LMFDB - Embedded newform 7448.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{3} +1.00000 q^{5} -0.381966 q^{9} +O(q^{10})$	$q-1.61803 q^{3} +1.00000 q^{5} -0.381966 q^{9} -3.85410 q^{11} +3.47214 q^{13} -1.61803 q^{15} +1.38197 q^{17} +1.00000 q^{19} -2.23607 q^{23} -4.00000 q^{25} +5.47214 q^{27} +5.09017 q^{29} -1.09017 q^{31} +6.23607 q^{33} -0.527864 q^{37} -5.61803 q^{39} +2.61803 q^{41} -2.00000 q^{43} -0.381966 q^{45} -12.7082 q^{47} -2.23607 q^{51} +0.0901699 q^{53} -3.85410 q^{55} -1.61803 q^{57} +9.94427 q^{59} +5.00000 q^{61} +3.47214 q^{65} +0.326238 q^{67} +3.61803 q^{69} +9.00000 q^{71} -0.854102 q^{73} +6.47214 q^{75} -2.94427 q^{79} -7.70820 q^{81} -7.61803 q^{83} +1.38197 q^{85} -8.23607 q^{87} +2.76393 q^{89} +1.76393 q^{93} +1.00000 q^{95} +13.4721 q^{97} +1.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} - 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 2 * q^5 - 3 * q^9	$ 2 q - q^{3} + 2 q^{5} - 3 q^{9} - q^{11} - 2 q^{13} - q^{15} + 5 q^{17} + 2 q^{19} - 8 q^{25} + 2 q^{27} - q^{29} + 9 q^{31} + 8 q^{33} - 10 q^{37} - 9 q^{39} + 3 q^{41} - 4 q^{43} - 3 q^{45} - 12 q^{47} - 11 q^{53} - q^{55} - q^{57} + 2 q^{59} + 10 q^{61} - 2 q^{65} - 15 q^{67} + 5 q^{69} + 18 q^{71} + 5 q^{73} + 4 q^{75} + 12 q^{79} - 2 q^{81} - 13 q^{83} + 5 q^{85} - 12 q^{87} + 10 q^{89} + 8 q^{93} + 2 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 - 3 * q^9 - q^11 - 2 * q^13 - q^15 + 5 * q^17 + 2 * q^19 - 8 * q^25 + 2 * q^27 - q^29 + 9 * q^31 + 8 * q^33 - 10 * q^37 - 9 * q^39 + 3 * q^41 - 4 * q^43 - 3 * q^45 - 12 * q^47 - 11 * q^53 - q^55 - q^57 + 2 * q^59 + 10 * q^61 - 2 * q^65 - 15 * q^67 + 5 * q^69 + 18 * q^71 + 5 * q^73 + 4 * q^75 + 12 * q^79 - 2 * q^81 - 13 * q^83 + 5 * q^85 - 12 * q^87 + 10 * q^89 + 8 * q^93 + 2 * q^95 + 18 * q^97 - 6 * 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
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−3.85410	−1.16206	−0.581028	−	0.813884i	$-0.697349\pi$
$11$	−3.85410	−1.16206	−0.581028	+	0.813884i	$0.697349\pi$
$12$	0	0
$13$	3.47214	0.962997	0.481499	−	0.876447i	$-0.340093\pi$
$13$	3.47214	0.962997	0.481499	+	0.876447i	$0.340093\pi$
$14$	0	0
$15$	−1.61803	−0.417775
$16$	0	0
$17$	1.38197	0.335176	0.167588	−	0.985857i	$-0.446402\pi$
$17$	1.38197	0.335176	0.167588	+	0.985857i	$0.446402\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.23607	−0.466252	−0.233126	−	0.972446i	$-0.574896\pi$
$23$	−2.23607	−0.466252	−0.233126	+	0.972446i	$0.574896\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	5.09017	0.945221	0.472610	−	0.881271i	$-0.343312\pi$
$29$	5.09017	0.945221	0.472610	+	0.881271i	$0.343312\pi$
$30$	0	0
$31$	−1.09017	−0.195800	−0.0979002	−	0.995196i	$-0.531213\pi$
$31$	−1.09017	−0.195800	−0.0979002	+	0.995196i	$0.531213\pi$
$32$	0	0
$33$	6.23607	1.08556
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.527864	−0.0867803	−0.0433902	−	0.999058i	$-0.513816\pi$
$37$	−0.527864	−0.0867803	−0.0433902	+	0.999058i	$0.513816\pi$
$38$	0	0
$39$	−5.61803	−0.899605
$40$	0	0
$41$	2.61803	0.408868	0.204434	−	0.978880i	$-0.434465\pi$
$41$	2.61803	0.408868	0.204434	+	0.978880i	$0.434465\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−0.381966	−0.0569401
$46$	0	0
$47$	−12.7082	−1.85368	−0.926841	−	0.375454i	$-0.877487\pi$
$47$	−12.7082	−1.85368	−0.926841	+	0.375454i	$0.877487\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.23607	−0.313112
$52$	0	0
$53$	0.0901699	0.0123858	0.00619290	−	0.999981i	$-0.498029\pi$
$53$	0.0901699	0.0123858	0.00619290	+	0.999981i	$0.498029\pi$
$54$	0	0
$55$	−3.85410	−0.519687
$56$	0	0
$57$	−1.61803	−0.214314
$58$	0	0
$59$	9.94427	1.29463	0.647317	−	0.762221i	$-0.275891\pi$
$59$	9.94427	1.29463	0.647317	+	0.762221i	$0.275891\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.47214	0.430665
$66$	0	0
$67$	0.326238	0.0398563	0.0199282	−	0.999801i	$-0.493656\pi$
$67$	0.326238	0.0398563	0.0199282	+	0.999801i	$0.493656\pi$
$68$	0	0
$69$	3.61803	0.435560
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	−0.854102	−0.0999651	−0.0499825	−	0.998750i	$-0.515917\pi$
$73$	−0.854102	−0.0999651	−0.0499825	+	0.998750i	$0.515917\pi$
$74$	0	0
$75$	6.47214	0.747338
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.94427	−0.331256	−0.165628	−	0.986188i	$-0.552965\pi$
$79$	−2.94427	−0.331256	−0.165628	+	0.986188i	$0.552965\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−7.61803	−0.836188	−0.418094	−	0.908404i	$-0.637302\pi$
$83$	−7.61803	−0.836188	−0.418094	+	0.908404i	$0.637302\pi$
$84$	0	0
$85$	1.38197	0.149895
$86$	0	0
$87$	−8.23607	−0.882999
$88$	0	0
$89$	2.76393	0.292976	0.146488	−	0.989212i	$-0.453203\pi$
$89$	2.76393	0.292976	0.146488	+	0.989212i	$0.453203\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.76393	0.182911
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	13.4721	1.36789	0.683944	−	0.729534i	$-0.260263\pi$
$97$	13.4721	1.36789	0.683944	+	0.729534i	$0.260263\pi$
$98$	0	0
$99$	1.47214	0.147955
$100$	0	0
$101$	4.70820	0.468484	0.234242	−	0.972178i	$-0.424739\pi$
$101$	4.70820	0.468484	0.234242	+	0.972178i	$0.424739\pi$
$102$	0	0
$103$	−6.52786	−0.643210	−0.321605	−	0.946874i	$-0.604222\pi$
$103$	−6.52786	−0.643210	−0.321605	+	0.946874i	$0.604222\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.94427	0.381307	0.190654	−	0.981657i	$-0.438939\pi$
$107$	3.94427	0.381307	0.190654	+	0.981657i	$0.438939\pi$
$108$	0	0
$109$	−10.2361	−0.980437	−0.490219	−	0.871599i	$-0.663083\pi$
$109$	−10.2361	−0.980437	−0.490219	+	0.871599i	$0.663083\pi$
$110$	0	0
$111$	0.854102	0.0810678
$112$	0	0
$113$	−1.61803	−0.152212	−0.0761059	−	0.997100i	$-0.524249\pi$
$113$	−1.61803	−0.152212	−0.0761059	+	0.997100i	$0.524249\pi$
$114$	0	0
$115$	−2.23607	−0.208514
$116$	0	0
$117$	−1.32624	−0.122611
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.85410	0.350373
$122$	0	0
$123$	−4.23607	−0.381953
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	10.1803	0.903359	0.451680	−	0.892180i	$-0.350825\pi$
$127$	10.1803	0.903359	0.451680	+	0.892180i	$0.350825\pi$
$128$	0	0
$129$	3.23607	0.284920
$130$	0	0
$131$	−18.0344	−1.57568	−0.787838	−	0.615882i	$-0.788800\pi$
$131$	−18.0344	−1.57568	−0.787838	+	0.615882i	$0.788800\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.47214	0.470966
$136$	0	0
$137$	−22.9443	−1.96026	−0.980131	−	0.198353i	$-0.936441\pi$
$137$	−22.9443	−1.96026	−0.980131	+	0.198353i	$0.936441\pi$
$138$	0	0
$139$	−11.1803	−0.948304	−0.474152	−	0.880443i	$-0.657245\pi$
$139$	−11.1803	−0.948304	−0.474152	+	0.880443i	$0.657245\pi$
$140$	0	0
$141$	20.5623	1.73166
$142$	0	0
$143$	−13.3820	−1.11906
$144$	0	0
$145$	5.09017	0.422716
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.05573	−0.332258	−0.166129	−	0.986104i	$-0.553127\pi$
$149$	−4.05573	−0.332258	−0.166129	+	0.986104i	$0.553127\pi$
$150$	0	0
$151$	−22.7984	−1.85531	−0.927653	−	0.373444i	$-0.878177\pi$
$151$	−22.7984	−1.85531	−0.927653	+	0.373444i	$0.878177\pi$
$152$	0	0
$153$	−0.527864	−0.0426753
$154$	0	0
$155$	−1.09017	−0.0875646
$156$	0	0
$157$	8.03444	0.641218	0.320609	−	0.947212i	$-0.396112\pi$
$157$	8.03444	0.641218	0.320609	+	0.947212i	$0.396112\pi$
$158$	0	0
$159$	−0.145898	−0.0115705
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.3262	−0.887139	−0.443570	−	0.896240i	$-0.646288\pi$
$163$	−11.3262	−0.887139	−0.443570	+	0.896240i	$0.646288\pi$
$164$	0	0
$165$	6.23607	0.485477
$166$	0	0
$167$	3.18034	0.246102	0.123051	−	0.992400i	$-0.460732\pi$
$167$	3.18034	0.246102	0.123051	+	0.992400i	$0.460732\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	0	0
$171$	−0.381966	−0.0292097
$172$	0	0
$173$	18.1803	1.38223	0.691113	−	0.722747i	$-0.257121\pi$
$173$	18.1803	1.38223	0.691113	+	0.722747i	$0.257121\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−16.0902	−1.20941
$178$	0	0
$179$	−3.85410	−0.288069	−0.144035	−	0.989573i	$-0.546008\pi$
$179$	−3.85410	−0.288069	−0.144035	+	0.989573i	$0.546008\pi$
$180$	0	0
$181$	−9.56231	−0.710761	−0.355380	−	0.934722i	$-0.615649\pi$
$181$	−9.56231	−0.710761	−0.355380	+	0.934722i	$0.615649\pi$
$182$	0	0
$183$	−8.09017	−0.598043
$184$	0	0
$185$	−0.527864	−0.0388093
$186$	0	0
$187$	−5.32624	−0.389493
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.5623	1.63255	0.816276	−	0.577663i	$-0.196035\pi$
$191$	22.5623	1.63255	0.816276	+	0.577663i	$0.196035\pi$
$192$	0	0
$193$	11.8541	0.853277	0.426638	−	0.904422i	$-0.359698\pi$
$193$	11.8541	0.853277	0.426638	+	0.904422i	$0.359698\pi$
$194$	0	0
$195$	−5.61803	−0.402316
$196$	0	0
$197$	−6.27051	−0.446755	−0.223378	−	0.974732i	$-0.571708\pi$
$197$	−6.27051	−0.446755	−0.223378	+	0.974732i	$0.571708\pi$
$198$	0	0
$199$	1.18034	0.0836721	0.0418360	−	0.999124i	$-0.486679\pi$
$199$	1.18034	0.0836721	0.0418360	+	0.999124i	$0.486679\pi$
$200$	0	0
$201$	−0.527864	−0.0372327
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.61803	0.182851
$206$	0	0
$207$	0.854102	0.0593642
$208$	0	0
$209$	−3.85410	−0.266594
$210$	0	0
$211$	−24.0902	−1.65844	−0.829218	−	0.558926i	$-0.811214\pi$
$211$	−24.0902	−1.65844	−0.829218	+	0.558926i	$0.811214\pi$
$212$	0	0
$213$	−14.5623	−0.997793
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.38197	0.0933846
$220$	0	0
$221$	4.79837	0.322774
$222$	0	0
$223$	−5.29180	−0.354365	−0.177182	−	0.984178i	$-0.556698\pi$
$223$	−5.29180	−0.354365	−0.177182	+	0.984178i	$0.556698\pi$
$224$	0	0
$225$	1.52786	0.101858
$226$	0	0
$227$	4.79837	0.318479	0.159240	−	0.987240i	$-0.449096\pi$
$227$	4.79837	0.318479	0.159240	+	0.987240i	$0.449096\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.3262	0.807519	0.403759	−	0.914865i	$-0.367703\pi$
$233$	12.3262	0.807519	0.403759	+	0.914865i	$0.367703\pi$
$234$	0	0
$235$	−12.7082	−0.828992
$236$	0	0
$237$	4.76393	0.309451
$238$	0	0
$239$	26.5967	1.72040	0.860200	−	0.509956i	$-0.170338\pi$
$239$	26.5967	1.72040	0.860200	+	0.509956i	$0.170338\pi$
$240$	0	0
$241$	−13.1246	−0.845431	−0.422715	−	0.906263i	$-0.638923\pi$
$241$	−13.1246	−0.845431	−0.422715	+	0.906263i	$0.638923\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.47214	0.220927
$248$	0	0
$249$	12.3262	0.781144
$250$	0	0
$251$	2.14590	0.135448	0.0677239	−	0.997704i	$-0.478426\pi$
$251$	2.14590	0.135448	0.0677239	+	0.997704i	$0.478426\pi$
$252$	0	0
$253$	8.61803	0.541811
$254$	0	0
$255$	−2.23607	−0.140028
$256$	0	0
$257$	19.9787	1.24624	0.623119	−	0.782127i	$-0.285865\pi$
$257$	19.9787	1.24624	0.623119	+	0.782127i	$0.285865\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.94427	−0.120347
$262$	0	0
$263$	6.38197	0.393529	0.196764	−	0.980451i	$-0.436957\pi$
$263$	6.38197	0.393529	0.196764	+	0.980451i	$0.436957\pi$
$264$	0	0
$265$	0.0901699	0.00553910
$266$	0	0
$267$	−4.47214	−0.273690
$268$	0	0
$269$	−24.8541	−1.51538	−0.757691	−	0.652614i	$-0.773672\pi$
$269$	−24.8541	−1.51538	−0.757691	+	0.652614i	$0.773672\pi$
$270$	0	0
$271$	0.437694	0.0265880	0.0132940	−	0.999912i	$-0.495768\pi$
$271$	0.437694	0.0265880	0.0132940	+	0.999912i	$0.495768\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	15.4164	0.929644
$276$	0	0
$277$	9.12461	0.548245	0.274122	−	0.961695i	$-0.411613\pi$
$277$	9.12461	0.548245	0.274122	+	0.961695i	$0.411613\pi$
$278$	0	0
$279$	0.416408	0.0249297
$280$	0	0
$281$	12.2361	0.729943	0.364971	−	0.931019i	$-0.381079\pi$
$281$	12.2361	0.729943	0.364971	+	0.931019i	$0.381079\pi$
$282$	0	0
$283$	15.6180	0.928396	0.464198	−	0.885732i	$-0.346343\pi$
$283$	15.6180	0.928396	0.464198	+	0.885732i	$0.346343\pi$
$284$	0	0
$285$	−1.61803	−0.0958441
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.0902	−0.887657
$290$	0	0
$291$	−21.7984	−1.27784
$292$	0	0
$293$	−0.347524	−0.0203026	−0.0101513	−	0.999948i	$-0.503231\pi$
$293$	−0.347524	−0.0203026	−0.0101513	+	0.999948i	$0.503231\pi$
$294$	0	0
$295$	9.94427	0.578978
$296$	0	0
$297$	−21.0902	−1.22378
$298$	0	0
$299$	−7.76393	−0.449000
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.61803	−0.437645
$304$	0	0
$305$	5.00000	0.286299
$306$	0	0
$307$	−26.3262	−1.50252	−0.751259	−	0.660008i	$-0.770553\pi$
$307$	−26.3262	−1.50252	−0.751259	+	0.660008i	$0.770553\pi$
$308$	0	0
$309$	10.5623	0.600869
$310$	0	0
$311$	−13.5623	−0.769048	−0.384524	−	0.923115i	$-0.625634\pi$
$311$	−13.5623	−0.769048	−0.384524	+	0.923115i	$0.625634\pi$
$312$	0	0
$313$	−23.5967	−1.33377	−0.666884	−	0.745162i	$-0.732372\pi$
$313$	−23.5967	−1.33377	−0.666884	+	0.745162i	$0.732372\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.41641	−0.0795534	−0.0397767	−	0.999209i	$-0.512665\pi$
$317$	−1.41641	−0.0795534	−0.0397767	+	0.999209i	$0.512665\pi$
$318$	0	0
$319$	−19.6180	−1.09840
$320$	0	0
$321$	−6.38197	−0.356207
$322$	0	0
$323$	1.38197	0.0768946
$324$	0	0
$325$	−13.8885	−0.770398
$326$	0	0
$327$	16.5623	0.915898
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8.85410	−0.486665	−0.243333	−	0.969943i	$-0.578241\pi$
$331$	−8.85410	−0.486665	−0.243333	+	0.969943i	$0.578241\pi$
$332$	0	0
$333$	0.201626	0.0110490
$334$	0	0
$335$	0.326238	0.0178243
$336$	0	0
$337$	−12.0344	−0.655558	−0.327779	−	0.944754i	$-0.606300\pi$
$337$	−12.0344	−0.655558	−0.327779	+	0.944754i	$0.606300\pi$
$338$	0	0
$339$	2.61803	0.142192
$340$	0	0
$341$	4.20163	0.227531
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.61803	0.194788
$346$	0	0
$347$	−26.0902	−1.40059	−0.700297	−	0.713852i	$-0.746949\pi$
$347$	−26.0902	−1.40059	−0.700297	+	0.713852i	$0.746949\pi$
$348$	0	0
$349$	−21.3262	−1.14157	−0.570784	−	0.821100i	$-0.693361\pi$
$349$	−21.3262	−1.14157	−0.570784	+	0.821100i	$0.693361\pi$
$350$	0	0
$351$	19.0000	1.01414
$352$	0	0
$353$	−29.5623	−1.57344	−0.786721	−	0.617308i	$-0.788223\pi$
$353$	−29.5623	−1.57344	−0.786721	+	0.617308i	$0.788223\pi$
$354$	0	0
$355$	9.00000	0.477670
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.14590	0.429924	0.214962	−	0.976622i	$-0.431037\pi$
$359$	8.14590	0.429924	0.214962	+	0.976622i	$0.431037\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−6.23607	−0.327309
$364$	0	0
$365$	−0.854102	−0.0447057
$366$	0	0
$367$	4.52786	0.236353	0.118176	−	0.992993i	$-0.462295\pi$
$367$	4.52786	0.236353	0.118176	+	0.992993i	$0.462295\pi$
$368$	0	0
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−30.9230	−1.60113	−0.800566	−	0.599245i	$-0.795468\pi$
$373$	−30.9230	−1.60113	−0.800566	+	0.599245i	$0.795468\pi$
$374$	0	0
$375$	14.5623	0.751994
$376$	0	0
$377$	17.6738	0.910245
$378$	0	0
$379$	−33.1803	−1.70436	−0.852180	−	0.523249i	$-0.824720\pi$
$379$	−33.1803	−1.70436	−0.852180	+	0.523249i	$0.824720\pi$
$380$	0	0
$381$	−16.4721	−0.843893
$382$	0	0
$383$	−14.4164	−0.736644	−0.368322	−	0.929698i	$-0.620068\pi$
$383$	−14.4164	−0.736644	−0.368322	+	0.929698i	$0.620068\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.763932	0.0388328
$388$	0	0
$389$	−30.5066	−1.54674	−0.773372	−	0.633952i	$-0.781432\pi$
$389$	−30.5066	−1.54674	−0.773372	+	0.633952i	$0.781432\pi$
$390$	0	0
$391$	−3.09017	−0.156277
$392$	0	0
$393$	29.1803	1.47195
$394$	0	0
$395$	−2.94427	−0.148142
$396$	0	0
$397$	27.1246	1.36135	0.680673	−	0.732588i	$-0.261688\pi$
$397$	27.1246	1.36135	0.680673	+	0.732588i	$0.261688\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.2016	−0.559383	−0.279691	−	0.960090i	$-0.590232\pi$
$401$	−11.2016	−0.559383	−0.279691	+	0.960090i	$0.590232\pi$
$402$	0	0
$403$	−3.78522	−0.188555
$404$	0	0
$405$	−7.70820	−0.383024
$406$	0	0
$407$	2.03444	0.100844
$408$	0	0
$409$	7.79837	0.385605	0.192802	−	0.981238i	$-0.438242\pi$
$409$	7.79837	0.385605	0.192802	+	0.981238i	$0.438242\pi$
$410$	0	0
$411$	37.1246	1.83122
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7.61803	−0.373955
$416$	0	0
$417$	18.0902	0.885879
$418$	0	0
$419$	−14.7082	−0.718543	−0.359271	−	0.933233i	$-0.616975\pi$
$419$	−14.7082	−0.718543	−0.359271	+	0.933233i	$0.616975\pi$
$420$	0	0
$421$	−0.527864	−0.0257265	−0.0128633	−	0.999917i	$-0.504095\pi$
$421$	−0.527864	−0.0257265	−0.0128633	+	0.999917i	$0.504095\pi$
$422$	0	0
$423$	4.85410	0.236015
$424$	0	0
$425$	−5.52786	−0.268141
$426$	0	0
$427$	0	0
$428$	0	0
$429$	21.6525	1.04539
$430$	0	0
$431$	−27.8885	−1.34334	−0.671672	−	0.740849i	$-0.734424\pi$
$431$	−27.8885	−1.34334	−0.671672	+	0.740849i	$0.734424\pi$
$432$	0	0
$433$	−14.1246	−0.678786	−0.339393	−	0.940645i	$-0.610222\pi$
$433$	−14.1246	−0.678786	−0.339393	+	0.940645i	$0.610222\pi$
$434$	0	0
$435$	−8.23607	−0.394889
$436$	0	0
$437$	−2.23607	−0.106966
$438$	0	0
$439$	19.4164	0.926695	0.463347	−	0.886177i	$-0.346648\pi$
$439$	19.4164	0.926695	0.463347	+	0.886177i	$0.346648\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.61803	−0.456967	−0.228483	−	0.973548i	$-0.573377\pi$
$443$	−9.61803	−0.456967	−0.228483	+	0.973548i	$0.573377\pi$
$444$	0	0
$445$	2.76393	0.131023
$446$	0	0
$447$	6.56231	0.310386
$448$	0	0
$449$	21.0344	0.992677	0.496338	−	0.868129i	$-0.334678\pi$
$449$	21.0344	0.992677	0.496338	+	0.868129i	$0.334678\pi$
$450$	0	0
$451$	−10.0902	−0.475128
$452$	0	0
$453$	36.8885	1.73317
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.5066	0.818923	0.409462	−	0.912327i	$-0.365717\pi$
$457$	17.5066	0.818923	0.409462	+	0.912327i	$0.365717\pi$
$458$	0	0
$459$	7.56231	0.352978
$460$	0	0
$461$	−4.09017	−0.190498	−0.0952491	−	0.995453i	$-0.530365\pi$
$461$	−4.09017	−0.190498	−0.0952491	+	0.995453i	$0.530365\pi$
$462$	0	0
$463$	−10.4164	−0.484092	−0.242046	−	0.970265i	$-0.577818\pi$
$463$	−10.4164	−0.484092	−0.242046	+	0.970265i	$0.577818\pi$
$464$	0	0
$465$	1.76393	0.0818004
$466$	0	0
$467$	0.562306	0.0260204	0.0130102	−	0.999915i	$-0.495859\pi$
$467$	0.562306	0.0260204	0.0130102	+	0.999915i	$0.495859\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	7.70820	0.354424
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−0.0344419	−0.00157698
$478$	0	0
$479$	30.0344	1.37231	0.686154	−	0.727456i	$-0.259297\pi$
$479$	30.0344	1.37231	0.686154	+	0.727456i	$0.259297\pi$
$480$	0	0
$481$	−1.83282	−0.0835692
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.4721	0.611738
$486$	0	0
$487$	−12.5279	−0.567692	−0.283846	−	0.958870i	$-0.591610\pi$
$487$	−12.5279	−0.567692	−0.283846	+	0.958870i	$0.591610\pi$
$488$	0	0
$489$	18.3262	0.828741
$490$	0	0
$491$	−23.4164	−1.05677	−0.528384	−	0.849006i	$-0.677202\pi$
$491$	−23.4164	−1.05677	−0.528384	+	0.849006i	$0.677202\pi$
$492$	0	0
$493$	7.03444	0.316815
$494$	0	0
$495$	1.47214	0.0661676
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−43.3951	−1.94263	−0.971316	−	0.237792i	$-0.923577\pi$
$499$	−43.3951	−1.94263	−0.971316	+	0.237792i	$0.923577\pi$
$500$	0	0
$501$	−5.14590	−0.229902
$502$	0	0
$503$	−38.8328	−1.73147	−0.865735	−	0.500503i	$-0.833148\pi$
$503$	−38.8328	−1.73147	−0.865735	+	0.500503i	$0.833148\pi$
$504$	0	0
$505$	4.70820	0.209512
$506$	0	0
$507$	1.52786	0.0678548
$508$	0	0
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.47214	0.241601
$514$	0	0
$515$	−6.52786	−0.287652
$516$	0	0
$517$	48.9787	2.15408
$518$	0	0
$519$	−29.4164	−1.29124
$520$	0	0
$521$	−12.8197	−0.561640	−0.280820	−	0.959761i	$-0.590606\pi$
$521$	−12.8197	−0.561640	−0.280820	+	0.959761i	$0.590606\pi$
$522$	0	0
$523$	1.76393	0.0771314	0.0385657	−	0.999256i	$-0.487721\pi$
$523$	1.76393	0.0771314	0.0385657	+	0.999256i	$0.487721\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.50658	−0.0656276
$528$	0	0
$529$	−18.0000	−0.782609
$530$	0	0
$531$	−3.79837	−0.164835
$532$	0	0
$533$	9.09017	0.393739
$534$	0	0
$535$	3.94427	0.170526
$536$	0	0
$537$	6.23607	0.269106
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23.8885	−1.02705	−0.513524	−	0.858075i	$-0.671660\pi$
$541$	−23.8885	−1.02705	−0.513524	+	0.858075i	$0.671660\pi$
$542$	0	0
$543$	15.4721	0.663973
$544$	0	0
$545$	−10.2361	−0.438465
$546$	0	0
$547$	−28.0902	−1.20105	−0.600524	−	0.799606i	$-0.705041\pi$
$547$	−28.0902	−1.20105	−0.600524	+	0.799606i	$0.705041\pi$
$548$	0	0
$549$	−1.90983	−0.0815096
$550$	0	0
$551$	5.09017	0.216849
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.854102	0.0362546
$556$	0	0
$557$	−24.4377	−1.03546	−0.517729	−	0.855545i	$-0.673223\pi$
$557$	−24.4377	−1.03546	−0.517729	+	0.855545i	$0.673223\pi$
$558$	0	0
$559$	−6.94427	−0.293711
$560$	0	0
$561$	8.61803	0.363854
$562$	0	0
$563$	−6.88854	−0.290318	−0.145159	−	0.989408i	$-0.546369\pi$
$563$	−6.88854	−0.290318	−0.145159	+	0.989408i	$0.546369\pi$
$564$	0	0
$565$	−1.61803	−0.0680712
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−32.7771	−1.37409	−0.687043	−	0.726616i	$-0.741092\pi$
$569$	−32.7771	−1.37409	−0.687043	+	0.726616i	$0.741092\pi$
$570$	0	0
$571$	0.416408	0.0174261	0.00871306	−	0.999962i	$-0.497227\pi$
$571$	0.416408	0.0174261	0.00871306	+	0.999962i	$0.497227\pi$
$572$	0	0
$573$	−36.5066	−1.52508
$574$	0	0
$575$	8.94427	0.373002
$576$	0	0
$577$	13.5066	0.562286	0.281143	−	0.959666i	$-0.409286\pi$
$577$	13.5066	0.562286	0.281143	+	0.959666i	$0.409286\pi$
$578$	0	0
$579$	−19.1803	−0.797108
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.347524	−0.0143930
$584$	0	0
$585$	−1.32624	−0.0548332
$586$	0	0
$587$	−9.41641	−0.388657	−0.194328	−	0.980937i	$-0.562253\pi$
$587$	−9.41641	−0.388657	−0.194328	+	0.980937i	$0.562253\pi$
$588$	0	0
$589$	−1.09017	−0.0449197
$590$	0	0
$591$	10.1459	0.417346
$592$	0	0
$593$	9.18034	0.376991	0.188496	−	0.982074i	$-0.439639\pi$
$593$	9.18034	0.376991	0.188496	+	0.982074i	$0.439639\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.90983	−0.0781641
$598$	0	0
$599$	12.2705	0.501359	0.250680	−	0.968070i	$-0.419346\pi$
$599$	12.2705	0.501359	0.250680	+	0.968070i	$0.419346\pi$
$600$	0	0
$601$	−21.6180	−0.881818	−0.440909	−	0.897552i	$-0.645344\pi$
$601$	−21.6180	−0.881818	−0.440909	+	0.897552i	$0.645344\pi$
$602$	0	0
$603$	−0.124612	−0.00507458
$604$	0	0
$605$	3.85410	0.156692
$606$	0	0
$607$	−32.8885	−1.33490	−0.667452	−	0.744652i	$-0.732615\pi$
$607$	−32.8885	−1.33490	−0.667452	+	0.744652i	$0.732615\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−44.1246	−1.78509
$612$	0	0
$613$	−8.56231	−0.345828	−0.172914	−	0.984937i	$-0.555318\pi$
$613$	−8.56231	−0.345828	−0.172914	+	0.984937i	$0.555318\pi$
$614$	0	0
$615$	−4.23607	−0.170815
$616$	0	0
$617$	−40.8673	−1.64525	−0.822627	−	0.568582i	$-0.807492\pi$
$617$	−40.8673	−1.64525	−0.822627	+	0.568582i	$0.807492\pi$
$618$	0	0
$619$	35.3262	1.41988	0.709941	−	0.704261i	$-0.248722\pi$
$619$	35.3262	1.41988	0.709941	+	0.704261i	$0.248722\pi$
$620$	0	0
$621$	−12.2361	−0.491016
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	6.23607	0.249045
$628$	0	0
$629$	−0.729490	−0.0290867
$630$	0	0
$631$	−26.0557	−1.03726	−0.518631	−	0.854998i	$-0.673558\pi$
$631$	−26.0557	−1.03726	−0.518631	+	0.854998i	$0.673558\pi$
$632$	0	0
$633$	38.9787	1.54926
$634$	0	0
$635$	10.1803	0.403994
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.43769	−0.135993
$640$	0	0
$641$	−5.38197	−0.212575	−0.106287	−	0.994335i	$-0.533896\pi$
$641$	−5.38197	−0.212575	−0.106287	+	0.994335i	$0.533896\pi$
$642$	0	0
$643$	3.00000	0.118308	0.0591542	−	0.998249i	$-0.481160\pi$
$643$	3.00000	0.118308	0.0591542	+	0.998249i	$0.481160\pi$
$644$	0	0
$645$	3.23607	0.127420
$646$	0	0
$647$	15.0689	0.592419	0.296209	−	0.955123i	$-0.404277\pi$
$647$	15.0689	0.592419	0.296209	+	0.955123i	$0.404277\pi$
$648$	0	0
$649$	−38.3262	−1.50444
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.6525	0.455997	0.227998	−	0.973662i	$-0.426782\pi$
$653$	11.6525	0.455997	0.227998	+	0.973662i	$0.426782\pi$
$654$	0	0
$655$	−18.0344	−0.704664
$656$	0	0
$657$	0.326238	0.0127278
$658$	0	0
$659$	11.4508	0.446062	0.223031	−	0.974811i	$-0.428405\pi$
$659$	11.4508	0.446062	0.223031	+	0.974811i	$0.428405\pi$
$660$	0	0
$661$	14.9443	0.581265	0.290632	−	0.956835i	$-0.406134\pi$
$661$	14.9443	0.581265	0.290632	+	0.956835i	$0.406134\pi$
$662$	0	0
$663$	−7.76393	−0.301526
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.3820	−0.440711
$668$	0	0
$669$	8.56231	0.331038
$670$	0	0
$671$	−19.2705	−0.743930
$672$	0	0
$673$	25.7984	0.994454	0.497227	−	0.867620i	$-0.334352\pi$
$673$	25.7984	0.994454	0.497227	+	0.867620i	$0.334352\pi$
$674$	0	0
$675$	−21.8885	−0.842490
$676$	0	0
$677$	−1.79837	−0.0691171	−0.0345586	−	0.999403i	$-0.511003\pi$
$677$	−1.79837	−0.0691171	−0.0345586	+	0.999403i	$0.511003\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.76393	−0.297515
$682$	0	0
$683$	25.9443	0.992730	0.496365	−	0.868114i	$-0.334668\pi$
$683$	25.9443	0.992730	0.496365	+	0.868114i	$0.334668\pi$
$684$	0	0
$685$	−22.9443	−0.876656
$686$	0	0
$687$	−22.6525	−0.864246
$688$	0	0
$689$	0.313082	0.0119275
$690$	0	0
$691$	12.7639	0.485563	0.242781	−	0.970081i	$-0.421940\pi$
$691$	12.7639	0.485563	0.242781	+	0.970081i	$0.421940\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.1803	−0.424094
$696$	0	0
$697$	3.61803	0.137043
$698$	0	0
$699$	−19.9443	−0.754362
$700$	0	0
$701$	37.8885	1.43103	0.715515	−	0.698597i	$-0.246192\pi$
$701$	37.8885	1.43103	0.715515	+	0.698597i	$0.246192\pi$
$702$	0	0
$703$	−0.527864	−0.0199088
$704$	0	0
$705$	20.5623	0.774421
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−14.8197	−0.556564	−0.278282	−	0.960499i	$-0.589765\pi$
$709$	−14.8197	−0.556564	−0.278282	+	0.960499i	$0.589765\pi$
$710$	0	0
$711$	1.12461	0.0421762
$712$	0	0
$713$	2.43769	0.0912924
$714$	0	0
$715$	−13.3820	−0.500457
$716$	0	0
$717$	−43.0344	−1.60715
$718$	0	0
$719$	−39.7771	−1.48344	−0.741718	−	0.670712i	$-0.765988\pi$
$719$	−39.7771	−1.48344	−0.741718	+	0.670712i	$0.765988\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	21.2361	0.789778
$724$	0	0
$725$	−20.3607	−0.756177
$726$	0	0
$727$	9.83282	0.364679	0.182339	−	0.983236i	$-0.441633\pi$
$727$	9.83282	0.364679	0.182339	+	0.983236i	$0.441633\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	−2.76393	−0.102228
$732$	0	0
$733$	−30.4721	−1.12551	−0.562757	−	0.826622i	$-0.690259\pi$
$733$	−30.4721	−1.12551	−0.562757	+	0.826622i	$0.690259\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.25735	−0.0463152
$738$	0	0
$739$	−19.0689	−0.701460	−0.350730	−	0.936477i	$-0.614067\pi$
$739$	−19.0689	−0.701460	−0.350730	+	0.936477i	$0.614067\pi$
$740$	0	0
$741$	−5.61803	−0.206384
$742$	0	0
$743$	5.58359	0.204842	0.102421	−	0.994741i	$-0.467341\pi$
$743$	5.58359	0.204842	0.102421	+	0.994741i	$0.467341\pi$
$744$	0	0
$745$	−4.05573	−0.148590
$746$	0	0
$747$	2.90983	0.106465
$748$	0	0
$749$	0	0
$750$	0	0
$751$	44.2705	1.61545	0.807727	−	0.589557i	$-0.200698\pi$
$751$	44.2705	1.61545	0.807727	+	0.589557i	$0.200698\pi$
$752$	0	0
$753$	−3.47214	−0.126532
$754$	0	0
$755$	−22.7984	−0.829718
$756$	0	0
$757$	−39.8885	−1.44977	−0.724887	−	0.688868i	$-0.758108\pi$
$757$	−39.8885	−1.44977	−0.724887	+	0.688868i	$0.758108\pi$
$758$	0	0
$759$	−13.9443	−0.506145
$760$	0	0
$761$	34.8885	1.26471	0.632354	−	0.774679i	$-0.282089\pi$
$761$	34.8885	1.26471	0.632354	+	0.774679i	$0.282089\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.527864	−0.0190850
$766$	0	0
$767$	34.5279	1.24673
$768$	0	0
$769$	39.4164	1.42139	0.710696	−	0.703499i	$-0.248380\pi$
$769$	39.4164	1.42139	0.710696	+	0.703499i	$0.248380\pi$
$770$	0	0
$771$	−32.3262	−1.16420
$772$	0	0
$773$	−7.81966	−0.281254	−0.140627	−	0.990063i	$-0.544912\pi$
$773$	−7.81966	−0.281254	−0.140627	+	0.990063i	$0.544912\pi$
$774$	0	0
$775$	4.36068	0.156640
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.61803	0.0938008
$780$	0	0
$781$	−34.6869	−1.24120
$782$	0	0
$783$	27.8541	0.995424
$784$	0	0
$785$	8.03444	0.286762
$786$	0	0
$787$	−18.8328	−0.671317	−0.335659	−	0.941984i	$-0.608959\pi$
$787$	−18.8328	−0.671317	−0.335659	+	0.941984i	$0.608959\pi$
$788$	0	0
$789$	−10.3262	−0.367624
$790$	0	0
$791$	0	0
$792$	0	0
$793$	17.3607	0.616496
$794$	0	0
$795$	−0.145898	−0.00517447
$796$	0	0
$797$	−2.03444	−0.0720636	−0.0360318	−	0.999351i	$-0.511472\pi$
$797$	−2.03444	−0.0720636	−0.0360318	+	0.999351i	$0.511472\pi$
$798$	0	0
$799$	−17.5623	−0.621310
$800$	0	0
$801$	−1.05573	−0.0373023
$802$	0	0
$803$	3.29180	0.116165
$804$	0	0
$805$	0	0
$806$	0	0
$807$	40.2148	1.41563
$808$	0	0
$809$	46.7082	1.64217	0.821086	−	0.570804i	$-0.193368\pi$
$809$	46.7082	1.64217	0.821086	+	0.570804i	$0.193368\pi$
$810$	0	0
$811$	−33.4164	−1.17341	−0.586704	−	0.809801i	$-0.699575\pi$
$811$	−33.4164	−1.17341	−0.586704	+	0.809801i	$0.699575\pi$
$812$	0	0
$813$	−0.708204	−0.0248378
$814$	0	0
$815$	−11.3262	−0.396741
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.00000	0.139601	0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000	0.139601	0.0698005	+	0.997561i	$0.477764\pi$
$822$	0	0
$823$	−36.5410	−1.27374	−0.636870	−	0.770971i	$-0.719771\pi$
$823$	−36.5410	−1.27374	−0.636870	+	0.770971i	$0.719771\pi$
$824$	0	0
$825$	−24.9443	−0.868448
$826$	0	0
$827$	−28.3050	−0.984260	−0.492130	−	0.870522i	$-0.663781\pi$
$827$	−28.3050	−0.984260	−0.492130	+	0.870522i	$0.663781\pi$
$828$	0	0
$829$	−39.7771	−1.38152	−0.690758	−	0.723086i	$-0.742723\pi$
$829$	−39.7771	−1.38152	−0.690758	+	0.723086i	$0.742723\pi$
$830$	0	0
$831$	−14.7639	−0.512155
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.18034	0.110060
$836$	0	0
$837$	−5.96556	−0.206200
$838$	0	0
$839$	30.2492	1.04432	0.522160	−	0.852848i	$-0.325127\pi$
$839$	30.2492	1.04432	0.522160	+	0.852848i	$0.325127\pi$
$840$	0	0
$841$	−3.09017	−0.106558
$842$	0	0
$843$	−19.7984	−0.681892
$844$	0	0
$845$	−0.944272	−0.0324839
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−25.2705	−0.867282
$850$	0	0
$851$	1.18034	0.0404615
$852$	0	0
$853$	40.8541	1.39882	0.699409	−	0.714722i	$-0.253447\pi$
$853$	40.8541	1.39882	0.699409	+	0.714722i	$0.253447\pi$
$854$	0	0
$855$	−0.381966	−0.0130630
$856$	0	0
$857$	−12.9787	−0.443344	−0.221672	−	0.975121i	$-0.571151\pi$
$857$	−12.9787	−0.443344	−0.221672	+	0.975121i	$0.571151\pi$
$858$	0	0
$859$	7.27051	0.248067	0.124033	−	0.992278i	$-0.460417\pi$
$859$	7.27051	0.248067	0.124033	+	0.992278i	$0.460417\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.450850	−0.0153471	−0.00767355	−	0.999971i	$-0.502443\pi$
$863$	−0.450850	−0.0153471	−0.00767355	+	0.999971i	$0.502443\pi$
$864$	0	0
$865$	18.1803	0.618150
$866$	0	0
$867$	24.4164	0.829225
$868$	0	0
$869$	11.3475	0.384938
$870$	0	0
$871$	1.13274	0.0383815
$872$	0	0
$873$	−5.14590	−0.174162
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−29.8197	−1.00694	−0.503469	−	0.864013i	$-0.667943\pi$
$877$	−29.8197	−1.00694	−0.503469	+	0.864013i	$0.667943\pi$
$878$	0	0
$879$	0.562306	0.0189661
$880$	0	0
$881$	8.20163	0.276320	0.138160	−	0.990410i	$-0.455881\pi$
$881$	8.20163	0.276320	0.138160	+	0.990410i	$0.455881\pi$
$882$	0	0
$883$	41.6525	1.40172	0.700859	−	0.713300i	$-0.252800\pi$
$883$	41.6525	1.40172	0.700859	+	0.713300i	$0.252800\pi$
$884$	0	0
$885$	−16.0902	−0.540865
$886$	0	0
$887$	−2.34752	−0.0788221	−0.0394111	−	0.999223i	$-0.512548\pi$
$887$	−2.34752	−0.0788221	−0.0394111	+	0.999223i	$0.512548\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	29.7082	0.995262
$892$	0	0
$893$	−12.7082	−0.425264
$894$	0	0
$895$	−3.85410	−0.128828
$896$	0	0
$897$	12.5623	0.419443
$898$	0	0
$899$	−5.54915	−0.185075
$900$	0	0
$901$	0.124612	0.00415142
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.56231	−0.317862
$906$	0	0
$907$	55.1803	1.83223	0.916117	−	0.400912i	$-0.131307\pi$
$907$	55.1803	1.83223	0.916117	+	0.400912i	$0.131307\pi$
$908$	0	0
$909$	−1.79837	−0.0596483
$910$	0	0
$911$	12.4721	0.413220	0.206610	−	0.978423i	$-0.433757\pi$
$911$	12.4721	0.413220	0.206610	+	0.978423i	$0.433757\pi$
$912$	0	0
$913$	29.3607	0.971697
$914$	0	0
$915$	−8.09017	−0.267453
$916$	0	0
$917$	0	0
$918$	0	0
$919$	41.3607	1.36436	0.682181	−	0.731183i	$-0.261031\pi$
$919$	41.3607	1.36436	0.682181	+	0.731183i	$0.261031\pi$
$920$	0	0
$921$	42.5967	1.40361
$922$	0	0
$923$	31.2492	1.02858
$924$	0	0
$925$	2.11146	0.0694243
$926$	0	0
$927$	2.49342	0.0818947
$928$	0	0
$929$	−47.2148	−1.54907	−0.774533	−	0.632533i	$-0.782015\pi$
$929$	−47.2148	−1.54907	−0.774533	+	0.632533i	$0.782015\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	21.9443	0.718423
$934$	0	0
$935$	−5.32624	−0.174187
$936$	0	0
$937$	7.85410	0.256582	0.128291	−	0.991737i	$-0.459051\pi$
$937$	7.85410	0.256582	0.128291	+	0.991737i	$0.459051\pi$
$938$	0	0
$939$	38.1803	1.24597
$940$	0	0
$941$	38.3607	1.25052	0.625261	−	0.780416i	$-0.284992\pi$
$941$	38.3607	1.25052	0.625261	+	0.780416i	$0.284992\pi$
$942$	0	0
$943$	−5.85410	−0.190636
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.1033	−1.33568	−0.667839	−	0.744305i	$-0.732781\pi$
$947$	−41.1033	−1.33568	−0.667839	+	0.744305i	$0.732781\pi$
$948$	0	0
$949$	−2.96556	−0.0962661
$950$	0	0
$951$	2.29180	0.0743166
$952$	0	0
$953$	31.5623	1.02240	0.511202	−	0.859461i	$-0.329201\pi$
$953$	31.5623	1.02240	0.511202	+	0.859461i	$0.329201\pi$
$954$	0	0
$955$	22.5623	0.730099
$956$	0	0
$957$	31.7426	1.02609
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.8115	−0.961662
$962$	0	0
$963$	−1.50658	−0.0485488
$964$	0	0
$965$	11.8541	0.381597
$966$	0	0
$967$	48.1033	1.54690	0.773449	−	0.633858i	$-0.218530\pi$
$967$	48.1033	1.54690	0.773449	+	0.633858i	$0.218530\pi$
$968$	0	0
$969$	−2.23607	−0.0718329
$970$	0	0
$971$	27.3607	0.878046	0.439023	−	0.898476i	$-0.355325\pi$
$971$	27.3607	0.878046	0.439023	+	0.898476i	$0.355325\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	22.4721	0.719684
$976$	0	0
$977$	−27.5967	−0.882898	−0.441449	−	0.897286i	$-0.645535\pi$
$977$	−27.5967	−0.882898	−0.441449	+	0.897286i	$0.645535\pi$
$978$	0	0
$979$	−10.6525	−0.340455
$980$	0	0
$981$	3.90983	0.124831
$982$	0	0
$983$	−21.5836	−0.688410	−0.344205	−	0.938895i	$-0.611851\pi$
$983$	−21.5836	−0.688410	−0.344205	+	0.938895i	$0.611851\pi$
$984$	0	0
$985$	−6.27051	−0.199795
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.47214	0.142206
$990$	0	0
$991$	−5.12461	−0.162789	−0.0813943	−	0.996682i	$-0.525937\pi$
$991$	−5.12461	−0.162789	−0.0813943	+	0.996682i	$0.525937\pi$
$992$	0	0
$993$	14.3262	0.454629
$994$	0	0
$995$	1.18034	0.0374193
$996$	0	0
$997$	−5.03444	−0.159442	−0.0797212	−	0.996817i	$-0.525403\pi$
$997$	−5.03444	−0.159442	−0.0797212	+	0.996817i	$0.525403\pi$
$998$	0	0
$999$	−2.88854	−0.0913895

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.z.1.1		2
7.6	odd	2		1064.2.a.d.1.2	✓	2
21.20	even	2		9576.2.a.bt.1.2		2
28.27	even	2		2128.2.a.f.1.1		2
56.13	odd	2		8512.2.a.q.1.1		2
56.27	even	2		8512.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.d.1.2	✓	2	7.6	odd	2
2128.2.a.f.1.1		2	28.27	even	2
7448.2.a.z.1.1		2	1.1	even	1	trivial
8512.2.a.q.1.1		2	56.13	odd	2
8512.2.a.y.1.2		2	56.27	even	2
9576.2.a.bt.1.2		2	21.20	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} +1.00000 q^{5} -0.381966 q^{9} +O(q^{10})\)	\(q-1.61803 q^{3} +1.00000 q^{5} -0.381966 q^{9} -3.85410 q^{11} +3.47214 q^{13} -1.61803 q^{15} +1.38197 q^{17} +1.00000 q^{19} -2.23607 q^{23} -4.00000 q^{25} +5.47214 q^{27} +5.09017 q^{29} -1.09017 q^{31} +6.23607 q^{33} -0.527864 q^{37} -5.61803 q^{39} +2.61803 q^{41} -2.00000 q^{43} -0.381966 q^{45} -12.7082 q^{47} -2.23607 q^{51} +0.0901699 q^{53} -3.85410 q^{55} -1.61803 q^{57} +9.94427 q^{59} +5.00000 q^{61} +3.47214 q^{65} +0.326238 q^{67} +3.61803 q^{69} +9.00000 q^{71} -0.854102 q^{73} +6.47214 q^{75} -2.94427 q^{79} -7.70820 q^{81} -7.61803 q^{83} +1.38197 q^{85} -8.23607 q^{87} +2.76393 q^{89} +1.76393 q^{93} +1.00000 q^{95} +13.4721 q^{97} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} - 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 2 * q^5 - 3 * q^9	\( 2 q - q^{3} + 2 q^{5} - 3 q^{9} - q^{11} - 2 q^{13} - q^{15} + 5 q^{17} + 2 q^{19} - 8 q^{25} + 2 q^{27} - q^{29} + 9 q^{31} + 8 q^{33} - 10 q^{37} - 9 q^{39} + 3 q^{41} - 4 q^{43} - 3 q^{45} - 12 q^{47} - 11 q^{53} - q^{55} - q^{57} + 2 q^{59} + 10 q^{61} - 2 q^{65} - 15 q^{67} + 5 q^{69} + 18 q^{71} + 5 q^{73} + 4 q^{75} + 12 q^{79} - 2 q^{81} - 13 q^{83} + 5 q^{85} - 12 q^{87} + 10 q^{89} + 8 q^{93} + 2 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 - 3 * q^9 - q^11 - 2 * q^13 - q^15 + 5 * q^17 + 2 * q^19 - 8 * q^25 + 2 * q^27 - q^29 + 9 * q^31 + 8 * q^33 - 10 * q^37 - 9 * q^39 + 3 * q^41 - 4 * q^43 - 3 * q^45 - 12 * q^47 - 11 * q^53 - q^55 - q^57 + 2 * q^59 + 10 * q^61 - 2 * q^65 - 15 * q^67 + 5 * q^69 + 18 * q^71 + 5 * q^73 + 4 * q^75 + 12 * q^79 - 2 * q^81 - 13 * q^83 + 5 * q^85 - 12 * q^87 + 10 * q^89 + 8 * q^93 + 2 * q^95 + 18 * q^97 - 6 * q^99

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