LMFDB - Embedded newform 7800.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.414214 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.414214 q^{7} +1.00000 q^{9} -2.41421 q^{11} -1.00000 q^{13} -1.82843 q^{17} -2.00000 q^{19} +0.414214 q^{21} +4.82843 q^{23} -1.00000 q^{27} +2.65685 q^{29} +1.58579 q^{31} +2.41421 q^{33} +3.65685 q^{37} +1.00000 q^{39} -5.65685 q^{41} +6.48528 q^{43} +12.0711 q^{47} -6.82843 q^{49} +1.82843 q^{51} -0.171573 q^{53} +2.00000 q^{57} -3.58579 q^{59} +3.82843 q^{61} -0.414214 q^{63} +9.24264 q^{67} -4.82843 q^{69} +3.65685 q^{71} -13.3137 q^{73} +1.00000 q^{77} -3.17157 q^{79} +1.00000 q^{81} -11.7279 q^{83} -2.65685 q^{87} +0.414214 q^{91} -1.58579 q^{93} -18.9706 q^{97} -2.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{17} - 4 q^{19} - 2 q^{21} + 4 q^{23} - 2 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 4 q^{37} + 2 q^{39} - 4 q^{43} + 10 q^{47} - 8 q^{49} - 2 q^{51} - 6 q^{53} + 4 q^{57} - 10 q^{59} + 2 q^{61} + 2 q^{63} + 10 q^{67} - 4 q^{69} - 4 q^{71} - 4 q^{73} + 2 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{83} + 6 q^{87} - 2 q^{91} - 6 q^{93} - 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^17 - 4 * q^19 - 2 * q^21 + 4 * q^23 - 2 * q^27 - 6 * q^29 + 6 * q^31 + 2 * q^33 - 4 * q^37 + 2 * q^39 - 4 * q^43 + 10 * q^47 - 8 * q^49 - 2 * q^51 - 6 * q^53 + 4 * q^57 - 10 * q^59 + 2 * q^61 + 2 * q^63 + 10 * q^67 - 4 * q^69 - 4 * q^71 - 4 * q^73 + 2 * q^77 - 12 * q^79 + 2 * q^81 + 2 * q^83 + 6 * q^87 - 2 * q^91 - 6 * q^93 - 4 * q^97 - 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.414214	−0.156558	−0.0782790	−	0.996931i	$-0.524942\pi$
$7$	−0.414214	−0.156558	−0.0782790	+	0.996931i	$0.524942\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.41421	−0.727913	−0.363956	−	0.931416i	$-0.618574\pi$
$11$	−2.41421	−0.727913	−0.363956	+	0.931416i	$0.618574\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.82843	−0.443459	−0.221729	−	0.975108i	$-0.571170\pi$
$17$	−1.82843	−0.443459	−0.221729	+	0.975108i	$0.571170\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0.414214	0.0903888
$22$	0	0
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.65685	0.493365	0.246683	−	0.969096i	$-0.420659\pi$
$29$	2.65685	0.493365	0.246683	+	0.969096i	$0.420659\pi$
$30$	0	0
$31$	1.58579	0.284816	0.142408	−	0.989808i	$-0.454516\pi$
$31$	1.58579	0.284816	0.142408	+	0.989808i	$0.454516\pi$
$32$	0	0
$33$	2.41421	0.420261
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.65685	0.601183	0.300592	−	0.953753i	$-0.402816\pi$
$37$	3.65685	0.601183	0.300592	+	0.953753i	$0.402816\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	6.48528	0.988996	0.494498	−	0.869179i	$-0.335352\pi$
$43$	6.48528	0.988996	0.494498	+	0.869179i	$0.335352\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0711	1.76075	0.880373	−	0.474282i	$-0.157292\pi$
$47$	12.0711	1.76075	0.880373	+	0.474282i	$0.157292\pi$
$48$	0	0
$49$	−6.82843	−0.975490
$50$	0	0
$51$	1.82843	0.256031
$52$	0	0
$53$	−0.171573	−0.0235673	−0.0117837	−	0.999931i	$-0.503751\pi$
$53$	−0.171573	−0.0235673	−0.0117837	+	0.999931i	$0.503751\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−3.58579	−0.466830	−0.233415	−	0.972377i	$-0.574990\pi$
$59$	−3.58579	−0.466830	−0.233415	+	0.972377i	$0.574990\pi$
$60$	0	0
$61$	3.82843	0.490180	0.245090	−	0.969500i	$-0.421183\pi$
$61$	3.82843	0.490180	0.245090	+	0.969500i	$0.421183\pi$
$62$	0	0
$63$	−0.414214	−0.0521860
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.24264	1.12917	0.564584	−	0.825376i	$-0.309037\pi$
$67$	9.24264	1.12917	0.564584	+	0.825376i	$0.309037\pi$
$68$	0	0
$69$	−4.82843	−0.581274
$70$	0	0
$71$	3.65685	0.433989	0.216994	−	0.976173i	$-0.430375\pi$
$71$	3.65685	0.433989	0.216994	+	0.976173i	$0.430375\pi$
$72$	0	0
$73$	−13.3137	−1.55825	−0.779126	−	0.626868i	$-0.784337\pi$
$73$	−13.3137	−1.55825	−0.779126	+	0.626868i	$0.784337\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−3.17157	−0.356830	−0.178415	−	0.983955i	$-0.557097\pi$
$79$	−3.17157	−0.356830	−0.178415	+	0.983955i	$0.557097\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.7279	−1.28731	−0.643653	−	0.765317i	$-0.722582\pi$
$83$	−11.7279	−1.28731	−0.643653	+	0.765317i	$0.722582\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.65685	−0.284845
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0.414214	0.0434214
$92$	0	0
$93$	−1.58579	−0.164438
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.9706	−1.92617	−0.963084	−	0.269200i	$-0.913241\pi$
$97$	−18.9706	−1.92617	−0.963084	+	0.269200i	$0.913241\pi$
$98$	0	0
$99$	−2.41421	−0.242638
$100$	0	0
$101$	−11.8284	−1.17697	−0.588486	−	0.808507i	$-0.700276\pi$
$101$	−11.8284	−1.17697	−0.588486	+	0.808507i	$0.700276\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.51472	0.146433	0.0732167	−	0.997316i	$-0.476674\pi$
$107$	1.51472	0.146433	0.0732167	+	0.997316i	$0.476674\pi$
$108$	0	0
$109$	3.31371	0.317396	0.158698	−	0.987327i	$-0.449270\pi$
$109$	3.31371	0.317396	0.158698	+	0.987327i	$0.449270\pi$
$110$	0	0
$111$	−3.65685	−0.347093
$112$	0	0
$113$	9.31371	0.876160	0.438080	−	0.898936i	$-0.355659\pi$
$113$	9.31371	0.876160	0.438080	+	0.898936i	$0.355659\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0.757359	0.0694270
$120$	0	0
$121$	−5.17157	−0.470143
$122$	0	0
$123$	5.65685	0.510061
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.8284	1.49328	0.746641	−	0.665228i	$-0.231665\pi$
$127$	16.8284	1.49328	0.746641	+	0.665228i	$0.231665\pi$
$128$	0	0
$129$	−6.48528	−0.570997
$130$	0	0
$131$	−6.34315	−0.554203	−0.277102	−	0.960841i	$-0.589374\pi$
$131$	−6.34315	−0.554203	−0.277102	+	0.960841i	$0.589374\pi$
$132$	0	0
$133$	0.828427	0.0718337
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.9706	1.44989	0.724947	−	0.688805i	$-0.241864\pi$
$137$	16.9706	1.44989	0.724947	+	0.688805i	$0.241864\pi$
$138$	0	0
$139$	−8.82843	−0.748817	−0.374409	−	0.927264i	$-0.622154\pi$
$139$	−8.82843	−0.748817	−0.374409	+	0.927264i	$0.622154\pi$
$140$	0	0
$141$	−12.0711	−1.01657
$142$	0	0
$143$	2.41421	0.201887
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.82843	0.563199
$148$	0	0
$149$	−5.31371	−0.435316	−0.217658	−	0.976025i	$-0.569842\pi$
$149$	−5.31371	−0.435316	−0.217658	+	0.976025i	$0.569842\pi$
$150$	0	0
$151$	1.24264	0.101125	0.0505623	−	0.998721i	$-0.483899\pi$
$151$	1.24264	0.101125	0.0505623	+	0.998721i	$0.483899\pi$
$152$	0	0
$153$	−1.82843	−0.147820
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.31371	0.344271	0.172136	−	0.985073i	$-0.444933\pi$
$157$	4.31371	0.344271	0.172136	+	0.985073i	$0.444933\pi$
$158$	0	0
$159$	0.171573	0.0136066
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−25.3137	−1.98272	−0.991361	−	0.131159i	$-0.958130\pi$
$163$	−25.3137	−1.98272	−0.991361	+	0.131159i	$0.958130\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.68629	0.207871	0.103936	−	0.994584i	$-0.466856\pi$
$167$	2.68629	0.207871	0.103936	+	0.994584i	$0.466856\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.00000	−0.152944
$172$	0	0
$173$	−13.4853	−1.02527	−0.512633	−	0.858608i	$-0.671330\pi$
$173$	−13.4853	−1.02527	−0.512633	+	0.858608i	$0.671330\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.58579	0.269524
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−14.6569	−1.08944	−0.544718	−	0.838619i	$-0.683363\pi$
$181$	−14.6569	−1.08944	−0.544718	+	0.838619i	$0.683363\pi$
$182$	0	0
$183$	−3.82843	−0.283005
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.41421	0.322799
$188$	0	0
$189$	0.414214	0.0301296
$190$	0	0
$191$	−2.48528	−0.179829	−0.0899143	−	0.995950i	$-0.528659\pi$
$191$	−2.48528	−0.179829	−0.0899143	+	0.995950i	$0.528659\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.00000	−0.284988	−0.142494	−	0.989796i	$-0.545512\pi$
$197$	−4.00000	−0.284988	−0.142494	+	0.989796i	$0.545512\pi$
$198$	0	0
$199$	4.82843	0.342278	0.171139	−	0.985247i	$-0.445255\pi$
$199$	4.82843	0.342278	0.171139	+	0.985247i	$0.445255\pi$
$200$	0	0
$201$	−9.24264	−0.651926
$202$	0	0
$203$	−1.10051	−0.0772403
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.82843	0.335599
$208$	0	0
$209$	4.82843	0.333989
$210$	0	0
$211$	−6.48528	−0.446465	−0.223233	−	0.974765i	$-0.571661\pi$
$211$	−6.48528	−0.446465	−0.223233	+	0.974765i	$0.571661\pi$
$212$	0	0
$213$	−3.65685	−0.250564
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.656854	−0.0445902
$218$	0	0
$219$	13.3137	0.899657
$220$	0	0
$221$	1.82843	0.122993
$222$	0	0
$223$	−12.3431	−0.826558	−0.413279	−	0.910604i	$-0.635617\pi$
$223$	−12.3431	−0.826558	−0.413279	+	0.910604i	$0.635617\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.4142	0.956705	0.478352	−	0.878168i	$-0.341234\pi$
$227$	14.4142	0.956705	0.478352	+	0.878168i	$0.341234\pi$
$228$	0	0
$229$	−15.6569	−1.03463	−0.517317	−	0.855794i	$-0.673069\pi$
$229$	−15.6569	−1.03463	−0.517317	+	0.855794i	$0.673069\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.17157	0.206016
$238$	0	0
$239$	3.24264	0.209749	0.104874	−	0.994485i	$-0.466556\pi$
$239$	3.24264	0.209749	0.104874	+	0.994485i	$0.466556\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	11.7279	0.743227
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	−11.6569	−0.732860
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.00000	0.0623783	0.0311891	−	0.999514i	$-0.490071\pi$
$257$	1.00000	0.0623783	0.0311891	+	0.999514i	$0.490071\pi$
$258$	0	0
$259$	−1.51472	−0.0941200
$260$	0	0
$261$	2.65685	0.164455
$262$	0	0
$263$	19.3137	1.19093	0.595467	−	0.803380i	$-0.296967\pi$
$263$	19.3137	1.19093	0.595467	+	0.803380i	$0.296967\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.3137	−0.628838	−0.314419	−	0.949284i	$-0.601810\pi$
$269$	−10.3137	−0.628838	−0.314419	+	0.949284i	$0.601810\pi$
$270$	0	0
$271$	14.8995	0.905080	0.452540	−	0.891744i	$-0.350518\pi$
$271$	14.8995	0.905080	0.452540	+	0.891744i	$0.350518\pi$
$272$	0	0
$273$	−0.414214	−0.0250693
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.3137	0.799943	0.399972	−	0.916528i	$-0.369020\pi$
$277$	13.3137	0.799943	0.399972	+	0.916528i	$0.369020\pi$
$278$	0	0
$279$	1.58579	0.0949386
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	−23.4558	−1.39431	−0.697153	−	0.716923i	$-0.745550\pi$
$283$	−23.4558	−1.39431	−0.697153	+	0.716923i	$0.745550\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.34315	0.138312
$288$	0	0
$289$	−13.6569	−0.803344
$290$	0	0
$291$	18.9706	1.11207
$292$	0	0
$293$	−17.3137	−1.01148	−0.505739	−	0.862687i	$-0.668780\pi$
$293$	−17.3137	−1.01148	−0.505739	+	0.862687i	$0.668780\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.41421	0.140087
$298$	0	0
$299$	−4.82843	−0.279235
$300$	0	0
$301$	−2.68629	−0.154835
$302$	0	0
$303$	11.8284	0.679525
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.51472	−0.0858918	−0.0429459	−	0.999077i	$-0.513674\pi$
$311$	−1.51472	−0.0858918	−0.0429459	+	0.999077i	$0.513674\pi$
$312$	0	0
$313$	−14.1716	−0.801025	−0.400512	−	0.916291i	$-0.631168\pi$
$313$	−14.1716	−0.801025	−0.400512	+	0.916291i	$0.631168\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.9706	−1.73948	−0.869740	−	0.493510i	$-0.835714\pi$
$317$	−30.9706	−1.73948	−0.869740	+	0.493510i	$0.835714\pi$
$318$	0	0
$319$	−6.41421	−0.359127
$320$	0	0
$321$	−1.51472	−0.0845433
$322$	0	0
$323$	3.65685	0.203473
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.31371	−0.183248
$328$	0	0
$329$	−5.00000	−0.275659
$330$	0	0
$331$	−14.9706	−0.822857	−0.411428	−	0.911442i	$-0.634970\pi$
$331$	−14.9706	−0.822857	−0.411428	+	0.911442i	$0.634970\pi$
$332$	0	0
$333$	3.65685	0.200394
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.48528	0.516696	0.258348	−	0.966052i	$-0.416822\pi$
$337$	9.48528	0.516696	0.258348	+	0.966052i	$0.416822\pi$
$338$	0	0
$339$	−9.31371	−0.505851
$340$	0	0
$341$	−3.82843	−0.207321
$342$	0	0
$343$	5.72792	0.309279
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.4853	1.63654	0.818268	−	0.574837i	$-0.194935\pi$
$347$	30.4853	1.63654	0.818268	+	0.574837i	$0.194935\pi$
$348$	0	0
$349$	7.65685	0.409862	0.204931	−	0.978776i	$-0.434303\pi$
$349$	7.65685	0.409862	0.204931	+	0.978776i	$0.434303\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	23.3137	1.24086	0.620432	−	0.784260i	$-0.286957\pi$
$353$	23.3137	1.24086	0.620432	+	0.784260i	$0.286957\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.757359	−0.0400837
$358$	0	0
$359$	−20.8995	−1.10303	−0.551517	−	0.834164i	$-0.685951\pi$
$359$	−20.8995	−1.10303	−0.551517	+	0.834164i	$0.685951\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	5.17157	0.271437
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.65685	0.0864871	0.0432435	−	0.999065i	$-0.486231\pi$
$367$	1.65685	0.0864871	0.0432435	+	0.999065i	$0.486231\pi$
$368$	0	0
$369$	−5.65685	−0.294484
$370$	0	0
$371$	0.0710678	0.00368966
$372$	0	0
$373$	−21.3431	−1.10511	−0.552553	−	0.833478i	$-0.686346\pi$
$373$	−21.3431	−1.10511	−0.552553	+	0.833478i	$0.686346\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.65685	−0.136835
$378$	0	0
$379$	−14.0711	−0.722782	−0.361391	−	0.932414i	$-0.617698\pi$
$379$	−14.0711	−0.722782	−0.361391	+	0.932414i	$0.617698\pi$
$380$	0	0
$381$	−16.8284	−0.862146
$382$	0	0
$383$	−7.65685	−0.391247	−0.195623	−	0.980679i	$-0.562673\pi$
$383$	−7.65685	−0.391247	−0.195623	+	0.980679i	$0.562673\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.48528	0.329665
$388$	0	0
$389$	28.6274	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$390$	0	0
$391$	−8.82843	−0.446473
$392$	0	0
$393$	6.34315	0.319969
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.2843	−1.11842	−0.559208	−	0.829028i	$-0.688895\pi$
$397$	−22.2843	−1.11842	−0.559208	+	0.829028i	$0.688895\pi$
$398$	0	0
$399$	−0.828427	−0.0414732
$400$	0	0
$401$	−6.68629	−0.333897	−0.166949	−	0.985966i	$-0.553391\pi$
$401$	−6.68629	−0.333897	−0.166949	+	0.985966i	$0.553391\pi$
$402$	0	0
$403$	−1.58579	−0.0789936
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.82843	−0.437609
$408$	0	0
$409$	−3.31371	−0.163852	−0.0819262	−	0.996638i	$-0.526107\pi$
$409$	−3.31371	−0.163852	−0.0819262	+	0.996638i	$0.526107\pi$
$410$	0	0
$411$	−16.9706	−0.837096
$412$	0	0
$413$	1.48528	0.0730859
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.82843	0.432330
$418$	0	0
$419$	13.6569	0.667181	0.333590	−	0.942718i	$-0.391740\pi$
$419$	13.6569	0.667181	0.333590	+	0.942718i	$0.391740\pi$
$420$	0	0
$421$	11.3137	0.551396	0.275698	−	0.961244i	$-0.411091\pi$
$421$	11.3137	0.551396	0.275698	+	0.961244i	$0.411091\pi$
$422$	0	0
$423$	12.0711	0.586915
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.58579	−0.0767416
$428$	0	0
$429$	−2.41421	−0.116559
$430$	0	0
$431$	−26.9706	−1.29913	−0.649563	−	0.760308i	$-0.725048\pi$
$431$	−26.9706	−1.29913	−0.649563	+	0.760308i	$0.725048\pi$
$432$	0	0
$433$	−2.68629	−0.129095	−0.0645475	−	0.997915i	$-0.520560\pi$
$433$	−2.68629	−0.129095	−0.0645475	+	0.997915i	$0.520560\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.65685	−0.461950
$438$	0	0
$439$	−1.51472	−0.0722936	−0.0361468	−	0.999346i	$-0.511508\pi$
$439$	−1.51472	−0.0722936	−0.0361468	+	0.999346i	$0.511508\pi$
$440$	0	0
$441$	−6.82843	−0.325163
$442$	0	0
$443$	−27.4558	−1.30447	−0.652233	−	0.758018i	$-0.726168\pi$
$443$	−27.4558	−1.30447	−0.652233	+	0.758018i	$0.726168\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.31371	0.251330
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	13.6569	0.643076
$452$	0	0
$453$	−1.24264	−0.0583844
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.3431	−0.858056	−0.429028	−	0.903291i	$-0.641144\pi$
$457$	−18.3431	−0.858056	−0.429028	+	0.903291i	$0.641144\pi$
$458$	0	0
$459$	1.82843	0.0853437
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	1.72792	0.0803033	0.0401517	−	0.999194i	$-0.487216\pi$
$463$	1.72792	0.0803033	0.0401517	+	0.999194i	$0.487216\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	−3.82843	−0.176780
$470$	0	0
$471$	−4.31371	−0.198765
$472$	0	0
$473$	−15.6569	−0.719903
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.171573	−0.00785578
$478$	0	0
$479$	−27.0416	−1.23556	−0.617782	−	0.786350i	$-0.711969\pi$
$479$	−27.0416	−1.23556	−0.617782	+	0.786350i	$0.711969\pi$
$480$	0	0
$481$	−3.65685	−0.166738
$482$	0	0
$483$	2.00000	0.0910032
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.24264	0.328195	0.164098	−	0.986444i	$-0.447529\pi$
$487$	7.24264	0.328195	0.164098	+	0.986444i	$0.447529\pi$
$488$	0	0
$489$	25.3137	1.14473
$490$	0	0
$491$	−5.51472	−0.248876	−0.124438	−	0.992227i	$-0.539713\pi$
$491$	−5.51472	−0.248876	−0.124438	+	0.992227i	$0.539713\pi$
$492$	0	0
$493$	−4.85786	−0.218787
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.51472	−0.0679444
$498$	0	0
$499$	−21.5858	−0.966313	−0.483156	−	0.875534i	$-0.660510\pi$
$499$	−21.5858	−0.966313	−0.483156	+	0.875534i	$0.660510\pi$
$500$	0	0
$501$	−2.68629	−0.120015
$502$	0	0
$503$	−36.2843	−1.61784	−0.808918	−	0.587922i	$-0.799946\pi$
$503$	−36.2843	−1.61784	−0.808918	+	0.587922i	$0.799946\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	5.51472	0.243957
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−29.1421	−1.28167
$518$	0	0
$519$	13.4853	0.591938
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−6.48528	−0.283582	−0.141791	−	0.989897i	$-0.545286\pi$
$523$	−6.48528	−0.283582	−0.141791	+	0.989897i	$0.545286\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.89949	−0.126304
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	−3.58579	−0.155610
$532$	0	0
$533$	5.65685	0.245026
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	16.4853	0.710071
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	14.6569	0.628986
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−22.6274	−0.967478	−0.483739	−	0.875212i	$-0.660722\pi$
$547$	−22.6274	−0.967478	−0.483739	+	0.875212i	$0.660722\pi$
$548$	0	0
$549$	3.82843	0.163393
$550$	0	0
$551$	−5.31371	−0.226372
$552$	0	0
$553$	1.31371	0.0558646
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.0000	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$558$	0	0
$559$	−6.48528	−0.274298
$560$	0	0
$561$	−4.41421	−0.186368
$562$	0	0
$563$	−6.48528	−0.273322	−0.136661	−	0.990618i	$-0.543637\pi$
$563$	−6.48528	−0.273322	−0.136661	+	0.990618i	$0.543637\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.414214	−0.0173953
$568$	0	0
$569$	44.3137	1.85773	0.928864	−	0.370422i	$-0.120787\pi$
$569$	44.3137	1.85773	0.928864	+	0.370422i	$0.120787\pi$
$570$	0	0
$571$	−1.51472	−0.0633890	−0.0316945	−	0.999498i	$-0.510090\pi$
$571$	−1.51472	−0.0633890	−0.0316945	+	0.999498i	$0.510090\pi$
$572$	0	0
$573$	2.48528	0.103824
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.2843	0.594662	0.297331	−	0.954774i	$-0.403904\pi$
$577$	14.2843	0.594662	0.297331	+	0.954774i	$0.403904\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	4.85786	0.201538
$582$	0	0
$583$	0.414214	0.0171550
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.7279	0.566612	0.283306	−	0.959030i	$-0.408569\pi$
$587$	13.7279	0.566612	0.283306	+	0.959030i	$0.408569\pi$
$588$	0	0
$589$	−3.17157	−0.130682
$590$	0	0
$591$	4.00000	0.164538
$592$	0	0
$593$	−28.6274	−1.17559	−0.587794	−	0.809011i	$-0.700003\pi$
$593$	−28.6274	−1.17559	−0.587794	+	0.809011i	$0.700003\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.82843	−0.197614
$598$	0	0
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	1.14214	0.0465887	0.0232943	−	0.999729i	$-0.492585\pi$
$601$	1.14214	0.0465887	0.0232943	+	0.999729i	$0.492585\pi$
$602$	0	0
$603$	9.24264	0.376389
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.1421	−0.492834	−0.246417	−	0.969164i	$-0.579253\pi$
$607$	−12.1421	−0.492834	−0.246417	+	0.969164i	$0.579253\pi$
$608$	0	0
$609$	1.10051	0.0445947
$610$	0	0
$611$	−12.0711	−0.488343
$612$	0	0
$613$	−35.6569	−1.44017	−0.720083	−	0.693888i	$-0.755896\pi$
$613$	−35.6569	−1.44017	−0.720083	+	0.693888i	$0.755896\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.02944	−0.202478	−0.101239	−	0.994862i	$-0.532281\pi$
$617$	−5.02944	−0.202478	−0.101239	+	0.994862i	$0.532281\pi$
$618$	0	0
$619$	0.627417	0.0252180	0.0126090	−	0.999921i	$-0.495986\pi$
$619$	0.627417	0.0252180	0.0126090	+	0.999921i	$0.495986\pi$
$620$	0	0
$621$	−4.82843	−0.193758
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.82843	−0.192829
$628$	0	0
$629$	−6.68629	−0.266600
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	0	0
$633$	6.48528	0.257767
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.82843	0.270552
$638$	0	0
$639$	3.65685	0.144663
$640$	0	0
$641$	−35.4853	−1.40158	−0.700792	−	0.713365i	$-0.747170\pi$
$641$	−35.4853	−1.40158	−0.700792	+	0.713365i	$0.747170\pi$
$642$	0	0
$643$	4.62742	0.182488	0.0912438	−	0.995829i	$-0.470916\pi$
$643$	4.62742	0.182488	0.0912438	+	0.995829i	$0.470916\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	45.9411	1.80613	0.903066	−	0.429502i	$-0.141311\pi$
$647$	45.9411	1.80613	0.903066	+	0.429502i	$0.141311\pi$
$648$	0	0
$649$	8.65685	0.339811
$650$	0	0
$651$	0.656854	0.0257441
$652$	0	0
$653$	34.6569	1.35623	0.678114	−	0.734957i	$-0.262798\pi$
$653$	34.6569	1.35623	0.678114	+	0.734957i	$0.262798\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.3137	−0.519417
$658$	0	0
$659$	−32.1421	−1.25208	−0.626040	−	0.779791i	$-0.715325\pi$
$659$	−32.1421	−1.25208	−0.626040	+	0.779791i	$0.715325\pi$
$660$	0	0
$661$	−21.9411	−0.853411	−0.426705	−	0.904391i	$-0.640326\pi$
$661$	−21.9411	−0.853411	−0.426705	+	0.904391i	$0.640326\pi$
$662$	0	0
$663$	−1.82843	−0.0710102
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.8284	0.496719
$668$	0	0
$669$	12.3431	0.477214
$670$	0	0
$671$	−9.24264	−0.356808
$672$	0	0
$673$	−21.1421	−0.814969	−0.407485	−	0.913212i	$-0.633594\pi$
$673$	−21.1421	−0.814969	−0.407485	+	0.913212i	$0.633594\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	7.85786	0.301557
$680$	0	0
$681$	−14.4142	−0.552354
$682$	0	0
$683$	33.1838	1.26974	0.634871	−	0.772618i	$-0.281053\pi$
$683$	33.1838	1.26974	0.634871	+	0.772618i	$0.281053\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.6569	0.597346
$688$	0	0
$689$	0.171573	0.00653641
$690$	0	0
$691$	0.757359	0.0288113	0.0144057	−	0.999896i	$-0.495414\pi$
$691$	0.757359	0.0288113	0.0144057	+	0.999896i	$0.495414\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.3431	0.391775
$698$	0	0
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−9.68629	−0.365846	−0.182923	−	0.983127i	$-0.558556\pi$
$701$	−9.68629	−0.365846	−0.182923	+	0.983127i	$0.558556\pi$
$702$	0	0
$703$	−7.31371	−0.275842
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.89949	0.184264
$708$	0	0
$709$	−10.6274	−0.399121	−0.199561	−	0.979886i	$-0.563951\pi$
$709$	−10.6274	−0.399121	−0.199561	+	0.979886i	$0.563951\pi$
$710$	0	0
$711$	−3.17157	−0.118943
$712$	0	0
$713$	7.65685	0.286751
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.24264	−0.121099
$718$	0	0
$719$	2.62742	0.0979861	0.0489931	−	0.998799i	$-0.484399\pi$
$719$	2.62742	0.0979861	0.0489931	+	0.998799i	$0.484399\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	4.00000	0.148762
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.3137	0.864658	0.432329	−	0.901716i	$-0.357692\pi$
$727$	23.3137	0.864658	0.432329	+	0.901716i	$0.357692\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.8579	−0.438579
$732$	0	0
$733$	−51.3137	−1.89532	−0.947658	−	0.319289i	$-0.896556\pi$
$733$	−51.3137	−1.89532	−0.947658	+	0.319289i	$0.896556\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22.3137	−0.821936
$738$	0	0
$739$	3.44365	0.126677	0.0633384	−	0.997992i	$-0.479825\pi$
$739$	3.44365	0.126677	0.0633384	+	0.997992i	$0.479825\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−15.5858	−0.571787	−0.285894	−	0.958261i	$-0.592290\pi$
$743$	−15.5858	−0.571787	−0.285894	+	0.958261i	$0.592290\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11.7279	−0.429102
$748$	0	0
$749$	−0.627417	−0.0229253
$750$	0	0
$751$	33.5147	1.22297	0.611485	−	0.791256i	$-0.290573\pi$
$751$	33.5147	1.22297	0.611485	+	0.791256i	$0.290573\pi$
$752$	0	0
$753$	8.00000	0.291536
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.79899	0.319805	0.159902	−	0.987133i	$-0.448882\pi$
$757$	8.79899	0.319805	0.159902	+	0.987133i	$0.448882\pi$
$758$	0	0
$759$	11.6569	0.423117
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	−1.37258	−0.0496908
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.58579	0.129475
$768$	0	0
$769$	−6.34315	−0.228740	−0.114370	−	0.993438i	$-0.536485\pi$
$769$	−6.34315	−0.228740	−0.114370	+	0.993438i	$0.536485\pi$
$770$	0	0
$771$	−1.00000	−0.0360141
$772$	0	0
$773$	16.6274	0.598047	0.299023	−	0.954246i	$-0.403339\pi$
$773$	16.6274	0.598047	0.299023	+	0.954246i	$0.403339\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.51472	0.0543402
$778$	0	0
$779$	11.3137	0.405356
$780$	0	0
$781$	−8.82843	−0.315906
$782$	0	0
$783$	−2.65685	−0.0949482
$784$	0	0
$785$	0	0
$786$	0	0
$787$	33.3848	1.19004	0.595019	−	0.803711i	$-0.297144\pi$
$787$	33.3848	1.19004	0.595019	+	0.803711i	$0.297144\pi$
$788$	0	0
$789$	−19.3137	−0.687586
$790$	0	0
$791$	−3.85786	−0.137170
$792$	0	0
$793$	−3.82843	−0.135951
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.6569	0.590016	0.295008	−	0.955495i	$-0.404678\pi$
$797$	16.6569	0.590016	0.295008	+	0.955495i	$0.404678\pi$
$798$	0	0
$799$	−22.0711	−0.780818
$800$	0	0
$801$	0	0
$802$	0	0
$803$	32.1421	1.13427
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.3137	0.363060
$808$	0	0
$809$	18.6863	0.656975	0.328488	−	0.944508i	$-0.393461\pi$
$809$	18.6863	0.656975	0.328488	+	0.944508i	$0.393461\pi$
$810$	0	0
$811$	53.7279	1.88664	0.943321	−	0.331881i	$-0.107683\pi$
$811$	53.7279	1.88664	0.943321	+	0.331881i	$0.107683\pi$
$812$	0	0
$813$	−14.8995	−0.522548
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−12.9706	−0.453783
$818$	0	0
$819$	0.414214	0.0144738
$820$	0	0
$821$	10.6863	0.372954	0.186477	−	0.982459i	$-0.440293\pi$
$821$	10.6863	0.372954	0.186477	+	0.982459i	$0.440293\pi$
$822$	0	0
$823$	−17.9411	−0.625388	−0.312694	−	0.949854i	$-0.601231\pi$
$823$	−17.9411	−0.625388	−0.312694	+	0.949854i	$0.601231\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19.3848	−0.674075	−0.337037	−	0.941491i	$-0.609425\pi$
$827$	−19.3848	−0.674075	−0.337037	+	0.941491i	$0.609425\pi$
$828$	0	0
$829$	1.82843	0.0635039	0.0317519	−	0.999496i	$-0.489891\pi$
$829$	1.82843	0.0635039	0.0317519	+	0.999496i	$0.489891\pi$
$830$	0	0
$831$	−13.3137	−0.461847
$832$	0	0
$833$	12.4853	0.432589
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.58579	−0.0548128
$838$	0	0
$839$	−3.65685	−0.126249	−0.0631243	−	0.998006i	$-0.520106\pi$
$839$	−3.65685	−0.126249	−0.0631243	+	0.998006i	$0.520106\pi$
$840$	0	0
$841$	−21.9411	−0.756591
$842$	0	0
$843$	14.0000	0.482186
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.14214	0.0736047
$848$	0	0
$849$	23.4558	0.805002
$850$	0	0
$851$	17.6569	0.605269
$852$	0	0
$853$	−29.6569	−1.01543	−0.507716	−	0.861525i	$-0.669510\pi$
$853$	−29.6569	−1.01543	−0.507716	+	0.861525i	$0.669510\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32.6274	−1.11453	−0.557266	−	0.830334i	$-0.688150\pi$
$857$	−32.6274	−1.11453	−0.557266	+	0.830334i	$0.688150\pi$
$858$	0	0
$859$	−44.8284	−1.52953	−0.764763	−	0.644312i	$-0.777144\pi$
$859$	−44.8284	−1.52953	−0.764763	+	0.644312i	$0.777144\pi$
$860$	0	0
$861$	−2.34315	−0.0798542
$862$	0	0
$863$	32.4142	1.10339	0.551696	−	0.834045i	$-0.313981\pi$
$863$	32.4142	1.10339	0.551696	+	0.834045i	$0.313981\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.6569	0.463811
$868$	0	0
$869$	7.65685	0.259741
$870$	0	0
$871$	−9.24264	−0.313175
$872$	0	0
$873$	−18.9706	−0.642056
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−25.6569	−0.866370	−0.433185	−	0.901305i	$-0.642610\pi$
$877$	−25.6569	−0.866370	−0.433185	+	0.901305i	$0.642610\pi$
$878$	0	0
$879$	17.3137	0.583977
$880$	0	0
$881$	24.5980	0.828727	0.414363	−	0.910111i	$-0.364004\pi$
$881$	24.5980	0.828727	0.414363	+	0.910111i	$0.364004\pi$
$882$	0	0
$883$	19.3137	0.649958	0.324979	−	0.945721i	$-0.394643\pi$
$883$	19.3137	0.649958	0.324979	+	0.945721i	$0.394643\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.82843	0.296430	0.148215	−	0.988955i	$-0.452647\pi$
$887$	8.82843	0.296430	0.148215	+	0.988955i	$0.452647\pi$
$888$	0	0
$889$	−6.97056	−0.233785
$890$	0	0
$891$	−2.41421	−0.0808792
$892$	0	0
$893$	−24.1421	−0.807886
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.82843	0.161216
$898$	0	0
$899$	4.21320	0.140518
$900$	0	0
$901$	0.313708	0.0104511
$902$	0	0
$903$	2.68629	0.0893942
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−33.9411	−1.12700	−0.563498	−	0.826117i	$-0.690545\pi$
$907$	−33.9411	−1.12700	−0.563498	+	0.826117i	$0.690545\pi$
$908$	0	0
$909$	−11.8284	−0.392324
$910$	0	0
$911$	−15.1716	−0.502657	−0.251328	−	0.967902i	$-0.580867\pi$
$911$	−15.1716	−0.502657	−0.251328	+	0.967902i	$0.580867\pi$
$912$	0	0
$913$	28.3137	0.937047
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.62742	0.0867650
$918$	0	0
$919$	−21.7990	−0.719082	−0.359541	−	0.933129i	$-0.617067\pi$
$919$	−21.7990	−0.719082	−0.359541	+	0.933129i	$0.617067\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	−3.65685	−0.120367
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	13.6569	0.447585
$932$	0	0
$933$	1.51472	0.0495897
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−32.5980	−1.06493	−0.532465	−	0.846452i	$-0.678734\pi$
$937$	−32.5980	−1.06493	−0.532465	+	0.846452i	$0.678734\pi$
$938$	0	0
$939$	14.1716	0.462472
$940$	0	0
$941$	22.6863	0.739552	0.369776	−	0.929121i	$-0.379434\pi$
$941$	22.6863	0.739552	0.369776	+	0.929121i	$0.379434\pi$
$942$	0	0
$943$	−27.3137	−0.889457
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.5858	0.571461	0.285731	−	0.958310i	$-0.407764\pi$
$947$	17.5858	0.571461	0.285731	+	0.958310i	$0.407764\pi$
$948$	0	0
$949$	13.3137	0.432181
$950$	0	0
$951$	30.9706	1.00429
$952$	0	0
$953$	23.3431	0.756159	0.378079	−	0.925773i	$-0.376585\pi$
$953$	23.3431	0.756159	0.378079	+	0.925773i	$0.376585\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.41421	0.207342
$958$	0	0
$959$	−7.02944	−0.226992
$960$	0	0
$961$	−28.4853	−0.918880
$962$	0	0
$963$	1.51472	0.0488111
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−58.0122	−1.86555	−0.932773	−	0.360464i	$-0.882618\pi$
$967$	−58.0122	−1.86555	−0.932773	+	0.360464i	$0.882618\pi$
$968$	0	0
$969$	−3.65685	−0.117475
$970$	0	0
$971$	−23.1716	−0.743611	−0.371806	−	0.928311i	$-0.621261\pi$
$971$	−23.1716	−0.743611	−0.371806	+	0.928311i	$0.621261\pi$
$972$	0	0
$973$	3.65685	0.117233
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.970563	−0.0310511	−0.0155255	−	0.999879i	$-0.504942\pi$
$977$	−0.970563	−0.0310511	−0.0155255	+	0.999879i	$0.504942\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	3.31371	0.105799
$982$	0	0
$983$	17.2426	0.549955	0.274977	−	0.961451i	$-0.411330\pi$
$983$	17.2426	0.549955	0.274977	+	0.961451i	$0.411330\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.00000	0.159152
$988$	0	0
$989$	31.3137	0.995718
$990$	0	0
$991$	44.1421	1.40222	0.701111	−	0.713053i	$-0.252688\pi$
$991$	44.1421	1.40222	0.701111	+	0.713053i	$0.252688\pi$
$992$	0	0
$993$	14.9706	0.475076
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−19.2010	−0.608102	−0.304051	−	0.952656i	$-0.598339\pi$
$997$	−19.2010	−0.608102	−0.304051	+	0.952656i	$0.598339\pi$
$998$	0	0
$999$	−3.65685	−0.115698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.z.1.1	✓	2
5.4	even	2		7800.2.a.ba.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.z.1.1	✓	2	1.1	even	1	trivial
7800.2.a.ba.1.2	yes	2	5.4	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.414214 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.414214 q^{7} +1.00000 q^{9} -2.41421 q^{11} -1.00000 q^{13} -1.82843 q^{17} -2.00000 q^{19} +0.414214 q^{21} +4.82843 q^{23} -1.00000 q^{27} +2.65685 q^{29} +1.58579 q^{31} +2.41421 q^{33} +3.65685 q^{37} +1.00000 q^{39} -5.65685 q^{41} +6.48528 q^{43} +12.0711 q^{47} -6.82843 q^{49} +1.82843 q^{51} -0.171573 q^{53} +2.00000 q^{57} -3.58579 q^{59} +3.82843 q^{61} -0.414214 q^{63} +9.24264 q^{67} -4.82843 q^{69} +3.65685 q^{71} -13.3137 q^{73} +1.00000 q^{77} -3.17157 q^{79} +1.00000 q^{81} -11.7279 q^{83} -2.65685 q^{87} +0.414214 q^{91} -1.58579 q^{93} -18.9706 q^{97} -2.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 2 q^{13} + 2 q^{17} - 4 q^{19} - 2 q^{21} + 4 q^{23} - 2 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 4 q^{37} + 2 q^{39} - 4 q^{43} + 10 q^{47} - 8 q^{49} - 2 q^{51} - 6 q^{53} + 4 q^{57} - 10 q^{59} + 2 q^{61} + 2 q^{63} + 10 q^{67} - 4 q^{69} - 4 q^{71} - 4 q^{73} + 2 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{83} + 6 q^{87} - 2 q^{91} - 6 q^{93} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 2 * q^11 - 2 * q^13 + 2 * q^17 - 4 * q^19 - 2 * q^21 + 4 * q^23 - 2 * q^27 - 6 * q^29 + 6 * q^31 + 2 * q^33 - 4 * q^37 + 2 * q^39 - 4 * q^43 + 10 * q^47 - 8 * q^49 - 2 * q^51 - 6 * q^53 + 4 * q^57 - 10 * q^59 + 2 * q^61 + 2 * q^63 + 10 * q^67 - 4 * q^69 - 4 * q^71 - 4 * q^73 + 2 * q^77 - 12 * q^79 + 2 * q^81 + 2 * q^83 + 6 * q^87 - 2 * q^91 - 6 * q^93 - 4 * q^97 - 2 * q^99

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