LMFDB - Embedded newform 552.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [552,2,Mod(1,552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(552, 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("552.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	552.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:	$4.40774219157$
Analytic rank:	$0$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	552.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.23607 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.23607 q^{5} +2.00000 q^{7} +1.00000 q^{9} -1.23607 q^{11} +4.47214 q^{13} -1.23607 q^{15} +2.47214 q^{17} +3.23607 q^{19} +2.00000 q^{21} +1.00000 q^{23} -3.47214 q^{25} +1.00000 q^{27} +0.472136 q^{29} +10.4721 q^{31} -1.23607 q^{33} -2.47214 q^{35} -5.70820 q^{37} +4.47214 q^{39} -2.00000 q^{41} +0.763932 q^{43} -1.23607 q^{45} +8.94427 q^{47} -3.00000 q^{49} +2.47214 q^{51} -10.1803 q^{53} +1.52786 q^{55} +3.23607 q^{57} -4.00000 q^{59} +1.70820 q^{61} +2.00000 q^{63} -5.52786 q^{65} -5.70820 q^{67} +1.00000 q^{69} -8.94427 q^{71} -4.47214 q^{73} -3.47214 q^{75} -2.47214 q^{77} +14.0000 q^{79} +1.00000 q^{81} -6.76393 q^{83} -3.05573 q^{85} +0.472136 q^{87} +0.944272 q^{89} +8.94427 q^{91} +10.4721 q^{93} -4.00000 q^{95} -4.47214 q^{97} -1.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 2 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 8 q^{29} + 12 q^{31} + 2 q^{33} + 4 q^{35} + 2 q^{37} - 4 q^{41} + 6 q^{43} + 2 q^{45} - 6 q^{49} - 4 q^{51} + 2 q^{53} + 12 q^{55} + 2 q^{57} - 8 q^{59} - 10 q^{61} + 4 q^{63} - 20 q^{65} + 2 q^{67} + 2 q^{69} + 2 q^{75} + 4 q^{77} + 28 q^{79} + 2 q^{81} - 18 q^{83} - 24 q^{85} - 8 q^{87} - 16 q^{89} + 12 q^{93} - 8 q^{95} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 + 2 * q^19 + 4 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 8 * q^29 + 12 * q^31 + 2 * q^33 + 4 * q^35 + 2 * q^37 - 4 * q^41 + 6 * q^43 + 2 * q^45 - 6 * q^49 - 4 * q^51 + 2 * q^53 + 12 * q^55 + 2 * q^57 - 8 * q^59 - 10 * q^61 + 4 * q^63 - 20 * q^65 + 2 * q^67 + 2 * q^69 + 2 * q^75 + 4 * q^77 + 28 * q^79 + 2 * q^81 - 18 * q^83 - 24 * q^85 - 8 * q^87 - 16 * q^89 + 12 * q^93 - 8 * q^95 + 2 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.23607	−0.372689	−0.186344	−	0.982485i	$-0.559664\pi$
$11$	−1.23607	−0.372689	−0.186344	+	0.982485i	$0.559664\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	3.23607	0.742405	0.371202	−	0.928552i	$-0.378946\pi$
$19$	3.23607	0.742405	0.371202	+	0.928552i	$0.378946\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	10.4721	1.88085	0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721	1.88085	0.940426	+	0.340000i	$0.110427\pi$
$32$	0	0
$33$	−1.23607	−0.215172
$34$	0	0
$35$	−2.47214	−0.417867
$36$	0	0
$37$	−5.70820	−0.938423	−0.469211	−	0.883086i	$-0.655462\pi$
$37$	−5.70820	−0.938423	−0.469211	+	0.883086i	$0.655462\pi$
$38$	0	0
$39$	4.47214	0.716115
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0.763932	0.116499	0.0582493	−	0.998302i	$-0.481448\pi$
$43$	0.763932	0.116499	0.0582493	+	0.998302i	$0.481448\pi$
$44$	0	0
$45$	−1.23607	−0.184262
$46$	0	0
$47$	8.94427	1.30466	0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427	1.30466	0.652328	+	0.757937i	$0.273792\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.47214	0.346168
$52$	0	0
$53$	−10.1803	−1.39838	−0.699189	−	0.714937i	$-0.746455\pi$
$53$	−10.1803	−1.39838	−0.699189	+	0.714937i	$0.746455\pi$
$54$	0	0
$55$	1.52786	0.206017
$56$	0	0
$57$	3.23607	0.428628
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	1.70820	0.218713	0.109357	−	0.994003i	$-0.465121\pi$
$61$	1.70820	0.218713	0.109357	+	0.994003i	$0.465121\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−5.52786	−0.685647
$66$	0	0
$67$	−5.70820	−0.697368	−0.348684	−	0.937240i	$-0.613371\pi$
$67$	−5.70820	−0.697368	−0.348684	+	0.937240i	$0.613371\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	0	0
$73$	−4.47214	−0.523424	−0.261712	−	0.965146i	$-0.584287\pi$
$73$	−4.47214	−0.523424	−0.261712	+	0.965146i	$0.584287\pi$
$74$	0	0
$75$	−3.47214	−0.400928
$76$	0	0
$77$	−2.47214	−0.281726
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.76393	−0.742438	−0.371219	−	0.928545i	$-0.621060\pi$
$83$	−6.76393	−0.742438	−0.371219	+	0.928545i	$0.621060\pi$
$84$	0	0
$85$	−3.05573	−0.331440
$86$	0	0
$87$	0.472136	0.0506183
$88$	0	0
$89$	0.944272	0.100093	0.0500463	−	0.998747i	$-0.484063\pi$
$89$	0.944272	0.100093	0.0500463	+	0.998747i	$0.484063\pi$
$90$	0	0
$91$	8.94427	0.937614
$92$	0	0
$93$	10.4721	1.08591
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−4.47214	−0.454077	−0.227038	−	0.973886i	$-0.572904\pi$
$97$	−4.47214	−0.454077	−0.227038	+	0.973886i	$0.572904\pi$
$98$	0	0
$99$	−1.23607	−0.124230
$100$	0	0
$101$	4.47214	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$101$	4.47214	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$102$	0	0
$103$	−4.47214	−0.440653	−0.220326	−	0.975426i	$-0.570712\pi$
$103$	−4.47214	−0.440653	−0.220326	+	0.975426i	$0.570712\pi$
$104$	0	0
$105$	−2.47214	−0.241256
$106$	0	0
$107$	−6.76393	−0.653894	−0.326947	−	0.945043i	$-0.606020\pi$
$107$	−6.76393	−0.653894	−0.326947	+	0.945043i	$0.606020\pi$
$108$	0	0
$109$	−7.23607	−0.693090	−0.346545	−	0.938033i	$-0.612645\pi$
$109$	−7.23607	−0.693090	−0.346545	+	0.938033i	$0.612645\pi$
$110$	0	0
$111$	−5.70820	−0.541799
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	−1.23607	−0.115264
$116$	0	0
$117$	4.47214	0.413449
$118$	0	0
$119$	4.94427	0.453241
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	16.9443	1.50356	0.751780	−	0.659413i	$-0.229195\pi$
$127$	16.9443	1.50356	0.751780	+	0.659413i	$0.229195\pi$
$128$	0	0
$129$	0.763932	0.0672605
$130$	0	0
$131$	−1.52786	−0.133490	−0.0667451	−	0.997770i	$-0.521261\pi$
$131$	−1.52786	−0.133490	−0.0667451	+	0.997770i	$0.521261\pi$
$132$	0	0
$133$	6.47214	0.561205
$134$	0	0
$135$	−1.23607	−0.106384
$136$	0	0
$137$	−19.4164	−1.65886	−0.829428	−	0.558614i	$-0.811333\pi$
$137$	−19.4164	−1.65886	−0.829428	+	0.558614i	$0.811333\pi$
$138$	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
$141$	8.94427	0.753244
$142$	0	0
$143$	−5.52786	−0.462263
$144$	0	0
$145$	−0.583592	−0.0484647
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	−14.1803	−1.16170	−0.580849	−	0.814011i	$-0.697279\pi$
$149$	−14.1803	−1.16170	−0.580849	+	0.814011i	$0.697279\pi$
$150$	0	0
$151$	1.52786	0.124336	0.0621679	−	0.998066i	$-0.480199\pi$
$151$	1.52786	0.124336	0.0621679	+	0.998066i	$0.480199\pi$
$152$	0	0
$153$	2.47214	0.199860
$154$	0	0
$155$	−12.9443	−1.03971
$156$	0	0
$157$	−13.7082	−1.09403	−0.547017	−	0.837122i	$-0.684237\pi$
$157$	−13.7082	−1.09403	−0.547017	+	0.837122i	$0.684237\pi$
$158$	0	0
$159$	−10.1803	−0.807353
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−5.52786	−0.432976	−0.216488	−	0.976285i	$-0.569460\pi$
$163$	−5.52786	−0.432976	−0.216488	+	0.976285i	$0.569460\pi$
$164$	0	0
$165$	1.52786	0.118944
$166$	0	0
$167$	−16.9443	−1.31119	−0.655594	−	0.755114i	$-0.727582\pi$
$167$	−16.9443	−1.31119	−0.655594	+	0.755114i	$0.727582\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	3.23607	0.247468
$172$	0	0
$173$	3.52786	0.268219	0.134109	−	0.990967i	$-0.457183\pi$
$173$	3.52786	0.268219	0.134109	+	0.990967i	$0.457183\pi$
$174$	0	0
$175$	−6.94427	−0.524938
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−9.52786	−0.712146	−0.356073	−	0.934458i	$-0.615885\pi$
$179$	−9.52786	−0.712146	−0.356073	+	0.934458i	$0.615885\pi$
$180$	0	0
$181$	−17.1246	−1.27286	−0.636431	−	0.771333i	$-0.719590\pi$
$181$	−17.1246	−1.27286	−0.636431	+	0.771333i	$0.719590\pi$
$182$	0	0
$183$	1.70820	0.126274
$184$	0	0
$185$	7.05573	0.518747
$186$	0	0
$187$	−3.05573	−0.223457
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	17.5279	1.26827	0.634136	−	0.773222i	$-0.281356\pi$
$191$	17.5279	1.26827	0.634136	+	0.773222i	$0.281356\pi$
$192$	0	0
$193$	23.8885	1.71954	0.859768	−	0.510686i	$-0.170608\pi$
$193$	23.8885	1.71954	0.859768	+	0.510686i	$0.170608\pi$
$194$	0	0
$195$	−5.52786	−0.395859
$196$	0	0
$197$	23.8885	1.70199	0.850994	−	0.525175i	$-0.176000\pi$
$197$	23.8885	1.70199	0.850994	+	0.525175i	$0.176000\pi$
$198$	0	0
$199$	−5.05573	−0.358391	−0.179196	−	0.983813i	$-0.557349\pi$
$199$	−5.05573	−0.358391	−0.179196	+	0.983813i	$0.557349\pi$
$200$	0	0
$201$	−5.70820	−0.402626
$202$	0	0
$203$	0.944272	0.0662749
$204$	0	0
$205$	2.47214	0.172661
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	7.41641	0.510567	0.255283	−	0.966866i	$-0.417831\pi$
$211$	7.41641	0.510567	0.255283	+	0.966866i	$0.417831\pi$
$212$	0	0
$213$	−8.94427	−0.612851
$214$	0	0
$215$	−0.944272	−0.0643988
$216$	0	0
$217$	20.9443	1.42179
$218$	0	0
$219$	−4.47214	−0.302199
$220$	0	0
$221$	11.0557	0.743689
$222$	0	0
$223$	9.52786	0.638033	0.319016	−	0.947749i	$-0.396647\pi$
$223$	9.52786	0.638033	0.319016	+	0.947749i	$0.396647\pi$
$224$	0	0
$225$	−3.47214	−0.231476
$226$	0	0
$227$	7.70820	0.511611	0.255806	−	0.966728i	$-0.417659\pi$
$227$	7.70820	0.511611	0.255806	+	0.966728i	$0.417659\pi$
$228$	0	0
$229$	−10.2918	−0.680101	−0.340051	−	0.940407i	$-0.610444\pi$
$229$	−10.2918	−0.680101	−0.340051	+	0.940407i	$0.610444\pi$
$230$	0	0
$231$	−2.47214	−0.162655
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	−11.0557	−0.721196
$236$	0	0
$237$	14.0000	0.909398
$238$	0	0
$239$	−4.94427	−0.319818	−0.159909	−	0.987132i	$-0.551120\pi$
$239$	−4.94427	−0.319818	−0.159909	+	0.987132i	$0.551120\pi$
$240$	0	0
$241$	8.47214	0.545738	0.272869	−	0.962051i	$-0.412027\pi$
$241$	8.47214	0.545738	0.272869	+	0.962051i	$0.412027\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.70820	0.236908
$246$	0	0
$247$	14.4721	0.920840
$248$	0	0
$249$	−6.76393	−0.428647
$250$	0	0
$251$	15.1246	0.954657	0.477329	−	0.878725i	$-0.341605\pi$
$251$	15.1246	0.954657	0.477329	+	0.878725i	$0.341605\pi$
$252$	0	0
$253$	−1.23607	−0.0777109
$254$	0	0
$255$	−3.05573	−0.191357
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	−11.4164	−0.709381
$260$	0	0
$261$	0.472136	0.0292245
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	12.5836	0.773004
$266$	0	0
$267$	0.944272	0.0577885
$268$	0	0
$269$	−11.8885	−0.724857	−0.362429	−	0.932012i	$-0.618052\pi$
$269$	−11.8885	−0.724857	−0.362429	+	0.932012i	$0.618052\pi$
$270$	0	0
$271$	−5.88854	−0.357704	−0.178852	−	0.983876i	$-0.557238\pi$
$271$	−5.88854	−0.357704	−0.178852	+	0.983876i	$0.557238\pi$
$272$	0	0
$273$	8.94427	0.541332
$274$	0	0
$275$	4.29180	0.258805
$276$	0	0
$277$	3.52786	0.211969	0.105984	−	0.994368i	$-0.466201\pi$
$277$	3.52786	0.211969	0.105984	+	0.994368i	$0.466201\pi$
$278$	0	0
$279$	10.4721	0.626950
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	−21.1246	−1.25573	−0.627864	−	0.778323i	$-0.716071\pi$
$283$	−21.1246	−1.25573	−0.627864	+	0.778323i	$0.716071\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	−4.47214	−0.262161
$292$	0	0
$293$	20.6525	1.20653	0.603265	−	0.797541i	$-0.293866\pi$
$293$	20.6525	1.20653	0.603265	+	0.797541i	$0.293866\pi$
$294$	0	0
$295$	4.94427	0.287867
$296$	0	0
$297$	−1.23607	−0.0717239
$298$	0	0
$299$	4.47214	0.258630
$300$	0	0
$301$	1.52786	0.0880646
$302$	0	0
$303$	4.47214	0.256917
$304$	0	0
$305$	−2.11146	−0.120902
$306$	0	0
$307$	−7.41641	−0.423277	−0.211638	−	0.977348i	$-0.567880\pi$
$307$	−7.41641	−0.423277	−0.211638	+	0.977348i	$0.567880\pi$
$308$	0	0
$309$	−4.47214	−0.254411
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	33.4164	1.88881	0.944404	−	0.328789i	$-0.106640\pi$
$313$	33.4164	1.88881	0.944404	+	0.328789i	$0.106640\pi$
$314$	0	0
$315$	−2.47214	−0.139289
$316$	0	0
$317$	34.3607	1.92989	0.964944	−	0.262456i	$-0.0845324\pi$
$317$	34.3607	1.92989	0.964944	+	0.262456i	$0.0845324\pi$
$318$	0	0
$319$	−0.583592	−0.0326749
$320$	0	0
$321$	−6.76393	−0.377526
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	−15.5279	−0.861331
$326$	0	0
$327$	−7.23607	−0.400155
$328$	0	0
$329$	17.8885	0.986227
$330$	0	0
$331$	26.4721	1.45504	0.727520	−	0.686086i	$-0.240673\pi$
$331$	26.4721	1.45504	0.727520	+	0.686086i	$0.240673\pi$
$332$	0	0
$333$	−5.70820	−0.312808
$334$	0	0
$335$	7.05573	0.385496
$336$	0	0
$337$	−32.8328	−1.78852	−0.894259	−	0.447550i	$-0.852297\pi$
$337$	−32.8328	−1.78852	−0.894259	+	0.447550i	$0.852297\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	−12.9443	−0.700972
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−1.23607	−0.0665477
$346$	0	0
$347$	−24.3607	−1.30775	−0.653875	−	0.756603i	$-0.726858\pi$
$347$	−24.3607	−1.30775	−0.653875	+	0.756603i	$0.726858\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	0	0
$353$	−6.94427	−0.369606	−0.184803	−	0.982776i	$-0.559165\pi$
$353$	−6.94427	−0.369606	−0.184803	+	0.982776i	$0.559165\pi$
$354$	0	0
$355$	11.0557	0.586777
$356$	0	0
$357$	4.94427	0.261679
$358$	0	0
$359$	24.3607	1.28571	0.642854	−	0.765989i	$-0.277750\pi$
$359$	24.3607	1.28571	0.642854	+	0.765989i	$0.277750\pi$
$360$	0	0
$361$	−8.52786	−0.448835
$362$	0	0
$363$	−9.47214	−0.497158
$364$	0	0
$365$	5.52786	0.289342
$366$	0	0
$367$	9.05573	0.472705	0.236353	−	0.971667i	$-0.424048\pi$
$367$	9.05573	0.472705	0.236353	+	0.971667i	$0.424048\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−20.3607	−1.05707
$372$	0	0
$373$	−0.763932	−0.0395549	−0.0197775	−	0.999804i	$-0.506296\pi$
$373$	−0.763932	−0.0395549	−0.0197775	+	0.999804i	$0.506296\pi$
$374$	0	0
$375$	10.4721	0.540779
$376$	0	0
$377$	2.11146	0.108746
$378$	0	0
$379$	−20.7639	−1.06657	−0.533286	−	0.845935i	$-0.679043\pi$
$379$	−20.7639	−1.06657	−0.533286	+	0.845935i	$0.679043\pi$
$380$	0	0
$381$	16.9443	0.868081
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	3.05573	0.155734
$386$	0	0
$387$	0.763932	0.0388328
$388$	0	0
$389$	−31.7082	−1.60767	−0.803835	−	0.594852i	$-0.797210\pi$
$389$	−31.7082	−1.60767	−0.803835	+	0.594852i	$0.797210\pi$
$390$	0	0
$391$	2.47214	0.125021
$392$	0	0
$393$	−1.52786	−0.0770705
$394$	0	0
$395$	−17.3050	−0.870707
$396$	0	0
$397$	−21.0557	−1.05676	−0.528378	−	0.849009i	$-0.677200\pi$
$397$	−21.0557	−1.05676	−0.528378	+	0.849009i	$0.677200\pi$
$398$	0	0
$399$	6.47214	0.324012
$400$	0	0
$401$	15.4164	0.769859	0.384929	−	0.922946i	$-0.374226\pi$
$401$	15.4164	0.769859	0.384929	+	0.922946i	$0.374226\pi$
$402$	0	0
$403$	46.8328	2.33291
$404$	0	0
$405$	−1.23607	−0.0614207
$406$	0	0
$407$	7.05573	0.349739
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	−19.4164	−0.957741
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	8.36068	0.410410
$416$	0	0
$417$	−8.94427	−0.438003
$418$	0	0
$419$	−27.7082	−1.35363	−0.676817	−	0.736151i	$-0.736641\pi$
$419$	−27.7082	−1.35363	−0.676817	+	0.736151i	$0.736641\pi$
$420$	0	0
$421$	−4.76393	−0.232180	−0.116090	−	0.993239i	$-0.537036\pi$
$421$	−4.76393	−0.232180	−0.116090	+	0.993239i	$0.537036\pi$
$422$	0	0
$423$	8.94427	0.434885
$424$	0	0
$425$	−8.58359	−0.416365
$426$	0	0
$427$	3.41641	0.165332
$428$	0	0
$429$	−5.52786	−0.266888
$430$	0	0
$431$	36.3607	1.75143	0.875716	−	0.482826i	$-0.160390\pi$
$431$	36.3607	1.75143	0.875716	+	0.482826i	$0.160390\pi$
$432$	0	0
$433$	−29.4164	−1.41366	−0.706831	−	0.707382i	$-0.749876\pi$
$433$	−29.4164	−1.41366	−0.706831	+	0.707382i	$0.749876\pi$
$434$	0	0
$435$	−0.583592	−0.0279811
$436$	0	0
$437$	3.23607	0.154802
$438$	0	0
$439$	−11.0557	−0.527661	−0.263831	−	0.964569i	$-0.584986\pi$
$439$	−11.0557	−0.527661	−0.263831	+	0.964569i	$0.584986\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	26.8328	1.27487	0.637433	−	0.770506i	$-0.279996\pi$
$443$	26.8328	1.27487	0.637433	+	0.770506i	$0.279996\pi$
$444$	0	0
$445$	−1.16718	−0.0553298
$446$	0	0
$447$	−14.1803	−0.670707
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	2.47214	0.116408
$452$	0	0
$453$	1.52786	0.0717853
$454$	0	0
$455$	−11.0557	−0.518301
$456$	0	0
$457$	42.3607	1.98155	0.990775	−	0.135521i	$-0.0432707\pi$
$457$	42.3607	1.98155	0.990775	+	0.135521i	$0.0432707\pi$
$458$	0	0
$459$	2.47214	0.115389
$460$	0	0
$461$	26.9443	1.25492	0.627460	−	0.778649i	$-0.284095\pi$
$461$	26.9443	1.25492	0.627460	+	0.778649i	$0.284095\pi$
$462$	0	0
$463$	−25.8885	−1.20314	−0.601571	−	0.798819i	$-0.705458\pi$
$463$	−25.8885	−1.20314	−0.601571	+	0.798819i	$0.705458\pi$
$464$	0	0
$465$	−12.9443	−0.600276
$466$	0	0
$467$	29.2361	1.35288	0.676442	−	0.736496i	$-0.263521\pi$
$467$	29.2361	1.35288	0.676442	+	0.736496i	$0.263521\pi$
$468$	0	0
$469$	−11.4164	−0.527161
$470$	0	0
$471$	−13.7082	−0.631641
$472$	0	0
$473$	−0.944272	−0.0434177
$474$	0	0
$475$	−11.2361	−0.515546
$476$	0	0
$477$	−10.1803	−0.466126
$478$	0	0
$479$	−3.05573	−0.139620	−0.0698099	−	0.997560i	$-0.522239\pi$
$479$	−3.05573	−0.139620	−0.0698099	+	0.997560i	$0.522239\pi$
$480$	0	0
$481$	−25.5279	−1.16397
$482$	0	0
$483$	2.00000	0.0910032
$484$	0	0
$485$	5.52786	0.251007
$486$	0	0
$487$	−33.8885	−1.53564	−0.767818	−	0.640668i	$-0.778658\pi$
$487$	−33.8885	−1.53564	−0.767818	+	0.640668i	$0.778658\pi$
$488$	0	0
$489$	−5.52786	−0.249979
$490$	0	0
$491$	16.3607	0.738347	0.369174	−	0.929360i	$-0.379641\pi$
$491$	16.3607	0.738347	0.369174	+	0.929360i	$0.379641\pi$
$492$	0	0
$493$	1.16718	0.0525673
$494$	0	0
$495$	1.52786	0.0686724
$496$	0	0
$497$	−17.8885	−0.802411
$498$	0	0
$499$	−18.4721	−0.826926	−0.413463	−	0.910521i	$-0.635681\pi$
$499$	−18.4721	−0.826926	−0.413463	+	0.910521i	$0.635681\pi$
$500$	0	0
$501$	−16.9443	−0.757014
$502$	0	0
$503$	−33.3050	−1.48499	−0.742497	−	0.669849i	$-0.766359\pi$
$503$	−33.3050	−1.48499	−0.742497	+	0.669849i	$0.766359\pi$
$504$	0	0
$505$	−5.52786	−0.245987
$506$	0	0
$507$	7.00000	0.310881
$508$	0	0
$509$	3.52786	0.156370	0.0781849	−	0.996939i	$-0.475088\pi$
$509$	3.52786	0.156370	0.0781849	+	0.996939i	$0.475088\pi$
$510$	0	0
$511$	−8.94427	−0.395671
$512$	0	0
$513$	3.23607	0.142876
$514$	0	0
$515$	5.52786	0.243587
$516$	0	0
$517$	−11.0557	−0.486230
$518$	0	0
$519$	3.52786	0.154856
$520$	0	0
$521$	−32.3607	−1.41775	−0.708874	−	0.705336i	$-0.750796\pi$
$521$	−32.3607	−1.41775	−0.708874	+	0.705336i	$0.750796\pi$
$522$	0	0
$523$	−5.70820	−0.249602	−0.124801	−	0.992182i	$-0.539829\pi$
$523$	−5.70820	−0.249602	−0.124801	+	0.992182i	$0.539829\pi$
$524$	0	0
$525$	−6.94427	−0.303073
$526$	0	0
$527$	25.8885	1.12772
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−8.94427	−0.387419
$534$	0	0
$535$	8.36068	0.361464
$536$	0	0
$537$	−9.52786	−0.411158
$538$	0	0
$539$	3.70820	0.159724
$540$	0	0
$541$	−15.8885	−0.683102	−0.341551	−	0.939863i	$-0.610952\pi$
$541$	−15.8885	−0.683102	−0.341551	+	0.939863i	$0.610952\pi$
$542$	0	0
$543$	−17.1246	−0.734887
$544$	0	0
$545$	8.94427	0.383131
$546$	0	0
$547$	0.583592	0.0249526	0.0124763	−	0.999922i	$-0.496029\pi$
$547$	0.583592	0.0249526	0.0124763	+	0.999922i	$0.496029\pi$
$548$	0	0
$549$	1.70820	0.0729044
$550$	0	0
$551$	1.52786	0.0650892
$552$	0	0
$553$	28.0000	1.19068
$554$	0	0
$555$	7.05573	0.299499
$556$	0	0
$557$	−2.76393	−0.117112	−0.0585558	−	0.998284i	$-0.518650\pi$
$557$	−2.76393	−0.117112	−0.0585558	+	0.998284i	$0.518650\pi$
$558$	0	0
$559$	3.41641	0.144499
$560$	0	0
$561$	−3.05573	−0.129013
$562$	0	0
$563$	−16.2918	−0.686617	−0.343309	−	0.939223i	$-0.611548\pi$
$563$	−16.2918	−0.686617	−0.343309	+	0.939223i	$0.611548\pi$
$564$	0	0
$565$	4.94427	0.208007
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−47.4164	−1.98780	−0.993900	−	0.110289i	$-0.964823\pi$
$569$	−47.4164	−1.98780	−0.993900	+	0.110289i	$0.964823\pi$
$570$	0	0
$571$	2.29180	0.0959087	0.0479543	−	0.998850i	$-0.484730\pi$
$571$	2.29180	0.0959087	0.0479543	+	0.998850i	$0.484730\pi$
$572$	0	0
$573$	17.5279	0.732237
$574$	0	0
$575$	−3.47214	−0.144798
$576$	0	0
$577$	27.3050	1.13672	0.568360	−	0.822780i	$-0.307578\pi$
$577$	27.3050	1.13672	0.568360	+	0.822780i	$0.307578\pi$
$578$	0	0
$579$	23.8885	0.992774
$580$	0	0
$581$	−13.5279	−0.561230
$582$	0	0
$583$	12.5836	0.521159
$584$	0	0
$585$	−5.52786	−0.228549
$586$	0	0
$587$	46.4721	1.91811	0.959055	−	0.283219i	$-0.0914024\pi$
$587$	46.4721	1.91811	0.959055	+	0.283219i	$0.0914024\pi$
$588$	0	0
$589$	33.8885	1.39635
$590$	0	0
$591$	23.8885	0.982643
$592$	0	0
$593$	−37.7771	−1.55132	−0.775660	−	0.631152i	$-0.782583\pi$
$593$	−37.7771	−1.55132	−0.775660	+	0.631152i	$0.782583\pi$
$594$	0	0
$595$	−6.11146	−0.250545
$596$	0	0
$597$	−5.05573	−0.206917
$598$	0	0
$599$	−17.8885	−0.730906	−0.365453	−	0.930830i	$-0.619086\pi$
$599$	−17.8885	−0.730906	−0.365453	+	0.930830i	$0.619086\pi$
$600$	0	0
$601$	−0.472136	−0.0192588	−0.00962941	−	0.999954i	$-0.503065\pi$
$601$	−0.472136	−0.0192588	−0.00962941	+	0.999954i	$0.503065\pi$
$602$	0	0
$603$	−5.70820	−0.232456
$604$	0	0
$605$	11.7082	0.476006
$606$	0	0
$607$	−11.4164	−0.463378	−0.231689	−	0.972790i	$-0.574425\pi$
$607$	−11.4164	−0.463378	−0.231689	+	0.972790i	$0.574425\pi$
$608$	0	0
$609$	0.944272	0.0382638
$610$	0	0
$611$	40.0000	1.61823
$612$	0	0
$613$	−14.2918	−0.577240	−0.288620	−	0.957444i	$-0.593196\pi$
$613$	−14.2918	−0.577240	−0.288620	+	0.957444i	$0.593196\pi$
$614$	0	0
$615$	2.47214	0.0996861
$616$	0	0
$617$	31.7771	1.27930	0.639649	−	0.768667i	$-0.279080\pi$
$617$	31.7771	1.27930	0.639649	+	0.768667i	$0.279080\pi$
$618$	0	0
$619$	−47.0132	−1.88962	−0.944809	−	0.327621i	$-0.893753\pi$
$619$	−47.0132	−1.88962	−0.944809	+	0.327621i	$0.893753\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	1.88854	0.0756629
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	−14.1115	−0.562661
$630$	0	0
$631$	11.5279	0.458917	0.229459	−	0.973318i	$-0.426304\pi$
$631$	11.5279	0.458917	0.229459	+	0.973318i	$0.426304\pi$
$632$	0	0
$633$	7.41641	0.294776
$634$	0	0
$635$	−20.9443	−0.831148
$636$	0	0
$637$	−13.4164	−0.531577
$638$	0	0
$639$	−8.94427	−0.353830
$640$	0	0
$641$	−31.4164	−1.24087	−0.620437	−	0.784256i	$-0.713045\pi$
$641$	−31.4164	−1.24087	−0.620437	+	0.784256i	$0.713045\pi$
$642$	0	0
$643$	24.7639	0.976594	0.488297	−	0.872677i	$-0.337618\pi$
$643$	24.7639	0.976594	0.488297	+	0.872677i	$0.337618\pi$
$644$	0	0
$645$	−0.944272	−0.0371807
$646$	0	0
$647$	−6.11146	−0.240266	−0.120133	−	0.992758i	$-0.538332\pi$
$647$	−6.11146	−0.240266	−0.120133	+	0.992758i	$0.538332\pi$
$648$	0	0
$649$	4.94427	0.194080
$650$	0	0
$651$	20.9443	0.820871
$652$	0	0
$653$	31.5279	1.23378	0.616890	−	0.787049i	$-0.288392\pi$
$653$	31.5279	1.23378	0.616890	+	0.787049i	$0.288392\pi$
$654$	0	0
$655$	1.88854	0.0737915
$656$	0	0
$657$	−4.47214	−0.174475
$658$	0	0
$659$	−43.4853	−1.69395	−0.846973	−	0.531636i	$-0.821578\pi$
$659$	−43.4853	−1.69395	−0.846973	+	0.531636i	$0.821578\pi$
$660$	0	0
$661$	4.76393	0.185295	0.0926477	−	0.995699i	$-0.470467\pi$
$661$	4.76393	0.185295	0.0926477	+	0.995699i	$0.470467\pi$
$662$	0	0
$663$	11.0557	0.429369
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	0.472136	0.0182812
$668$	0	0
$669$	9.52786	0.368369
$670$	0	0
$671$	−2.11146	−0.0815119
$672$	0	0
$673$	−5.41641	−0.208787	−0.104394	−	0.994536i	$-0.533290\pi$
$673$	−5.41641	−0.208787	−0.104394	+	0.994536i	$0.533290\pi$
$674$	0	0
$675$	−3.47214	−0.133643
$676$	0	0
$677$	−19.1246	−0.735019	−0.367509	−	0.930020i	$-0.619789\pi$
$677$	−19.1246	−0.735019	−0.367509	+	0.930020i	$0.619789\pi$
$678$	0	0
$679$	−8.94427	−0.343250
$680$	0	0
$681$	7.70820	0.295379
$682$	0	0
$683$	16.3607	0.626024	0.313012	−	0.949749i	$-0.398662\pi$
$683$	16.3607	0.626024	0.313012	+	0.949749i	$0.398662\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	0	0
$687$	−10.2918	−0.392657
$688$	0	0
$689$	−45.5279	−1.73447
$690$	0	0
$691$	37.5279	1.42763	0.713814	−	0.700336i	$-0.246966\pi$
$691$	37.5279	1.42763	0.713814	+	0.700336i	$0.246966\pi$
$692$	0	0
$693$	−2.47214	−0.0939087
$694$	0	0
$695$	11.0557	0.419368
$696$	0	0
$697$	−4.94427	−0.187278
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	−25.8197	−0.975195	−0.487598	−	0.873069i	$-0.662127\pi$
$701$	−25.8197	−0.975195	−0.487598	+	0.873069i	$0.662127\pi$
$702$	0	0
$703$	−18.4721	−0.696690
$704$	0	0
$705$	−11.0557	−0.416383
$706$	0	0
$707$	8.94427	0.336384
$708$	0	0
$709$	−14.6525	−0.550285	−0.275143	−	0.961403i	$-0.588725\pi$
$709$	−14.6525	−0.550285	−0.275143	+	0.961403i	$0.588725\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	10.4721	0.392185
$714$	0	0
$715$	6.83282	0.255533
$716$	0	0
$717$	−4.94427	−0.184647
$718$	0	0
$719$	−41.8885	−1.56218	−0.781090	−	0.624419i	$-0.785336\pi$
$719$	−41.8885	−1.56218	−0.781090	+	0.624419i	$0.785336\pi$
$720$	0	0
$721$	−8.94427	−0.333102
$722$	0	0
$723$	8.47214	0.315082
$724$	0	0
$725$	−1.63932	−0.0608828
$726$	0	0
$727$	16.8328	0.624295	0.312147	−	0.950034i	$-0.398952\pi$
$727$	16.8328	0.624295	0.312147	+	0.950034i	$0.398952\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.88854	0.0698503
$732$	0	0
$733$	22.2918	0.823366	0.411683	−	0.911327i	$-0.364941\pi$
$733$	22.2918	0.823366	0.411683	+	0.911327i	$0.364941\pi$
$734$	0	0
$735$	3.70820	0.136779
$736$	0	0
$737$	7.05573	0.259901
$738$	0	0
$739$	−23.0557	−0.848119	−0.424059	−	0.905634i	$-0.639395\pi$
$739$	−23.0557	−0.848119	−0.424059	+	0.905634i	$0.639395\pi$
$740$	0	0
$741$	14.4721	0.531647
$742$	0	0
$743$	40.9443	1.50210	0.751050	−	0.660246i	$-0.229548\pi$
$743$	40.9443	1.50210	0.751050	+	0.660246i	$0.229548\pi$
$744$	0	0
$745$	17.5279	0.642171
$746$	0	0
$747$	−6.76393	−0.247479
$748$	0	0
$749$	−13.5279	−0.494297
$750$	0	0
$751$	18.5836	0.678125	0.339062	−	0.940764i	$-0.389890\pi$
$751$	18.5836	0.678125	0.339062	+	0.940764i	$0.389890\pi$
$752$	0	0
$753$	15.1246	0.551171
$754$	0	0
$755$	−1.88854	−0.0687311
$756$	0	0
$757$	−23.2361	−0.844529	−0.422265	−	0.906473i	$-0.638765\pi$
$757$	−23.2361	−0.844529	−0.422265	+	0.906473i	$0.638765\pi$
$758$	0	0
$759$	−1.23607	−0.0448664
$760$	0	0
$761$	7.88854	0.285959	0.142980	−	0.989726i	$-0.454332\pi$
$761$	7.88854	0.285959	0.142980	+	0.989726i	$0.454332\pi$
$762$	0	0
$763$	−14.4721	−0.523926
$764$	0	0
$765$	−3.05573	−0.110480
$766$	0	0
$767$	−17.8885	−0.645918
$768$	0	0
$769$	52.4721	1.89219	0.946097	−	0.323884i	$-0.104989\pi$
$769$	52.4721	1.89219	0.946097	+	0.323884i	$0.104989\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	0	0
$773$	45.0132	1.61901	0.809505	−	0.587113i	$-0.199735\pi$
$773$	45.0132	1.61901	0.809505	+	0.587113i	$0.199735\pi$
$774$	0	0
$775$	−36.3607	−1.30611
$776$	0	0
$777$	−11.4164	−0.409561
$778$	0	0
$779$	−6.47214	−0.231888
$780$	0	0
$781$	11.0557	0.395605
$782$	0	0
$783$	0.472136	0.0168728
$784$	0	0
$785$	16.9443	0.604767
$786$	0	0
$787$	−7.23607	−0.257938	−0.128969	−	0.991649i	$-0.541167\pi$
$787$	−7.23607	−0.257938	−0.128969	+	0.991649i	$0.541167\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	7.63932	0.271280
$794$	0	0
$795$	12.5836	0.446294
$796$	0	0
$797$	18.1803	0.643981	0.321990	−	0.946743i	$-0.395648\pi$
$797$	18.1803	0.643981	0.321990	+	0.946743i	$0.395648\pi$
$798$	0	0
$799$	22.1115	0.782247
$800$	0	0
$801$	0.944272	0.0333642
$802$	0	0
$803$	5.52786	0.195074
$804$	0	0
$805$	−2.47214	−0.0871313
$806$	0	0
$807$	−11.8885	−0.418497
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−9.30495	−0.326741	−0.163371	−	0.986565i	$-0.552237\pi$
$811$	−9.30495	−0.326741	−0.163371	+	0.986565i	$0.552237\pi$
$812$	0	0
$813$	−5.88854	−0.206520
$814$	0	0
$815$	6.83282	0.239343
$816$	0	0
$817$	2.47214	0.0864891
$818$	0	0
$819$	8.94427	0.312538
$820$	0	0
$821$	−2.36068	−0.0823883	−0.0411941	−	0.999151i	$-0.513116\pi$
$821$	−2.36068	−0.0823883	−0.0411941	+	0.999151i	$0.513116\pi$
$822$	0	0
$823$	12.3607	0.430866	0.215433	−	0.976519i	$-0.430884\pi$
$823$	12.3607	0.430866	0.215433	+	0.976519i	$0.430884\pi$
$824$	0	0
$825$	4.29180	0.149421
$826$	0	0
$827$	29.2361	1.01664	0.508319	−	0.861169i	$-0.330267\pi$
$827$	29.2361	1.01664	0.508319	+	0.861169i	$0.330267\pi$
$828$	0	0
$829$	53.7771	1.86776	0.933878	−	0.357592i	$-0.116402\pi$
$829$	53.7771	1.86776	0.933878	+	0.357592i	$0.116402\pi$
$830$	0	0
$831$	3.52786	0.122380
$832$	0	0
$833$	−7.41641	−0.256963
$834$	0	0
$835$	20.9443	0.724806
$836$	0	0
$837$	10.4721	0.361970
$838$	0	0
$839$	−7.05573	−0.243591	−0.121795	−	0.992555i	$-0.538865\pi$
$839$	−7.05573	−0.243591	−0.121795	+	0.992555i	$0.538865\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	4.00000	0.137767
$844$	0	0
$845$	−8.65248	−0.297654
$846$	0	0
$847$	−18.9443	−0.650933
$848$	0	0
$849$	−21.1246	−0.724995
$850$	0	0
$851$	−5.70820	−0.195675
$852$	0	0
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−41.7771	−1.42708	−0.713539	−	0.700615i	$-0.752909\pi$
$857$	−41.7771	−1.42708	−0.713539	+	0.700615i	$0.752909\pi$
$858$	0	0
$859$	15.7771	0.538307	0.269154	−	0.963097i	$-0.413256\pi$
$859$	15.7771	0.538307	0.269154	+	0.963097i	$0.413256\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	−1.88854	−0.0642868	−0.0321434	−	0.999483i	$-0.510233\pi$
$863$	−1.88854	−0.0642868	−0.0321434	+	0.999483i	$0.510233\pi$
$864$	0	0
$865$	−4.36068	−0.148268
$866$	0	0
$867$	−10.8885	−0.369794
$868$	0	0
$869$	−17.3050	−0.587030
$870$	0	0
$871$	−25.5279	−0.864979
$872$	0	0
$873$	−4.47214	−0.151359
$874$	0	0
$875$	20.9443	0.708046
$876$	0	0
$877$	−46.3607	−1.56549	−0.782744	−	0.622343i	$-0.786181\pi$
$877$	−46.3607	−1.56549	−0.782744	+	0.622343i	$0.786181\pi$
$878$	0	0
$879$	20.6525	0.696591
$880$	0	0
$881$	−32.9443	−1.10992	−0.554960	−	0.831877i	$-0.687267\pi$
$881$	−32.9443	−1.10992	−0.554960	+	0.831877i	$0.687267\pi$
$882$	0	0
$883$	44.3607	1.49286	0.746428	−	0.665466i	$-0.231767\pi$
$883$	44.3607	1.49286	0.746428	+	0.665466i	$0.231767\pi$
$884$	0	0
$885$	4.94427	0.166200
$886$	0	0
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	33.8885	1.13659
$890$	0	0
$891$	−1.23607	−0.0414098
$892$	0	0
$893$	28.9443	0.968583
$894$	0	0
$895$	11.7771	0.393665
$896$	0	0
$897$	4.47214	0.149320
$898$	0	0
$899$	4.94427	0.164901
$900$	0	0
$901$	−25.1672	−0.838440
$902$	0	0
$903$	1.52786	0.0508441
$904$	0	0
$905$	21.1672	0.703621
$906$	0	0
$907$	54.6525	1.81471	0.907353	−	0.420370i	$-0.138100\pi$
$907$	54.6525	1.81471	0.907353	+	0.420370i	$0.138100\pi$
$908$	0	0
$909$	4.47214	0.148331
$910$	0	0
$911$	−33.3050	−1.10344	−0.551721	−	0.834029i	$-0.686029\pi$
$911$	−33.3050	−1.10344	−0.551721	+	0.834029i	$0.686029\pi$
$912$	0	0
$913$	8.36068	0.276698
$914$	0	0
$915$	−2.11146	−0.0698026
$916$	0	0
$917$	−3.05573	−0.100909
$918$	0	0
$919$	9.05573	0.298721	0.149360	−	0.988783i	$-0.452279\pi$
$919$	9.05573	0.298721	0.149360	+	0.988783i	$0.452279\pi$
$920$	0	0
$921$	−7.41641	−0.244379
$922$	0	0
$923$	−40.0000	−1.31662
$924$	0	0
$925$	19.8197	0.651666
$926$	0	0
$927$	−4.47214	−0.146884
$928$	0	0
$929$	−5.05573	−0.165873	−0.0829365	−	0.996555i	$-0.526430\pi$
$929$	−5.05573	−0.165873	−0.0829365	+	0.996555i	$0.526430\pi$
$930$	0	0
$931$	−9.70820	−0.318174
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	3.77709	0.123524
$936$	0	0
$937$	53.7771	1.75682	0.878410	−	0.477907i	$-0.158604\pi$
$937$	53.7771	1.75682	0.878410	+	0.477907i	$0.158604\pi$
$938$	0	0
$939$	33.4164	1.09050
$940$	0	0
$941$	13.5967	0.443241	0.221621	−	0.975133i	$-0.428865\pi$
$941$	13.5967	0.443241	0.221621	+	0.975133i	$0.428865\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	−2.47214	−0.0804186
$946$	0	0
$947$	27.4164	0.890914	0.445457	−	0.895303i	$-0.353041\pi$
$947$	27.4164	0.890914	0.445457	+	0.895303i	$0.353041\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	34.3607	1.11422
$952$	0	0
$953$	36.3607	1.17784	0.588919	−	0.808192i	$-0.299554\pi$
$953$	36.3607	1.17784	0.588919	+	0.808192i	$0.299554\pi$
$954$	0	0
$955$	−21.6656	−0.701083
$956$	0	0
$957$	−0.583592	−0.0188649
$958$	0	0
$959$	−38.8328	−1.25398
$960$	0	0
$961$	78.6656	2.53760
$962$	0	0
$963$	−6.76393	−0.217965
$964$	0	0
$965$	−29.5279	−0.950536
$966$	0	0
$967$	−49.3050	−1.58554	−0.792770	−	0.609521i	$-0.791362\pi$
$967$	−49.3050	−1.58554	−0.792770	+	0.609521i	$0.791362\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	23.7082	0.760832	0.380416	−	0.924815i	$-0.375781\pi$
$971$	23.7082	0.760832	0.380416	+	0.924815i	$0.375781\pi$
$972$	0	0
$973$	−17.8885	−0.573480
$974$	0	0
$975$	−15.5279	−0.497290
$976$	0	0
$977$	−27.4164	−0.877129	−0.438564	−	0.898700i	$-0.644513\pi$
$977$	−27.4164	−0.877129	−0.438564	+	0.898700i	$0.644513\pi$
$978$	0	0
$979$	−1.16718	−0.0373034
$980$	0	0
$981$	−7.23607	−0.231030
$982$	0	0
$983$	−33.3050	−1.06226	−0.531131	−	0.847289i	$-0.678233\pi$
$983$	−33.3050	−1.06226	−0.531131	+	0.847289i	$0.678233\pi$
$984$	0	0
$985$	−29.5279	−0.940836
$986$	0	0
$987$	17.8885	0.569399
$988$	0	0
$989$	0.763932	0.0242916
$990$	0	0
$991$	−33.3050	−1.05797	−0.528983	−	0.848632i	$-0.677427\pi$
$991$	−33.3050	−1.05797	−0.528983	+	0.848632i	$0.677427\pi$
$992$	0	0
$993$	26.4721	0.840068
$994$	0	0
$995$	6.24922	0.198114
$996$	0	0
$997$	−44.8328	−1.41987	−0.709935	−	0.704267i	$-0.751276\pi$
$997$	−44.8328	−1.41987	−0.709935	+	0.704267i	$0.751276\pi$
$998$	0	0
$999$	−5.70820	−0.180600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.a.f.1.1	✓	2
3.2	odd	2		1656.2.a.l.1.2		2
4.3	odd	2		1104.2.a.k.1.1		2
8.3	odd	2		4416.2.a.bj.1.2		2
8.5	even	2		4416.2.a.bd.1.2		2
12.11	even	2		3312.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.a.f.1.1	✓	2	1.1	even	1	trivial
1104.2.a.k.1.1		2	4.3	odd	2
1656.2.a.l.1.2		2	3.2	odd	2
3312.2.a.w.1.2		2	12.11	even	2
4416.2.a.bd.1.2		2	8.5	even	2
4416.2.a.bj.1.2		2	8.3	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.23607 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.23607 q^{5} +2.00000 q^{7} +1.00000 q^{9} -1.23607 q^{11} +4.47214 q^{13} -1.23607 q^{15} +2.47214 q^{17} +3.23607 q^{19} +2.00000 q^{21} +1.00000 q^{23} -3.47214 q^{25} +1.00000 q^{27} +0.472136 q^{29} +10.4721 q^{31} -1.23607 q^{33} -2.47214 q^{35} -5.70820 q^{37} +4.47214 q^{39} -2.00000 q^{41} +0.763932 q^{43} -1.23607 q^{45} +8.94427 q^{47} -3.00000 q^{49} +2.47214 q^{51} -10.1803 q^{53} +1.52786 q^{55} +3.23607 q^{57} -4.00000 q^{59} +1.70820 q^{61} +2.00000 q^{63} -5.52786 q^{65} -5.70820 q^{67} +1.00000 q^{69} -8.94427 q^{71} -4.47214 q^{73} -3.47214 q^{75} -2.47214 q^{77} +14.0000 q^{79} +1.00000 q^{81} -6.76393 q^{83} -3.05573 q^{85} +0.472136 q^{87} +0.944272 q^{89} +8.94427 q^{91} +10.4721 q^{93} -4.00000 q^{95} -4.47214 q^{97} -1.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 2 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 8 q^{29} + 12 q^{31} + 2 q^{33} + 4 q^{35} + 2 q^{37} - 4 q^{41} + 6 q^{43} + 2 q^{45} - 6 q^{49} - 4 q^{51} + 2 q^{53} + 12 q^{55} + 2 q^{57} - 8 q^{59} - 10 q^{61} + 4 q^{63} - 20 q^{65} + 2 q^{67} + 2 q^{69} + 2 q^{75} + 4 q^{77} + 28 q^{79} + 2 q^{81} - 18 q^{83} - 24 q^{85} - 8 q^{87} - 16 q^{89} + 12 q^{93} - 8 q^{95} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 + 2 * q^19 + 4 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 8 * q^29 + 12 * q^31 + 2 * q^33 + 4 * q^35 + 2 * q^37 - 4 * q^41 + 6 * q^43 + 2 * q^45 - 6 * q^49 - 4 * q^51 + 2 * q^53 + 12 * q^55 + 2 * q^57 - 8 * q^59 - 10 * q^61 + 4 * q^63 - 20 * q^65 + 2 * q^67 + 2 * q^69 + 2 * q^75 + 4 * q^77 + 28 * q^79 + 2 * q^81 - 18 * q^83 - 24 * q^85 - 8 * q^87 - 16 * q^89 + 12 * q^93 - 8 * q^95 + 2 * q^99

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