LMFDB - Embedded newform 4416.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4416,2,Mod(1,4416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4416, 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("4416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4416 = 2^{6} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4416.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:	$35.2619375326$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.23607 q^{5} -1.23607 q^{7} +1.00000 q^{9} -4.00000 q^{11} -4.47214 q^{13} +3.23607 q^{15} -7.23607 q^{17} -2.76393 q^{19} -1.23607 q^{21} +1.00000 q^{23} +5.47214 q^{25} +1.00000 q^{27} +4.47214 q^{29} +2.47214 q^{31} -4.00000 q^{33} -4.00000 q^{35} +4.47214 q^{37} -4.47214 q^{39} +6.94427 q^{41} -7.70820 q^{43} +3.23607 q^{45} -4.00000 q^{47} -5.47214 q^{49} -7.23607 q^{51} +0.763932 q^{53} -12.9443 q^{55} -2.76393 q^{57} -12.9443 q^{59} +4.47214 q^{61} -1.23607 q^{63} -14.4721 q^{65} -5.23607 q^{67} +1.00000 q^{69} -8.00000 q^{71} -10.9443 q^{73} +5.47214 q^{75} +4.94427 q^{77} -3.70820 q^{79} +1.00000 q^{81} -4.00000 q^{83} -23.4164 q^{85} +4.47214 q^{87} +3.23607 q^{89} +5.52786 q^{91} +2.47214 q^{93} -8.94427 q^{95} -0.472136 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 2 q^{15} - 10 q^{17} - 10 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{31} - 8 q^{33} - 8 q^{35} - 4 q^{41} - 2 q^{43} + 2 q^{45} - 8 q^{47}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 2 * q^15 - 10 * q^17 - 10 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^31 - 8 * q^33 - 8 * q^35 - 4 * q^41 - 2 * q^43 + 2 * q^45 - 8 * q^47 - 2 * q^49 - 10 * q^51 + 6 * q^53 - 8 * q^55 - 10 * q^57 - 8 * q^59 + 2 * q^63 - 20 * q^65 - 6 * q^67 + 2 * q^69 - 16 * q^71 - 4 * q^73 + 2 * q^75 - 8 * q^77 + 6 * q^79 + 2 * q^81 - 8 * q^83 - 20 * q^85 + 2 * q^89 + 20 * q^91 - 4 * q^93 + 8 * q^97 - 8 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	3.23607	0.835549
$16$	0	0
$17$	−7.23607	−1.75500	−0.877502	−	0.479573i	$-0.840792\pi$
$17$	−7.23607	−1.75500	−0.877502	+	0.479573i	$0.840792\pi$
$18$	0	0
$19$	−2.76393	−0.634089	−0.317045	−	0.948411i	$-0.602691\pi$
$19$	−2.76393	−0.634089	−0.317045	+	0.948411i	$0.602691\pi$
$20$	0	0
$21$	−1.23607	−0.269732
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	0	0
$39$	−4.47214	−0.716115
$40$	0	0
$41$	6.94427	1.08451	0.542257	−	0.840213i	$-0.317570\pi$
$41$	6.94427	1.08451	0.542257	+	0.840213i	$0.317570\pi$
$42$	0	0
$43$	−7.70820	−1.17549	−0.587745	−	0.809046i	$-0.699984\pi$
$43$	−7.70820	−1.17549	−0.587745	+	0.809046i	$0.699984\pi$
$44$	0	0
$45$	3.23607	0.482405
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−7.23607	−1.01325
$52$	0	0
$53$	0.763932	0.104934	0.0524671	−	0.998623i	$-0.483292\pi$
$53$	0.763932	0.104934	0.0524671	+	0.998623i	$0.483292\pi$
$54$	0	0
$55$	−12.9443	−1.74541
$56$	0	0
$57$	−2.76393	−0.366092
$58$	0	0
$59$	−12.9443	−1.68520	−0.842600	−	0.538539i	$-0.818976\pi$
$59$	−12.9443	−1.68520	−0.842600	+	0.538539i	$0.818976\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	0	0
$63$	−1.23607	−0.155730
$64$	0	0
$65$	−14.4721	−1.79505
$66$	0	0
$67$	−5.23607	−0.639688	−0.319844	−	0.947470i	$-0.603630\pi$
$67$	−5.23607	−0.639688	−0.319844	+	0.947470i	$0.603630\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−10.9443	−1.28093	−0.640465	−	0.767987i	$-0.721258\pi$
$73$	−10.9443	−1.28093	−0.640465	+	0.767987i	$0.721258\pi$
$74$	0	0
$75$	5.47214	0.631868
$76$	0	0
$77$	4.94427	0.563452
$78$	0	0
$79$	−3.70820	−0.417206	−0.208603	−	0.978000i	$-0.566892\pi$
$79$	−3.70820	−0.417206	−0.208603	+	0.978000i	$0.566892\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−23.4164	−2.53987
$86$	0	0
$87$	4.47214	0.479463
$88$	0	0
$89$	3.23607	0.343023	0.171511	−	0.985182i	$-0.445135\pi$
$89$	3.23607	0.343023	0.171511	+	0.985182i	$0.445135\pi$
$90$	0	0
$91$	5.52786	0.579478
$92$	0	0
$93$	2.47214	0.256349
$94$	0	0
$95$	−8.94427	−0.917663
$96$	0	0
$97$	−0.472136	−0.0479381	−0.0239691	−	0.999713i	$-0.507630\pi$
$97$	−0.472136	−0.0479381	−0.0239691	+	0.999713i	$0.507630\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−10.9443	−1.08900	−0.544498	−	0.838762i	$-0.683280\pi$
$101$	−10.9443	−1.08900	−0.544498	+	0.838762i	$0.683280\pi$
$102$	0	0
$103$	−6.76393	−0.666470	−0.333235	−	0.942844i	$-0.608140\pi$
$103$	−6.76393	−0.666470	−0.333235	+	0.942844i	$0.608140\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	0.944272	0.0912862	0.0456431	−	0.998958i	$-0.485466\pi$
$107$	0.944272	0.0912862	0.0456431	+	0.998958i	$0.485466\pi$
$108$	0	0
$109$	−2.94427	−0.282010	−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427	−0.282010	−0.141005	+	0.990009i	$0.545033\pi$
$110$	0	0
$111$	4.47214	0.424476
$112$	0	0
$113$	16.1803	1.52212	0.761059	−	0.648682i	$-0.224680\pi$
$113$	16.1803	1.52212	0.761059	+	0.648682i	$0.224680\pi$
$114$	0	0
$115$	3.23607	0.301765
$116$	0	0
$117$	−4.47214	−0.413449
$118$	0	0
$119$	8.94427	0.819920
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	6.94427	0.626144
$124$	0	0
$125$	1.52786	0.136656
$126$	0	0
$127$	10.4721	0.929252	0.464626	−	0.885507i	$-0.346189\pi$
$127$	10.4721	0.929252	0.464626	+	0.885507i	$0.346189\pi$
$128$	0	0
$129$	−7.70820	−0.678670
$130$	0	0
$131$	0.944272	0.0825014	0.0412507	−	0.999149i	$-0.486866\pi$
$131$	0.944272	0.0825014	0.0412507	+	0.999149i	$0.486866\pi$
$132$	0	0
$133$	3.41641	0.296240
$134$	0	0
$135$	3.23607	0.278516
$136$	0	0
$137$	3.23607	0.276476	0.138238	−	0.990399i	$-0.455856\pi$
$137$	3.23607	0.276476	0.138238	+	0.990399i	$0.455856\pi$
$138$	0	0
$139$	−8.94427	−0.758643	−0.379322	−	0.925265i	$-0.623843\pi$
$139$	−8.94427	−0.758643	−0.379322	+	0.925265i	$0.623843\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	17.8885	1.49592
$144$	0	0
$145$	14.4721	1.20185
$146$	0	0
$147$	−5.47214	−0.451334
$148$	0	0
$149$	−1.70820	−0.139942	−0.0699708	−	0.997549i	$-0.522291\pi$
$149$	−1.70820	−0.139942	−0.0699708	+	0.997549i	$0.522291\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−7.23607	−0.585001
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	12.4721	0.995385	0.497692	−	0.867354i	$-0.334181\pi$
$157$	12.4721	0.995385	0.497692	+	0.867354i	$0.334181\pi$
$158$	0	0
$159$	0.763932	0.0605838
$160$	0	0
$161$	−1.23607	−0.0974158
$162$	0	0
$163$	19.4164	1.52081	0.760405	−	0.649449i	$-0.225000\pi$
$163$	19.4164	1.52081	0.760405	+	0.649449i	$0.225000\pi$
$164$	0	0
$165$	−12.9443	−1.00771
$166$	0	0
$167$	−4.94427	−0.382599	−0.191300	−	0.981532i	$-0.561270\pi$
$167$	−4.94427	−0.382599	−0.191300	+	0.981532i	$0.561270\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	−2.76393	−0.211363
$172$	0	0
$173$	4.47214	0.340010	0.170005	−	0.985443i	$-0.445622\pi$
$173$	4.47214	0.340010	0.170005	+	0.985443i	$0.445622\pi$
$174$	0	0
$175$	−6.76393	−0.511305
$176$	0	0
$177$	−12.9443	−0.972951
$178$	0	0
$179$	−3.05573	−0.228396	−0.114198	−	0.993458i	$-0.536430\pi$
$179$	−3.05573	−0.228396	−0.114198	+	0.993458i	$0.536430\pi$
$180$	0	0
$181$	−23.8885	−1.77562	−0.887811	−	0.460209i	$-0.847775\pi$
$181$	−23.8885	−1.77562	−0.887811	+	0.460209i	$0.847775\pi$
$182$	0	0
$183$	4.47214	0.330590
$184$	0	0
$185$	14.4721	1.06401
$186$	0	0
$187$	28.9443	2.11661
$188$	0	0
$189$	−1.23607	−0.0899107
$190$	0	0
$191$	10.4721	0.757737	0.378869	−	0.925450i	$-0.376313\pi$
$191$	10.4721	0.757737	0.378869	+	0.925450i	$0.376313\pi$
$192$	0	0
$193$	−9.41641	−0.677808	−0.338904	−	0.940821i	$-0.610056\pi$
$193$	−9.41641	−0.677808	−0.338904	+	0.940821i	$0.610056\pi$
$194$	0	0
$195$	−14.4721	−1.03637
$196$	0	0
$197$	9.41641	0.670891	0.335446	−	0.942060i	$-0.391113\pi$
$197$	9.41641	0.670891	0.335446	+	0.942060i	$0.391113\pi$
$198$	0	0
$199$	−3.70820	−0.262868	−0.131434	−	0.991325i	$-0.541958\pi$
$199$	−3.70820	−0.262868	−0.131434	+	0.991325i	$0.541958\pi$
$200$	0	0
$201$	−5.23607	−0.369324
$202$	0	0
$203$	−5.52786	−0.387980
$204$	0	0
$205$	22.4721	1.56952
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	11.0557	0.764741
$210$	0	0
$211$	−22.4721	−1.54705	−0.773523	−	0.633768i	$-0.781507\pi$
$211$	−22.4721	−1.54705	−0.773523	+	0.633768i	$0.781507\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−24.9443	−1.70119
$216$	0	0
$217$	−3.05573	−0.207436
$218$	0	0
$219$	−10.9443	−0.739545
$220$	0	0
$221$	32.3607	2.17681
$222$	0	0
$223$	9.88854	0.662186	0.331093	−	0.943598i	$-0.392583\pi$
$223$	9.88854	0.662186	0.331093	+	0.943598i	$0.392583\pi$
$224$	0	0
$225$	5.47214	0.364809
$226$	0	0
$227$	22.4721	1.49153	0.745764	−	0.666210i	$-0.232085\pi$
$227$	22.4721	1.49153	0.745764	+	0.666210i	$0.232085\pi$
$228$	0	0
$229$	14.9443	0.987545	0.493773	−	0.869591i	$-0.335618\pi$
$229$	14.9443	0.987545	0.493773	+	0.869591i	$0.335618\pi$
$230$	0	0
$231$	4.94427	0.325309
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	−12.9443	−0.844391
$236$	0	0
$237$	−3.70820	−0.240874
$238$	0	0
$239$	−12.9443	−0.837295	−0.418648	−	0.908149i	$-0.637496\pi$
$239$	−12.9443	−0.837295	−0.418648	+	0.908149i	$0.637496\pi$
$240$	0	0
$241$	28.4721	1.83405	0.917026	−	0.398828i	$-0.130583\pi$
$241$	28.4721	1.83405	0.917026	+	0.398828i	$0.130583\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−17.7082	−1.13134
$246$	0	0
$247$	12.3607	0.786491
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	1.52786	0.0964379	0.0482190	−	0.998837i	$-0.484645\pi$
$251$	1.52786	0.0964379	0.0482190	+	0.998837i	$0.484645\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	−23.4164	−1.46639
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	−5.52786	−0.343485
$260$	0	0
$261$	4.47214	0.276818
$262$	0	0
$263$	−2.47214	−0.152438	−0.0762192	−	0.997091i	$-0.524285\pi$
$263$	−2.47214	−0.152438	−0.0762192	+	0.997091i	$0.524285\pi$
$264$	0	0
$265$	2.47214	0.151862
$266$	0	0
$267$	3.23607	0.198044
$268$	0	0
$269$	−8.47214	−0.516555	−0.258278	−	0.966071i	$-0.583155\pi$
$269$	−8.47214	−0.516555	−0.258278	+	0.966071i	$0.583155\pi$
$270$	0	0
$271$	−26.4721	−1.60807	−0.804034	−	0.594583i	$-0.797317\pi$
$271$	−26.4721	−1.60807	−0.804034	+	0.594583i	$0.797317\pi$
$272$	0	0
$273$	5.52786	0.334562
$274$	0	0
$275$	−21.8885	−1.31993
$276$	0	0
$277$	19.8885	1.19499	0.597493	−	0.801874i	$-0.296163\pi$
$277$	19.8885	1.19499	0.597493	+	0.801874i	$0.296163\pi$
$278$	0	0
$279$	2.47214	0.148003
$280$	0	0
$281$	−6.65248	−0.396853	−0.198427	−	0.980116i	$-0.563583\pi$
$281$	−6.65248	−0.396853	−0.198427	+	0.980116i	$0.563583\pi$
$282$	0	0
$283$	7.70820	0.458205	0.229103	−	0.973402i	$-0.426421\pi$
$283$	7.70820	0.458205	0.229103	+	0.973402i	$0.426421\pi$
$284$	0	0
$285$	−8.94427	−0.529813
$286$	0	0
$287$	−8.58359	−0.506673
$288$	0	0
$289$	35.3607	2.08004
$290$	0	0
$291$	−0.472136	−0.0276771
$292$	0	0
$293$	5.70820	0.333477	0.166738	−	0.986001i	$-0.446676\pi$
$293$	5.70820	0.333477	0.166738	+	0.986001i	$0.446676\pi$
$294$	0	0
$295$	−41.8885	−2.43885
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	−4.47214	−0.258630
$300$	0	0
$301$	9.52786	0.549177
$302$	0	0
$303$	−10.9443	−0.628732
$304$	0	0
$305$	14.4721	0.828672
$306$	0	0
$307$	6.47214	0.369384	0.184692	−	0.982796i	$-0.440871\pi$
$307$	6.47214	0.369384	0.184692	+	0.982796i	$0.440871\pi$
$308$	0	0
$309$	−6.76393	−0.384787
$310$	0	0
$311$	24.9443	1.41446	0.707230	−	0.706984i	$-0.249945\pi$
$311$	24.9443	1.41446	0.707230	+	0.706984i	$0.249945\pi$
$312$	0	0
$313$	22.9443	1.29689	0.648443	−	0.761263i	$-0.275420\pi$
$313$	22.9443	1.29689	0.648443	+	0.761263i	$0.275420\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	−29.4164	−1.65219	−0.826095	−	0.563531i	$-0.809443\pi$
$317$	−29.4164	−1.65219	−0.826095	+	0.563531i	$0.809443\pi$
$318$	0	0
$319$	−17.8885	−1.00157
$320$	0	0
$321$	0.944272	0.0527041
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	−24.4721	−1.35747
$326$	0	0
$327$	−2.94427	−0.162819
$328$	0	0
$329$	4.94427	0.272587
$330$	0	0
$331$	3.41641	0.187783	0.0938914	−	0.995582i	$-0.470069\pi$
$331$	3.41641	0.187783	0.0938914	+	0.995582i	$0.470069\pi$
$332$	0	0
$333$	4.47214	0.245072
$334$	0	0
$335$	−16.9443	−0.925764
$336$	0	0
$337$	30.3607	1.65385	0.826926	−	0.562311i	$-0.190088\pi$
$337$	30.3607	1.65385	0.826926	+	0.562311i	$0.190088\pi$
$338$	0	0
$339$	16.1803	0.878795
$340$	0	0
$341$	−9.88854	−0.535495
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	3.23607	0.174224
$346$	0	0
$347$	−25.8885	−1.38977	−0.694885	−	0.719121i	$-0.744545\pi$
$347$	−25.8885	−1.38977	−0.694885	+	0.719121i	$0.744545\pi$
$348$	0	0
$349$	32.4721	1.73819	0.869097	−	0.494642i	$-0.164701\pi$
$349$	32.4721	1.73819	0.869097	+	0.494642i	$0.164701\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	0	0
$353$	9.41641	0.501185	0.250592	−	0.968093i	$-0.419375\pi$
$353$	9.41641	0.501185	0.250592	+	0.968093i	$0.419375\pi$
$354$	0	0
$355$	−25.8885	−1.37402
$356$	0	0
$357$	8.94427	0.473381
$358$	0	0
$359$	−2.47214	−0.130474	−0.0652372	−	0.997870i	$-0.520780\pi$
$359$	−2.47214	−0.130474	−0.0652372	+	0.997870i	$0.520780\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−35.4164	−1.85378
$366$	0	0
$367$	12.2918	0.641627	0.320813	−	0.947142i	$-0.396044\pi$
$367$	12.2918	0.641627	0.320813	+	0.947142i	$0.396044\pi$
$368$	0	0
$369$	6.94427	0.361504
$370$	0	0
$371$	−0.944272	−0.0490242
$372$	0	0
$373$	−11.5279	−0.596890	−0.298445	−	0.954427i	$-0.596468\pi$
$373$	−11.5279	−0.596890	−0.298445	+	0.954427i	$0.596468\pi$
$374$	0	0
$375$	1.52786	0.0788986
$376$	0	0
$377$	−20.0000	−1.03005
$378$	0	0
$379$	28.0689	1.44180	0.720901	−	0.693038i	$-0.243728\pi$
$379$	28.0689	1.44180	0.720901	+	0.693038i	$0.243728\pi$
$380$	0	0
$381$	10.4721	0.536504
$382$	0	0
$383$	34.4721	1.76144	0.880722	−	0.473634i	$-0.157058\pi$
$383$	34.4721	1.76144	0.880722	+	0.473634i	$0.157058\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	−7.70820	−0.391830
$388$	0	0
$389$	10.6525	0.540102	0.270051	−	0.962846i	$-0.412959\pi$
$389$	10.6525	0.540102	0.270051	+	0.962846i	$0.412959\pi$
$390$	0	0
$391$	−7.23607	−0.365944
$392$	0	0
$393$	0.944272	0.0476322
$394$	0	0
$395$	−12.0000	−0.603786
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	3.41641	0.171034
$400$	0	0
$401$	−38.0689	−1.90107	−0.950535	−	0.310618i	$-0.899464\pi$
$401$	−38.0689	−1.90107	−0.950535	+	0.310618i	$0.899464\pi$
$402$	0	0
$403$	−11.0557	−0.550725
$404$	0	0
$405$	3.23607	0.160802
$406$	0	0
$407$	−17.8885	−0.886702
$408$	0	0
$409$	−9.41641	−0.465611	−0.232806	−	0.972523i	$-0.574791\pi$
$409$	−9.41641	−0.465611	−0.232806	+	0.972523i	$0.574791\pi$
$410$	0	0
$411$	3.23607	0.159623
$412$	0	0
$413$	16.0000	0.787309
$414$	0	0
$415$	−12.9443	−0.635409
$416$	0	0
$417$	−8.94427	−0.438003
$418$	0	0
$419$	21.8885	1.06933	0.534663	−	0.845066i	$-0.320439\pi$
$419$	21.8885	1.06933	0.534663	+	0.845066i	$0.320439\pi$
$420$	0	0
$421$	13.0557	0.636297	0.318149	−	0.948041i	$-0.396939\pi$
$421$	13.0557	0.636297	0.318149	+	0.948041i	$0.396939\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	−39.5967	−1.92072
$426$	0	0
$427$	−5.52786	−0.267512
$428$	0	0
$429$	17.8885	0.863667
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	−7.88854	−0.379099	−0.189550	−	0.981871i	$-0.560703\pi$
$433$	−7.88854	−0.379099	−0.189550	+	0.981871i	$0.560703\pi$
$434$	0	0
$435$	14.4721	0.693886
$436$	0	0
$437$	−2.76393	−0.132217
$438$	0	0
$439$	25.8885	1.23559	0.617796	−	0.786338i	$-0.288026\pi$
$439$	25.8885	1.23559	0.617796	+	0.786338i	$0.288026\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	0	0
$443$	−18.8328	−0.894774	−0.447387	−	0.894340i	$-0.647645\pi$
$443$	−18.8328	−0.894774	−0.447387	+	0.894340i	$0.647645\pi$
$444$	0	0
$445$	10.4721	0.496427
$446$	0	0
$447$	−1.70820	−0.0807953
$448$	0	0
$449$	−2.94427	−0.138949	−0.0694744	−	0.997584i	$-0.522132\pi$
$449$	−2.94427	−0.138949	−0.0694744	+	0.997584i	$0.522132\pi$
$450$	0	0
$451$	−27.7771	−1.30797
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	17.8885	0.838628
$456$	0	0
$457$	−0.472136	−0.0220856	−0.0110428	−	0.999939i	$-0.503515\pi$
$457$	−0.472136	−0.0220856	−0.0110428	+	0.999939i	$0.503515\pi$
$458$	0	0
$459$	−7.23607	−0.337751
$460$	0	0
$461$	−21.4164	−0.997462	−0.498731	−	0.866757i	$-0.666200\pi$
$461$	−21.4164	−0.997462	−0.498731	+	0.866757i	$0.666200\pi$
$462$	0	0
$463$	4.94427	0.229780	0.114890	−	0.993378i	$-0.463348\pi$
$463$	4.94427	0.229780	0.114890	+	0.993378i	$0.463348\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	24.3607	1.12728	0.563639	−	0.826021i	$-0.309401\pi$
$467$	24.3607	1.12728	0.563639	+	0.826021i	$0.309401\pi$
$468$	0	0
$469$	6.47214	0.298855
$470$	0	0
$471$	12.4721	0.574686
$472$	0	0
$473$	30.8328	1.41769
$474$	0	0
$475$	−15.1246	−0.693965
$476$	0	0
$477$	0.763932	0.0349780
$478$	0	0
$479$	1.88854	0.0862898	0.0431449	−	0.999069i	$-0.486262\pi$
$479$	1.88854	0.0862898	0.0431449	+	0.999069i	$0.486262\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	−1.23607	−0.0562430
$484$	0	0
$485$	−1.52786	−0.0693767
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	0	0
$489$	19.4164	0.878040
$490$	0	0
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	0	0
$493$	−32.3607	−1.45745
$494$	0	0
$495$	−12.9443	−0.581802
$496$	0	0
$497$	9.88854	0.443562
$498$	0	0
$499$	−32.3607	−1.44866	−0.724331	−	0.689452i	$-0.757851\pi$
$499$	−32.3607	−1.44866	−0.724331	+	0.689452i	$0.757851\pi$
$500$	0	0
$501$	−4.94427	−0.220894
$502$	0	0
$503$	−36.9443	−1.64726	−0.823632	−	0.567125i	$-0.808056\pi$
$503$	−36.9443	−1.64726	−0.823632	+	0.567125i	$0.808056\pi$
$504$	0	0
$505$	−35.4164	−1.57601
$506$	0	0
$507$	7.00000	0.310881
$508$	0	0
$509$	7.52786	0.333667	0.166833	−	0.985985i	$-0.446646\pi$
$509$	7.52786	0.333667	0.166833	+	0.985985i	$0.446646\pi$
$510$	0	0
$511$	13.5279	0.598437
$512$	0	0
$513$	−2.76393	−0.122031
$514$	0	0
$515$	−21.8885	−0.964524
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	4.47214	0.196305
$520$	0	0
$521$	−28.7639	−1.26017	−0.630085	−	0.776526i	$-0.716980\pi$
$521$	−28.7639	−1.26017	−0.630085	+	0.776526i	$0.716980\pi$
$522$	0	0
$523$	33.5967	1.46908	0.734542	−	0.678564i	$-0.237397\pi$
$523$	33.5967	1.46908	0.734542	+	0.678564i	$0.237397\pi$
$524$	0	0
$525$	−6.76393	−0.295202
$526$	0	0
$527$	−17.8885	−0.779237
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−12.9443	−0.561734
$532$	0	0
$533$	−31.0557	−1.34517
$534$	0	0
$535$	3.05573	0.132111
$536$	0	0
$537$	−3.05573	−0.131864
$538$	0	0
$539$	21.8885	0.942806
$540$	0	0
$541$	24.4721	1.05214	0.526070	−	0.850441i	$-0.323665\pi$
$541$	24.4721	1.05214	0.526070	+	0.850441i	$0.323665\pi$
$542$	0	0
$543$	−23.8885	−1.02516
$544$	0	0
$545$	−9.52786	−0.408129
$546$	0	0
$547$	−22.4721	−0.960839	−0.480420	−	0.877039i	$-0.659516\pi$
$547$	−22.4721	−0.960839	−0.480420	+	0.877039i	$0.659516\pi$
$548$	0	0
$549$	4.47214	0.190866
$550$	0	0
$551$	−12.3607	−0.526583
$552$	0	0
$553$	4.58359	0.194914
$554$	0	0
$555$	14.4721	0.614308
$556$	0	0
$557$	19.8197	0.839786	0.419893	−	0.907574i	$-0.362068\pi$
$557$	19.8197	0.839786	0.419893	+	0.907574i	$0.362068\pi$
$558$	0	0
$559$	34.4721	1.45802
$560$	0	0
$561$	28.9443	1.22203
$562$	0	0
$563$	−19.4164	−0.818304	−0.409152	−	0.912466i	$-0.634175\pi$
$563$	−19.4164	−0.818304	−0.409152	+	0.912466i	$0.634175\pi$
$564$	0	0
$565$	52.3607	2.20283
$566$	0	0
$567$	−1.23607	−0.0519100
$568$	0	0
$569$	−15.2361	−0.638729	−0.319365	−	0.947632i	$-0.603469\pi$
$569$	−15.2361	−0.638729	−0.319365	+	0.947632i	$0.603469\pi$
$570$	0	0
$571$	−16.2918	−0.681790	−0.340895	−	0.940101i	$-0.610730\pi$
$571$	−16.2918	−0.681790	−0.340895	+	0.940101i	$0.610730\pi$
$572$	0	0
$573$	10.4721	0.437480
$574$	0	0
$575$	5.47214	0.228204
$576$	0	0
$577$	16.4721	0.685744	0.342872	−	0.939382i	$-0.388600\pi$
$577$	16.4721	0.685744	0.342872	+	0.939382i	$0.388600\pi$
$578$	0	0
$579$	−9.41641	−0.391333
$580$	0	0
$581$	4.94427	0.205123
$582$	0	0
$583$	−3.05573	−0.126555
$584$	0	0
$585$	−14.4721	−0.598349
$586$	0	0
$587$	−16.9443	−0.699365	−0.349682	−	0.936868i	$-0.613711\pi$
$587$	−16.9443	−0.699365	−0.349682	+	0.936868i	$0.613711\pi$
$588$	0	0
$589$	−6.83282	−0.281541
$590$	0	0
$591$	9.41641	0.387339
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	28.9443	1.18660
$596$	0	0
$597$	−3.70820	−0.151767
$598$	0	0
$599$	−20.9443	−0.855760	−0.427880	−	0.903836i	$-0.640739\pi$
$599$	−20.9443	−0.855760	−0.427880	+	0.903836i	$0.640739\pi$
$600$	0	0
$601$	2.36068	0.0962941	0.0481471	−	0.998840i	$-0.484668\pi$
$601$	2.36068	0.0962941	0.0481471	+	0.998840i	$0.484668\pi$
$602$	0	0
$603$	−5.23607	−0.213229
$604$	0	0
$605$	16.1803	0.657824
$606$	0	0
$607$	38.8328	1.57618	0.788088	−	0.615563i	$-0.211071\pi$
$607$	38.8328	1.57618	0.788088	+	0.615563i	$0.211071\pi$
$608$	0	0
$609$	−5.52786	−0.224000
$610$	0	0
$611$	17.8885	0.723693
$612$	0	0
$613$	31.5279	1.27340	0.636699	−	0.771112i	$-0.280299\pi$
$613$	31.5279	1.27340	0.636699	+	0.771112i	$0.280299\pi$
$614$	0	0
$615$	22.4721	0.906164
$616$	0	0
$617$	−33.7082	−1.35704	−0.678521	−	0.734581i	$-0.737379\pi$
$617$	−33.7082	−1.35704	−0.678521	+	0.734581i	$0.737379\pi$
$618$	0	0
$619$	−31.1246	−1.25100	−0.625502	−	0.780223i	$-0.715106\pi$
$619$	−31.1246	−1.25100	−0.625502	+	0.780223i	$0.715106\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	11.0557	0.441523
$628$	0	0
$629$	−32.3607	−1.29030
$630$	0	0
$631$	6.18034	0.246035	0.123018	−	0.992404i	$-0.460743\pi$
$631$	6.18034	0.246035	0.123018	+	0.992404i	$0.460743\pi$
$632$	0	0
$633$	−22.4721	−0.893187
$634$	0	0
$635$	33.8885	1.34483
$636$	0	0
$637$	24.4721	0.969621
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−27.0132	−1.06696	−0.533478	−	0.845814i	$-0.679115\pi$
$641$	−27.0132	−1.06696	−0.533478	+	0.845814i	$0.679115\pi$
$642$	0	0
$643$	20.0689	0.791440	0.395720	−	0.918371i	$-0.370495\pi$
$643$	20.0689	0.791440	0.395720	+	0.918371i	$0.370495\pi$
$644$	0	0
$645$	−24.9443	−0.982180
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	51.7771	2.03243
$650$	0	0
$651$	−3.05573	−0.119763
$652$	0	0
$653$	5.05573	0.197846	0.0989230	−	0.995095i	$-0.468460\pi$
$653$	5.05573	0.197846	0.0989230	+	0.995095i	$0.468460\pi$
$654$	0	0
$655$	3.05573	0.119397
$656$	0	0
$657$	−10.9443	−0.426977
$658$	0	0
$659$	8.36068	0.325686	0.162843	−	0.986652i	$-0.447934\pi$
$659$	8.36068	0.325686	0.162843	+	0.986652i	$0.447934\pi$
$660$	0	0
$661$	20.4721	0.796274	0.398137	−	0.917326i	$-0.369657\pi$
$661$	20.4721	0.796274	0.398137	+	0.917326i	$0.369657\pi$
$662$	0	0
$663$	32.3607	1.25678
$664$	0	0
$665$	11.0557	0.428723
$666$	0	0
$667$	4.47214	0.173162
$668$	0	0
$669$	9.88854	0.382313
$670$	0	0
$671$	−17.8885	−0.690580
$672$	0	0
$673$	29.4164	1.13392	0.566960	−	0.823746i	$-0.308120\pi$
$673$	29.4164	1.13392	0.566960	+	0.823746i	$0.308120\pi$
$674$	0	0
$675$	5.47214	0.210623
$676$	0	0
$677$	−37.4853	−1.44068	−0.720338	−	0.693623i	$-0.756013\pi$
$677$	−37.4853	−1.44068	−0.720338	+	0.693623i	$0.756013\pi$
$678$	0	0
$679$	0.583592	0.0223962
$680$	0	0
$681$	22.4721	0.861134
$682$	0	0
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	10.4721	0.400120
$686$	0	0
$687$	14.9443	0.570160
$688$	0	0
$689$	−3.41641	−0.130155
$690$	0	0
$691$	40.3607	1.53539	0.767696	−	0.640814i	$-0.221403\pi$
$691$	40.3607	1.53539	0.767696	+	0.640814i	$0.221403\pi$
$692$	0	0
$693$	4.94427	0.187817
$694$	0	0
$695$	−28.9443	−1.09792
$696$	0	0
$697$	−50.2492	−1.90333
$698$	0	0
$699$	−14.0000	−0.529529
$700$	0	0
$701$	−20.7639	−0.784243	−0.392121	−	0.919913i	$-0.628259\pi$
$701$	−20.7639	−0.784243	−0.392121	+	0.919913i	$0.628259\pi$
$702$	0	0
$703$	−12.3607	−0.466192
$704$	0	0
$705$	−12.9443	−0.487509
$706$	0	0
$707$	13.5279	0.508768
$708$	0	0
$709$	−20.8328	−0.782393	−0.391196	−	0.920307i	$-0.627939\pi$
$709$	−20.8328	−0.782393	−0.391196	+	0.920307i	$0.627939\pi$
$710$	0	0
$711$	−3.70820	−0.139069
$712$	0	0
$713$	2.47214	0.0925822
$714$	0	0
$715$	57.8885	2.16491
$716$	0	0
$717$	−12.9443	−0.483413
$718$	0	0
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	0	0
$721$	8.36068	0.311368
$722$	0	0
$723$	28.4721	1.05889
$724$	0	0
$725$	24.4721	0.908872
$726$	0	0
$727$	32.6525	1.21101	0.605507	−	0.795840i	$-0.292971\pi$
$727$	32.6525	1.21101	0.605507	+	0.795840i	$0.292971\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	55.7771	2.06299
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	−17.7082	−0.653177
$736$	0	0
$737$	20.9443	0.771492
$738$	0	0
$739$	−26.8328	−0.987061	−0.493531	−	0.869728i	$-0.664294\pi$
$739$	−26.8328	−0.987061	−0.493531	+	0.869728i	$0.664294\pi$
$740$	0	0
$741$	12.3607	0.454081
$742$	0	0
$743$	20.3607	0.746961	0.373480	−	0.927638i	$-0.378164\pi$
$743$	20.3607	0.746961	0.373480	+	0.927638i	$0.378164\pi$
$744$	0	0
$745$	−5.52786	−0.202525
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−1.16718	−0.0426480
$750$	0	0
$751$	−8.06888	−0.294438	−0.147219	−	0.989104i	$-0.547032\pi$
$751$	−8.06888	−0.294438	−0.147219	+	0.989104i	$0.547032\pi$
$752$	0	0
$753$	1.52786	0.0556785
$754$	0	0
$755$	−51.7771	−1.88436
$756$	0	0
$757$	−4.11146	−0.149433	−0.0747167	−	0.997205i	$-0.523805\pi$
$757$	−4.11146	−0.149433	−0.0747167	+	0.997205i	$0.523805\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−2.36068	−0.0855746	−0.0427873	−	0.999084i	$-0.513624\pi$
$761$	−2.36068	−0.0855746	−0.0427873	+	0.999084i	$0.513624\pi$
$762$	0	0
$763$	3.63932	0.131752
$764$	0	0
$765$	−23.4164	−0.846622
$766$	0	0
$767$	57.8885	2.09023
$768$	0	0
$769$	−36.8328	−1.32823	−0.664113	−	0.747633i	$-0.731190\pi$
$769$	−36.8328	−1.32823	−0.664113	+	0.747633i	$0.731190\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	0	0
$773$	−49.7082	−1.78788	−0.893940	−	0.448187i	$-0.852070\pi$
$773$	−49.7082	−1.78788	−0.893940	+	0.448187i	$0.852070\pi$
$774$	0	0
$775$	13.5279	0.485935
$776$	0	0
$777$	−5.52786	−0.198311
$778$	0	0
$779$	−19.1935	−0.687678
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	4.47214	0.159821
$784$	0	0
$785$	40.3607	1.44053
$786$	0	0
$787$	0.291796	0.0104014	0.00520070	−	0.999986i	$-0.498345\pi$
$787$	0.291796	0.0104014	0.00520070	+	0.999986i	$0.498345\pi$
$788$	0	0
$789$	−2.47214	−0.0880104
$790$	0	0
$791$	−20.0000	−0.711118
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	2.47214	0.0876776
$796$	0	0
$797$	18.6525	0.660705	0.330352	−	0.943858i	$-0.392832\pi$
$797$	18.6525	0.660705	0.330352	+	0.943858i	$0.392832\pi$
$798$	0	0
$799$	28.9443	1.02397
$800$	0	0
$801$	3.23607	0.114341
$802$	0	0
$803$	43.7771	1.54486
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	−8.47214	−0.298233
$808$	0	0
$809$	19.3050	0.678726	0.339363	−	0.940655i	$-0.389789\pi$
$809$	19.3050	0.678726	0.339363	+	0.940655i	$0.389789\pi$
$810$	0	0
$811$	−3.41641	−0.119966	−0.0599832	−	0.998199i	$-0.519105\pi$
$811$	−3.41641	−0.119966	−0.0599832	+	0.998199i	$0.519105\pi$
$812$	0	0
$813$	−26.4721	−0.928418
$814$	0	0
$815$	62.8328	2.20094
$816$	0	0
$817$	21.3050	0.745366
$818$	0	0
$819$	5.52786	0.193159
$820$	0	0
$821$	−47.8885	−1.67132	−0.835661	−	0.549246i	$-0.814915\pi$
$821$	−47.8885	−1.67132	−0.835661	+	0.549246i	$0.814915\pi$
$822$	0	0
$823$	−34.4721	−1.20162	−0.600812	−	0.799391i	$-0.705156\pi$
$823$	−34.4721	−1.20162	−0.600812	+	0.799391i	$0.705156\pi$
$824$	0	0
$825$	−21.8885	−0.762061
$826$	0	0
$827$	−35.4164	−1.23155	−0.615775	−	0.787922i	$-0.711157\pi$
$827$	−35.4164	−1.23155	−0.615775	+	0.787922i	$0.711157\pi$
$828$	0	0
$829$	34.3607	1.19340	0.596698	−	0.802466i	$-0.296479\pi$
$829$	34.3607	1.19340	0.596698	+	0.802466i	$0.296479\pi$
$830$	0	0
$831$	19.8885	0.689926
$832$	0	0
$833$	39.5967	1.37195
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	2.47214	0.0854495
$838$	0	0
$839$	−39.4164	−1.36081	−0.680403	−	0.732838i	$-0.738195\pi$
$839$	−39.4164	−1.36081	−0.680403	+	0.732838i	$0.738195\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	−6.65248	−0.229123
$844$	0	0
$845$	22.6525	0.779269
$846$	0	0
$847$	−6.18034	−0.212359
$848$	0	0
$849$	7.70820	0.264545
$850$	0	0
$851$	4.47214	0.153303
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	−8.94427	−0.305888
$856$	0	0
$857$	−25.0557	−0.855887	−0.427944	−	0.903805i	$-0.640762\pi$
$857$	−25.0557	−0.855887	−0.427944	+	0.903805i	$0.640762\pi$
$858$	0	0
$859$	24.9443	0.851088	0.425544	−	0.904938i	$-0.360083\pi$
$859$	24.9443	0.851088	0.425544	+	0.904938i	$0.360083\pi$
$860$	0	0
$861$	−8.58359	−0.292528
$862$	0	0
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	14.4721	0.492067
$866$	0	0
$867$	35.3607	1.20091
$868$	0	0
$869$	14.8328	0.503169
$870$	0	0
$871$	23.4164	0.793435
$872$	0	0
$873$	−0.472136	−0.0159794
$874$	0	0
$875$	−1.88854	−0.0638444
$876$	0	0
$877$	−50.9443	−1.72027	−0.860133	−	0.510070i	$-0.829619\pi$
$877$	−50.9443	−1.72027	−0.860133	+	0.510070i	$0.829619\pi$
$878$	0	0
$879$	5.70820	0.192533
$880$	0	0
$881$	−40.5410	−1.36586	−0.682931	−	0.730483i	$-0.739295\pi$
$881$	−40.5410	−1.36586	−0.682931	+	0.730483i	$0.739295\pi$
$882$	0	0
$883$	−19.4164	−0.653414	−0.326707	−	0.945126i	$-0.605939\pi$
$883$	−19.4164	−0.653414	−0.326707	+	0.945126i	$0.605939\pi$
$884$	0	0
$885$	−41.8885	−1.40807
$886$	0	0
$887$	48.9443	1.64339	0.821694	−	0.569929i	$-0.193029\pi$
$887$	48.9443	1.64339	0.821694	+	0.569929i	$0.193029\pi$
$888$	0	0
$889$	−12.9443	−0.434137
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	11.0557	0.369966
$894$	0	0
$895$	−9.88854	−0.330538
$896$	0	0
$897$	−4.47214	−0.149320
$898$	0	0
$899$	11.0557	0.368729
$900$	0	0
$901$	−5.52786	−0.184160
$902$	0	0
$903$	9.52786	0.317067
$904$	0	0
$905$	−77.3050	−2.56970
$906$	0	0
$907$	−39.1246	−1.29911	−0.649556	−	0.760314i	$-0.725045\pi$
$907$	−39.1246	−1.29911	−0.649556	+	0.760314i	$0.725045\pi$
$908$	0	0
$909$	−10.9443	−0.362999
$910$	0	0
$911$	48.7214	1.61421	0.807105	−	0.590407i	$-0.201033\pi$
$911$	48.7214	1.61421	0.807105	+	0.590407i	$0.201033\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	0	0
$915$	14.4721	0.478434
$916$	0	0
$917$	−1.16718	−0.0385438
$918$	0	0
$919$	53.0132	1.74874	0.874371	−	0.485257i	$-0.161274\pi$
$919$	53.0132	1.74874	0.874371	+	0.485257i	$0.161274\pi$
$920$	0	0
$921$	6.47214	0.213264
$922$	0	0
$923$	35.7771	1.17762
$924$	0	0
$925$	24.4721	0.804639
$926$	0	0
$927$	−6.76393	−0.222157
$928$	0	0
$929$	24.8328	0.814738	0.407369	−	0.913264i	$-0.366446\pi$
$929$	24.8328	0.814738	0.407369	+	0.913264i	$0.366446\pi$
$930$	0	0
$931$	15.1246	0.495689
$932$	0	0
$933$	24.9443	0.816639
$934$	0	0
$935$	93.6656	3.06319
$936$	0	0
$937$	40.8328	1.33395	0.666975	−	0.745080i	$-0.267589\pi$
$937$	40.8328	1.33395	0.666975	+	0.745080i	$0.267589\pi$
$938$	0	0
$939$	22.9443	0.748758
$940$	0	0
$941$	−24.5410	−0.800014	−0.400007	−	0.916512i	$-0.630992\pi$
$941$	−24.5410	−0.800014	−0.400007	+	0.916512i	$0.630992\pi$
$942$	0	0
$943$	6.94427	0.226137
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	54.8328	1.78183	0.890914	−	0.454173i	$-0.150065\pi$
$947$	54.8328	1.78183	0.890914	+	0.454173i	$0.150065\pi$
$948$	0	0
$949$	48.9443	1.58880
$950$	0	0
$951$	−29.4164	−0.953892
$952$	0	0
$953$	−46.0689	−1.49232	−0.746159	−	0.665768i	$-0.768104\pi$
$953$	−46.0689	−1.49232	−0.746159	+	0.665768i	$0.768104\pi$
$954$	0	0
$955$	33.8885	1.09661
$956$	0	0
$957$	−17.8885	−0.578254
$958$	0	0
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	0.944272	0.0304287
$964$	0	0
$965$	−30.4721	−0.980933
$966$	0	0
$967$	−52.3607	−1.68381	−0.841903	−	0.539629i	$-0.818565\pi$
$967$	−52.3607	−1.68381	−0.841903	+	0.539629i	$0.818565\pi$
$968$	0	0
$969$	20.0000	0.642493
$970$	0	0
$971$	45.3050	1.45391	0.726953	−	0.686688i	$-0.240936\pi$
$971$	45.3050	1.45391	0.726953	+	0.686688i	$0.240936\pi$
$972$	0	0
$973$	11.0557	0.354430
$974$	0	0
$975$	−24.4721	−0.783736
$976$	0	0
$977$	26.0689	0.834017	0.417009	−	0.908902i	$-0.363078\pi$
$977$	26.0689	0.834017	0.417009	+	0.908902i	$0.363078\pi$
$978$	0	0
$979$	−12.9443	−0.413701
$980$	0	0
$981$	−2.94427	−0.0940034
$982$	0	0
$983$	−30.8328	−0.983414	−0.491707	−	0.870761i	$-0.663627\pi$
$983$	−30.8328	−0.983414	−0.491707	+	0.870761i	$0.663627\pi$
$984$	0	0
$985$	30.4721	0.970923
$986$	0	0
$987$	4.94427	0.157378
$988$	0	0
$989$	−7.70820	−0.245107
$990$	0	0
$991$	−10.4721	−0.332658	−0.166329	−	0.986070i	$-0.553191\pi$
$991$	−10.4721	−0.332658	−0.166329	+	0.986070i	$0.553191\pi$
$992$	0	0
$993$	3.41641	0.108416
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	2.36068	0.0747635	0.0373817	−	0.999301i	$-0.488098\pi$
$997$	2.36068	0.0747635	0.0373817	+	0.999301i	$0.488098\pi$
$998$	0	0
$999$	4.47214	0.141492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4416.2.a.bm.1.2		2
4.3	odd	2		4416.2.a.bg.1.2		2
8.3	odd	2		1104.2.a.m.1.1		2
8.5	even	2		69.2.a.b.1.2	✓	2
24.5	odd	2		207.2.a.c.1.1		2
24.11	even	2		3312.2.a.bb.1.2		2
40.13	odd	4		1725.2.b.o.1174.1		4
40.29	even	2		1725.2.a.ba.1.1		2
40.37	odd	4		1725.2.b.o.1174.4		4
56.13	odd	2		3381.2.a.t.1.2		2
88.21	odd	2		8349.2.a.i.1.1		2
120.29	odd	2		5175.2.a.bk.1.2		2
184.45	odd	2		1587.2.a.i.1.2		2
552.413	even	2		4761.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.a.b.1.2	✓	2	8.5	even	2
207.2.a.c.1.1		2	24.5	odd	2
1104.2.a.m.1.1		2	8.3	odd	2
1587.2.a.i.1.2		2	184.45	odd	2
1725.2.a.ba.1.1		2	40.29	even	2
1725.2.b.o.1174.1		4	40.13	odd	4
1725.2.b.o.1174.4		4	40.37	odd	4
3312.2.a.bb.1.2		2	24.11	even	2
3381.2.a.t.1.2		2	56.13	odd	2
4416.2.a.bg.1.2		2	4.3	odd	2
4416.2.a.bm.1.2		2	1.1	even	1	trivial
4761.2.a.v.1.1		2	552.413	even	2
5175.2.a.bk.1.2		2	120.29	odd	2
8349.2.a.i.1.1		2	88.21	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4416 = 2^{6} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4416.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.23607 q^{5} -1.23607 q^{7} +1.00000 q^{9} -4.00000 q^{11} -4.47214 q^{13} +3.23607 q^{15} -7.23607 q^{17} -2.76393 q^{19} -1.23607 q^{21} +1.00000 q^{23} +5.47214 q^{25} +1.00000 q^{27} +4.47214 q^{29} +2.47214 q^{31} -4.00000 q^{33} -4.00000 q^{35} +4.47214 q^{37} -4.47214 q^{39} +6.94427 q^{41} -7.70820 q^{43} +3.23607 q^{45} -4.00000 q^{47} -5.47214 q^{49} -7.23607 q^{51} +0.763932 q^{53} -12.9443 q^{55} -2.76393 q^{57} -12.9443 q^{59} +4.47214 q^{61} -1.23607 q^{63} -14.4721 q^{65} -5.23607 q^{67} +1.00000 q^{69} -8.00000 q^{71} -10.9443 q^{73} +5.47214 q^{75} +4.94427 q^{77} -3.70820 q^{79} +1.00000 q^{81} -4.00000 q^{83} -23.4164 q^{85} +4.47214 q^{87} +3.23607 q^{89} +5.52786 q^{91} +2.47214 q^{93} -8.94427 q^{95} -0.472136 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 2 q^{15} - 10 q^{17} - 10 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{31} - 8 q^{33} - 8 q^{35} - 4 q^{41} - 2 q^{43} + 2 q^{45} - 8 q^{47}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 2 * q^15 - 10 * q^17 - 10 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^31 - 8 * q^33 - 8 * q^35 - 4 * q^41 - 2 * q^43 + 2 * q^45 - 8 * q^47 - 2 * q^49 - 10 * q^51 + 6 * q^53 - 8 * q^55 - 10 * q^57 - 8 * q^59 + 2 * q^63 - 20 * q^65 - 6 * q^67 + 2 * q^69 - 16 * q^71 - 4 * q^73 + 2 * q^75 - 8 * q^77 + 6 * q^79 + 2 * q^81 - 8 * q^83 - 20 * q^85 + 2 * q^89 + 20 * q^91 - 4 * q^93 + 8 * q^97 - 8 * q^99

Introduction

L-functions

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Database

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