LMFDB - Embedded newform 2448.4.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,4,Mod(1,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, 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("2448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2448.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:	$144.436675694$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.9706 q^{5} -12.9706 q^{7} +49.9706 q^{11} +32.9411 q^{13} +17.0000 q^{17} -54.8823 q^{19} -82.0294 q^{23} +70.1766 q^{25} -289.882 q^{29} -232.059 q^{31} -181.206 q^{35} -227.529 q^{37} -437.735 q^{41} +158.882 q^{43} +159.088 q^{47} -174.765 q^{49} -376.087 q^{53} +698.117 q^{55} -185.294 q^{59} +861.852 q^{61} +460.206 q^{65} +178.530 q^{67} -1161.59 q^{71} +383.088 q^{73} -648.146 q^{77} -254.000 q^{79} +447.088 q^{83} +237.500 q^{85} +1213.15 q^{89} -427.265 q^{91} -766.736 q^{95} +291.383 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} + 8 q^{7} + 66 q^{11} - 2 q^{13} + 34 q^{17} + 26 q^{19} - 198 q^{23} + 344 q^{25} - 444 q^{29} - 532 q^{31} - 600 q^{35} + 88 q^{37} - 570 q^{41} + 182 q^{43} + 420 q^{47} - 78 q^{49} + 300 q^{53}+ \cdots + 1024 q^{97}+O(q^{100}) $ 2 * q - 6 * q^5 + 8 * q^7 + 66 * q^11 - 2 * q^13 + 34 * q^17 + 26 * q^19 - 198 * q^23 + 344 * q^25 - 444 * q^29 - 532 * q^31 - 600 * q^35 + 88 * q^37 - 570 * q^41 + 182 * q^43 + 420 * q^47 - 78 * q^49 + 300 * q^53 + 378 * q^55 + 444 * q^59 + 400 * q^61 + 1158 * q^65 + 968 * q^67 - 1848 * q^71 + 868 * q^73 - 312 * q^77 - 508 * q^79 + 996 * q^83 - 102 * q^85 + 288 * q^89 - 1160 * q^91 - 2382 * q^95 + 1024 * 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
$5$	13.9706	1.24957	0.624783	−	0.780799i	$-0.285188\pi$
$5$	13.9706	1.24957	0.624783	+	0.780799i	$0.285188\pi$
$6$	0	0
$7$	−12.9706	−0.700345	−0.350172	−	0.936685i	$-0.613877\pi$
$7$	−12.9706	−0.700345	−0.350172	+	0.936685i	$0.613877\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	49.9706	1.36970	0.684850	−	0.728684i	$-0.259868\pi$
$11$	49.9706	1.36970	0.684850	+	0.728684i	$0.259868\pi$
$12$	0	0
$13$	32.9411	0.702786	0.351393	−	0.936228i	$-0.385708\pi$
$13$	32.9411	0.702786	0.351393	+	0.936228i	$0.385708\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−54.8823	−0.662676	−0.331338	−	0.943512i	$-0.607500\pi$
$19$	−54.8823	−0.662676	−0.331338	+	0.943512i	$0.607500\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−82.0294	−0.743666	−0.371833	−	0.928300i	$-0.621271\pi$
$23$	−82.0294	−0.743666	−0.371833	+	0.928300i	$0.621271\pi$
$24$	0	0
$25$	70.1766	0.561413
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−289.882	−1.85620	−0.928100	−	0.372332i	$-0.878558\pi$
$29$	−289.882	−1.85620	−0.928100	+	0.372332i	$0.878558\pi$
$30$	0	0
$31$	−232.059	−1.34448	−0.672242	−	0.740331i	$-0.734669\pi$
$31$	−232.059	−1.34448	−0.672242	+	0.740331i	$0.734669\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−181.206	−0.875126
$36$	0	0
$37$	−227.529	−1.01096	−0.505480	−	0.862838i	$-0.668685\pi$
$37$	−227.529	−1.01096	−0.505480	+	0.862838i	$0.668685\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−437.735	−1.66738	−0.833692	−	0.552230i	$-0.813777\pi$
$41$	−437.735	−1.66738	−0.833692	+	0.552230i	$0.813777\pi$
$42$	0	0
$43$	158.882	0.563472	0.281736	−	0.959492i	$-0.409090\pi$
$43$	158.882	0.563472	0.281736	+	0.959492i	$0.409090\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	159.088	0.493732	0.246866	−	0.969050i	$-0.420599\pi$
$47$	159.088	0.493732	0.246866	+	0.969050i	$0.420599\pi$
$48$	0	0
$49$	−174.765	−0.509517
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−376.087	−0.974709	−0.487355	−	0.873204i	$-0.662038\pi$
$53$	−376.087	−0.974709	−0.487355	+	0.873204i	$0.662038\pi$
$54$	0	0
$55$	698.117	1.71153
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−185.294	−0.408867	−0.204434	−	0.978880i	$-0.565535\pi$
$59$	−185.294	−0.408867	−0.204434	+	0.978880i	$0.565535\pi$
$60$	0	0
$61$	861.852	1.80900	0.904499	−	0.426477i	$-0.140245\pi$
$61$	861.852	1.80900	0.904499	+	0.426477i	$0.140245\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	460.206	0.878177
$66$	0	0
$67$	178.530	0.325536	0.162768	−	0.986664i	$-0.447958\pi$
$67$	178.530	0.325536	0.162768	+	0.986664i	$0.447958\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1161.59	−1.94162	−0.970811	−	0.239847i	$-0.922903\pi$
$71$	−1161.59	−1.94162	−0.970811	+	0.239847i	$0.922903\pi$
$72$	0	0
$73$	383.088	0.614207	0.307103	−	0.951676i	$-0.400640\pi$
$73$	383.088	0.614207	0.307103	+	0.951676i	$0.400640\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−648.146	−0.959261
$78$	0	0
$79$	−254.000	−0.361737	−0.180869	−	0.983507i	$-0.557891\pi$
$79$	−254.000	−0.361737	−0.180869	+	0.983507i	$0.557891\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	447.088	0.591257	0.295628	−	0.955303i	$-0.404471\pi$
$83$	447.088	0.591257	0.295628	+	0.955303i	$0.404471\pi$
$84$	0	0
$85$	237.500	0.303064
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1213.15	1.44487	0.722433	−	0.691440i	$-0.243024\pi$
$89$	1213.15	1.44487	0.722433	+	0.691440i	$0.243024\pi$
$90$	0	0
$91$	−427.265	−0.492193
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−766.736	−0.828057
$96$	0	0
$97$	291.383	0.305004	0.152502	−	0.988303i	$-0.451267\pi$
$97$	291.383	0.305004	0.152502	+	0.988303i	$0.451267\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1568.53	−1.54529	−0.772645	−	0.634838i	$-0.781067\pi$
$101$	−1568.53	−1.54529	−0.772645	+	0.634838i	$0.781067\pi$
$102$	0	0
$103$	−412.647	−0.394750	−0.197375	−	0.980328i	$-0.563242\pi$
$103$	−412.647	−0.394750	−0.197375	+	0.980328i	$0.563242\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−239.382	−0.216280	−0.108140	−	0.994136i	$-0.534489\pi$
$107$	−239.382	−0.216280	−0.108140	+	0.994136i	$0.534489\pi$
$108$	0	0
$109$	−759.647	−0.667532	−0.333766	−	0.942656i	$-0.608319\pi$
$109$	−759.647	−0.667532	−0.333766	+	0.942656i	$0.608319\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	292.383	0.243408	0.121704	−	0.992566i	$-0.461164\pi$
$113$	292.383	0.243408	0.121704	+	0.992566i	$0.461164\pi$
$114$	0	0
$115$	−1146.00	−0.929259
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−220.500	−0.169859
$120$	0	0
$121$	1166.06	0.876076
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−765.913	−0.548043
$126$	0	0
$127$	2646.41	1.84906	0.924531	−	0.381107i	$-0.124457\pi$
$127$	2646.41	1.84906	0.924531	+	0.381107i	$0.124457\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1964.15	−1.30999	−0.654994	−	0.755634i	$-0.727329\pi$
$131$	−1964.15	−1.30999	−0.654994	+	0.755634i	$0.727329\pi$
$132$	0	0
$133$	711.854	0.464102
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2083.15	1.29909	0.649544	−	0.760324i	$-0.274960\pi$
$137$	2083.15	1.29909	0.649544	+	0.760324i	$0.274960\pi$
$138$	0	0
$139$	−1715.73	−1.04695	−0.523477	−	0.852040i	$-0.675365\pi$
$139$	−1715.73	−1.04695	−0.523477	+	0.852040i	$0.675365\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1646.09	0.962606
$144$	0	0
$145$	−4049.82	−2.31944
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1679.62	0.923487	0.461743	−	0.887014i	$-0.347224\pi$
$149$	1679.62	0.923487	0.461743	+	0.887014i	$0.347224\pi$
$150$	0	0
$151$	−644.353	−0.347263	−0.173632	−	0.984811i	$-0.555550\pi$
$151$	−644.353	−0.347263	−0.173632	+	0.984811i	$0.555550\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3241.99	−1.68002
$156$	0	0
$157$	2130.53	1.08302	0.541512	−	0.840693i	$-0.317852\pi$
$157$	2130.53	1.08302	0.541512	+	0.840693i	$0.317852\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1063.97	0.520822
$162$	0	0
$163$	−3582.41	−1.72145	−0.860724	−	0.509072i	$-0.829989\pi$
$163$	−3582.41	−1.72145	−0.860724	+	0.509072i	$0.829989\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1861.79	−0.862694	−0.431347	−	0.902186i	$-0.641962\pi$
$167$	−1861.79	−0.862694	−0.431347	+	0.902186i	$0.641962\pi$
$168$	0	0
$169$	−1111.88	−0.506091
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2612.50	−1.14812	−0.574059	−	0.818814i	$-0.694632\pi$
$173$	−2612.50	−1.14812	−0.574059	+	0.818814i	$0.694632\pi$
$174$	0	0
$175$	−910.230	−0.393183
$176$	0	0
$177$	0	0
$178$	0	0
$179$	126.646	0.0528824	0.0264412	−	0.999650i	$-0.491583\pi$
$179$	126.646	0.0528824	0.0264412	+	0.999650i	$0.491583\pi$
$180$	0	0
$181$	1783.26	0.732314	0.366157	−	0.930553i	$-0.380673\pi$
$181$	1783.26	0.732314	0.366157	+	0.930553i	$0.380673\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3178.71	−1.26326
$186$	0	0
$187$	849.500	0.332201
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2144.44	0.812388	0.406194	−	0.913787i	$-0.366856\pi$
$191$	2144.44	0.812388	0.406194	+	0.913787i	$0.366856\pi$
$192$	0	0
$193$	3205.82	1.19565	0.597823	−	0.801628i	$-0.296032\pi$
$193$	3205.82	1.19565	0.597823	+	0.801628i	$0.296032\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2768.32	−1.00119	−0.500596	−	0.865681i	$-0.666886\pi$
$197$	−2768.32	−1.00119	−0.500596	+	0.865681i	$0.666886\pi$
$198$	0	0
$199$	712.792	0.253912	0.126956	−	0.991908i	$-0.459479\pi$
$199$	712.792	0.253912	0.126956	+	0.991908i	$0.459479\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3759.94	1.29998
$204$	0	0
$205$	−6115.41	−2.08350
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2742.50	−0.907667
$210$	0	0
$211$	−787.591	−0.256967	−0.128483	−	0.991712i	$-0.541011\pi$
$211$	−787.591	−0.256967	−0.128483	+	0.991712i	$0.541011\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2219.67	0.704096
$216$	0	0
$217$	3009.93	0.941602
$218$	0	0
$219$	0	0
$220$	0	0
$221$	559.999	0.170451
$222$	0	0
$223$	2926.53	0.878811	0.439405	−	0.898289i	$-0.355189\pi$
$223$	2926.53	0.878811	0.439405	+	0.898289i	$0.355189\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6212.03	−1.81633	−0.908164	−	0.418614i	$-0.862516\pi$
$227$	−6212.03	−1.81633	−0.908164	+	0.418614i	$0.862516\pi$
$228$	0	0
$229$	−4516.35	−1.30327	−0.651635	−	0.758533i	$-0.725916\pi$
$229$	−4516.35	−1.30327	−0.651635	+	0.758533i	$0.725916\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3547.26	−0.997376	−0.498688	−	0.866782i	$-0.666185\pi$
$233$	−3547.26	−0.997376	−0.498688	+	0.866782i	$0.666185\pi$
$234$	0	0
$235$	2222.55	0.616951
$236$	0	0
$237$	0	0
$238$	0	0
$239$	726.969	0.196752	0.0983760	−	0.995149i	$-0.468635\pi$
$239$	726.969	0.196752	0.0983760	+	0.995149i	$0.468635\pi$
$240$	0	0
$241$	1689.67	0.451623	0.225812	−	0.974171i	$-0.427497\pi$
$241$	1689.67	0.451623	0.225812	+	0.974171i	$0.427497\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2441.56	−0.636675
$246$	0	0
$247$	−1807.88	−0.465720
$248$	0	0
$249$	0	0
$250$	0	0
$251$	911.707	0.229269	0.114634	−	0.993408i	$-0.463430\pi$
$251$	911.707	0.229269	0.114634	+	0.993408i	$0.463430\pi$
$252$	0	0
$253$	−4099.06	−1.01860
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3123.56	0.758141	0.379070	−	0.925368i	$-0.376244\pi$
$257$	3123.56	0.758141	0.379070	+	0.925368i	$0.376244\pi$
$258$	0	0
$259$	2951.18	0.708021
$260$	0	0
$261$	0	0
$262$	0	0
$263$	137.288	0.0321884	0.0160942	−	0.999870i	$-0.494877\pi$
$263$	137.288	0.0321884	0.0160942	+	0.999870i	$0.494877\pi$
$264$	0	0
$265$	−5254.15	−1.21796
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2030.26	0.460175	0.230087	−	0.973170i	$-0.426099\pi$
$269$	2030.26	0.460175	0.230087	+	0.973170i	$0.426099\pi$
$270$	0	0
$271$	1187.23	0.266123	0.133061	−	0.991108i	$-0.457519\pi$
$271$	1187.23	0.266123	0.133061	+	0.991108i	$0.457519\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3506.77	0.768967
$276$	0	0
$277$	3027.91	0.656786	0.328393	−	0.944541i	$-0.393493\pi$
$277$	3027.91	0.656786	0.328393	+	0.944541i	$0.393493\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5519.90	1.17185	0.585925	−	0.810365i	$-0.300731\pi$
$281$	5519.90	1.17185	0.585925	+	0.810365i	$0.300731\pi$
$282$	0	0
$283$	5888.17	1.23680	0.618402	−	0.785862i	$-0.287780\pi$
$283$	5888.17	1.23680	0.618402	+	0.785862i	$0.287780\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5677.67	1.16774
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2873.06	0.572852	0.286426	−	0.958102i	$-0.407533\pi$
$293$	2873.06	0.572852	0.286426	+	0.958102i	$0.407533\pi$
$294$	0	0
$295$	−2588.65	−0.510906
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2702.14	−0.522638
$300$	0	0
$301$	−2060.79	−0.394625
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12040.6	2.26046
$306$	0	0
$307$	−318.234	−0.0591614	−0.0295807	−	0.999562i	$-0.509417\pi$
$307$	−318.234	−0.0591614	−0.0295807	+	0.999562i	$0.509417\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1940.05	−0.353731	−0.176866	−	0.984235i	$-0.556596\pi$
$311$	−1940.05	−0.353731	−0.176866	+	0.984235i	$0.556596\pi$
$312$	0	0
$313$	−5487.84	−0.991026	−0.495513	−	0.868600i	$-0.665020\pi$
$313$	−5487.84	−0.991026	−0.495513	+	0.868600i	$0.665020\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1337.29	−0.236940	−0.118470	−	0.992958i	$-0.537799\pi$
$317$	−1337.29	−0.236940	−0.118470	+	0.992958i	$0.537799\pi$
$318$	0	0
$319$	−14485.6	−2.54243
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−932.998	−0.160723
$324$	0	0
$325$	2311.70	0.394553
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2063.46	−0.345783
$330$	0	0
$331$	4826.46	0.801470	0.400735	−	0.916194i	$-0.368755\pi$
$331$	4826.46	0.801470	0.400735	+	0.916194i	$0.368755\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2494.16	0.406778
$336$	0	0
$337$	−8265.21	−1.33601	−0.668004	−	0.744158i	$-0.732851\pi$
$337$	−8265.21	−1.33601	−0.668004	+	0.744158i	$0.732851\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11596.1	−1.84154
$342$	0	0
$343$	6715.70	1.05718
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5841.35	0.903688	0.451844	−	0.892097i	$-0.350766\pi$
$347$	5841.35	0.903688	0.451844	+	0.892097i	$0.350766\pi$
$348$	0	0
$349$	−3873.11	−0.594048	−0.297024	−	0.954870i	$-0.595994\pi$
$349$	−3873.11	−0.594048	−0.297024	+	0.954870i	$0.595994\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4020.29	−0.606171	−0.303085	−	0.952963i	$-0.598017\pi$
$353$	−4020.29	−0.606171	−0.303085	+	0.952963i	$0.598017\pi$
$354$	0	0
$355$	−16228.0	−2.42618
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2272.06	−0.334024	−0.167012	−	0.985955i	$-0.553412\pi$
$359$	−2272.06	−0.334024	−0.167012	+	0.985955i	$0.553412\pi$
$360$	0	0
$361$	−3846.94	−0.560860
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5351.96	0.767491
$366$	0	0
$367$	−7353.58	−1.04592	−0.522962	−	0.852356i	$-0.675173\pi$
$367$	−7353.58	−1.04592	−0.522962	+	0.852356i	$0.675173\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4878.07	0.682632
$372$	0	0
$373$	320.701	0.0445182	0.0222591	−	0.999752i	$-0.492914\pi$
$373$	320.701	0.0445182	0.0222591	+	0.999752i	$0.492914\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9549.05	−1.30451
$378$	0	0
$379$	700.903	0.0949946	0.0474973	−	0.998871i	$-0.484875\pi$
$379$	700.903	0.0949946	0.0474973	+	0.998871i	$0.484875\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8217.97	1.09639	0.548196	−	0.836350i	$-0.315315\pi$
$383$	8217.97	1.09639	0.548196	+	0.836350i	$0.315315\pi$
$384$	0	0
$385$	−9054.97	−1.19866
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3800.59	0.495366	0.247683	−	0.968841i	$-0.420331\pi$
$389$	3800.59	0.495366	0.247683	+	0.968841i	$0.420331\pi$
$390$	0	0
$391$	−1394.50	−0.180366
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3548.52	−0.452014
$396$	0	0
$397$	−5955.38	−0.752876	−0.376438	−	0.926442i	$-0.622851\pi$
$397$	−5955.38	−0.752876	−0.376438	+	0.926442i	$0.622851\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−581.967	−0.0724739	−0.0362370	−	0.999343i	$-0.511537\pi$
$401$	−581.967	−0.0724739	−0.0362370	+	0.999343i	$0.511537\pi$
$402$	0	0
$403$	−7644.28	−0.944885
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11369.8	−1.38471
$408$	0	0
$409$	−11357.7	−1.37311	−0.686555	−	0.727078i	$-0.740878\pi$
$409$	−11357.7	−1.37311	−0.686555	+	0.727078i	$0.740878\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2403.36	0.286348
$414$	0	0
$415$	6246.08	0.738814
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.9420	−0.00244173	−0.00122086	−	0.999999i	$-0.500389\pi$
$419$	−20.9420	−0.00244173	−0.00122086	+	0.999999i	$0.500389\pi$
$420$	0	0
$421$	4455.28	0.515765	0.257882	−	0.966176i	$-0.416975\pi$
$421$	4455.28	0.515765	0.257882	+	0.966176i	$0.416975\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1193.00	0.136163
$426$	0	0
$427$	−11178.7	−1.26692
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2410.99	−0.269451	−0.134725	−	0.990883i	$-0.543015\pi$
$431$	−2410.99	−0.269451	−0.134725	+	0.990883i	$0.543015\pi$
$432$	0	0
$433$	−1653.71	−0.183539	−0.0917693	−	0.995780i	$-0.529252\pi$
$433$	−1653.71	−0.183539	−0.0917693	+	0.995780i	$0.529252\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4501.96	0.492810
$438$	0	0
$439$	−14852.5	−1.61474	−0.807371	−	0.590044i	$-0.799110\pi$
$439$	−14852.5	−1.61474	−0.807371	+	0.590044i	$0.799110\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5393.26	−0.578423	−0.289212	−	0.957265i	$-0.593393\pi$
$443$	−5393.26	−0.578423	−0.289212	+	0.957265i	$0.593393\pi$
$444$	0	0
$445$	16948.3	1.80546
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6144.82	−0.645862	−0.322931	−	0.946422i	$-0.604668\pi$
$449$	−6144.82	−0.645862	−0.322931	+	0.946422i	$0.604668\pi$
$450$	0	0
$451$	−21873.9	−2.28381
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5969.13	−0.615027
$456$	0	0
$457$	6041.18	0.618369	0.309184	−	0.951002i	$-0.399944\pi$
$457$	6041.18	0.618369	0.309184	+	0.951002i	$0.399944\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13829.8	−1.39722	−0.698610	−	0.715503i	$-0.746198\pi$
$461$	−13829.8	−1.39722	−0.698610	+	0.715503i	$0.746198\pi$
$462$	0	0
$463$	−12585.3	−1.26326	−0.631629	−	0.775271i	$-0.717613\pi$
$463$	−12585.3	−1.26326	−0.631629	+	0.775271i	$0.717613\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6561.91	−0.650212	−0.325106	−	0.945678i	$-0.605400\pi$
$467$	−6561.91	−0.650212	−0.325106	+	0.945678i	$0.605400\pi$
$468$	0	0
$469$	−2315.63	−0.227987
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7939.44	0.771788
$474$	0	0
$475$	−3851.45	−0.372035
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16609.4	−1.58435	−0.792175	−	0.610294i	$-0.791051\pi$
$479$	−16609.4	−1.58435	−0.792175	+	0.610294i	$0.791051\pi$
$480$	0	0
$481$	−7495.06	−0.710489
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4070.78	0.381123
$486$	0	0
$487$	7999.46	0.744333	0.372166	−	0.928166i	$-0.378615\pi$
$487$	7999.46	0.744333	0.372166	+	0.928166i	$0.378615\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11669.1	−1.07255	−0.536274	−	0.844044i	$-0.680169\pi$
$491$	−11669.1	−1.07255	−0.536274	+	0.844044i	$0.680169\pi$
$492$	0	0
$493$	−4928.00	−0.450194
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15066.4	1.35980
$498$	0	0
$499$	20896.2	1.87464	0.937319	−	0.348473i	$-0.113300\pi$
$499$	20896.2	1.87464	0.937319	+	0.348473i	$0.113300\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17429.1	1.54498	0.772490	−	0.635028i	$-0.219011\pi$
$503$	17429.1	1.54498	0.772490	+	0.635028i	$0.219011\pi$
$504$	0	0
$505$	−21913.2	−1.93094
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1020.29	−0.0888481	−0.0444240	−	0.999013i	$-0.514145\pi$
$509$	−1020.29	−0.0888481	−0.0444240	+	0.999013i	$0.514145\pi$
$510$	0	0
$511$	−4968.87	−0.430156
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5764.91	−0.493266
$516$	0	0
$517$	7949.73	0.676265
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5281.92	0.444155	0.222078	−	0.975029i	$-0.428716\pi$
$521$	5281.92	0.444155	0.222078	+	0.975029i	$0.428716\pi$
$522$	0	0
$523$	15906.1	1.32988	0.664938	−	0.746898i	$-0.268458\pi$
$523$	15906.1	1.32988	0.664938	+	0.746898i	$0.268458\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3945.00	−0.326085
$528$	0	0
$529$	−5438.17	−0.446961
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14419.5	−1.17181
$534$	0	0
$535$	−3344.30	−0.270255
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−8733.08	−0.697886
$540$	0	0
$541$	21923.7	1.74228	0.871139	−	0.491036i	$-0.163382\pi$
$541$	21923.7	1.74228	0.871139	+	0.491036i	$0.163382\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10612.7	−0.834124
$546$	0	0
$547$	−4960.74	−0.387762	−0.193881	−	0.981025i	$-0.562108\pi$
$547$	−4960.74	−0.387762	−0.193881	+	0.981025i	$0.562108\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15909.4	1.23006
$552$	0	0
$553$	3294.52	0.253341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22404.1	−1.70429	−0.852146	−	0.523305i	$-0.824699\pi$
$557$	−22404.1	−1.70429	−0.852146	+	0.523305i	$0.824699\pi$
$558$	0	0
$559$	5233.76	0.396001
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17361.5	−1.29965	−0.649824	−	0.760085i	$-0.725157\pi$
$563$	−17361.5	−1.29965	−0.649824	+	0.760085i	$0.725157\pi$
$564$	0	0
$565$	4084.75	0.304154
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1980.78	−0.145938	−0.0729690	−	0.997334i	$-0.523247\pi$
$569$	−1980.78	−0.145938	−0.0729690	+	0.997334i	$0.523247\pi$
$570$	0	0
$571$	19164.5	1.40457	0.702284	−	0.711897i	$-0.252164\pi$
$571$	19164.5	1.40457	0.702284	+	0.711897i	$0.252164\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5756.55	−0.417504
$576$	0	0
$577$	8729.71	0.629848	0.314924	−	0.949117i	$-0.398021\pi$
$577$	8729.71	0.629848	0.314924	+	0.949117i	$0.398021\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5798.99	−0.414084
$582$	0	0
$583$	−18793.3	−1.33506
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2927.87	0.205871	0.102935	−	0.994688i	$-0.467177\pi$
$587$	2927.87	0.205871	0.102935	+	0.994688i	$0.467177\pi$
$588$	0	0
$589$	12735.9	0.890958
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2154.97	−0.149231	−0.0746157	−	0.997212i	$-0.523773\pi$
$593$	−2154.97	−0.149231	−0.0746157	+	0.997212i	$0.523773\pi$
$594$	0	0
$595$	−3080.50	−0.212249
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7065.12	0.481925	0.240963	−	0.970534i	$-0.422537\pi$
$599$	7065.12	0.481925	0.240963	+	0.970534i	$0.422537\pi$
$600$	0	0
$601$	10656.5	0.723272	0.361636	−	0.932319i	$-0.382218\pi$
$601$	10656.5	0.723272	0.361636	+	0.932319i	$0.382218\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16290.5	1.09471
$606$	0	0
$607$	−82.8667	−0.00554111	−0.00277056	−	0.999996i	$-0.500882\pi$
$607$	−82.8667	−0.00554111	−0.00277056	+	0.999996i	$0.500882\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5240.55	0.346988
$612$	0	0
$613$	−15588.4	−1.02710	−0.513549	−	0.858060i	$-0.671670\pi$
$613$	−15588.4	−1.02710	−0.513549	+	0.858060i	$0.671670\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4430.58	0.289090	0.144545	−	0.989498i	$-0.453828\pi$
$617$	4430.58	0.289090	0.144545	+	0.989498i	$0.453828\pi$
$618$	0	0
$619$	−3111.78	−0.202056	−0.101028	−	0.994884i	$-0.532213\pi$
$619$	−3111.78	−0.202056	−0.101028	+	0.994884i	$0.532213\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15735.2	−1.01190
$624$	0	0
$625$	−19472.3	−1.24623
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3867.99	−0.245194
$630$	0	0
$631$	14808.2	0.934236	0.467118	−	0.884195i	$-0.345292\pi$
$631$	14808.2	0.934236	0.467118	+	0.884195i	$0.345292\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	36971.8	2.31052
$636$	0	0
$637$	−5756.94	−0.358082
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9233.91	0.568982	0.284491	−	0.958679i	$-0.408175\pi$
$641$	9233.91	0.568982	0.284491	+	0.958679i	$0.408175\pi$
$642$	0	0
$643$	−11712.7	−0.718355	−0.359177	−	0.933269i	$-0.616943\pi$
$643$	−11712.7	−0.718355	−0.359177	+	0.933269i	$0.616943\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12099.6	−0.735216	−0.367608	−	0.929981i	$-0.619823\pi$
$647$	−12099.6	−0.735216	−0.367608	+	0.929981i	$0.619823\pi$
$648$	0	0
$649$	−9259.22	−0.560025
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8335.37	0.499523	0.249761	−	0.968307i	$-0.419648\pi$
$653$	8335.37	0.499523	0.249761	+	0.968307i	$0.419648\pi$
$654$	0	0
$655$	−27440.2	−1.63691
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7751.68	0.458213	0.229107	−	0.973401i	$-0.426420\pi$
$659$	7751.68	0.458213	0.229107	+	0.973401i	$0.426420\pi$
$660$	0	0
$661$	13808.0	0.812510	0.406255	−	0.913760i	$-0.366834\pi$
$661$	13808.0	0.812510	0.406255	+	0.913760i	$0.366834\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9945.00	0.579925
$666$	0	0
$667$	23778.9	1.38039
$668$	0	0
$669$	0	0
$670$	0	0
$671$	43067.2	2.47778
$672$	0	0
$673$	−8406.44	−0.481493	−0.240746	−	0.970588i	$-0.577392\pi$
$673$	−8406.44	−0.481493	−0.240746	+	0.970588i	$0.577392\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19257.9	1.09327	0.546633	−	0.837372i	$-0.315909\pi$
$677$	19257.9	1.09327	0.546633	+	0.837372i	$0.315909\pi$
$678$	0	0
$679$	−3779.40	−0.213608
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3451.93	−0.193388	−0.0966942	−	0.995314i	$-0.530827\pi$
$683$	−3451.93	−0.193388	−0.0966942	+	0.995314i	$0.530827\pi$
$684$	0	0
$685$	29102.7	1.62330
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12388.7	−0.685012
$690$	0	0
$691$	26090.5	1.43637	0.718184	−	0.695853i	$-0.244974\pi$
$691$	26090.5	1.43637	0.718184	+	0.695853i	$0.244974\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−23969.8	−1.30824
$696$	0	0
$697$	−7441.50	−0.404400
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22283.9	−1.20064	−0.600321	−	0.799759i	$-0.704961\pi$
$701$	−22283.9	−1.20064	−0.600321	+	0.799759i	$0.704961\pi$
$702$	0	0
$703$	12487.3	0.669940
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20344.7	1.08224
$708$	0	0
$709$	−4561.83	−0.241640	−0.120820	−	0.992674i	$-0.538552\pi$
$709$	−4561.83	−0.241640	−0.120820	+	0.992674i	$0.538552\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19035.7	0.999847
$714$	0	0
$715$	22996.8	1.20284
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12458.7	0.646218	0.323109	−	0.946362i	$-0.395272\pi$
$719$	12458.7	0.646218	0.323109	+	0.946362i	$0.395272\pi$
$720$	0	0
$721$	5352.26	0.276461
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−20343.0	−1.04209
$726$	0	0
$727$	−19361.4	−0.987721	−0.493860	−	0.869541i	$-0.664415\pi$
$727$	−19361.4	−0.987721	−0.493860	+	0.869541i	$0.664415\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2701.00	0.136662
$732$	0	0
$733$	21638.4	1.09036	0.545180	−	0.838319i	$-0.316461\pi$
$733$	21638.4	1.09036	0.545180	+	0.838319i	$0.316461\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8921.24	0.445886
$738$	0	0
$739$	33520.7	1.66858	0.834290	−	0.551326i	$-0.185878\pi$
$739$	33520.7	1.66858	0.834290	+	0.551326i	$0.185878\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28486.4	−1.40655	−0.703274	−	0.710919i	$-0.748279\pi$
$743$	−28486.4	−1.40655	−0.703274	+	0.710919i	$0.748279\pi$
$744$	0	0
$745$	23465.2	1.15396
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3104.92	0.151470
$750$	0	0
$751$	29427.4	1.42985	0.714927	−	0.699199i	$-0.246460\pi$
$751$	29427.4	1.42985	0.714927	+	0.699199i	$0.246460\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9001.98	−0.433928
$756$	0	0
$757$	30790.8	1.47835	0.739174	−	0.673514i	$-0.235216\pi$
$757$	30790.8	1.47835	0.739174	+	0.673514i	$0.235216\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17677.5	0.842062	0.421031	−	0.907046i	$-0.361668\pi$
$761$	17677.5	0.842062	0.421031	+	0.907046i	$0.361668\pi$
$762$	0	0
$763$	9853.05	0.467502
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6103.78	−0.287346
$768$	0	0
$769$	6309.56	0.295876	0.147938	−	0.988997i	$-0.452736\pi$
$769$	6309.56	0.295876	0.147938	+	0.988997i	$0.452736\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10323.3	−0.480343	−0.240171	−	0.970731i	$-0.577204\pi$
$773$	−10323.3	−0.480343	−0.240171	+	0.970731i	$0.577204\pi$
$774$	0	0
$775$	−16285.1	−0.754811
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24023.9	1.10494
$780$	0	0
$781$	−58045.2	−2.65944
$782$	0	0
$783$	0	0
$784$	0	0
$785$	29764.7	1.35331
$786$	0	0
$787$	−11118.3	−0.503589	−0.251795	−	0.967781i	$-0.581021\pi$
$787$	−11118.3	−0.503589	−0.251795	+	0.967781i	$0.581021\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3792.37	−0.170469
$792$	0	0
$793$	28390.4	1.27134
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32556.2	−1.44692	−0.723462	−	0.690364i	$-0.757450\pi$
$797$	−32556.2	−1.44692	−0.723462	+	0.690364i	$0.757450\pi$
$798$	0	0
$799$	2704.50	0.119748
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19143.1	0.841279
$804$	0	0
$805$	14864.2	0.650802
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39644.3	−1.72289	−0.861445	−	0.507851i	$-0.830440\pi$
$809$	−39644.3	−1.72289	−0.861445	+	0.507851i	$0.830440\pi$
$810$	0	0
$811$	7839.62	0.339440	0.169720	−	0.985492i	$-0.445714\pi$
$811$	7839.62	0.339440	0.169720	+	0.985492i	$0.445714\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−50048.3	−2.15106
$816$	0	0
$817$	−8719.82	−0.373400
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−344.345	−0.0146379	−0.00731895	−	0.999973i	$-0.502330\pi$
$821$	−344.345	−0.0146379	−0.00731895	+	0.999973i	$0.502330\pi$
$822$	0	0
$823$	43172.9	1.82857	0.914284	−	0.405074i	$-0.132754\pi$
$823$	43172.9	1.82857	0.914284	+	0.405074i	$0.132754\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21158.3	−0.889656	−0.444828	−	0.895616i	$-0.646735\pi$
$827$	−21158.3	−0.889656	−0.444828	+	0.895616i	$0.646735\pi$
$828$	0	0
$829$	−10514.0	−0.440490	−0.220245	−	0.975445i	$-0.570686\pi$
$829$	−10514.0	−0.440490	−0.220245	+	0.975445i	$0.570686\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2971.00	−0.123576
$834$	0	0
$835$	−26010.3	−1.07799
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10036.2	−0.412978	−0.206489	−	0.978449i	$-0.566204\pi$
$839$	−10036.2	−0.412978	−0.206489	+	0.978449i	$0.566204\pi$
$840$	0	0
$841$	59642.7	2.44548
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−15533.6	−0.632394
$846$	0	0
$847$	−15124.4	−0.613555
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18664.1	0.751817
$852$	0	0
$853$	9343.14	0.375033	0.187516	−	0.982261i	$-0.439956\pi$
$853$	9343.14	0.375033	0.187516	+	0.982261i	$0.439956\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25235.0	−1.00585	−0.502923	−	0.864331i	$-0.667742\pi$
$857$	−25235.0	−1.00585	−0.502923	+	0.864331i	$0.667742\pi$
$858$	0	0
$859$	39717.7	1.57759	0.788795	−	0.614656i	$-0.210705\pi$
$859$	39717.7	1.57759	0.788795	+	0.614656i	$0.210705\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26812.4	−1.05760	−0.528798	−	0.848748i	$-0.677357\pi$
$863$	−26812.4	−1.05760	−0.528798	+	0.848748i	$0.677357\pi$
$864$	0	0
$865$	−36498.0	−1.43465
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12692.5	−0.495471
$870$	0	0
$871$	5880.97	0.228782
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9934.33	0.383819
$876$	0	0
$877$	−33850.8	−1.30338	−0.651688	−	0.758487i	$-0.725939\pi$
$877$	−33850.8	−1.30338	−0.651688	+	0.758487i	$0.725939\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19939.4	0.762514	0.381257	−	0.924469i	$-0.375491\pi$
$881$	19939.4	0.762514	0.381257	+	0.924469i	$0.375491\pi$
$882$	0	0
$883$	−31421.5	−1.19753	−0.598765	−	0.800925i	$-0.704342\pi$
$883$	−31421.5	−1.19753	−0.598765	+	0.800925i	$0.704342\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14713.6	0.556971	0.278486	−	0.960440i	$-0.410168\pi$
$887$	14713.6	0.556971	0.278486	+	0.960440i	$0.410168\pi$
$888$	0	0
$889$	−34325.4	−1.29498
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8731.12	−0.327185
$894$	0	0
$895$	1769.31	0.0660801
$896$	0	0
$897$	0	0
$898$	0	0
$899$	67269.7	2.49563
$900$	0	0
$901$	−6393.49	−0.236402
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24913.2	0.915075
$906$	0	0
$907$	−36405.1	−1.33276	−0.666380	−	0.745612i	$-0.732157\pi$
$907$	−36405.1	−1.33276	−0.666380	+	0.745612i	$0.732157\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1574.12	0.0572481	0.0286241	−	0.999590i	$-0.490887\pi$
$911$	1574.12	0.0572481	0.0286241	+	0.999590i	$0.490887\pi$
$912$	0	0
$913$	22341.3	0.809844
$914$	0	0
$915$	0	0
$916$	0	0
$917$	25476.1	0.917442
$918$	0	0
$919$	−44823.7	−1.60892	−0.804460	−	0.594007i	$-0.797545\pi$
$919$	−44823.7	−1.60892	−0.804460	+	0.594007i	$0.797545\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−38264.0	−1.36455
$924$	0	0
$925$	−15967.2	−0.567566
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10654.7	0.376287	0.188143	−	0.982142i	$-0.439753\pi$
$929$	10654.7	0.376287	0.188143	+	0.982142i	$0.439753\pi$
$930$	0	0
$931$	9591.47	0.337645
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11868.0	0.415107
$936$	0	0
$937$	46738.5	1.62954	0.814770	−	0.579784i	$-0.196863\pi$
$937$	46738.5	1.62954	0.814770	+	0.579784i	$0.196863\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31346.7	1.08594	0.542972	−	0.839751i	$-0.317299\pi$
$941$	31346.7	1.08594	0.542972	+	0.839751i	$0.317299\pi$
$942$	0	0
$943$	35907.2	1.23998
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18191.9	−0.624242	−0.312121	−	0.950042i	$-0.601039\pi$
$947$	−18191.9	−0.624242	−0.312121	+	0.950042i	$0.601039\pi$
$948$	0	0
$949$	12619.4	0.431656
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−47969.6	−1.63052	−0.815261	−	0.579094i	$-0.803407\pi$
$953$	−47969.6	−1.63052	−0.815261	+	0.579094i	$0.803407\pi$
$954$	0	0
$955$	29959.0	1.01513
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−27019.6	−0.909810
$960$	0	0
$961$	24060.3	0.807637
$962$	0	0
$963$	0	0
$964$	0	0
$965$	44787.1	1.49404
$966$	0	0
$967$	−51466.6	−1.71154	−0.855768	−	0.517360i	$-0.826915\pi$
$967$	−51466.6	−1.71154	−0.855768	+	0.517360i	$0.826915\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4007.62	−0.132452	−0.0662259	−	0.997805i	$-0.521096\pi$
$971$	−4007.62	−0.132452	−0.0662259	+	0.997805i	$0.521096\pi$
$972$	0	0
$973$	22254.0	0.733229
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37362.2	1.22346	0.611731	−	0.791066i	$-0.290474\pi$
$977$	37362.2	1.22346	0.611731	+	0.791066i	$0.290474\pi$
$978$	0	0
$979$	60621.6	1.97903
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38659.1	−1.25436	−0.627179	−	0.778875i	$-0.715790\pi$
$983$	−38659.1	−1.25436	−0.627179	+	0.778875i	$0.715790\pi$
$984$	0	0
$985$	−38675.0	−1.25106
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13033.0	−0.419035
$990$	0	0
$991$	−46.8701	−0.00150240	−0.000751200	−	1.00000i	$-0.500239\pi$
$991$	−46.8701	−0.00150240	−0.000751200		1.00000i	$0.500239\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9958.11	0.317280
$996$	0	0
$997$	−16913.4	−0.537265	−0.268633	−	0.963243i	$-0.586572\pi$
$997$	−16913.4	−0.537265	−0.268633	+	0.963243i	$0.586572\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.4.a.v.1.2		2
3.2	odd	2		816.4.a.o.1.1		2
4.3	odd	2		153.4.a.e.1.2		2
12.11	even	2		51.4.a.d.1.1	✓	2
60.59	even	2		1275.4.a.m.1.2		2
84.83	odd	2		2499.4.a.l.1.1		2
204.203	even	2		867.4.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.d.1.1	✓	2	12.11	even	2
153.4.a.e.1.2		2	4.3	odd	2
816.4.a.o.1.1		2	3.2	odd	2
867.4.a.j.1.1		2	204.203	even	2
1275.4.a.m.1.2		2	60.59	even	2
2448.4.a.v.1.2		2	1.1	even	1	trivial
2499.4.a.l.1.1		2	84.83	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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2448.a (trivial)

\(f(q)\)	\(=\)	\(q+13.9706 q^{5} -12.9706 q^{7} +49.9706 q^{11} +32.9411 q^{13} +17.0000 q^{17} -54.8823 q^{19} -82.0294 q^{23} +70.1766 q^{25} -289.882 q^{29} -232.059 q^{31} -181.206 q^{35} -227.529 q^{37} -437.735 q^{41} +158.882 q^{43} +159.088 q^{47} -174.765 q^{49} -376.087 q^{53} +698.117 q^{55} -185.294 q^{59} +861.852 q^{61} +460.206 q^{65} +178.530 q^{67} -1161.59 q^{71} +383.088 q^{73} -648.146 q^{77} -254.000 q^{79} +447.088 q^{83} +237.500 q^{85} +1213.15 q^{89} -427.265 q^{91} -766.736 q^{95} +291.383 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 8 q^{7} + 66 q^{11} - 2 q^{13} + 34 q^{17} + 26 q^{19} - 198 q^{23} + 344 q^{25} - 444 q^{29} - 532 q^{31} - 600 q^{35} + 88 q^{37} - 570 q^{41} + 182 q^{43} + 420 q^{47} - 78 q^{49} + 300 q^{53}+ \cdots + 1024 q^{97}+O(q^{100}) \) 2 * q - 6 * q^5 + 8 * q^7 + 66 * q^11 - 2 * q^13 + 34 * q^17 + 26 * q^19 - 198 * q^23 + 344 * q^25 - 444 * q^29 - 532 * q^31 - 600 * q^35 + 88 * q^37 - 570 * q^41 + 182 * q^43 + 420 * q^47 - 78 * q^49 + 300 * q^53 + 378 * q^55 + 444 * q^59 + 400 * q^61 + 1158 * q^65 + 968 * q^67 - 1848 * q^71 + 868 * q^73 - 312 * q^77 - 508 * q^79 + 996 * q^83 - 102 * q^85 + 288 * q^89 - 1160 * q^91 - 2382 * q^95 + 1024 * q^97

Introduction

L-functions

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