−389.000	−0.0291197	−0.0145599	−	0.999894i	$-0.504635\pi$
$563$	−389.000	−0.0291197	−0.0145599	+	0.999894i	$0.504635\pi$
$564$	0	0
$565$	−6367.69	−0.474143
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2217.56	−0.163383	−0.0816914	−	0.996658i	$-0.526032\pi$
$569$	−2217.56	−0.163383	−0.0816914	+	0.996658i	$0.526032\pi$
$570$	0	0
$571$	17087.3	1.25233	0.626167	−	0.779689i	$-0.284623\pi$
$571$	17087.3	1.25233	0.626167	+	0.779689i	$0.284623\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11401.4	−0.826906
$576$	0	0
$577$	−3977.26	−0.286959	−0.143480	−	0.989653i	$-0.545829\pi$
$577$	−3977.26	−0.286959	−0.143480	+	0.989653i	$0.545829\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5806.92	0.414650
$582$	0	0
$583$	14067.1	0.999312
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16880.3	−1.18693	−0.593463	−	0.804861i	$-0.702240\pi$
$587$	−16880.3	−1.18693	−0.593463	+	0.804861i	$0.702240\pi$
$588$	0	0
$589$	−6911.75	−0.483521
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2423.25	−0.167810	−0.0839048	−	0.996474i	$-0.526739\pi$
$593$	−2423.25	−0.167810	−0.0839048	+	0.996474i	$0.526739\pi$
$594$	0	0
$595$	−11936.1	−0.822408
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3900.55	−0.266064	−0.133032	−	0.991112i	$-0.542471\pi$
$599$	−3900.55	−0.266064	−0.133032	+	0.991112i	$0.542471\pi$
$600$	0	0
$601$	28653.4	1.94476	0.972378	−	0.233413i	$-0.0749893\pi$
$601$	28653.4	1.94476	0.972378	+	0.233413i	$0.0749893\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3761.54	−0.252774
$606$	0	0
$607$	−214.736	−0.0143589	−0.00717946	−	0.999974i	$-0.502285\pi$
$607$	−214.736	−0.0143589	−0.00717946	+	0.999974i	$0.502285\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1555.49	0.102993
$612$	0	0
$613$	26438.5	1.74199	0.870996	−	0.491290i	$-0.163475\pi$
$613$	26438.5	1.74199	0.870996	+	0.491290i	$0.163475\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6700.96	−0.437229	−0.218615	−	0.975811i	$-0.570154\pi$
$617$	−6700.96	−0.437229	−0.218615	+	0.975811i	$0.570154\pi$
$618$	0	0
$619$	27319.1	1.77391	0.886953	−	0.461860i	$-0.152818\pi$
$619$	27319.1	1.77391	0.886953	+	0.461860i	$0.152818\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	38886.6	2.50074
$624$	0	0
$625$	10131.7	0.648429
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13422.4	−0.850855
$630$	0	0
$631$	7126.87	0.449629	0.224815	−	0.974402i	$-0.427822\pi$
$631$	7126.87	0.449629	0.224815	+	0.974402i	$0.427822\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1078.98	−0.0674301
$636$	0	0
$637$	12777.7	0.794775
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23615.0	−1.45513	−0.727565	−	0.686039i	$-0.759348\pi$
$641$	−23615.0	−1.45513	−0.727565	+	0.686039i	$0.759348\pi$
$642$	0	0
$643$	8144.41	0.499509	0.249755	−	0.968309i	$-0.419650\pi$
$643$	8144.41	0.499509	0.249755	+	0.968309i	$0.419650\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9682.00	0.588313	0.294157	−	0.955757i	$-0.404961\pi$
$647$	9682.00	0.588313	0.294157	+	0.955757i	$0.404961\pi$
$648$	0	0
$649$	−4410.50	−0.266760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18193.6	1.09030	0.545152	−	0.838337i	$-0.316472\pi$
$653$	18193.6	1.09030	0.545152	+	0.838337i	$0.316472\pi$
$654$	0	0
$655$	−376.020	−0.0224310
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9300.88	0.549789	0.274895	−	0.961474i	$-0.411357\pi$
$659$	9300.88	0.549789	0.274895	+	0.961474i	$0.411357\pi$
$660$	0	0
$661$	−5437.29	−0.319949	−0.159974	−	0.987121i	$-0.551141\pi$
$661$	−5437.29	−0.319949	−0.159974	+	0.987121i	$0.551141\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6666.32	0.388735
$666$	0	0
$667$	−11257.1	−0.653489
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2091.08	0.120306
$672$	0	0
$673$	−8682.75	−0.497319	−0.248659	−	0.968591i	$-0.579990\pi$
$673$	−8682.75	−0.497319	−0.248659	+	0.968591i	$0.579990\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13300.1	−0.755041	−0.377521	−	0.926001i	$-0.623223\pi$
$677$	−13300.1	−0.755041	−0.377521	+	0.926001i	$0.623223\pi$
$678$	0	0
$679$	−31274.4	−1.76760
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−504.175	−0.0282455	−0.0141228	−	0.999900i	$-0.504496\pi$
$683$	−504.175	−0.0282455	−0.0141228	+	0.999900i	$0.504496\pi$
$684$	0	0
$685$	−10232.0	−0.570722
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9527.43	0.526802
$690$	0	0
$691$	−13443.8	−0.740124	−0.370062	−	0.929007i	$-0.620664\pi$
$691$	−13443.8	−0.740124	−0.370062	+	0.929007i	$0.620664\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7672.27	−0.418742
$696$	0	0
$697$	−19418.3	−1.05527
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28735.6	−1.54826	−0.774128	−	0.633030i	$-0.781811\pi$
$701$	−28735.6	−1.54826	−0.774128	+	0.633030i	$0.781811\pi$
$702$	0	0
$703$	7496.44	0.402181
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−57319.9	−3.04913
$708$	0	0
$709$	17610.2	0.932812	0.466406	−	0.884571i	$-0.345549\pi$
$709$	17610.2	0.932812	0.466406	+	0.884571i	$0.345549\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15329.3	0.805168
$714$	0	0
$715$	975.087	0.0510016
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9226.04	−0.478544	−0.239272	−	0.970953i	$-0.576909\pi$
$719$	−9226.04	−0.478544	−0.239272	+	0.970953i	$0.576909\pi$
$720$	0	0
$721$	−4723.63	−0.243990
$722$	0	0
$723$	0	0
$724$	0	0
$725$	11888.2	0.608987
$726$	0	0
$727$	−33246.0	−1.69604	−0.848022	−	0.529961i	$-0.822207\pi$
$727$	−33246.0	−1.69604	−0.848022	+	0.529961i	$0.822207\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	28546.9	1.44438
$732$	0	0
$733$	−4423.26	−0.222888	−0.111444	−	0.993771i	$-0.535548\pi$
$733$	−4423.26	−0.222888	−0.111444	+	0.993771i	$0.535548\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−199.535	−0.00997281
$738$	0	0
$739$	4529.56	0.225470	0.112735	−	0.993625i	$-0.464039\pi$
$739$	4529.56	0.225470	0.112735	+	0.993625i	$0.464039\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10851.5	0.535803	0.267901	−	0.963446i	$-0.413670\pi$
$743$	10851.5	0.535803	0.267901	+	0.963446i	$0.413670\pi$
$744$	0	0
$745$	1176.67	0.0578656
$746$	0	0
$747$	0	0
$748$	0	0
$749$	71275.9	3.47712
$750$	0	0
$751$	33022.6	1.60454	0.802272	−	0.596958i	$-0.203624\pi$
$751$	33022.6	1.60454	0.802272	+	0.596958i	$0.203624\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1340.26	−0.0646053
$756$	0	0
$757$	−3443.77	−0.165345	−0.0826724	−	0.996577i	$-0.526346\pi$
$757$	−3443.77	−0.165345	−0.0826724	+	0.996577i	$0.526346\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19562.6	−0.931858	−0.465929	−	0.884822i	$-0.654280\pi$
$761$	−19562.6	−0.931858	−0.465929	+	0.884822i	$0.654280\pi$
$762$	0	0
$763$	−44743.2	−2.12295
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2987.17	−0.140626
$768$	0	0
$769$	−17061.1	−0.800049	−0.400025	−	0.916504i	$-0.630998\pi$
$769$	−17061.1	−0.800049	−0.400025	+	0.916504i	$0.630998\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10798.4	−0.502448	−0.251224	−	0.967929i	$-0.580833\pi$
$773$	−10798.4	−0.502448	−0.251224	+	0.967929i	$0.580833\pi$
$774$	0	0
$775$	−16188.6	−0.750337
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10845.1	0.498802
$780$	0	0
$781$	−16683.7	−0.764391
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5029.16	−0.228661
$786$	0	0
$787$	35607.0	1.61277	0.806386	−	0.591390i	$-0.201420\pi$
$787$	35607.0	1.61277	0.806386	+	0.591390i	$0.201420\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	59334.8	2.66713
$792$	0	0
$793$	1416.26	0.0634210
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22155.3	−0.984668	−0.492334	−	0.870406i	$-0.663856\pi$
$797$	−22155.3	−0.984668	−0.492334	+	0.870406i	$0.663856\pi$
$798$	0	0
$799$	10037.0	0.444410
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−21095.9	−0.927094
$804$	0	0
$805$	−14784.9	−0.647329
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22524.6	0.978889	0.489445	−	0.872034i	$-0.337200\pi$
$809$	22524.6	0.978889	0.489445	+	0.872034i	$0.337200\pi$
$810$	0	0
$811$	−4452.39	−0.192780	−0.0963900	−	0.995344i	$-0.530730\pi$
$811$	−4452.39	−0.192780	−0.0963900	+	0.995344i	$0.530730\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2081.12	−0.0894460
$816$	0	0
$817$	−15943.4	−0.682729
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7097.27	0.301701	0.150851	−	0.988557i	$-0.451799\pi$
$821$	7097.27	0.301701	0.150851	+	0.988557i	$0.451799\pi$
$822$	0	0
$823$	12193.8	0.516463	0.258231	−	0.966083i	$-0.416860\pi$
$823$	12193.8	0.516463	0.258231	+	0.966083i	$0.416860\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7427.97	0.312329	0.156164	−	0.987731i	$-0.450087\pi$
$827$	7427.97	0.312329	0.156164	+	0.987731i	$0.450087\pi$
$828$	0	0
$829$	16966.2	0.710810	0.355405	−	0.934712i	$-0.384343\pi$
$829$	16966.2	0.710810	0.355405	+	0.934712i	$0.384343\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	82449.7	3.42943
$834$	0	0
$835$	163.947	0.00679477
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12025.7	0.494844	0.247422	−	0.968908i	$-0.420417\pi$
$839$	12025.7	0.494844	0.247422	+	0.968908i	$0.420417\pi$
$840$	0	0
$841$	−12651.3	−0.518728
$842$	0	0
$843$	0	0
$844$	0	0
$845$	660.412	0.0268862
$846$	0	0
$847$	35050.4	1.42189
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−16626.0	−0.669720
$852$	0	0
$853$	−22187.2	−0.890593	−0.445297	−	0.895383i	$-0.646902\pi$
$853$	−22187.2	−0.890593	−0.445297	+	0.895383i	$0.646902\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5746.19	−0.229038	−0.114519	−	0.993421i	$-0.536533\pi$
$857$	−5746.19	−0.229038	−0.114519	+	0.993421i	$0.536533\pi$
$858$	0	0
$859$	−8305.66	−0.329902	−0.164951	−	0.986302i	$-0.552747\pi$
$859$	−8305.66	−0.329902	−0.164951	+	0.986302i	$0.552747\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38086.4	1.50229	0.751146	−	0.660137i	$-0.229502\pi$
$863$	38086.4	1.50229	0.751146	+	0.660137i	$0.229502\pi$
$864$	0	0
$865$	4167.61	0.163819
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2695.07	−0.105206
$870$	0	0
$871$	−135.142	−0.00525731
$872$	0	0
$873$	0	0
$874$	0	0
$875$	33400.4	1.29044
$876$	0	0
$877$	−2098.53	−0.0808009	−0.0404005	−	0.999184i	$-0.512863\pi$
$877$	−2098.53	−0.0808009	−0.0404005	+	0.999184i	$0.512863\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14555.3	−0.556619	−0.278309	−	0.960491i	$-0.589774\pi$
$881$	−14555.3	−0.556619	−0.278309	+	0.960491i	$0.589774\pi$
$882$	0	0
$883$	−2122.88	−0.0809066	−0.0404533	−	0.999181i	$-0.512880\pi$
$883$	−2122.88	−0.0809066	−0.0404533	+	0.999181i	$0.512880\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12487.3	−0.472696	−0.236348	−	0.971668i	$-0.575951\pi$
$887$	−12487.3	−0.472696	−0.236348	+	0.971668i	$0.575951\pi$
$888$	0	0
$889$	10054.1	0.379305
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5605.66	−0.210063
$894$	0	0
$895$	12405.6	0.463323
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15983.7	−0.592978
$900$	0	0
$901$	61476.8	2.27313
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10650.3	−0.391193
$906$	0	0
$907$	−29679.8	−1.08655	−0.543275	−	0.839555i	$-0.682816\pi$
$907$	−29679.8	−1.08655	−0.543275	+	0.839555i	$0.682816\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24800.0	0.901934	0.450967	−	0.892541i	$-0.351079\pi$
$911$	24800.0	0.901934	0.450967	+	0.892541i	$0.351079\pi$
$912$	0	0
$913$	−3060.99	−0.110957
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3503.80	0.126178
$918$	0	0
$919$	6597.90	0.236828	0.118414	−	0.992964i	$-0.462219\pi$
$919$	6597.90	0.236828	0.118414	+	0.992964i	$0.462219\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11299.6	−0.402959
$924$	0	0
$925$	17558.0	0.624113
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15056.0	−0.531724	−0.265862	−	0.964011i	$-0.585656\pi$
$929$	−15056.0	−0.531724	−0.265862	+	0.964011i	$0.585656\pi$
$930$	0	0
$931$	−46048.1	−1.62102
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6291.85	0.220070
$936$	0	0
$937$	−35777.0	−1.24737	−0.623683	−	0.781677i	$-0.714365\pi$
$937$	−35777.0	−1.24737	−0.623683	+	0.781677i	$0.714365\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22973.6	0.795873	0.397937	−	0.917413i	$-0.369726\pi$
$941$	22973.6	0.795873	0.397937	+	0.917413i	$0.369726\pi$
$942$	0	0
$943$	−24052.9	−0.830615
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−51038.3	−1.75134	−0.875671	−	0.482908i	$-0.839580\pi$
$947$	−51038.3	−1.75134	−0.875671	+	0.482908i	$0.839580\pi$
$948$	0	0
$949$	−14287.9	−0.488731
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22586.5	0.767733	0.383866	−	0.923389i	$-0.374592\pi$
$953$	22586.5	0.767733	0.383866	+	0.923389i	$0.374592\pi$
$954$	0	0
$955$	3067.16	0.103928
$956$	0	0
$957$	0	0
$958$	0	0
$959$	95342.8	3.21041
$960$	0	0
$961$	−8025.32	−0.269387
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4907.64	−0.163712
$966$	0	0
$967$	36678.7	1.21976	0.609880	−	0.792494i	$-0.291218\pi$
$967$	36678.7	1.21976	0.609880	+	0.792494i	$0.291218\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32635.2	1.07859	0.539296	−	0.842116i	$-0.318690\pi$
$971$	32635.2	1.07859	0.539296	+	0.842116i	$0.318690\pi$
$972$	0	0
$973$	71491.0	2.35549
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44432.6	−1.45499	−0.727496	−	0.686112i	$-0.759316\pi$
$977$	−44432.6	−1.45499	−0.727496	+	0.686112i	$0.759316\pi$
$978$	0	0
$979$	−20498.2	−0.669178
$980$	0	0
$981$	0	0
$982$	0	0
$983$	484.485	0.0157199	0.00785996	−	0.999969i	$-0.497498\pi$
$983$	484.485	0.0157199	0.00785996	+	0.999969i	$0.497498\pi$
$984$	0	0
$985$	10853.2	0.351079
$986$	0	0
$987$	0	0
$988$	0	0
$989$	35360.2	1.13689
$990$	0	0
$991$	48017.1	1.53917	0.769583	−	0.638546i	$-0.220464\pi$
$991$	48017.1	1.53917	0.769583	+	0.638546i	$0.220464\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5824.25	−0.185569
$996$	0	0
$997$	−26561.9	−0.843755	−0.421877	−	0.906653i	$-0.638629\pi$
$997$	−26561.9	−0.843755	−0.421877	+	0.906653i	$0.638629\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.bk.1.2		3
3.2	odd	2		624.4.a.t.1.2		3
4.3	odd	2		117.4.a.f.1.3		3
12.11	even	2		39.4.a.c.1.1	✓	3
24.5	odd	2		2496.4.a.bp.1.2		3
24.11	even	2		2496.4.a.bl.1.2		3
52.51	odd	2		1521.4.a.u.1.1		3
60.59	even	2		975.4.a.l.1.3		3
84.83	odd	2		1911.4.a.k.1.1		3
156.47	odd	4		507.4.b.g.337.2		6
156.83	odd	4		507.4.b.g.337.5		6
156.155	even	2		507.4.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.1	✓	3	12.11	even	2
117.4.a.f.1.3		3	4.3	odd	2
507.4.a.h.1.3		3	156.155	even	2
507.4.b.g.337.2		6	156.47	odd	4
507.4.b.g.337.5		6	156.83	odd	4
624.4.a.t.1.2		3	3.2	odd	2
975.4.a.l.1.3		3	60.59	even	2
1521.4.a.u.1.1		3	52.51	odd	2
1872.4.a.bk.1.2		3	1.1	even	1	trivial
1911.4.a.k.1.1		3	84.83	odd	2
2496.4.a.bl.1.2		3	24.11	even	2
2496.4.a.bp.1.2		3	24.5	odd	2

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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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q+3.90776 q^{5} -36.4129 q^{7} +O(q^{10})\)	\(q+3.90776 q^{5} -36.4129 q^{7} +19.1943 q^{11} +13.0000 q^{13} +83.8839 q^{17} -46.8492 q^{19} +103.905 q^{23} -109.729 q^{25} -108.341 q^{29} +147.532 q^{31} -142.293 q^{35} -160.012 q^{37} -231.490 q^{41} +340.314 q^{43} +119.653 q^{47} +982.902 q^{49} +732.879 q^{53} +75.0067 q^{55} -229.782 q^{59} +108.943 q^{61} +50.8009 q^{65} -10.3955 q^{67} -869.201 q^{71} -1099.07 q^{73} -698.920 q^{77} -140.410 q^{79} -159.474 q^{83} +327.799 q^{85} -1067.93 q^{89} -473.368 q^{91} -183.075 q^{95} +858.881 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{5} - 30 q^{7}+O(q^{10}) \) 3 * q - 4 * q^5 - 30 * q^7	\( 3 q - 4 q^{5} - 30 q^{7} - 16 q^{11} + 39 q^{13} + 146 q^{17} - 94 q^{19} - 48 q^{23} + 145 q^{25} + 2 q^{29} - 302 q^{31} + 80 q^{35} + 374 q^{37} - 480 q^{41} + 260 q^{43} - 24 q^{47} + 447 q^{49} + 678 q^{53} + 1552 q^{55} - 1788 q^{59} + 230 q^{61} - 52 q^{65} - 74 q^{67} - 948 q^{71} - 222 q^{73} - 112 q^{77} + 24 q^{79} - 796 q^{83} - 248 q^{85} - 1436 q^{89} - 390 q^{91} - 4032 q^{95} + 3242 q^{97}+O(q^{100}) \) 3 * q - 4 * q^5 - 30 * q^7 - 16 * q^11 + 39 * q^13 + 146 * q^17 - 94 * q^19 - 48 * q^23 + 145 * q^25 + 2 * q^29 - 302 * q^31 + 80 * q^35 + 374 * q^37 - 480 * q^41 + 260 * q^43 - 24 * q^47 + 447 * q^49 + 678 * q^53 + 1552 * q^55 - 1788 * q^59 + 230 * q^61 - 52 * q^65 - 74 * q^67 - 948 * q^71 - 222 * q^73 - 112 * q^77 + 24 * q^79 - 796 * q^83 - 248 * q^85 - 1436 * q^89 - 390 * q^91 - 4032 * q^95 + 3242 * q^97

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