LMFDB - Embedded newform 1872.4.a.bi.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q+4.87689 q^{5} +25.6155 q^{7} +59.7235 q^{11} +13.0000 q^{13} +75.4773 q^{17} +116.354 q^{19} -90.5227 q^{23} -101.216 q^{25} +187.261 q^{29} +225.062 q^{31} +124.924 q^{35} +290.648 q^{37} +191.339 q^{41} -326.833 q^{43} -406.773 q^{47} +313.155 q^{49} +426.985 q^{53} +291.265 q^{55} +331.015 q^{59} -524.678 q^{61} +63.3996 q^{65} -968.172 q^{67} -8.39776 q^{71} -903.329 q^{73} +1529.85 q^{77} -1157.98 q^{79} +952.371 q^{83} +368.095 q^{85} -1059.96 q^{89} +333.002 q^{91} +567.447 q^{95} -90.7689 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{5} + 10 q^{7} + 4 q^{11} + 26 q^{13} + 52 q^{17} + 142 q^{19} - 280 q^{23} - 54 q^{25} + 424 q^{29} + 178 q^{31} - 80 q^{35} + 136 q^{37} + 226 q^{41} + 72 q^{43} + 176 q^{47} + 214 q^{49} + 788 q^{53}+ \cdots - 264 q^{97}+O(q^{100}) $ 2 * q + 18 * q^5 + 10 * q^7 + 4 * q^11 + 26 * q^13 + 52 * q^17 + 142 * q^19 - 280 * q^23 - 54 * q^25 + 424 * q^29 + 178 * q^31 - 80 * q^35 + 136 * q^37 + 226 * q^41 + 72 * q^43 + 176 * q^47 + 214 * q^49 + 788 * q^53 - 440 * q^55 + 728 * q^59 - 736 * q^61 + 234 * q^65 - 1054 * q^67 - 660 * q^71 + 24 * q^73 + 2400 * q^77 - 40 * q^79 + 1344 * q^83 + 60 * q^85 - 314 * q^89 + 130 * q^91 + 904 * q^95 - 264 * 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
$5$	4.87689	0.436203	0.218101	−	0.975926i	$-0.430014\pi$
$5$	4.87689	0.436203	0.218101	+	0.975926i	$0.430014\pi$
$6$	0	0
$7$	25.6155	1.38311	0.691554	−	0.722325i	$-0.256926\pi$
$7$	25.6155	1.38311	0.691554	+	0.722325i	$0.256926\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	59.7235	1.63703	0.818514	−	0.574487i	$-0.194798\pi$
$11$	59.7235	1.63703	0.818514	+	0.574487i	$0.194798\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	75.4773	1.07682	0.538410	−	0.842683i	$-0.319025\pi$
$17$	75.4773	1.07682	0.538410	+	0.842683i	$0.319025\pi$
$18$	0	0
$19$	116.354	1.40492	0.702460	−	0.711723i	$-0.252085\pi$
$19$	116.354	1.40492	0.702460	+	0.711723i	$0.252085\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−90.5227	−0.820665	−0.410332	−	0.911936i	$-0.634587\pi$
$23$	−90.5227	−0.820665	−0.410332	+	0.911936i	$0.634587\pi$
$24$	0	0
$25$	−101.216	−0.809727
$26$	0	0
$27$	0	0
$28$	0	0
$29$	187.261	1.19909	0.599544	−	0.800342i	$-0.295349\pi$
$29$	187.261	1.19909	0.599544	+	0.800342i	$0.295349\pi$
$30$	0	0
$31$	225.062	1.30395	0.651974	−	0.758241i	$-0.273941\pi$
$31$	225.062	1.30395	0.651974	+	0.758241i	$0.273941\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	124.924	0.603316
$36$	0	0
$37$	290.648	1.29141	0.645705	−	0.763587i	$-0.276563\pi$
$37$	290.648	1.29141	0.645705	+	0.763587i	$0.276563\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	191.339	0.728833	0.364416	−	0.931236i	$-0.381269\pi$
$41$	191.339	0.728833	0.364416	+	0.931236i	$0.381269\pi$
$42$	0	0
$43$	−326.833	−1.15911	−0.579554	−	0.814934i	$-0.696773\pi$
$43$	−326.833	−1.15911	−0.579554	+	0.814934i	$0.696773\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−406.773	−1.26242	−0.631212	−	0.775611i	$-0.717442\pi$
$47$	−406.773	−1.26242	−0.631212	+	0.775611i	$0.717442\pi$
$48$	0	0
$49$	313.155	0.912989
$50$	0	0
$51$	0	0
$52$	0	0
$53$	426.985	1.10662	0.553310	−	0.832975i	$-0.313364\pi$
$53$	426.985	1.10662	0.553310	+	0.832975i	$0.313364\pi$
$54$	0	0
$55$	291.265	0.714076
$56$	0	0
$57$	0	0
$58$	0	0
$59$	331.015	0.730415	0.365208	−	0.930926i	$-0.380998\pi$
$59$	331.015	0.730415	0.365208	+	0.930926i	$0.380998\pi$
$60$	0	0
$61$	−524.678	−1.10128	−0.550640	−	0.834743i	$-0.685616\pi$
$61$	−524.678	−1.10128	−0.550640	+	0.834743i	$0.685616\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	63.3996	0.120981
$66$	0	0
$67$	−968.172	−1.76539	−0.882695	−	0.469947i	$-0.844273\pi$
$67$	−968.172	−1.76539	−0.882695	+	0.469947i	$0.844273\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.39776	−0.0140371	−0.00701853	−	0.999975i	$-0.502234\pi$
$71$	−8.39776	−0.0140371	−0.00701853	+	0.999975i	$0.502234\pi$
$72$	0	0
$73$	−903.329	−1.44831	−0.724156	−	0.689637i	$-0.757770\pi$
$73$	−903.329	−1.44831	−0.724156	+	0.689637i	$0.757770\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1529.85	2.26419
$78$	0	0
$79$	−1157.98	−1.64915	−0.824573	−	0.565755i	$-0.808585\pi$
$79$	−1157.98	−1.64915	−0.824573	+	0.565755i	$0.808585\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	952.371	1.25947	0.629737	−	0.776809i	$-0.283163\pi$
$83$	952.371	1.25947	0.629737	+	0.776809i	$0.283163\pi$
$84$	0	0
$85$	368.095	0.469711
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1059.96	−1.26242	−0.631211	−	0.775611i	$-0.717442\pi$
$89$	−1059.96	−1.26242	−0.631211	+	0.775611i	$0.717442\pi$
$90$	0	0
$91$	333.002	0.383605
$92$	0	0
$93$	0	0
$94$	0	0
$95$	567.447	0.612830
$96$	0	0
$97$	−90.7689	−0.0950123	−0.0475061	−	0.998871i	$-0.515127\pi$
$97$	−90.7689	−0.0950123	−0.0475061	+	0.998871i	$0.515127\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−39.4507	−0.0388662	−0.0194331	−	0.999811i	$-0.506186\pi$
$101$	−39.4507	−0.0388662	−0.0194331	+	0.999811i	$0.506186\pi$
$102$	0	0
$103$	−445.599	−0.426273	−0.213137	−	0.977022i	$-0.568368\pi$
$103$	−445.599	−0.426273	−0.213137	+	0.977022i	$0.568368\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1079.89	−0.975669	−0.487834	−	0.872936i	$-0.662213\pi$
$107$	−1079.89	−0.975669	−0.487834	+	0.872936i	$0.662213\pi$
$108$	0	0
$109$	−551.788	−0.484878	−0.242439	−	0.970167i	$-0.577947\pi$
$109$	−551.788	−0.484878	−0.242439	+	0.970167i	$0.577947\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2173.98	1.80983	0.904916	−	0.425591i	$-0.139934\pi$
$113$	2173.98	1.80983	0.904916	+	0.425591i	$0.139934\pi$
$114$	0	0
$115$	−441.470	−0.357976
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1933.39	1.48936
$120$	0	0
$121$	2235.89	1.67986
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1103.23	−0.789408
$126$	0	0
$127$	959.360	0.670310	0.335155	−	0.942163i	$-0.391211\pi$
$127$	959.360	0.670310	0.335155	+	0.942163i	$0.391211\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1859.58	−1.24025	−0.620124	−	0.784504i	$-0.712918\pi$
$131$	−1859.58	−1.24025	−0.620124	+	0.784504i	$0.712918\pi$
$132$	0	0
$133$	2980.47	1.94316
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1601.07	0.998455	0.499227	−	0.866471i	$-0.333617\pi$
$137$	1601.07	0.998455	0.499227	+	0.866471i	$0.333617\pi$
$138$	0	0
$139$	−1278.56	−0.780185	−0.390093	−	0.920776i	$-0.627557\pi$
$139$	−1278.56	−0.780185	−0.390093	+	0.920776i	$0.627557\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	776.405	0.454030
$144$	0	0
$145$	913.254	0.523046
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1928.65	−1.06041	−0.530205	−	0.847869i	$-0.677885\pi$
$149$	−1928.65	−1.06041	−0.530205	+	0.847869i	$0.677885\pi$
$150$	0	0
$151$	−2822.23	−1.52099	−0.760496	−	0.649343i	$-0.775044\pi$
$151$	−2822.23	−1.52099	−0.760496	+	0.649343i	$0.775044\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1097.61	0.568786
$156$	0	0
$157$	−690.345	−0.350927	−0.175463	−	0.984486i	$-0.556142\pi$
$157$	−690.345	−0.350927	−0.175463	+	0.984486i	$0.556142\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2318.79	−1.13507
$162$	0	0
$163$	−3606.51	−1.73303	−0.866515	−	0.499151i	$-0.833645\pi$
$163$	−3606.51	−1.73303	−0.866515	+	0.499151i	$0.833645\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1019.70	−0.472494	−0.236247	−	0.971693i	$-0.575917\pi$
$167$	−1019.70	−0.472494	−0.236247	+	0.971693i	$0.575917\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2395.16	1.05260	0.526301	−	0.850298i	$-0.323578\pi$
$173$	2395.16	1.05260	0.526301	+	0.850298i	$0.323578\pi$
$174$	0	0
$175$	−2592.70	−1.11994
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3609.61	1.50724	0.753618	−	0.657313i	$-0.228307\pi$
$179$	3609.61	1.50724	0.753618	+	0.657313i	$0.228307\pi$
$180$	0	0
$181$	−1848.55	−0.759125	−0.379562	−	0.925166i	$-0.623925\pi$
$181$	−1848.55	−0.759125	−0.379562	+	0.925166i	$0.623925\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1417.46	0.563317
$186$	0	0
$187$	4507.76	1.76278
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2364.55	−0.895772	−0.447886	−	0.894091i	$-0.647823\pi$
$191$	−2364.55	−0.895772	−0.447886	+	0.894091i	$0.647823\pi$
$192$	0	0
$193$	4199.18	1.56613	0.783067	−	0.621938i	$-0.213654\pi$
$193$	4199.18	1.56613	0.783067	+	0.621938i	$0.213654\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	848.869	0.307002	0.153501	−	0.988148i	$-0.450945\pi$
$197$	848.869	0.307002	0.153501	+	0.988148i	$0.450945\pi$
$198$	0	0
$199$	−620.958	−0.221199	−0.110599	−	0.993865i	$-0.535277\pi$
$199$	−620.958	−0.221199	−0.110599	+	0.993865i	$0.535277\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4796.80	1.65847
$204$	0	0
$205$	933.140	0.317919
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6949.08	2.29989
$210$	0	0
$211$	2911.89	0.950059	0.475030	−	0.879970i	$-0.342437\pi$
$211$	2911.89	0.950059	0.475030	+	0.879970i	$0.342437\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1593.93	−0.505606
$216$	0	0
$217$	5765.09	1.80350
$218$	0	0
$219$	0	0
$220$	0	0
$221$	981.204	0.298656
$222$	0	0
$223$	5433.63	1.63167	0.815836	−	0.578283i	$-0.196277\pi$
$223$	5433.63	1.63167	0.815836	+	0.578283i	$0.196277\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4797.03	−1.40260	−0.701300	−	0.712866i	$-0.747397\pi$
$227$	−4797.03	−1.40260	−0.701300	+	0.712866i	$0.747397\pi$
$228$	0	0
$229$	932.947	0.269218	0.134609	−	0.990899i	$-0.457022\pi$
$229$	932.947	0.269218	0.134609	+	0.990899i	$0.457022\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	169.095	0.0475441	0.0237720	−	0.999717i	$-0.492432\pi$
$233$	169.095	0.0475441	0.0237720	+	0.999717i	$0.492432\pi$
$234$	0	0
$235$	−1983.79	−0.550672
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1467.25	−0.397106	−0.198553	−	0.980090i	$-0.563624\pi$
$239$	−1467.25	−0.397106	−0.198553	+	0.980090i	$0.563624\pi$
$240$	0	0
$241$	361.064	0.0965070	0.0482535	−	0.998835i	$-0.484634\pi$
$241$	361.064	0.0965070	0.0482535	+	0.998835i	$0.484634\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1527.23	0.398248
$246$	0	0
$247$	1512.60	0.389655
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1271.80	0.319823	0.159911	−	0.987131i	$-0.448879\pi$
$251$	1271.80	0.319823	0.159911	+	0.987131i	$0.448879\pi$
$252$	0	0
$253$	−5406.33	−1.34345
$254$	0	0
$255$	0	0
$256$	0	0
$257$	842.178	0.204411	0.102205	−	0.994763i	$-0.467410\pi$
$257$	842.178	0.204411	0.102205	+	0.994763i	$0.467410\pi$
$258$	0	0
$259$	7445.09	1.78616
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3988.33	0.935099	0.467550	−	0.883967i	$-0.345137\pi$
$263$	3988.33	0.935099	0.467550	+	0.883967i	$0.345137\pi$
$264$	0	0
$265$	2082.36	0.482711
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4850.14	−1.09932	−0.549662	−	0.835387i	$-0.685244\pi$
$269$	−4850.14	−1.09932	−0.549662	+	0.835387i	$0.685244\pi$
$270$	0	0
$271$	−1997.17	−0.447674	−0.223837	−	0.974627i	$-0.571858\pi$
$271$	−1997.17	−0.447674	−0.223837	+	0.974627i	$0.571858\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6044.97	−1.32555
$276$	0	0
$277$	−615.909	−0.133597	−0.0667985	−	0.997766i	$-0.521278\pi$
$277$	−615.909	−0.133597	−0.0667985	+	0.997766i	$0.521278\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4322.77	−0.917703	−0.458852	−	0.888513i	$-0.651739\pi$
$281$	−4322.77	−0.917703	−0.458852	+	0.888513i	$0.651739\pi$
$282$	0	0
$283$	1033.91	0.217172	0.108586	−	0.994087i	$-0.465368\pi$
$283$	1033.91	0.217172	0.108586	+	0.994087i	$0.465368\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4901.25	1.00805
$288$	0	0
$289$	783.818	0.159540
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8391.93	1.67325	0.836625	−	0.547777i	$-0.184526\pi$
$293$	8391.93	1.67325	0.836625	+	0.547777i	$0.184526\pi$
$294$	0	0
$295$	1614.33	0.318609
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1176.80	−0.227612
$300$	0	0
$301$	−8372.01	−1.60317
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2558.80	−0.480382
$306$	0	0
$307$	4374.08	0.813166	0.406583	−	0.913614i	$-0.366720\pi$
$307$	4374.08	0.813166	0.406583	+	0.913614i	$0.366720\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5981.42	1.09060	0.545298	−	0.838242i	$-0.316416\pi$
$311$	5981.42	1.09060	0.545298	+	0.838242i	$0.316416\pi$
$312$	0	0
$313$	1159.92	0.209466	0.104733	−	0.994500i	$-0.466601\pi$
$313$	1159.92	0.209466	0.104733	+	0.994500i	$0.466601\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1502.60	0.266229	0.133115	−	0.991101i	$-0.457502\pi$
$317$	1502.60	0.266229	0.133115	+	0.991101i	$0.457502\pi$
$318$	0	0
$319$	11183.9	1.96294
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8782.09	1.51284
$324$	0	0
$325$	−1315.81	−0.224578
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10419.7	−1.74607
$330$	0	0
$331$	472.260	0.0784221	0.0392111	−	0.999231i	$-0.487516\pi$
$331$	472.260	0.0784221	0.0392111	+	0.999231i	$0.487516\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4721.67	−0.770067
$336$	0	0
$337$	−8318.44	−1.34461	−0.672306	−	0.740273i	$-0.734696\pi$
$337$	−8318.44	−1.34461	−0.672306	+	0.740273i	$0.734696\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13441.5	2.13460
$342$	0	0
$343$	−764.488	−0.120345
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11115.7	−1.71967	−0.859834	−	0.510574i	$-0.829433\pi$
$347$	−11115.7	−1.71967	−0.859834	+	0.510574i	$0.829433\pi$
$348$	0	0
$349$	6057.81	0.929133	0.464566	−	0.885538i	$-0.346210\pi$
$349$	6057.81	0.929133	0.464566	+	0.885538i	$0.346210\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−675.403	−0.101836	−0.0509179	−	0.998703i	$-0.516215\pi$
$353$	−675.403	−0.101836	−0.0509179	+	0.998703i	$0.516215\pi$
$354$	0	0
$355$	−40.9550	−0.00612300
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−351.424	−0.0516642	−0.0258321	−	0.999666i	$-0.508224\pi$
$359$	−351.424	−0.0516642	−0.0258321	+	0.999666i	$0.508224\pi$
$360$	0	0
$361$	6679.29	0.973800
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4405.44	−0.631757
$366$	0	0
$367$	6707.82	0.954075	0.477037	−	0.878883i	$-0.341711\pi$
$367$	6707.82	0.954075	0.477037	+	0.878883i	$0.341711\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10937.4	1.53058
$372$	0	0
$373$	7236.36	1.00452	0.502258	−	0.864718i	$-0.332503\pi$
$373$	7236.36	1.00452	0.502258	+	0.864718i	$0.332503\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2434.40	0.332567
$378$	0	0
$379$	−4084.92	−0.553636	−0.276818	−	0.960922i	$-0.589280\pi$
$379$	−4084.92	−0.553636	−0.276818	+	0.960922i	$0.589280\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7764.76	1.03593	0.517965	−	0.855402i	$-0.326690\pi$
$383$	7764.76	1.03593	0.517965	+	0.855402i	$0.326690\pi$
$384$	0	0
$385$	7460.91	0.987645
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1927.25	0.251197	0.125598	−	0.992081i	$-0.459915\pi$
$389$	1927.25	0.251197	0.125598	+	0.992081i	$0.459915\pi$
$390$	0	0
$391$	−6832.41	−0.883708
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5647.33	−0.719362
$396$	0	0
$397$	−2607.17	−0.329597	−0.164799	−	0.986327i	$-0.552697\pi$
$397$	−2607.17	−0.329597	−0.164799	+	0.986327i	$0.552697\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3890.24	0.484462	0.242231	−	0.970219i	$-0.422121\pi$
$401$	3890.24	0.484462	0.242231	+	0.970219i	$0.422121\pi$
$402$	0	0
$403$	2925.81	0.361650
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17358.5	2.11407
$408$	0	0
$409$	5916.54	0.715291	0.357646	−	0.933857i	$-0.383580\pi$
$409$	5916.54	0.715291	0.357646	+	0.933857i	$0.383580\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8479.13	1.01024
$414$	0	0
$415$	4644.61	0.549386
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1729.83	0.201689	0.100845	−	0.994902i	$-0.467845\pi$
$419$	1729.83	0.201689	0.100845	+	0.994902i	$0.467845\pi$
$420$	0	0
$421$	−5546.89	−0.642135	−0.321067	−	0.947056i	$-0.604042\pi$
$421$	−5546.89	−0.642135	−0.321067	+	0.947056i	$0.604042\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7639.50	−0.871930
$426$	0	0
$427$	−13439.9	−1.52319
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11184.4	−1.24996	−0.624979	−	0.780641i	$-0.714893\pi$
$431$	−11184.4	−1.24996	−0.624979	+	0.780641i	$0.714893\pi$
$432$	0	0
$433$	4298.00	0.477017	0.238509	−	0.971140i	$-0.423341\pi$
$433$	4298.00	0.477017	0.238509	+	0.971140i	$0.423341\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10532.7	−1.15297
$438$	0	0
$439$	12481.1	1.35692	0.678461	−	0.734636i	$-0.262647\pi$
$439$	12481.1	1.35692	0.678461	+	0.734636i	$0.262647\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12999.3	−1.39417	−0.697084	−	0.716989i	$-0.745520\pi$
$443$	−12999.3	−1.39417	−0.697084	+	0.716989i	$0.745520\pi$
$444$	0	0
$445$	−5169.31	−0.550672
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14636.6	−1.53840	−0.769201	−	0.639007i	$-0.779346\pi$
$449$	−14636.6	−1.53840	−0.769201	+	0.639007i	$0.779346\pi$
$450$	0	0
$451$	11427.4	1.19312
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1624.01	0.167330
$456$	0	0
$457$	−8803.22	−0.901088	−0.450544	−	0.892754i	$-0.648770\pi$
$457$	−8803.22	−0.901088	−0.450544	+	0.892754i	$0.648770\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13223.9	1.33601	0.668004	−	0.744158i	$-0.267149\pi$
$461$	13223.9	1.33601	0.668004	+	0.744158i	$0.267149\pi$
$462$	0	0
$463$	14136.4	1.41895	0.709476	−	0.704729i	$-0.248932\pi$
$463$	14136.4	1.41895	0.709476	+	0.704729i	$0.248932\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8654.11	0.857525	0.428763	−	0.903417i	$-0.358950\pi$
$467$	8654.11	0.857525	0.428763	+	0.903417i	$0.358950\pi$
$468$	0	0
$469$	−24800.2	−2.44172
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−19519.6	−1.89749
$474$	0	0
$475$	−11776.9	−1.13760
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3622.59	0.345554	0.172777	−	0.984961i	$-0.444726\pi$
$479$	3622.59	0.345554	0.172777	+	0.984961i	$0.444726\pi$
$480$	0	0
$481$	3778.42	0.358173
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−442.671	−0.0414446
$486$	0	0
$487$	−20857.4	−1.94074	−0.970368	−	0.241632i	$-0.922317\pi$
$487$	−20857.4	−1.94074	−0.970368	+	0.241632i	$0.922317\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5060.09	−0.465089	−0.232545	−	0.972586i	$-0.574705\pi$
$491$	−5060.09	−0.465089	−0.232545	+	0.972586i	$0.574705\pi$
$492$	0	0
$493$	14134.0	1.29120
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−215.113	−0.0194148
$498$	0	0
$499$	1150.05	0.103173	0.0515864	−	0.998669i	$-0.483572\pi$
$499$	1150.05	0.103173	0.0515864	+	0.998669i	$0.483572\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7145.59	0.633412	0.316706	−	0.948524i	$-0.397423\pi$
$503$	7145.59	0.633412	0.316706	+	0.948524i	$0.397423\pi$
$504$	0	0
$505$	−192.397	−0.0169536
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7240.54	−0.630513	−0.315257	−	0.949006i	$-0.602091\pi$
$509$	−7240.54	−0.630513	−0.315257	+	0.949006i	$0.602091\pi$
$510$	0	0
$511$	−23139.3	−2.00317
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2173.14	−0.185941
$516$	0	0
$517$	−24293.9	−2.06662
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2690.90	−0.226277	−0.113139	−	0.993579i	$-0.536090\pi$
$521$	−2690.90	−0.226277	−0.113139	+	0.993579i	$0.536090\pi$
$522$	0	0
$523$	−507.043	−0.0423928	−0.0211964	−	0.999775i	$-0.506748\pi$
$523$	−507.043	−0.0423928	−0.0211964	+	0.999775i	$0.506748\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16987.1	1.40412
$528$	0	0
$529$	−3972.63	−0.326509
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2487.41	0.202142
$534$	0	0
$535$	−5266.49	−0.425589
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18702.7	1.49459
$540$	0	0
$541$	997.477	0.0792696	0.0396348	−	0.999214i	$-0.487381\pi$
$541$	997.477	0.0792696	0.0396348	+	0.999214i	$0.487381\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2691.01	−0.211505
$546$	0	0
$547$	15191.1	1.18743	0.593716	−	0.804675i	$-0.297660\pi$
$547$	15191.1	1.18743	0.593716	+	0.804675i	$0.297660\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	21788.6	1.68462
$552$	0	0
$553$	−29662.2	−2.28095
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1426.17	0.108489	0.0542447	−	0.998528i	$-0.482725\pi$
$557$	1426.17	0.108489	0.0542447	+	0.998528i	$0.482725\pi$
$558$	0	0
$559$	−4248.83	−0.321478
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14066.7	1.05300	0.526502	−	0.850174i	$-0.323503\pi$
$563$	14066.7	1.05300	0.526502	+	0.850174i	$0.323503\pi$
$564$	0	0
$565$	10602.3	0.789453
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20020.8	1.47507	0.737537	−	0.675307i	$-0.235989\pi$
$569$	20020.8	1.47507	0.737537	+	0.675307i	$0.235989\pi$
$570$	0	0
$571$	6023.04	0.441430	0.220715	−	0.975338i	$-0.429161\pi$
$571$	6023.04	0.441430	0.220715	+	0.975338i	$0.429161\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9162.34	0.664515
$576$	0	0
$577$	−6825.67	−0.492472	−0.246236	−	0.969210i	$-0.579194\pi$
$577$	−6825.67	−0.492472	−0.246236	+	0.969210i	$0.579194\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24395.5	1.74199
$582$	0	0
$583$	25501.0	1.81157
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24638.4	−1.73243	−0.866213	−	0.499675i	$-0.833453\pi$
$587$	−24638.4	−1.73243	−0.866213	+	0.499675i	$0.833453\pi$
$588$	0	0
$589$	26187.0	1.83194
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4694.14	0.325068	0.162534	−	0.986703i	$-0.448033\pi$
$593$	4694.14	0.325068	0.162534	+	0.986703i	$0.448033\pi$
$594$	0	0
$595$	9428.94	0.649662
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10043.5	0.685087	0.342544	−	0.939502i	$-0.388712\pi$
$599$	10043.5	0.685087	0.342544	+	0.939502i	$0.388712\pi$
$600$	0	0
$601$	−20677.8	−1.40343	−0.701717	−	0.712456i	$-0.747583\pi$
$601$	−20677.8	−1.40343	−0.701717	+	0.712456i	$0.747583\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10904.2	0.732760
$606$	0	0
$607$	−4990.49	−0.333703	−0.166851	−	0.985982i	$-0.553360\pi$
$607$	−4990.49	−0.333703	−0.166851	+	0.985982i	$0.553360\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5288.04	−0.350133
$612$	0	0
$613$	−27970.0	−1.84290	−0.921450	−	0.388496i	$-0.872995\pi$
$613$	−27970.0	−1.84290	−0.921450	+	0.388496i	$0.872995\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2978.15	0.194321	0.0971603	−	0.995269i	$-0.469024\pi$
$617$	2978.15	0.194321	0.0971603	+	0.995269i	$0.469024\pi$
$618$	0	0
$619$	−29605.9	−1.92239	−0.961196	−	0.275865i	$-0.911036\pi$
$619$	−29605.9	−1.92239	−0.961196	+	0.275865i	$0.911036\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−27151.4	−1.74607
$624$	0	0
$625$	7271.65	0.465385
$626$	0	0
$627$	0	0
$628$	0	0
$629$	21937.3	1.39062
$630$	0	0
$631$	937.256	0.0591309	0.0295654	−	0.999563i	$-0.490588\pi$
$631$	937.256	0.0591309	0.0295654	+	0.999563i	$0.490588\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4678.70	0.292391
$636$	0	0
$637$	4071.02	0.253218
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8879.08	0.547118	0.273559	−	0.961855i	$-0.411799\pi$
$641$	8879.08	0.547118	0.273559	+	0.961855i	$0.411799\pi$
$642$	0	0
$643$	8098.91	0.496718	0.248359	−	0.968668i	$-0.420109\pi$
$643$	8098.91	0.496718	0.248359	+	0.968668i	$0.420109\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16069.3	−0.976429	−0.488214	−	0.872724i	$-0.662352\pi$
$647$	−16069.3	−0.976429	−0.488214	+	0.872724i	$0.662352\pi$
$648$	0	0
$649$	19769.4	1.19571
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4610.46	−0.276296	−0.138148	−	0.990412i	$-0.544115\pi$
$653$	−4610.46	−0.276296	−0.138148	+	0.990412i	$0.544115\pi$
$654$	0	0
$655$	−9068.99	−0.541000
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24960.1	−1.47543	−0.737714	−	0.675113i	$-0.764095\pi$
$659$	−24960.1	−1.47543	−0.737714	+	0.675113i	$0.764095\pi$
$660$	0	0
$661$	24437.3	1.43798	0.718988	−	0.695023i	$-0.244606\pi$
$661$	24437.3	1.43798	0.718988	+	0.695023i	$0.244606\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	14535.5	0.847610
$666$	0	0
$667$	−16951.4	−0.984050
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−31335.6	−1.80283
$672$	0	0
$673$	2702.96	0.154816	0.0774081	−	0.996999i	$-0.475336\pi$
$673$	2702.96	0.154816	0.0774081	+	0.996999i	$0.475336\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11071.8	−0.628544	−0.314272	−	0.949333i	$-0.601760\pi$
$677$	−11071.8	−0.628544	−0.314272	+	0.949333i	$0.601760\pi$
$678$	0	0
$679$	−2325.09	−0.131412
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11778.4	−0.659864	−0.329932	−	0.944005i	$-0.607026\pi$
$683$	−11778.4	−0.659864	−0.329932	+	0.944005i	$0.607026\pi$
$684$	0	0
$685$	7808.23	0.435529
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5550.80	0.306921
$690$	0	0
$691$	−30715.1	−1.69097	−0.845483	−	0.534002i	$-0.820687\pi$
$691$	−30715.1	−1.69097	−0.845483	+	0.534002i	$0.820687\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6235.39	−0.340319
$696$	0	0
$697$	14441.7	0.784821
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−741.115	−0.0399308	−0.0199654	−	0.999801i	$-0.506356\pi$
$701$	−741.115	−0.0399308	−0.0199654	+	0.999801i	$0.506356\pi$
$702$	0	0
$703$	33818.1	1.81433
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1010.55	−0.0537562
$708$	0	0
$709$	−24269.6	−1.28556	−0.642781	−	0.766050i	$-0.722219\pi$
$709$	−24269.6	−1.28556	−0.642781	+	0.766050i	$0.722219\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−20373.3	−1.07011
$714$	0	0
$715$	3786.45	0.198049
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15598.6	0.809083	0.404541	−	0.914520i	$-0.367431\pi$
$719$	15598.6	0.809083	0.404541	+	0.914520i	$0.367431\pi$
$720$	0	0
$721$	−11414.2	−0.589582
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−18953.8	−0.970934
$726$	0	0
$727$	32399.2	1.65285	0.826423	−	0.563050i	$-0.190372\pi$
$727$	32399.2	1.65285	0.826423	+	0.563050i	$0.190372\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24668.5	−1.24815
$732$	0	0
$733$	7851.92	0.395658	0.197829	−	0.980237i	$-0.436611\pi$
$733$	7851.92	0.395658	0.197829	+	0.980237i	$0.436611\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−57822.6	−2.88999
$738$	0	0
$739$	−13348.4	−0.664451	−0.332226	−	0.943200i	$-0.607800\pi$
$739$	−13348.4	−0.664451	−0.332226	+	0.943200i	$0.607800\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9303.61	−0.459376	−0.229688	−	0.973264i	$-0.573771\pi$
$743$	−9303.61	−0.459376	−0.229688	+	0.973264i	$0.573771\pi$
$744$	0	0
$745$	−9405.82	−0.462554
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−27661.9	−1.34946
$750$	0	0
$751$	9774.38	0.474930	0.237465	−	0.971396i	$-0.423684\pi$
$751$	9774.38	0.474930	0.237465	+	0.971396i	$0.423684\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−13763.7	−0.663461
$756$	0	0
$757$	24541.6	1.17831	0.589154	−	0.808021i	$-0.299461\pi$
$757$	24541.6	1.17831	0.589154	+	0.808021i	$0.299461\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14409.4	0.686389	0.343194	−	0.939264i	$-0.388491\pi$
$761$	14409.4	0.686389	0.343194	+	0.939264i	$0.388491\pi$
$762$	0	0
$763$	−14134.3	−0.670639
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4303.20	0.202581
$768$	0	0
$769$	4426.67	0.207581	0.103791	−	0.994599i	$-0.466903\pi$
$769$	4426.67	0.207581	0.103791	+	0.994599i	$0.466903\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4608.64	0.214439	0.107219	−	0.994235i	$-0.465805\pi$
$773$	4608.64	0.214439	0.107219	+	0.994235i	$0.465805\pi$
$774$	0	0
$775$	−22779.9	−1.05584
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22263.1	1.02395
$780$	0	0
$781$	−501.544	−0.0229790
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3366.74	−0.153075
$786$	0	0
$787$	−2743.83	−0.124278	−0.0621392	−	0.998067i	$-0.519792\pi$
$787$	−2743.83	−0.124278	−0.0621392	+	0.998067i	$0.519792\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	55687.7	2.50319
$792$	0	0
$793$	−6820.81	−0.305440
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2474.62	0.109982	0.0549910	−	0.998487i	$-0.482487\pi$
$797$	2474.62	0.109982	0.0549910	+	0.998487i	$0.482487\pi$
$798$	0	0
$799$	−30702.1	−1.35940
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−53950.0	−2.37093
$804$	0	0
$805$	−11308.5	−0.495120
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−13410.2	−0.582790	−0.291395	−	0.956603i	$-0.594119\pi$
$809$	−13410.2	−0.582790	−0.291395	+	0.956603i	$0.594119\pi$
$810$	0	0
$811$	−24145.8	−1.04547	−0.522733	−	0.852497i	$-0.675087\pi$
$811$	−24145.8	−1.04547	−0.522733	+	0.852497i	$0.675087\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17588.6	−0.755952
$816$	0	0
$817$	−38028.4	−1.62845
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5569.07	0.236738	0.118369	−	0.992970i	$-0.462233\pi$
$821$	5569.07	0.236738	0.118369	+	0.992970i	$0.462233\pi$
$822$	0	0
$823$	−17139.2	−0.725921	−0.362961	−	0.931804i	$-0.618234\pi$
$823$	−17139.2	−0.725921	−0.362961	+	0.931804i	$0.618234\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3756.54	0.157954	0.0789769	−	0.996876i	$-0.474835\pi$
$827$	3756.54	0.157954	0.0789769	+	0.996876i	$0.474835\pi$
$828$	0	0
$829$	−15534.3	−0.650819	−0.325410	−	0.945573i	$-0.605502\pi$
$829$	−15534.3	−0.650819	−0.325410	+	0.945573i	$0.605502\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23636.1	0.983124
$834$	0	0
$835$	−4972.95	−0.206103
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31953.0	−1.31483	−0.657413	−	0.753531i	$-0.728349\pi$
$839$	−31953.0	−1.31483	−0.657413	+	0.753531i	$0.728349\pi$
$840$	0	0
$841$	10677.8	0.437813
$842$	0	0
$843$	0	0
$844$	0	0
$845$	824.195	0.0335541
$846$	0	0
$847$	57273.6	2.32343
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−26310.2	−1.05982
$852$	0	0
$853$	41587.2	1.66931	0.834654	−	0.550775i	$-0.185668\pi$
$853$	41587.2	1.66931	0.834654	+	0.550775i	$0.185668\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8080.58	0.322086	0.161043	−	0.986947i	$-0.448514\pi$
$857$	8080.58	0.322086	0.161043	+	0.986947i	$0.448514\pi$
$858$	0	0
$859$	−33848.0	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$859$	−33848.0	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27002.6	−1.06510	−0.532548	−	0.846400i	$-0.678766\pi$
$863$	−27002.6	−1.06510	−0.532548	+	0.846400i	$0.678766\pi$
$864$	0	0
$865$	11680.9	0.459148
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−69158.4	−2.69970
$870$	0	0
$871$	−12586.2	−0.489631
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−28259.8	−1.09184
$876$	0	0
$877$	45535.6	1.75328	0.876641	−	0.481145i	$-0.159779\pi$
$877$	45535.6	1.75328	0.876641	+	0.481145i	$0.159779\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−864.335	−0.0330536	−0.0165268	−	0.999863i	$-0.505261\pi$
$881$	−864.335	−0.0330536	−0.0165268	+	0.999863i	$0.505261\pi$
$882$	0	0
$883$	1075.10	0.0409741	0.0204871	−	0.999790i	$-0.493478\pi$
$883$	1075.10	0.0409741	0.0204871	+	0.999790i	$0.493478\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36604.8	−1.38565	−0.692824	−	0.721107i	$-0.743634\pi$
$887$	−36604.8	−1.38565	−0.692824	+	0.721107i	$0.743634\pi$
$888$	0	0
$889$	24574.5	0.927112
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−47329.7	−1.77360
$894$	0	0
$895$	17603.7	0.657460
$896$	0	0
$897$	0	0
$898$	0	0
$899$	42145.5	1.56355
$900$	0	0
$901$	32227.6	1.19163
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9015.18	−0.331132
$906$	0	0
$907$	−19657.7	−0.719649	−0.359825	−	0.933020i	$-0.617163\pi$
$907$	−19657.7	−0.719649	−0.359825	+	0.933020i	$0.617163\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8278.09	−0.301060	−0.150530	−	0.988605i	$-0.548098\pi$
$911$	−8278.09	−0.301060	−0.150530	+	0.988605i	$0.548098\pi$
$912$	0	0
$913$	56878.9	2.06179
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−47634.2	−1.71540
$918$	0	0
$919$	−20750.6	−0.744829	−0.372414	−	0.928067i	$-0.621470\pi$
$919$	−20750.6	−0.744829	−0.372414	+	0.928067i	$0.621470\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−109.171	−0.00389318
$924$	0	0
$925$	−29418.2	−1.04569
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27905.6	0.985526	0.492763	−	0.870164i	$-0.335987\pi$
$929$	27905.6	0.985526	0.492763	+	0.870164i	$0.335987\pi$
$930$	0	0
$931$	36436.9	1.28268
$932$	0	0
$933$	0	0
$934$	0	0
$935$	21983.9	0.768931
$936$	0	0
$937$	1734.90	0.0604873	0.0302437	−	0.999543i	$-0.490372\pi$
$937$	1734.90	0.0604873	0.0302437	+	0.999543i	$0.490372\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18761.2	−0.649943	−0.324972	−	0.945724i	$-0.605355\pi$
$941$	−18761.2	−0.649943	−0.324972	+	0.945724i	$0.605355\pi$
$942$	0	0
$943$	−17320.5	−0.598127
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20003.3	−0.686400	−0.343200	−	0.939262i	$-0.611511\pi$
$947$	−20003.3	−0.686400	−0.343200	+	0.939262i	$0.611511\pi$
$948$	0	0
$949$	−11743.3	−0.401689
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14639.9	0.497620	0.248810	−	0.968552i	$-0.419961\pi$
$953$	14639.9	0.497620	0.248810	+	0.968552i	$0.419961\pi$
$954$	0	0
$955$	−11531.6	−0.390738
$956$	0	0
$957$	0	0
$958$	0	0
$959$	41012.2	1.38097
$960$	0	0
$961$	20862.1	0.700283
$962$	0	0
$963$	0	0
$964$	0	0
$965$	20479.0	0.683152
$966$	0	0
$967$	29052.1	0.966136	0.483068	−	0.875583i	$-0.339522\pi$
$967$	29052.1	0.966136	0.483068	+	0.875583i	$0.339522\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30911.6	−1.02163	−0.510814	−	0.859691i	$-0.670656\pi$
$971$	−30911.6	−1.02163	−0.510814	+	0.859691i	$0.670656\pi$
$972$	0	0
$973$	−32750.9	−1.07908
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34458.8	1.12839	0.564194	−	0.825642i	$-0.309187\pi$
$977$	34458.8	1.12839	0.564194	+	0.825642i	$0.309187\pi$
$978$	0	0
$979$	−63304.5	−2.06662
$980$	0	0
$981$	0	0
$982$	0	0
$983$	41353.9	1.34180	0.670898	−	0.741550i	$-0.265909\pi$
$983$	41353.9	1.34180	0.670898	+	0.741550i	$0.265909\pi$
$984$	0	0
$985$	4139.85	0.133915
$986$	0	0
$987$	0	0
$988$	0	0
$989$	29585.8	0.951239
$990$	0	0
$991$	−4649.74	−0.149045	−0.0745227	−	0.997219i	$-0.523743\pi$
$991$	−4649.74	−0.149045	−0.0745227	+	0.997219i	$0.523743\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3028.35	−0.0964875
$996$	0	0
$997$	−45125.0	−1.43342	−0.716712	−	0.697369i	$-0.754354\pi$
$997$	−45125.0	−1.43342	−0.716712	+	0.697369i	$0.754354\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.bi.1.1		2
3.2	odd	2		624.4.a.j.1.2		2
4.3	odd	2		936.4.a.j.1.1		2
12.11	even	2		312.4.a.d.1.2	✓	2
24.5	odd	2		2496.4.a.bj.1.1		2
24.11	even	2		2496.4.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.d.1.2	✓	2	12.11	even	2
624.4.a.j.1.2		2	3.2	odd	2
936.4.a.j.1.1		2	4.3	odd	2
1872.4.a.bi.1.1		2	1.1	even	1	trivial
2496.4.a.ba.1.1		2	24.11	even	2
2496.4.a.bj.1.1		2	24.5	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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q+4.87689 q^{5} +25.6155 q^{7} +59.7235 q^{11} +13.0000 q^{13} +75.4773 q^{17} +116.354 q^{19} -90.5227 q^{23} -101.216 q^{25} +187.261 q^{29} +225.062 q^{31} +124.924 q^{35} +290.648 q^{37} +191.339 q^{41} -326.833 q^{43} -406.773 q^{47} +313.155 q^{49} +426.985 q^{53} +291.265 q^{55} +331.015 q^{59} -524.678 q^{61} +63.3996 q^{65} -968.172 q^{67} -8.39776 q^{71} -903.329 q^{73} +1529.85 q^{77} -1157.98 q^{79} +952.371 q^{83} +368.095 q^{85} -1059.96 q^{89} +333.002 q^{91} +567.447 q^{95} -90.7689 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{5} + 10 q^{7} + 4 q^{11} + 26 q^{13} + 52 q^{17} + 142 q^{19} - 280 q^{23} - 54 q^{25} + 424 q^{29} + 178 q^{31} - 80 q^{35} + 136 q^{37} + 226 q^{41} + 72 q^{43} + 176 q^{47} + 214 q^{49} + 788 q^{53}+ \cdots - 264 q^{97}+O(q^{100}) \) 2 * q + 18 * q^5 + 10 * q^7 + 4 * q^11 + 26 * q^13 + 52 * q^17 + 142 * q^19 - 280 * q^23 - 54 * q^25 + 424 * q^29 + 178 * q^31 - 80 * q^35 + 136 * q^37 + 226 * q^41 + 72 * q^43 + 176 * q^47 + 214 * q^49 + 788 * q^53 - 440 * q^55 + 728 * q^59 - 736 * q^61 + 234 * q^65 - 1054 * q^67 - 660 * q^71 + 24 * q^73 + 2400 * q^77 - 40 * q^79 + 1344 * q^83 + 60 * q^85 - 314 * q^89 + 130 * q^91 + 904 * q^95 - 264 * q^97

Introduction

L-functions

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