LMFDB - Embedded newform 3744.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.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:	$29.8959905168$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	3744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.34017 q^{5} -1.07838 q^{7} +O(q^{10})$	$q-4.34017 q^{5} -1.07838 q^{7} +3.41855 q^{11} +1.00000 q^{13} -2.00000 q^{17} +1.07838 q^{19} -2.15676 q^{23} +13.8371 q^{25} -2.00000 q^{29} +5.75872 q^{31} +4.68035 q^{35} -6.68035 q^{37} +0.340173 q^{41} +10.8371 q^{43} +7.41855 q^{47} -5.83710 q^{49} +2.68035 q^{53} -14.8371 q^{55} -9.26180 q^{59} -4.52359 q^{61} -4.34017 q^{65} -15.9155 q^{67} +5.26180 q^{71} +14.6803 q^{73} -3.68649 q^{77} -12.0000 q^{79} +1.26180 q^{83} +8.68035 q^{85} +13.0205 q^{89} -1.07838 q^{91} -4.68035 q^{95} +6.68035 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5}+O(q^{10}) $ 3 * q - 2 * q^5	$ 3 q - 2 q^{5} - 4 q^{11} + 3 q^{13} - 6 q^{17} + 13 q^{25} - 6 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{37} - 10 q^{41} + 4 q^{43} + 8 q^{47} + 11 q^{49} - 14 q^{53} - 16 q^{55} - 20 q^{59} + 2 q^{61} - 2 q^{65} - 16 q^{67} + 8 q^{71} + 22 q^{73} - 24 q^{77} - 36 q^{79} - 4 q^{83} + 4 q^{85} + 6 q^{89} + 8 q^{95} - 2 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 - 4 * q^11 + 3 * q^13 - 6 * q^17 + 13 * q^25 - 6 * q^29 - 8 * q^31 - 8 * q^35 + 2 * q^37 - 10 * q^41 + 4 * q^43 + 8 * q^47 + 11 * q^49 - 14 * q^53 - 16 * q^55 - 20 * q^59 + 2 * q^61 - 2 * q^65 - 16 * q^67 + 8 * q^71 + 22 * q^73 - 24 * q^77 - 36 * q^79 - 4 * q^83 + 4 * q^85 + 6 * q^89 + 8 * q^95 - 2 * 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
$5$	−4.34017	−1.94098	−0.970492	−	0.241133i	$-0.922481\pi$
$5$	−4.34017	−1.94098	−0.970492	+	0.241133i	$0.922481\pi$
$6$	0	0
$7$	−1.07838	−0.407588	−0.203794	−	0.979014i	$-0.565327\pi$
$7$	−1.07838	−0.407588	−0.203794	+	0.979014i	$0.565327\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.41855	1.03073	0.515366	−	0.856970i	$-0.327656\pi$
$11$	3.41855	1.03073	0.515366	+	0.856970i	$0.327656\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	1.07838	0.247397	0.123698	−	0.992320i	$-0.460524\pi$
$19$	1.07838	0.247397	0.123698	+	0.992320i	$0.460524\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.15676	−0.449715	−0.224857	−	0.974392i	$-0.572192\pi$
$23$	−2.15676	−0.449715	−0.224857	+	0.974392i	$0.572192\pi$
$24$	0	0
$25$	13.8371	2.76742
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	5.75872	1.03430	0.517149	−	0.855896i	$-0.326993\pi$
$31$	5.75872	1.03430	0.517149	+	0.855896i	$0.326993\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.68035	0.791123
$36$	0	0
$37$	−6.68035	−1.09824	−0.549121	−	0.835743i	$-0.685037\pi$
$37$	−6.68035	−1.09824	−0.549121	+	0.835743i	$0.685037\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.340173	0.0531261	0.0265630	−	0.999647i	$-0.491544\pi$
$41$	0.340173	0.0531261	0.0265630	+	0.999647i	$0.491544\pi$
$42$	0	0
$43$	10.8371	1.65264	0.826321	−	0.563199i	$-0.190430\pi$
$43$	10.8371	1.65264	0.826321	+	0.563199i	$0.190430\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.41855	1.08211	0.541053	−	0.840988i	$-0.318026\pi$
$47$	7.41855	1.08211	0.541053	+	0.840988i	$0.318026\pi$
$48$	0	0
$49$	−5.83710	−0.833872
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.68035	0.368174	0.184087	−	0.982910i	$-0.441067\pi$
$53$	2.68035	0.368174	0.184087	+	0.982910i	$0.441067\pi$
$54$	0	0
$55$	−14.8371	−2.00063
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.26180	−1.20578	−0.602892	−	0.797823i	$-0.705985\pi$
$59$	−9.26180	−1.20578	−0.602892	+	0.797823i	$0.705985\pi$
$60$	0	0
$61$	−4.52359	−0.579186	−0.289593	−	0.957150i	$-0.593520\pi$
$61$	−4.52359	−0.579186	−0.289593	+	0.957150i	$0.593520\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.34017	−0.538332
$66$	0	0
$67$	−15.9155	−1.94439	−0.972193	−	0.234183i	$-0.924759\pi$
$67$	−15.9155	−1.94439	−0.972193	+	0.234183i	$0.924759\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.26180	0.624460	0.312230	−	0.950007i	$-0.398924\pi$
$71$	5.26180	0.624460	0.312230	+	0.950007i	$0.398924\pi$
$72$	0	0
$73$	14.6803	1.71820	0.859102	−	0.511804i	$-0.171023\pi$
$73$	14.6803	1.71820	0.859102	+	0.511804i	$0.171023\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.68649	−0.420114
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.26180	0.138500	0.0692500	−	0.997599i	$-0.477939\pi$
$83$	1.26180	0.138500	0.0692500	+	0.997599i	$0.477939\pi$
$84$	0	0
$85$	8.68035	0.941516
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.0205	1.38017	0.690086	−	0.723727i	$-0.257573\pi$
$89$	13.0205	1.38017	0.690086	+	0.723727i	$0.257573\pi$
$90$	0	0
$91$	−1.07838	−0.113045
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.68035	−0.480193
$96$	0	0
$97$	6.68035	0.678286	0.339143	−	0.940735i	$-0.389863\pi$
$97$	6.68035	0.678286	0.339143	+	0.940735i	$0.389863\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.31351	−0.628218	−0.314109	−	0.949387i	$-0.601706\pi$
$101$	−6.31351	−0.628218	−0.314109	+	0.949387i	$0.601706\pi$
$102$	0	0
$103$	−14.5236	−1.43105	−0.715526	−	0.698586i	$-0.753813\pi$
$103$	−14.5236	−1.43105	−0.715526	+	0.698586i	$0.753813\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.2039	−1.85651	−0.928257	−	0.371939i	$-0.878693\pi$
$107$	−19.2039	−1.85651	−0.928257	+	0.371939i	$0.878693\pi$
$108$	0	0
$109$	−6.31351	−0.604725	−0.302362	−	0.953193i	$-0.597775\pi$
$109$	−6.31351	−0.604725	−0.302362	+	0.953193i	$0.597775\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.9939	1.41050	0.705252	−	0.708957i	$-0.250834\pi$
$113$	14.9939	1.41050	0.705252	+	0.708957i	$0.250834\pi$
$114$	0	0
$115$	9.36069	0.872889
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.15676	0.197709
$120$	0	0
$121$	0.686489	0.0624081
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−38.3545	−3.43054
$126$	0	0
$127$	−9.84324	−0.873447	−0.436723	−	0.899596i	$-0.643861\pi$
$127$	−9.84324	−0.873447	−0.436723	+	0.899596i	$0.643861\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.5174	−1.35577	−0.677883	−	0.735170i	$-0.737102\pi$
$131$	−15.5174	−1.35577	−0.677883	+	0.735170i	$0.737102\pi$
$132$	0	0
$133$	−1.16290	−0.100836
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.3340	−1.13920	−0.569602	−	0.821921i	$-0.692902\pi$
$137$	−13.3340	−1.13920	−0.569602	+	0.821921i	$0.692902\pi$
$138$	0	0
$139$	−0.680346	−0.0577062	−0.0288531	−	0.999584i	$-0.509186\pi$
$139$	−0.680346	−0.0577062	−0.0288531	+	0.999584i	$0.509186\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.41855	0.285874
$144$	0	0
$145$	8.68035	0.720863
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.02052	−0.738990	−0.369495	−	0.929233i	$-0.620469\pi$
$149$	−9.02052	−0.738990	−0.369495	+	0.929233i	$0.620469\pi$
$150$	0	0
$151$	3.60197	0.293124	0.146562	−	0.989201i	$-0.453179\pi$
$151$	3.60197	0.293124	0.146562	+	0.989201i	$0.453179\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−24.9939	−2.00755
$156$	0	0
$157$	−18.1978	−1.45234	−0.726171	−	0.687514i	$-0.758702\pi$
$157$	−18.1978	−1.45234	−0.726171	+	0.687514i	$0.758702\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.32580	0.183298
$162$	0	0
$163$	−14.1256	−1.10640	−0.553200	−	0.833049i	$-0.686593\pi$
$163$	−14.1256	−1.10640	−0.553200	+	0.833049i	$0.686593\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.7792	1.29842	0.649208	−	0.760611i	$-0.275100\pi$
$167$	16.7792	1.29842	0.649208	+	0.760611i	$0.275100\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.36069	0.559623	0.279812	−	0.960055i	$-0.409728\pi$
$173$	7.36069	0.559623	0.279812	+	0.960055i	$0.409728\pi$
$174$	0	0
$175$	−14.9216	−1.12797
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.31965	−0.547097	−0.273548	−	0.961858i	$-0.588197\pi$
$179$	−7.31965	−0.547097	−0.273548	+	0.961858i	$0.588197\pi$
$180$	0	0
$181$	0.523590	0.0389182	0.0194591	−	0.999811i	$-0.493806\pi$
$181$	0.523590	0.0389182	0.0194591	+	0.999811i	$0.493806\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	28.9939	2.13167
$186$	0	0
$187$	−6.83710	−0.499978
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.0410	−1.59483	−0.797417	−	0.603429i	$-0.793801\pi$
$191$	−22.0410	−1.59483	−0.797417	+	0.603429i	$0.793801\pi$
$192$	0	0
$193$	−19.0472	−1.37105	−0.685523	−	0.728051i	$-0.740426\pi$
$193$	−19.0472	−1.37105	−0.685523	+	0.728051i	$0.740426\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.3340	1.23500	0.617499	−	0.786571i	$-0.288146\pi$
$197$	17.3340	1.23500	0.617499	+	0.786571i	$0.288146\pi$
$198$	0	0
$199$	−15.5174	−1.10000	−0.550001	−	0.835164i	$-0.685373\pi$
$199$	−15.5174	−1.10000	−0.550001	+	0.835164i	$0.685373\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.15676	0.151375
$204$	0	0
$205$	−1.47641	−0.103117
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.68649	0.255000
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−47.0349	−3.20775
$216$	0	0
$217$	−6.21008	−0.421568
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	23.1194	1.54819	0.774095	−	0.633069i	$-0.218205\pi$
$223$	23.1194	1.54819	0.774095	+	0.633069i	$0.218205\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.41855	−0.226897	−0.113449	−	0.993544i	$-0.536190\pi$
$227$	−3.41855	−0.226897	−0.113449	+	0.993544i	$0.536190\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.6803	−0.961741	−0.480871	−	0.876792i	$-0.659679\pi$
$233$	−14.6803	−0.961741	−0.480871	+	0.876792i	$0.659679\pi$
$234$	0	0
$235$	−32.1978	−2.10035
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.57531	0.619375	0.309688	−	0.950838i	$-0.399776\pi$
$239$	9.57531	0.619375	0.309688	+	0.950838i	$0.399776\pi$
$240$	0	0
$241$	−28.0410	−1.80628	−0.903141	−	0.429344i	$-0.858745\pi$
$241$	−28.0410	−1.80628	−0.903141	+	0.429344i	$0.858745\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	25.3340	1.61853
$246$	0	0
$247$	1.07838	0.0686155
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.16290	−0.325879	−0.162940	−	0.986636i	$-0.552098\pi$
$251$	−5.16290	−0.325879	−0.162940	+	0.986636i	$0.552098\pi$
$252$	0	0
$253$	−7.37298	−0.463535
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.6803	1.16525	0.582624	−	0.812742i	$-0.302026\pi$
$257$	18.6803	1.16525	0.582624	+	0.812742i	$0.302026\pi$
$258$	0	0
$259$	7.20394	0.447631
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.8371	−1.40820	−0.704098	−	0.710103i	$-0.748648\pi$
$263$	−22.8371	−1.40820	−0.704098	+	0.710103i	$0.748648\pi$
$264$	0	0
$265$	−11.6332	−0.714620
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.36069	0.448789	0.224395	−	0.974498i	$-0.427960\pi$
$269$	7.36069	0.448789	0.224395	+	0.974498i	$0.427960\pi$
$270$	0	0
$271$	−26.4391	−1.60606	−0.803030	−	0.595939i	$-0.796780\pi$
$271$	−26.4391	−1.60606	−0.803030	+	0.595939i	$0.796780\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	47.3028	2.85247
$276$	0	0
$277$	15.3607	0.922935	0.461467	−	0.887157i	$-0.347323\pi$
$277$	15.3607	0.922935	0.461467	+	0.887157i	$0.347323\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.6537	0.754855	0.377428	−	0.926039i	$-0.376809\pi$
$281$	12.6537	0.754855	0.377428	+	0.926039i	$0.376809\pi$
$282$	0	0
$283$	27.2039	1.61711	0.808553	−	0.588423i	$-0.200251\pi$
$283$	27.2039	1.61711	0.808553	+	0.588423i	$0.200251\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.366835	−0.0216536
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.6537	0.739236	0.369618	−	0.929184i	$-0.379489\pi$
$293$	12.6537	0.739236	0.369618	+	0.929184i	$0.379489\pi$
$294$	0	0
$295$	40.1978	2.34041
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.15676	−0.124728
$300$	0	0
$301$	−11.6865	−0.673598
$302$	0	0
$303$	0	0
$304$	0	0
$305$	19.6332	1.12419
$306$	0	0
$307$	−8.08452	−0.461408	−0.230704	−	0.973024i	$-0.574103\pi$
$307$	−8.08452	−0.461408	−0.230704	+	0.973024i	$0.574103\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.8310	1.35133	0.675665	−	0.737209i	$-0.263857\pi$
$311$	23.8310	1.35133	0.675665	+	0.737209i	$0.263857\pi$
$312$	0	0
$313$	5.68649	0.321419	0.160710	−	0.987002i	$-0.448622\pi$
$313$	5.68649	0.321419	0.160710	+	0.987002i	$0.448622\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.65368	−0.486039	−0.243020	−	0.970021i	$-0.578138\pi$
$317$	−8.65368	−0.486039	−0.243020	+	0.970021i	$0.578138\pi$
$318$	0	0
$319$	−6.83710	−0.382804
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.15676	−0.120005
$324$	0	0
$325$	13.8371	0.767544
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	5.75872	0.316528	0.158264	−	0.987397i	$-0.449410\pi$
$331$	5.75872	0.316528	0.158264	+	0.987397i	$0.449410\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	69.0759	3.77402
$336$	0	0
$337$	22.3135	1.21549	0.607747	−	0.794131i	$-0.292073\pi$
$337$	22.3135	1.21549	0.607747	+	0.794131i	$0.292073\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.6865	1.06608
$342$	0	0
$343$	13.8432	0.747465
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.0472	0.915141	0.457570	−	0.889173i	$-0.348720\pi$
$347$	17.0472	0.915141	0.457570	+	0.889173i	$0.348720\pi$
$348$	0	0
$349$	−5.31965	−0.284755	−0.142377	−	0.989812i	$-0.545475\pi$
$349$	−5.31965	−0.284755	−0.142377	+	0.989812i	$0.545475\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.65368	0.247691	0.123845	−	0.992302i	$-0.460477\pi$
$353$	4.65368	0.247691	0.123845	+	0.992302i	$0.460477\pi$
$354$	0	0
$355$	−22.8371	−1.21207
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−34.5692	−1.82449	−0.912245	−	0.409644i	$-0.865653\pi$
$359$	−34.5692	−1.82449	−0.912245	+	0.409644i	$0.865653\pi$
$360$	0	0
$361$	−17.8371	−0.938795
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−63.7152	−3.33501
$366$	0	0
$367$	−4.36683	−0.227947	−0.113973	−	0.993484i	$-0.536358\pi$
$367$	−4.36683	−0.227947	−0.113973	+	0.993484i	$0.536358\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.89043	−0.150063
$372$	0	0
$373$	32.7214	1.69425	0.847125	−	0.531394i	$-0.178332\pi$
$373$	32.7214	1.69425	0.847125	+	0.531394i	$0.178332\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	13.3919	0.687895	0.343948	−	0.938989i	$-0.388236\pi$
$379$	13.3919	0.687895	0.343948	+	0.938989i	$0.388236\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.5814	0.847272	0.423636	−	0.905832i	$-0.360753\pi$
$383$	16.5814	0.847272	0.423636	+	0.905832i	$0.360753\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.63317	−0.0828048	−0.0414024	−	0.999143i	$-0.513183\pi$
$389$	−1.63317	−0.0828048	−0.0414024	+	0.999143i	$0.513183\pi$
$390$	0	0
$391$	4.31351	0.218144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	52.0821	2.62053
$396$	0	0
$397$	10.6803	0.536031	0.268016	−	0.963415i	$-0.413632\pi$
$397$	10.6803	0.536031	0.268016	+	0.963415i	$0.413632\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10.0144	−0.500094	−0.250047	−	0.968234i	$-0.580446\pi$
$401$	−10.0144	−0.500094	−0.250047	+	0.968234i	$0.580446\pi$
$402$	0	0
$403$	5.75872	0.286862
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22.8371	−1.13199
$408$	0	0
$409$	−1.31965	−0.0652527	−0.0326263	−	0.999468i	$-0.510387\pi$
$409$	−1.31965	−0.0652527	−0.0326263	+	0.999468i	$0.510387\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.98771	0.491463
$414$	0	0
$415$	−5.47641	−0.268826
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16.8781	−0.824551	−0.412276	−	0.911059i	$-0.635266\pi$
$419$	−16.8781	−0.824551	−0.412276	+	0.911059i	$0.635266\pi$
$420$	0	0
$421$	−30.3135	−1.47739	−0.738695	−	0.674040i	$-0.764558\pi$
$421$	−30.3135	−1.47739	−0.738695	+	0.674040i	$0.764558\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−27.6742	−1.34240
$426$	0	0
$427$	4.87814	0.236070
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.37137	−0.114225	−0.0571124	−	0.998368i	$-0.518189\pi$
$431$	−2.37137	−0.114225	−0.0571124	+	0.998368i	$0.518189\pi$
$432$	0	0
$433$	−6.19779	−0.297847	−0.148923	−	0.988849i	$-0.547581\pi$
$433$	−6.19779	−0.297847	−0.148923	+	0.988849i	$0.547581\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.32580	−0.111258
$438$	0	0
$439$	15.8843	0.758115	0.379058	−	0.925373i	$-0.376248\pi$
$439$	15.8843	0.758115	0.379058	+	0.925373i	$0.376248\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.5174	−0.737256	−0.368628	−	0.929577i	$-0.620172\pi$
$443$	−15.5174	−0.737256	−0.368628	+	0.929577i	$0.620172\pi$
$444$	0	0
$445$	−56.5113	−2.67889
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.97334	0.376285	0.188143	−	0.982142i	$-0.439753\pi$
$449$	7.97334	0.376285	0.188143	+	0.982142i	$0.439753\pi$
$450$	0	0
$451$	1.16290	0.0547588
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.68035	0.219418
$456$	0	0
$457$	34.9939	1.63694	0.818472	−	0.574547i	$-0.194822\pi$
$457$	34.9939	1.63694	0.818472	+	0.574547i	$0.194822\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.0205	1.35162	0.675810	−	0.737076i	$-0.263794\pi$
$461$	29.0205	1.35162	0.675810	+	0.737076i	$0.263794\pi$
$462$	0	0
$463$	−33.4740	−1.55567	−0.777834	−	0.628470i	$-0.783681\pi$
$463$	−33.4740	−1.55567	−0.777834	+	0.628470i	$0.783681\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.52359	−0.301876	−0.150938	−	0.988543i	$-0.548229\pi$
$467$	−6.52359	−0.301876	−0.150938	+	0.988543i	$0.548229\pi$
$468$	0	0
$469$	17.1629	0.792509
$470$	0	0
$471$	0	0
$472$	0	0
$473$	37.0472	1.70343
$474$	0	0
$475$	14.9216	0.684651
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.47187	0.158634	0.0793170	−	0.996849i	$-0.474726\pi$
$479$	3.47187	0.158634	0.0793170	+	0.996849i	$0.474726\pi$
$480$	0	0
$481$	−6.68035	−0.304598
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−28.9939	−1.31654
$486$	0	0
$487$	−20.2290	−0.916663	−0.458332	−	0.888781i	$-0.651553\pi$
$487$	−20.2290	−0.916663	−0.458332	+	0.888781i	$0.651553\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.21008	−0.0997395	−0.0498697	−	0.998756i	$-0.515881\pi$
$491$	−2.21008	−0.0997395	−0.0498697	+	0.998756i	$0.515881\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.67420	−0.254523
$498$	0	0
$499$	13.7587	0.615925	0.307963	−	0.951398i	$-0.400353\pi$
$499$	13.7587	0.615925	0.307963	+	0.951398i	$0.400353\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	28.5113	1.27126	0.635628	−	0.771995i	$-0.280741\pi$
$503$	28.5113	1.27126	0.635628	+	0.771995i	$0.280741\pi$
$504$	0	0
$505$	27.4017	1.21936
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.0144	−1.50766	−0.753830	−	0.657069i	$-0.771796\pi$
$509$	−34.0144	−1.50766	−0.753830	+	0.657069i	$0.771796\pi$
$510$	0	0
$511$	−15.8310	−0.700320
$512$	0	0
$513$	0	0
$514$	0	0
$515$	63.0349	2.77765
$516$	0	0
$517$	25.3607	1.11536
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.35455	0.366019	0.183010	−	0.983111i	$-0.441416\pi$
$521$	8.35455	0.366019	0.183010	+	0.983111i	$0.441416\pi$
$522$	0	0
$523$	−20.3668	−0.890580	−0.445290	−	0.895387i	$-0.646899\pi$
$523$	−20.3668	−0.890580	−0.445290	+	0.895387i	$0.646899\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.5174	−0.501708
$528$	0	0
$529$	−18.3484	−0.797757
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.340173	0.0147345
$534$	0	0
$535$	83.3484	3.60347
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−19.9544	−0.859498
$540$	0	0
$541$	4.63931	0.199459	0.0997297	−	0.995015i	$-0.468202\pi$
$541$	4.63931	0.199459	0.0997297	+	0.995015i	$0.468202\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	27.4017	1.17376
$546$	0	0
$547$	18.0410	0.771379	0.385690	−	0.922629i	$-0.373964\pi$
$547$	18.0410	0.771379	0.385690	+	0.922629i	$0.373964\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.15676	−0.0918809
$552$	0	0
$553$	12.9405	0.550287
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−17.0205	−0.721183	−0.360591	−	0.932724i	$-0.617425\pi$
$557$	−17.0205	−0.721183	−0.360591	+	0.932724i	$0.617425\pi$
$558$	0	0
$559$	10.8371	0.458361
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.7152	−1.33664	−0.668319	−	0.743875i	$-0.732986\pi$
$563$	−31.7152	−1.33664	−0.668319	+	0.743875i	$0.732986\pi$
$564$	0	0
$565$	−65.0759	−2.73777
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−45.7152	−1.91648	−0.958241	−	0.285961i	$-0.907687\pi$
$569$	−45.7152	−1.91648	−0.958241	+	0.285961i	$0.907687\pi$
$570$	0	0
$571$	−16.8781	−0.706328	−0.353164	−	0.935561i	$-0.614894\pi$
$571$	−16.8781	−0.706328	−0.353164	+	0.935561i	$0.614894\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−29.8432	−1.24455
$576$	0	0
$577$	−41.3484	−1.72136	−0.860678	−	0.509149i	$-0.829960\pi$
$577$	−41.3484	−1.72136	−0.860678	+	0.509149i	$0.829960\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.36069	−0.0564510
$582$	0	0
$583$	9.16290	0.379488
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.5753	1.55090	0.775449	−	0.631410i	$-0.217523\pi$
$587$	37.5753	1.55090	0.775449	+	0.631410i	$0.217523\pi$
$588$	0	0
$589$	6.21008	0.255882
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.9672	−1.18954	−0.594770	−	0.803896i	$-0.702757\pi$
$593$	−28.9672	−1.18954	−0.594770	+	0.803896i	$0.702757\pi$
$594$	0	0
$595$	−9.36069	−0.383751
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.21008	−0.253737	−0.126868	−	0.991920i	$-0.540493\pi$
$599$	−6.21008	−0.253737	−0.126868	+	0.991920i	$0.540493\pi$
$600$	0	0
$601$	−26.3135	−1.07335	−0.536675	−	0.843789i	$-0.680320\pi$
$601$	−26.3135	−1.07335	−0.536675	+	0.843789i	$0.680320\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.97948	−0.121133
$606$	0	0
$607$	24.6803	1.00174	0.500872	−	0.865521i	$-0.333013\pi$
$607$	24.6803	1.00174	0.500872	+	0.865521i	$0.333013\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.41855	0.300122
$612$	0	0
$613$	−20.3545	−0.822112	−0.411056	−	0.911610i	$-0.634840\pi$
$613$	−20.3545	−0.822112	−0.411056	+	0.911610i	$0.634840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.0616	1.08946	0.544729	−	0.838612i	$-0.316633\pi$
$617$	27.0616	1.08946	0.544729	+	0.838612i	$0.316633\pi$
$618$	0	0
$619$	15.9155	0.639697	0.319849	−	0.947469i	$-0.396368\pi$
$619$	15.9155	0.639697	0.319849	+	0.947469i	$0.396368\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.0410	−0.562542
$624$	0	0
$625$	97.2799	3.89119
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.3607	0.532726
$630$	0	0
$631$	10.2413	0.407699	0.203849	−	0.979002i	$-0.434655\pi$
$631$	10.2413	0.407699	0.203849	+	0.979002i	$0.434655\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	42.7214	1.69535
$636$	0	0
$637$	−5.83710	−0.231274
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11.3607	−0.448720	−0.224360	−	0.974506i	$-0.572029\pi$
$641$	−11.3607	−0.448720	−0.224360	+	0.974506i	$0.572029\pi$
$642$	0	0
$643$	3.77101	0.148714	0.0743571	−	0.997232i	$-0.476310\pi$
$643$	3.77101	0.148714	0.0743571	+	0.997232i	$0.476310\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.9877	−1.33619	−0.668097	−	0.744074i	$-0.732891\pi$
$647$	−33.9877	−1.33619	−0.668097	+	0.744074i	$0.732891\pi$
$648$	0	0
$649$	−31.6619	−1.24284
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.3135	0.403599	0.201799	−	0.979427i	$-0.435321\pi$
$653$	10.3135	0.403599	0.201799	+	0.979427i	$0.435321\pi$
$654$	0	0
$655$	67.3484	2.63152
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13.3607	−0.520459	−0.260229	−	0.965547i	$-0.583798\pi$
$659$	−13.3607	−0.520459	−0.260229	+	0.965547i	$0.583798\pi$
$660$	0	0
$661$	27.3074	1.06213	0.531067	−	0.847330i	$-0.321791\pi$
$661$	27.3074	1.06213	0.531067	+	0.847330i	$0.321791\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.04718	0.195721
$666$	0	0
$667$	4.31351	0.167020
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15.4641	−0.596986
$672$	0	0
$673$	46.5113	1.79288	0.896440	−	0.443166i	$-0.146145\pi$
$673$	46.5113	1.79288	0.896440	+	0.443166i	$0.146145\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−46.6803	−1.79407	−0.897036	−	0.441958i	$-0.854284\pi$
$677$	−46.6803	−1.79407	−0.897036	+	0.441958i	$0.854284\pi$
$678$	0	0
$679$	−7.20394	−0.276462
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−49.2905	−1.88605	−0.943025	−	0.332721i	$-0.892033\pi$
$683$	−49.2905	−1.88605	−0.943025	+	0.332721i	$0.892033\pi$
$684$	0	0
$685$	57.8720	2.21118
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.68035	0.102113
$690$	0	0
$691$	−17.2762	−0.657217	−0.328608	−	0.944466i	$-0.606580\pi$
$691$	−17.2762	−0.657217	−0.328608	+	0.944466i	$0.606580\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.95282	0.112007
$696$	0	0
$697$	−0.680346	−0.0257699
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.68035	−0.252313	−0.126157	−	0.992010i	$-0.540264\pi$
$701$	−6.68035	−0.252313	−0.126157	+	0.992010i	$0.540264\pi$
$702$	0	0
$703$	−7.20394	−0.271702
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.80835	0.256054
$708$	0	0
$709$	35.6742	1.33977	0.669886	−	0.742464i	$-0.266343\pi$
$709$	35.6742	1.33977	0.669886	+	0.742464i	$0.266343\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.4202	−0.465139
$714$	0	0
$715$	−14.8371	−0.554876
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−15.2039	−0.567011	−0.283506	−	0.958971i	$-0.591497\pi$
$719$	−15.2039	−0.567011	−0.283506	+	0.958971i	$0.591497\pi$
$720$	0	0
$721$	15.6619	0.583280
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−27.6742	−1.02779
$726$	0	0
$727$	−30.8904	−1.14566	−0.572831	−	0.819673i	$-0.694155\pi$
$727$	−30.8904	−1.14566	−0.572831	+	0.819673i	$0.694155\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21.6742	−0.801649
$732$	0	0
$733$	−9.37298	−0.346199	−0.173099	−	0.984904i	$-0.555378\pi$
$733$	−9.37298	−0.346199	−0.173099	+	0.984904i	$0.555378\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−54.4079	−2.00414
$738$	0	0
$739$	−26.8059	−0.986071	−0.493036	−	0.870009i	$-0.664113\pi$
$739$	−26.8059	−0.986071	−0.493036	+	0.870009i	$0.664113\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.0989	−0.443865	−0.221933	−	0.975062i	$-0.571237\pi$
$743$	−12.0989	−0.443865	−0.221933	+	0.975062i	$0.571237\pi$
$744$	0	0
$745$	39.1506	1.43437
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20.7091	0.756694
$750$	0	0
$751$	−25.4764	−0.929647	−0.464824	−	0.885403i	$-0.653882\pi$
$751$	−25.4764	−0.929647	−0.464824	+	0.885403i	$0.653882\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.6332	−0.568949
$756$	0	0
$757$	18.5113	0.672805	0.336402	−	0.941718i	$-0.390790\pi$
$757$	18.5113	0.672805	0.336402	+	0.941718i	$0.390790\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.96719	0.325061	0.162530	−	0.986704i	$-0.448034\pi$
$761$	8.96719	0.325061	0.162530	+	0.986704i	$0.448034\pi$
$762$	0	0
$763$	6.80835	0.246479
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.26180	−0.334424
$768$	0	0
$769$	32.4079	1.16866	0.584329	−	0.811517i	$-0.301358\pi$
$769$	32.4079	1.16866	0.584329	+	0.811517i	$0.301358\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.29299	0.118441	0.0592203	−	0.998245i	$-0.481139\pi$
$773$	3.29299	0.118441	0.0592203	+	0.998245i	$0.481139\pi$
$774$	0	0
$775$	79.6840	2.86234
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.366835	0.0131432
$780$	0	0
$781$	17.9877	0.643651
$782$	0	0
$783$	0	0
$784$	0	0
$785$	78.9816	2.81897
$786$	0	0
$787$	−18.8059	−0.670358	−0.335179	−	0.942154i	$-0.608797\pi$
$787$	−18.8059	−0.670358	−0.335179	+	0.942154i	$0.608797\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.1690	−0.574905
$792$	0	0
$793$	−4.52359	−0.160637
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.6803	−1.08675	−0.543377	−	0.839489i	$-0.682854\pi$
$797$	−30.6803	−1.08675	−0.543377	+	0.839489i	$0.682854\pi$
$798$	0	0
$799$	−14.8371	−0.524899
$800$	0	0
$801$	0	0
$802$	0	0
$803$	50.1855	1.77101
$804$	0	0
$805$	−10.0944	−0.355780
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22.3668	0.786376	0.393188	−	0.919458i	$-0.371372\pi$
$809$	22.3668	0.786376	0.393188	+	0.919458i	$0.371372\pi$
$810$	0	0
$811$	43.7998	1.53802	0.769009	−	0.639238i	$-0.220750\pi$
$811$	43.7998	1.53802	0.769009	+	0.639238i	$0.220750\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	61.3074	2.14750
$816$	0	0
$817$	11.6865	0.408858
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.06770	−0.211764	−0.105882	−	0.994379i	$-0.533767\pi$
$821$	−6.06770	−0.211764	−0.105882	+	0.994379i	$0.533767\pi$
$822$	0	0
$823$	36.8248	1.28363	0.641816	−	0.766859i	$-0.278181\pi$
$823$	36.8248	1.28363	0.641816	+	0.766859i	$0.278181\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.9893	−1.21670	−0.608349	−	0.793670i	$-0.708168\pi$
$827$	−34.9893	−1.21670	−0.608349	+	0.793670i	$0.708168\pi$
$828$	0	0
$829$	−14.5113	−0.503998	−0.251999	−	0.967727i	$-0.581088\pi$
$829$	−14.5113	−0.503998	−0.251999	+	0.967727i	$0.581088\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	11.6742	0.404487
$834$	0	0
$835$	−72.8248	−2.52021
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.4124	1.39519	0.697596	−	0.716492i	$-0.254253\pi$
$839$	40.4124	1.39519	0.697596	+	0.716492i	$0.254253\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.34017	−0.149306
$846$	0	0
$847$	−0.740294	−0.0254368
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.4079	0.493896
$852$	0	0
$853$	−29.7152	−1.01743	−0.508715	−	0.860935i	$-0.669879\pi$
$853$	−29.7152	−1.01743	−0.508715	+	0.860935i	$0.669879\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−52.3545	−1.78840	−0.894199	−	0.447670i	$-0.852254\pi$
$857$	−52.3545	−1.78840	−0.894199	+	0.447670i	$0.852254\pi$
$858$	0	0
$859$	11.3730	0.388041	0.194021	−	0.980997i	$-0.437847\pi$
$859$	11.3730	0.388041	0.194021	+	0.980997i	$0.437847\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.20847	0.313460	0.156730	−	0.987641i	$-0.449905\pi$
$863$	9.20847	0.313460	0.156730	+	0.987641i	$0.449905\pi$
$864$	0	0
$865$	−31.9467	−1.08622
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−41.0226	−1.39160
$870$	0	0
$871$	−15.9155	−0.539275
$872$	0	0
$873$	0	0
$874$	0	0
$875$	41.3607	1.39825
$876$	0	0
$877$	−23.3074	−0.787034	−0.393517	−	0.919317i	$-0.628742\pi$
$877$	−23.3074	−0.787034	−0.393517	+	0.919317i	$0.628742\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.9939	1.31374	0.656868	−	0.754005i	$-0.271881\pi$
$881$	38.9939	1.31374	0.656868	+	0.754005i	$0.271881\pi$
$882$	0	0
$883$	36.5646	1.23050	0.615249	−	0.788333i	$-0.289056\pi$
$883$	36.5646	1.23050	0.615249	+	0.788333i	$0.289056\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.0410	−1.00868	−0.504340	−	0.863505i	$-0.668264\pi$
$887$	−30.0410	−1.00868	−0.504340	+	0.863505i	$0.668264\pi$
$888$	0	0
$889$	10.6147	0.356007
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	31.7686	1.06191
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11.5174	−0.384128
$900$	0	0
$901$	−5.36069	−0.178591
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.27247	−0.0755396
$906$	0	0
$907$	11.8310	0.392841	0.196420	−	0.980520i	$-0.437068\pi$
$907$	11.8310	0.392841	0.196420	+	0.980520i	$0.437068\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44.3423	−1.46912	−0.734562	−	0.678541i	$-0.762612\pi$
$911$	−44.3423	−1.46912	−0.734562	+	0.678541i	$0.762612\pi$
$912$	0	0
$913$	4.31351	0.142756
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.7337	0.552594
$918$	0	0
$919$	−32.0821	−1.05829	−0.529145	−	0.848531i	$-0.677487\pi$
$919$	−32.0821	−1.05829	−0.529145	+	0.848531i	$0.677487\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.26180	0.173194
$924$	0	0
$925$	−92.4366	−3.03930
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.4345	1.12976	0.564880	−	0.825173i	$-0.308922\pi$
$929$	34.4345	1.12976	0.564880	+	0.825173i	$0.308922\pi$
$930$	0	0
$931$	−6.29460	−0.206297
$932$	0	0
$933$	0	0
$934$	0	0
$935$	29.6742	0.970450
$936$	0	0
$937$	34.1978	1.11719	0.558597	−	0.829439i	$-0.311340\pi$
$937$	34.1978	1.11719	0.558597	+	0.829439i	$0.311340\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17.7009	0.577032	0.288516	−	0.957475i	$-0.406838\pi$
$941$	17.7009	0.577032	0.288516	+	0.957475i	$0.406838\pi$
$942$	0	0
$943$	−0.733670	−0.0238916
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.7259	0.543519	0.271760	−	0.962365i	$-0.412394\pi$
$947$	16.7259	0.543519	0.271760	+	0.962365i	$0.412394\pi$
$948$	0	0
$949$	14.6803	0.476544
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−10.9939	−0.356126	−0.178063	−	0.984019i	$-0.556983\pi$
$953$	−10.9939	−0.356126	−0.178063	+	0.984019i	$0.556983\pi$
$954$	0	0
$955$	95.6619	3.09555
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.3791	0.464326
$960$	0	0
$961$	2.16290	0.0697709
$962$	0	0
$963$	0	0
$964$	0	0
$965$	82.6681	2.66118
$966$	0	0
$967$	−40.3111	−1.29632	−0.648158	−	0.761506i	$-0.724460\pi$
$967$	−40.3111	−1.29632	−0.648158	+	0.761506i	$0.724460\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.67420	−0.0537277	−0.0268639	−	0.999639i	$-0.508552\pi$
$971$	−1.67420	−0.0537277	−0.0268639	+	0.999639i	$0.508552\pi$
$972$	0	0
$973$	0.733670	0.0235204
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.0328	0.352971	0.176485	−	0.984303i	$-0.443527\pi$
$977$	11.0328	0.352971	0.176485	+	0.984303i	$0.443527\pi$
$978$	0	0
$979$	44.5113	1.42259
$980$	0	0
$981$	0	0
$982$	0	0
$983$	50.4001	1.60751	0.803757	−	0.594958i	$-0.202831\pi$
$983$	50.4001	1.60751	0.803757	+	0.594958i	$0.202831\pi$
$984$	0	0
$985$	−75.2327	−2.39711
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−23.3730	−0.743217
$990$	0	0
$991$	23.7152	0.753339	0.376670	−	0.926348i	$-0.377069\pi$
$991$	23.7152	0.753339	0.376670	+	0.926348i	$0.377069\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	67.3484	2.13509
$996$	0	0
$997$	5.37298	0.170164	0.0850820	−	0.996374i	$-0.472885\pi$
$997$	5.37298	0.170164	0.0850820	+	0.996374i	$0.472885\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.a.z.1.1		3
3.2	odd	2		1248.2.a.p.1.3	yes	3
4.3	odd	2		3744.2.a.ba.1.1		3
8.3	odd	2		7488.2.a.cx.1.3		3
8.5	even	2		7488.2.a.cy.1.3		3
12.11	even	2		1248.2.a.o.1.3	✓	3
24.5	odd	2		2496.2.a.bk.1.1		3
24.11	even	2		2496.2.a.bl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.a.o.1.3	✓	3	12.11	even	2
1248.2.a.p.1.3	yes	3	3.2	odd	2
2496.2.a.bk.1.1		3	24.5	odd	2
2496.2.a.bl.1.1		3	24.11	even	2
3744.2.a.z.1.1		3	1.1	even	1	trivial
3744.2.a.ba.1.1		3	4.3	odd	2
7488.2.a.cx.1.3		3	8.3	odd	2
7488.2.a.cy.1.3		3	8.5	even	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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.a (trivial)

\(f(q)\)	\(=\)	\(q-4.34017 q^{5} -1.07838 q^{7} +O(q^{10})\)	\(q-4.34017 q^{5} -1.07838 q^{7} +3.41855 q^{11} +1.00000 q^{13} -2.00000 q^{17} +1.07838 q^{19} -2.15676 q^{23} +13.8371 q^{25} -2.00000 q^{29} +5.75872 q^{31} +4.68035 q^{35} -6.68035 q^{37} +0.340173 q^{41} +10.8371 q^{43} +7.41855 q^{47} -5.83710 q^{49} +2.68035 q^{53} -14.8371 q^{55} -9.26180 q^{59} -4.52359 q^{61} -4.34017 q^{65} -15.9155 q^{67} +5.26180 q^{71} +14.6803 q^{73} -3.68649 q^{77} -12.0000 q^{79} +1.26180 q^{83} +8.68035 q^{85} +13.0205 q^{89} -1.07838 q^{91} -4.68035 q^{95} +6.68035 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5}+O(q^{10}) \) 3 * q - 2 * q^5	\( 3 q - 2 q^{5} - 4 q^{11} + 3 q^{13} - 6 q^{17} + 13 q^{25} - 6 q^{29} - 8 q^{31} - 8 q^{35} + 2 q^{37} - 10 q^{41} + 4 q^{43} + 8 q^{47} + 11 q^{49} - 14 q^{53} - 16 q^{55} - 20 q^{59} + 2 q^{61} - 2 q^{65} - 16 q^{67} + 8 q^{71} + 22 q^{73} - 24 q^{77} - 36 q^{79} - 4 q^{83} + 4 q^{85} + 6 q^{89} + 8 q^{95} - 2 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 - 4 * q^11 + 3 * q^13 - 6 * q^17 + 13 * q^25 - 6 * q^29 - 8 * q^31 - 8 * q^35 + 2 * q^37 - 10 * q^41 + 4 * q^43 + 8 * q^47 + 11 * q^49 - 14 * q^53 - 16 * q^55 - 20 * q^59 + 2 * q^61 - 2 * q^65 - 16 * q^67 + 8 * q^71 + 22 * q^73 - 24 * q^77 - 36 * q^79 - 4 * q^83 + 4 * q^85 + 6 * q^89 + 8 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more