LMFDB - Embedded newform 578.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [578,2,Mod(1,578)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(578, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("578.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 578 = 2 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	578.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:	$4.61535323683$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 34)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	578.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} -1.00000 q^{8} -3.00000 q^{9} +1.41421 q^{10} +5.65685 q^{11} -4.00000 q^{13} +1.00000 q^{16} +3.00000 q^{18} -4.00000 q^{19} -1.41421 q^{20} -5.65685 q^{22} -5.65685 q^{23} -3.00000 q^{25} +4.00000 q^{26} -4.24264 q^{29} +5.65685 q^{31} -1.00000 q^{32} -3.00000 q^{36} -4.24264 q^{37} +4.00000 q^{38} +1.41421 q^{40} +1.41421 q^{41} -4.00000 q^{43} +5.65685 q^{44} +4.24264 q^{45} +5.65685 q^{46} -8.00000 q^{47} -7.00000 q^{49} +3.00000 q^{50} -4.00000 q^{52} +4.00000 q^{53} -8.00000 q^{55} +4.24264 q^{58} +4.00000 q^{59} -12.7279 q^{61} -5.65685 q^{62} +1.00000 q^{64} +5.65685 q^{65} -12.0000 q^{67} +5.65685 q^{71} +3.00000 q^{72} +7.07107 q^{73} +4.24264 q^{74} -4.00000 q^{76} -11.3137 q^{79} -1.41421 q^{80} +9.00000 q^{81} -1.41421 q^{82} +4.00000 q^{83} +4.00000 q^{86} -5.65685 q^{88} -4.24264 q^{90} -5.65685 q^{92} +8.00000 q^{94} +5.65685 q^{95} +4.24264 q^{97} +7.00000 q^{98} -16.9706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9} - 8 q^{13} + 2 q^{16} + 6 q^{18} - 8 q^{19} - 6 q^{25} + 8 q^{26} - 2 q^{32} - 6 q^{36} + 8 q^{38} - 8 q^{43} - 16 q^{47} - 14 q^{49} + 6 q^{50} - 8 q^{52} + 8 q^{53} - 16 q^{55} + 8 q^{59} + 2 q^{64} - 24 q^{67} + 6 q^{72} - 8 q^{76} + 18 q^{81} + 8 q^{83} + 8 q^{86} + 16 q^{94} + 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9 - 8 * q^13 + 2 * q^16 + 6 * q^18 - 8 * q^19 - 6 * q^25 + 8 * q^26 - 2 * q^32 - 6 * q^36 + 8 * q^38 - 8 * q^43 - 16 * q^47 - 14 * q^49 + 6 * q^50 - 8 * q^52 + 8 * q^53 - 16 * q^55 + 8 * q^59 + 2 * q^64 - 24 * q^67 + 6 * q^72 - 8 * q^76 + 18 * q^81 + 8 * q^83 + 8 * q^86 + 16 * q^94 + 14 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	−3.00000	−1.00000
$10$	1.41421	0.447214
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	3.00000	0.707107
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.41421	−0.316228
$21$	0	0
$22$	−5.65685	−1.20605
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	4.00000	0.784465
$27$	0	0
$28$	0	0
$29$	−4.24264	−0.787839	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264	−0.787839	−0.393919	+	0.919145i	$0.628881\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−3.00000	−0.500000
$37$	−4.24264	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$37$	−4.24264	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	1.41421	0.223607
$41$	1.41421	0.220863	0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421	0.220863	0.110432	+	0.993884i	$0.464777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	5.65685	0.852803
$45$	4.24264	0.632456
$46$	5.65685	0.834058
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	3.00000	0.424264
$51$	0	0
$52$	−4.00000	−0.554700
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	0	0
$58$	4.24264	0.557086
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−12.7279	−1.62964	−0.814822	−	0.579712i	$-0.803165\pi$
$61$	−12.7279	−1.62964	−0.814822	+	0.579712i	$0.803165\pi$
$62$	−5.65685	−0.718421
$63$	0	0
$64$	1.00000	0.125000
$65$	5.65685	0.701646
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	3.00000	0.353553
$73$	7.07107	0.827606	0.413803	−	0.910366i	$-0.364200\pi$
$73$	7.07107	0.827606	0.413803	+	0.910366i	$0.364200\pi$
$74$	4.24264	0.493197
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	−1.41421	−0.158114
$81$	9.00000	1.00000
$82$	−1.41421	−0.156174
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	−5.65685	−0.603023
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−4.24264	−0.447214
$91$	0	0
$92$	−5.65685	−0.589768
$93$	0	0
$94$	8.00000	0.825137
$95$	5.65685	0.580381
$96$	0	0
$97$	4.24264	0.430775	0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264	0.430775	0.215387	+	0.976529i	$0.430899\pi$
$98$	7.00000	0.707107
$99$	−16.9706	−1.70561
$100$	−3.00000	−0.300000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−4.00000	−0.388514
$107$	11.3137	1.09374	0.546869	−	0.837218i	$-0.315820\pi$
$107$	11.3137	1.09374	0.546869	+	0.837218i	$0.315820\pi$
$108$	0	0
$109$	4.24264	0.406371	0.203186	−	0.979140i	$-0.434871\pi$
$109$	4.24264	0.406371	0.203186	+	0.979140i	$0.434871\pi$
$110$	8.00000	0.762770
$111$	0	0
$112$	0	0
$113$	1.41421	0.133038	0.0665190	−	0.997785i	$-0.478811\pi$
$113$	1.41421	0.133038	0.0665190	+	0.997785i	$0.478811\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	−4.24264	−0.393919
$117$	12.0000	1.10940
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	21.0000	1.90909
$122$	12.7279	1.15233
$123$	0	0
$124$	5.65685	0.508001
$125$	11.3137	1.01193
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−5.65685	−0.496139
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	0	0
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	11.3137	0.959616	0.479808	−	0.877373i	$-0.340706\pi$
$139$	11.3137	0.959616	0.479808	+	0.877373i	$0.340706\pi$
$140$	0	0
$141$	0	0
$142$	−5.65685	−0.474713
$143$	−22.6274	−1.89220
$144$	−3.00000	−0.250000
$145$	6.00000	0.498273
$146$	−7.07107	−0.585206
$147$	0	0
$148$	−4.24264	−0.348743
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	11.3137	0.900070
$159$	0	0
$160$	1.41421	0.111803
$161$	0	0
$162$	−9.00000	−0.707107
$163$	5.65685	0.443079	0.221540	−	0.975151i	$-0.428892\pi$
$163$	5.65685	0.443079	0.221540	+	0.975151i	$0.428892\pi$
$164$	1.41421	0.110432
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	12.0000	0.917663
$172$	−4.00000	−0.304997
$173$	−7.07107	−0.537603	−0.268802	−	0.963196i	$-0.586628\pi$
$173$	−7.07107	−0.537603	−0.268802	+	0.963196i	$0.586628\pi$
$174$	0	0
$175$	0	0
$176$	5.65685	0.426401
$177$	0	0
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	4.24264	0.316228
$181$	−15.5563	−1.15629	−0.578147	−	0.815933i	$-0.696224\pi$
$181$	−15.5563	−1.15629	−0.578147	+	0.815933i	$0.696224\pi$
$182$	0	0
$183$	0	0
$184$	5.65685	0.417029
$185$	6.00000	0.441129
$186$	0	0
$187$	0	0
$188$	−8.00000	−0.583460
$189$	0	0
$190$	−5.65685	−0.410391
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	15.5563	1.11977	0.559885	−	0.828570i	$-0.310845\pi$
$193$	15.5563	1.11977	0.559885	+	0.828570i	$0.310845\pi$
$194$	−4.24264	−0.304604
$195$	0	0
$196$	−7.00000	−0.500000
$197$	7.07107	0.503793	0.251896	−	0.967754i	$-0.418946\pi$
$197$	7.07107	0.503793	0.251896	+	0.967754i	$0.418946\pi$
$198$	16.9706	1.20605
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	−12.0000	−0.844317
$203$	0	0
$204$	0	0
$205$	−2.00000	−0.139686
$206$	16.0000	1.11477
$207$	16.9706	1.17954
$208$	−4.00000	−0.277350
$209$	−22.6274	−1.56517
$210$	0	0
$211$	22.6274	1.55774	0.778868	−	0.627188i	$-0.215794\pi$
$211$	22.6274	1.55774	0.778868	+	0.627188i	$0.215794\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	−11.3137	−0.773389
$215$	5.65685	0.385794
$216$	0	0
$217$	0	0
$218$	−4.24264	−0.287348
$219$	0	0
$220$	−8.00000	−0.539360
$221$	0	0
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	9.00000	0.600000
$226$	−1.41421	−0.0940721
$227$	−28.2843	−1.87729	−0.938647	−	0.344881i	$-0.887919\pi$
$227$	−28.2843	−1.87729	−0.938647	+	0.344881i	$0.887919\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	4.24264	0.278543
$233$	−7.07107	−0.463241	−0.231621	−	0.972806i	$-0.574403\pi$
$233$	−7.07107	−0.463241	−0.231621	+	0.972806i	$0.574403\pi$
$234$	−12.0000	−0.784465
$235$	11.3137	0.738025
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−29.6985	−1.91305	−0.956524	−	0.291654i	$-0.905794\pi$
$241$	−29.6985	−1.91305	−0.956524	+	0.291654i	$0.905794\pi$
$242$	−21.0000	−1.34993
$243$	0	0
$244$	−12.7279	−0.814822
$245$	9.89949	0.632456
$246$	0	0
$247$	16.0000	1.01806
$248$	−5.65685	−0.359211
$249$	0	0
$250$	−11.3137	−0.715542
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−32.0000	−2.01182
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	0	0
$260$	5.65685	0.350823
$261$	12.7279	0.787839
$262$	5.65685	0.349482
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−5.65685	−0.347498
$266$	0	0
$267$	0	0
$268$	−12.0000	−0.733017
$269$	18.3848	1.12094	0.560470	−	0.828175i	$-0.310621\pi$
$269$	18.3848	1.12094	0.560470	+	0.828175i	$0.310621\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	−8.00000	−0.483298
$275$	−16.9706	−1.02336
$276$	0	0
$277$	−9.89949	−0.594803	−0.297402	−	0.954753i	$-0.596120\pi$
$277$	−9.89949	−0.594803	−0.297402	+	0.954753i	$0.596120\pi$
$278$	−11.3137	−0.678551
$279$	−16.9706	−1.01600
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−5.65685	−0.336265	−0.168133	−	0.985764i	$-0.553774\pi$
$283$	−5.65685	−0.336265	−0.168133	+	0.985764i	$0.553774\pi$
$284$	5.65685	0.335673
$285$	0	0
$286$	22.6274	1.33799
$287$	0	0
$288$	3.00000	0.176777
$289$	0	0
$290$	−6.00000	−0.352332
$291$	0	0
$292$	7.07107	0.413803
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−5.65685	−0.329355
$296$	4.24264	0.246598
$297$	0	0
$298$	−10.0000	−0.579284
$299$	22.6274	1.30858
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−4.00000	−0.229416
$305$	18.0000	1.03068
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	8.00000	0.454369
$311$	−33.9411	−1.92462	−0.962312	−	0.271947i	$-0.912333\pi$
$311$	−33.9411	−1.92462	−0.962312	+	0.271947i	$0.912333\pi$
$312$	0	0
$313$	12.7279	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	12.7279	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−11.3137	−0.636446
$317$	21.2132	1.19145	0.595726	−	0.803188i	$-0.296864\pi$
$317$	21.2132	1.19145	0.595726	+	0.803188i	$0.296864\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	−1.41421	−0.0790569
$321$	0	0
$322$	0	0
$323$	0	0
$324$	9.00000	0.500000
$325$	12.0000	0.665640
$326$	−5.65685	−0.313304
$327$	0	0
$328$	−1.41421	−0.0780869
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	4.00000	0.219529
$333$	12.7279	0.697486
$334$	11.3137	0.619059
$335$	16.9706	0.927201
$336$	0	0
$337$	9.89949	0.539260	0.269630	−	0.962964i	$-0.413099\pi$
$337$	9.89949	0.539260	0.269630	+	0.962964i	$0.413099\pi$
$338$	−3.00000	−0.163178
$339$	0	0
$340$	0	0
$341$	32.0000	1.73290
$342$	−12.0000	−0.648886
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	7.07107	0.380143
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	0	0
$351$	0	0
$352$	−5.65685	−0.301511
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	−4.24264	−0.223607
$361$	−3.00000	−0.157895
$362$	15.5563	0.817624
$363$	0	0
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−5.65685	−0.294884
$369$	−4.24264	−0.220863
$370$	−6.00000	−0.311925
$371$	0	0
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	16.9706	0.874028
$378$	0	0
$379$	16.9706	0.871719	0.435860	−	0.900015i	$-0.356444\pi$
$379$	16.9706	0.871719	0.435860	+	0.900015i	$0.356444\pi$
$380$	5.65685	0.290191
$381$	0	0
$382$	−8.00000	−0.409316
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	−15.5563	−0.791797
$387$	12.0000	0.609994
$388$	4.24264	0.215387
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	7.00000	0.353553
$393$	0	0
$394$	−7.07107	−0.356235
$395$	16.0000	0.805047
$396$	−16.9706	−0.852803
$397$	−21.2132	−1.06466	−0.532330	−	0.846537i	$-0.678683\pi$
$397$	−21.2132	−1.06466	−0.532330	+	0.846537i	$0.678683\pi$
$398$	−16.9706	−0.850657
$399$	0	0
$400$	−3.00000	−0.150000
$401$	−26.8701	−1.34183	−0.670913	−	0.741536i	$-0.734098\pi$
$401$	−26.8701	−1.34183	−0.670913	+	0.741536i	$0.734098\pi$
$402$	0	0
$403$	−22.6274	−1.12715
$404$	12.0000	0.597022
$405$	−12.7279	−0.632456
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	2.00000	0.0987730
$411$	0	0
$412$	−16.0000	−0.788263
$413$	0	0
$414$	−16.9706	−0.834058
$415$	−5.65685	−0.277684
$416$	4.00000	0.196116
$417$	0	0
$418$	22.6274	1.10674
$419$	16.9706	0.829066	0.414533	−	0.910034i	$-0.363945\pi$
$419$	16.9706	0.829066	0.414533	+	0.910034i	$0.363945\pi$
$420$	0	0
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	−22.6274	−1.10149
$423$	24.0000	1.16692
$424$	−4.00000	−0.194257
$425$	0	0
$426$	0	0
$427$	0	0
$428$	11.3137	0.546869
$429$	0	0
$430$	−5.65685	−0.272798
$431$	33.9411	1.63489	0.817443	−	0.576009i	$-0.195391\pi$
$431$	33.9411	1.63489	0.817443	+	0.576009i	$0.195391\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	4.24264	0.203186
$437$	22.6274	1.08242
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	8.00000	0.381385
$441$	21.0000	1.00000
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	0	0
$446$	16.0000	0.757622
$447$	0	0
$448$	0	0
$449$	4.24264	0.200223	0.100111	−	0.994976i	$-0.468080\pi$
$449$	4.24264	0.200223	0.100111	+	0.994976i	$0.468080\pi$
$450$	−9.00000	−0.424264
$451$	8.00000	0.376705
$452$	1.41421	0.0665190
$453$	0	0
$454$	28.2843	1.32745
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	8.00000	0.373002
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−4.24264	−0.196960
$465$	0	0
$466$	7.07107	0.327561
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	12.0000	0.554700
$469$	0	0
$470$	−11.3137	−0.521862
$471$	0	0
$472$	−4.00000	−0.184115
$473$	−22.6274	−1.04041
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	16.9706	0.773791
$482$	29.6985	1.35273
$483$	0	0
$484$	21.0000	0.954545
$485$	−6.00000	−0.272446
$486$	0	0
$487$	−28.2843	−1.28168	−0.640841	−	0.767673i	$-0.721414\pi$
$487$	−28.2843	−1.28168	−0.640841	+	0.767673i	$0.721414\pi$
$488$	12.7279	0.576166
$489$	0	0
$490$	−9.89949	−0.447214
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	0	0
$494$	−16.0000	−0.719874
$495$	24.0000	1.07872
$496$	5.65685	0.254000
$497$	0	0
$498$	0	0
$499$	−11.3137	−0.506471	−0.253236	−	0.967405i	$-0.581495\pi$
$499$	−11.3137	−0.506471	−0.253236	+	0.967405i	$0.581495\pi$
$500$	11.3137	0.505964
$501$	0	0
$502$	12.0000	0.535586
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	−16.9706	−0.755180
$506$	32.0000	1.42257
$507$	0	0
$508$	8.00000	0.354943
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	22.6274	0.997083
$516$	0	0
$517$	−45.2548	−1.99031
$518$	0	0
$519$	0	0
$520$	−5.65685	−0.248069
$521$	−1.41421	−0.0619578	−0.0309789	−	0.999520i	$-0.509862\pi$
$521$	−1.41421	−0.0619578	−0.0309789	+	0.999520i	$0.509862\pi$
$522$	−12.7279	−0.557086
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−5.65685	−0.247121
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	0	0
$529$	9.00000	0.391304
$530$	5.65685	0.245718
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−5.65685	−0.245026
$534$	0	0
$535$	−16.0000	−0.691740
$536$	12.0000	0.518321
$537$	0	0
$538$	−18.3848	−0.792624
$539$	−39.5980	−1.70561
$540$	0	0
$541$	12.7279	0.547216	0.273608	−	0.961841i	$-0.411783\pi$
$541$	12.7279	0.547216	0.273608	+	0.961841i	$0.411783\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	16.9706	0.725609	0.362804	−	0.931865i	$-0.381819\pi$
$547$	16.9706	0.725609	0.362804	+	0.931865i	$0.381819\pi$
$548$	8.00000	0.341743
$549$	38.1838	1.62964
$550$	16.9706	0.723627
$551$	16.9706	0.722970
$552$	0	0
$553$	0	0
$554$	9.89949	0.420589
$555$	0	0
$556$	11.3137	0.479808
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	16.9706	0.718421
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	5.65685	0.237775
$567$	0	0
$568$	−5.65685	−0.237356
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−33.9411	−1.42039	−0.710196	−	0.704004i	$-0.751394\pi$
$571$	−33.9411	−1.42039	−0.710196	+	0.704004i	$0.751394\pi$
$572$	−22.6274	−0.946100
$573$	0	0
$574$	0	0
$575$	16.9706	0.707721
$576$	−3.00000	−0.125000
$577$	−32.0000	−1.33218	−0.666089	−	0.745873i	$-0.732033\pi$
$577$	−32.0000	−1.33218	−0.666089	+	0.745873i	$0.732033\pi$
$578$	0	0
$579$	0	0
$580$	6.00000	0.249136
$581$	0	0
$582$	0	0
$583$	22.6274	0.937132
$584$	−7.07107	−0.292603
$585$	−16.9706	−0.701646
$586$	−6.00000	−0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−22.6274	−0.932346
$590$	5.65685	0.232889
$591$	0	0
$592$	−4.24264	−0.174371
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	−22.6274	−0.925304
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	26.8701	1.09605	0.548026	−	0.836461i	$-0.315379\pi$
$601$	26.8701	1.09605	0.548026	+	0.836461i	$0.315379\pi$
$602$	0	0
$603$	36.0000	1.46603
$604$	0	0
$605$	−29.6985	−1.20742
$606$	0	0
$607$	39.5980	1.60723	0.803616	−	0.595148i	$-0.202907\pi$
$607$	39.5980	1.60723	0.803616	+	0.595148i	$0.202907\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−18.0000	−0.728799
$611$	32.0000	1.29458
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	4.24264	0.170802	0.0854011	−	0.996347i	$-0.472783\pi$
$617$	4.24264	0.170802	0.0854011	+	0.996347i	$0.472783\pi$
$618$	0	0
$619$	11.3137	0.454736	0.227368	−	0.973809i	$-0.426988\pi$
$619$	11.3137	0.454736	0.227368	+	0.973809i	$0.426988\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	33.9411	1.36092
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	−12.7279	−0.508710
$627$	0	0
$628$	2.00000	0.0798087
$629$	0	0
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	11.3137	0.450035
$633$	0	0
$634$	−21.2132	−0.842484
$635$	−11.3137	−0.448971
$636$	0	0
$637$	28.0000	1.10940
$638$	24.0000	0.950169
$639$	−16.9706	−0.671345
$640$	1.41421	0.0559017
$641$	15.5563	0.614439	0.307219	−	0.951639i	$-0.400601\pi$
$641$	15.5563	0.614439	0.307219	+	0.951639i	$0.400601\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	−9.00000	−0.353553
$649$	22.6274	0.888204
$650$	−12.0000	−0.470679
$651$	0	0
$652$	5.65685	0.221540
$653$	41.0122	1.60493	0.802466	−	0.596698i	$-0.203521\pi$
$653$	41.0122	1.60493	0.802466	+	0.596698i	$0.203521\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	1.41421	0.0552158
$657$	−21.2132	−0.827606
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−12.7279	−0.493197
$667$	24.0000	0.929284
$668$	−11.3137	−0.437741
$669$	0	0
$670$	−16.9706	−0.655630
$671$	−72.0000	−2.77953
$672$	0	0
$673$	26.8701	1.03576	0.517882	−	0.855452i	$-0.326721\pi$
$673$	26.8701	1.03576	0.517882	+	0.855452i	$0.326721\pi$
$674$	−9.89949	−0.381314
$675$	0	0
$676$	3.00000	0.115385
$677$	−18.3848	−0.706584	−0.353292	−	0.935513i	$-0.614938\pi$
$677$	−18.3848	−0.706584	−0.353292	+	0.935513i	$0.614938\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−32.0000	−1.22534
$683$	−28.2843	−1.08227	−0.541134	−	0.840937i	$-0.682005\pi$
$683$	−28.2843	−1.08227	−0.541134	+	0.840937i	$0.682005\pi$
$684$	12.0000	0.458831
$685$	−11.3137	−0.432275
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−16.0000	−0.609551
$690$	0	0
$691$	33.9411	1.29118	0.645591	−	0.763684i	$-0.276611\pi$
$691$	33.9411	1.29118	0.645591	+	0.763684i	$0.276611\pi$
$692$	−7.07107	−0.268802
$693$	0	0
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	0	0
$698$	−4.00000	−0.151402
$699$	0	0
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	16.9706	0.640057
$704$	5.65685	0.213201
$705$	0	0
$706$	14.0000	0.526897
$707$	0	0
$708$	0	0
$709$	−32.5269	−1.22157	−0.610787	−	0.791795i	$-0.709147\pi$
$709$	−32.5269	−1.22157	−0.610787	+	0.791795i	$0.709147\pi$
$710$	8.00000	0.300235
$711$	33.9411	1.27289
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	32.0000	1.19673
$716$	4.00000	0.149487
$717$	0	0
$718$	24.0000	0.895672
$719$	−11.3137	−0.421930	−0.210965	−	0.977494i	$-0.567661\pi$
$719$	−11.3137	−0.421930	−0.210965	+	0.977494i	$0.567661\pi$
$720$	4.24264	0.158114
$721$	0	0
$722$	3.00000	0.111648
$723$	0	0
$724$	−15.5563	−0.578147
$725$	12.7279	0.472703
$726$	0	0
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	10.0000	0.370117
$731$	0	0
$732$	0	0
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	0	0
$735$	0	0
$736$	5.65685	0.208514
$737$	−67.8823	−2.50047
$738$	4.24264	0.156174
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	6.00000	0.220564
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−14.1421	−0.518128
$746$	−4.00000	−0.146450
$747$	−12.0000	−0.439057
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−22.6274	−0.825686	−0.412843	−	0.910802i	$-0.635464\pi$
$751$	−22.6274	−0.825686	−0.412843	+	0.910802i	$0.635464\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	−16.9706	−0.618031
$755$	0	0
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−16.9706	−0.616399
$759$	0	0
$760$	−5.65685	−0.205196
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	0	0
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	24.0000	0.867155
$767$	−16.0000	−0.577727
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	0	0
$772$	15.5563	0.559885
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−12.0000	−0.431331
$775$	−16.9706	−0.609601
$776$	−4.24264	−0.152302
$777$	0	0
$778$	−6.00000	−0.215110
$779$	−5.65685	−0.202678
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	0	0
$784$	−7.00000	−0.250000
$785$	−2.82843	−0.100951
$786$	0	0
$787$	−16.9706	−0.604935	−0.302468	−	0.953160i	$-0.597810\pi$
$787$	−16.9706	−0.604935	−0.302468	+	0.953160i	$0.597810\pi$
$788$	7.07107	0.251896
$789$	0	0
$790$	−16.0000	−0.569254
$791$	0	0
$792$	16.9706	0.603023
$793$	50.9117	1.80793
$794$	21.2132	0.752828
$795$	0	0
$796$	16.9706	0.601506
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	0	0
$800$	3.00000	0.106066
$801$	0	0
$802$	26.8701	0.948815
$803$	40.0000	1.41157
$804$	0	0
$805$	0	0
$806$	22.6274	0.797017
$807$	0	0
$808$	−12.0000	−0.422159
$809$	32.5269	1.14359	0.571793	−	0.820398i	$-0.306248\pi$
$809$	32.5269	1.14359	0.571793	+	0.820398i	$0.306248\pi$
$810$	12.7279	0.447214
$811$	−5.65685	−0.198639	−0.0993195	−	0.995056i	$-0.531667\pi$
$811$	−5.65685	−0.198639	−0.0993195	+	0.995056i	$0.531667\pi$
$812$	0	0
$813$	0	0
$814$	24.0000	0.841200
$815$	−8.00000	−0.280228
$816$	0	0
$817$	16.0000	0.559769
$818$	10.0000	0.349642
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	−26.8701	−0.937771	−0.468886	−	0.883259i	$-0.655344\pi$
$821$	−26.8701	−0.937771	−0.468886	+	0.883259i	$0.655344\pi$
$822$	0	0
$823$	−50.9117	−1.77467	−0.887335	−	0.461125i	$-0.847446\pi$
$823$	−50.9117	−1.77467	−0.887335	+	0.461125i	$0.847446\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	28.2843	0.983540	0.491770	−	0.870725i	$-0.336350\pi$
$827$	28.2843	0.983540	0.491770	+	0.870725i	$0.336350\pi$
$828$	16.9706	0.589768
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	5.65685	0.196352
$831$	0	0
$832$	−4.00000	−0.138675
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	−22.6274	−0.782586
$837$	0	0
$838$	−16.9706	−0.586238
$839$	45.2548	1.56237	0.781185	−	0.624299i	$-0.214615\pi$
$839$	45.2548	1.56237	0.781185	+	0.624299i	$0.214615\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	−28.0000	−0.964944
$843$	0	0
$844$	22.6274	0.778868
$845$	−4.24264	−0.145951
$846$	−24.0000	−0.825137
$847$	0	0
$848$	4.00000	0.137361
$849$	0	0
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	35.3553	1.21054	0.605272	−	0.796019i	$-0.293064\pi$
$853$	35.3553	1.21054	0.605272	+	0.796019i	$0.293064\pi$
$854$	0	0
$855$	−16.9706	−0.580381
$856$	−11.3137	−0.386695
$857$	−21.2132	−0.724629	−0.362315	−	0.932056i	$-0.618013\pi$
$857$	−21.2132	−0.724629	−0.362315	+	0.932056i	$0.618013\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	5.65685	0.192897
$861$	0	0
$862$	−33.9411	−1.15604
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	14.0000	0.475739
$867$	0	0
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	48.0000	1.62642
$872$	−4.24264	−0.143674
$873$	−12.7279	−0.430775
$874$	−22.6274	−0.765384
$875$	0	0
$876$	0	0
$877$	49.4975	1.67141	0.835705	−	0.549178i	$-0.185059\pi$
$877$	49.4975	1.67141	0.835705	+	0.549178i	$0.185059\pi$
$878$	16.9706	0.572729
$879$	0	0
$880$	−8.00000	−0.269680
$881$	55.1543	1.85820	0.929098	−	0.369833i	$-0.120585\pi$
$881$	55.1543	1.85820	0.929098	+	0.369833i	$0.120585\pi$
$882$	−21.0000	−0.707107
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−39.5980	−1.32957	−0.664785	−	0.747035i	$-0.731477\pi$
$887$	−39.5980	−1.32957	−0.664785	+	0.747035i	$0.731477\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	50.9117	1.70561
$892$	−16.0000	−0.535720
$893$	32.0000	1.07084
$894$	0	0
$895$	−5.65685	−0.189088
$896$	0	0
$897$	0	0
$898$	−4.24264	−0.141579
$899$	−24.0000	−0.800445
$900$	9.00000	0.300000
$901$	0	0
$902$	−8.00000	−0.266371
$903$	0	0
$904$	−1.41421	−0.0470360
$905$	22.0000	0.731305
$906$	0	0
$907$	−28.2843	−0.939164	−0.469582	−	0.882889i	$-0.655595\pi$
$907$	−28.2843	−0.939164	−0.469582	+	0.882889i	$0.655595\pi$
$908$	−28.2843	−0.938647
$909$	−36.0000	−1.19404
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	22.6274	0.748858
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	−20.0000	−0.658665
$923$	−22.6274	−0.744791
$924$	0	0
$925$	12.7279	0.418491
$926$	24.0000	0.788689
$927$	48.0000	1.57653
$928$	4.24264	0.139272
$929$	−18.3848	−0.603185	−0.301592	−	0.953437i	$-0.597518\pi$
$929$	−18.3848	−0.603185	−0.301592	+	0.953437i	$0.597518\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	−7.07107	−0.231621
$933$	0	0
$934$	−28.0000	−0.916188
$935$	0	0
$936$	−12.0000	−0.392232
$937$	−48.0000	−1.56809	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−48.0000	−1.56809	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	0	0
$940$	11.3137	0.369012
$941$	−1.41421	−0.0461020	−0.0230510	−	0.999734i	$-0.507338\pi$
$941$	−1.41421	−0.0461020	−0.0230510	+	0.999734i	$0.507338\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	4.00000	0.130189
$945$	0	0
$946$	22.6274	0.735681
$947$	−16.9706	−0.551469	−0.275735	−	0.961234i	$-0.588921\pi$
$947$	−16.9706	−0.551469	−0.275735	+	0.961234i	$0.588921\pi$
$948$	0	0
$949$	−28.2843	−0.918146
$950$	−12.0000	−0.389331
$951$	0	0
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	12.0000	0.388514
$955$	−11.3137	−0.366103
$956$	0	0
$957$	0	0
$958$	−11.3137	−0.365529
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	−16.9706	−0.547153
$963$	−33.9411	−1.09374
$964$	−29.6985	−0.956524
$965$	−22.0000	−0.708205
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	6.00000	0.192648
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	0	0
$974$	28.2843	0.906287
$975$	0	0
$976$	−12.7279	−0.407411
$977$	8.00000	0.255943	0.127971	−	0.991778i	$-0.459153\pi$
$977$	8.00000	0.255943	0.127971	+	0.991778i	$0.459153\pi$
$978$	0	0
$979$	0	0
$980$	9.89949	0.316228
$981$	−12.7279	−0.406371
$982$	20.0000	0.638226
$983$	28.2843	0.902128	0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843	0.902128	0.451064	+	0.892492i	$0.351045\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	0	0
$988$	16.0000	0.509028
$989$	22.6274	0.719510
$990$	−24.0000	−0.762770
$991$	−33.9411	−1.07818	−0.539088	−	0.842250i	$-0.681231\pi$
$991$	−33.9411	−1.07818	−0.539088	+	0.842250i	$0.681231\pi$
$992$	−5.65685	−0.179605
$993$	0	0
$994$	0	0
$995$	−24.0000	−0.760851
$996$	0	0
$997$	49.4975	1.56760	0.783800	−	0.621013i	$-0.213279\pi$
$997$	49.4975	1.56760	0.783800	+	0.621013i	$0.213279\pi$
$998$	11.3137	0.358129
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	578.2.a.b.1.1		2
3.2	odd	2		5202.2.a.bb.1.2		2
4.3	odd	2		4624.2.a.l.1.1		2
17.2	even	8		578.2.c.b.327.1		2
17.3	odd	16		578.2.d.f.179.2		8
17.4	even	4		578.2.b.c.577.2		2
17.5	odd	16		578.2.d.f.399.2		8
17.6	odd	16		578.2.d.f.155.2		8
17.7	odd	16		578.2.d.f.423.2		8
17.8	even	8		34.2.c.b.13.1	✓	2
17.9	even	8		578.2.c.b.251.1		2
17.10	odd	16		578.2.d.f.423.1		8
17.11	odd	16		578.2.d.f.155.1		8
17.12	odd	16		578.2.d.f.399.1		8
17.13	even	4		578.2.b.c.577.1		2
17.14	odd	16		578.2.d.f.179.1		8
17.15	even	8		34.2.c.b.21.1	yes	2
17.16	even	2	inner	578.2.a.b.1.2		2
51.8	odd	8		306.2.g.d.217.1		2
51.32	odd	8		306.2.g.d.55.1		2
51.50	odd	2		5202.2.a.bb.1.1		2
68.15	odd	8		272.2.o.c.225.1		2
68.59	odd	8		272.2.o.c.81.1		2
68.67	odd	2		4624.2.a.l.1.2		2
85.8	odd	8		850.2.g.d.149.1		2
85.32	odd	8		850.2.g.d.599.1		2
85.42	odd	8		850.2.g.a.149.1		2
85.49	even	8		850.2.h.c.701.1		2
85.59	even	8		850.2.h.c.251.1		2
85.83	odd	8		850.2.g.a.599.1		2
136.59	odd	8		1088.2.o.i.897.1		2
136.83	odd	8		1088.2.o.i.769.1		2
136.93	even	8		1088.2.o.k.897.1		2
136.117	even	8		1088.2.o.k.769.1		2
204.59	even	8		2448.2.be.j.1441.1		2
204.83	even	8		2448.2.be.j.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.c.b.13.1	✓	2	17.8	even	8
34.2.c.b.21.1	yes	2	17.15	even	8
272.2.o.c.81.1		2	68.59	odd	8
272.2.o.c.225.1		2	68.15	odd	8
306.2.g.d.55.1		2	51.32	odd	8
306.2.g.d.217.1		2	51.8	odd	8
578.2.a.b.1.1		2	1.1	even	1	trivial
578.2.a.b.1.2		2	17.16	even	2	inner
578.2.b.c.577.1		2	17.13	even	4
578.2.b.c.577.2		2	17.4	even	4
578.2.c.b.251.1		2	17.9	even	8
578.2.c.b.327.1		2	17.2	even	8
578.2.d.f.155.1		8	17.11	odd	16
578.2.d.f.155.2		8	17.6	odd	16
578.2.d.f.179.1		8	17.14	odd	16
578.2.d.f.179.2		8	17.3	odd	16
578.2.d.f.399.1		8	17.12	odd	16
578.2.d.f.399.2		8	17.5	odd	16
578.2.d.f.423.1		8	17.10	odd	16
578.2.d.f.423.2		8	17.7	odd	16
850.2.g.a.149.1		2	85.42	odd	8
850.2.g.a.599.1		2	85.83	odd	8
850.2.g.d.149.1		2	85.8	odd	8
850.2.g.d.599.1		2	85.32	odd	8
850.2.h.c.251.1		2	85.59	even	8
850.2.h.c.701.1		2	85.49	even	8
1088.2.o.i.769.1		2	136.83	odd	8
1088.2.o.i.897.1		2	136.59	odd	8
1088.2.o.k.769.1		2	136.117	even	8
1088.2.o.k.897.1		2	136.93	even	8
2448.2.be.j.1441.1		2	204.59	even	8
2448.2.be.j.1585.1		2	204.83	even	8
4624.2.a.l.1.1		2	4.3	odd	2
4624.2.a.l.1.2		2	68.67	odd	2
5202.2.a.bb.1.1		2	51.50	odd	2
5202.2.a.bb.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 578 = 2 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	578.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} -1.00000 q^{8} -3.00000 q^{9} +1.41421 q^{10} +5.65685 q^{11} -4.00000 q^{13} +1.00000 q^{16} +3.00000 q^{18} -4.00000 q^{19} -1.41421 q^{20} -5.65685 q^{22} -5.65685 q^{23} -3.00000 q^{25} +4.00000 q^{26} -4.24264 q^{29} +5.65685 q^{31} -1.00000 q^{32} -3.00000 q^{36} -4.24264 q^{37} +4.00000 q^{38} +1.41421 q^{40} +1.41421 q^{41} -4.00000 q^{43} +5.65685 q^{44} +4.24264 q^{45} +5.65685 q^{46} -8.00000 q^{47} -7.00000 q^{49} +3.00000 q^{50} -4.00000 q^{52} +4.00000 q^{53} -8.00000 q^{55} +4.24264 q^{58} +4.00000 q^{59} -12.7279 q^{61} -5.65685 q^{62} +1.00000 q^{64} +5.65685 q^{65} -12.0000 q^{67} +5.65685 q^{71} +3.00000 q^{72} +7.07107 q^{73} +4.24264 q^{74} -4.00000 q^{76} -11.3137 q^{79} -1.41421 q^{80} +9.00000 q^{81} -1.41421 q^{82} +4.00000 q^{83} +4.00000 q^{86} -5.65685 q^{88} -4.24264 q^{90} -5.65685 q^{92} +8.00000 q^{94} +5.65685 q^{95} +4.24264 q^{97} +7.00000 q^{98} -16.9706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9} - 8 q^{13} + 2 q^{16} + 6 q^{18} - 8 q^{19} - 6 q^{25} + 8 q^{26} - 2 q^{32} - 6 q^{36} + 8 q^{38} - 8 q^{43} - 16 q^{47} - 14 q^{49} + 6 q^{50} - 8 q^{52} + 8 q^{53} - 16 q^{55} + 8 q^{59} + 2 q^{64} - 24 q^{67} + 6 q^{72} - 8 q^{76} + 18 q^{81} + 8 q^{83} + 8 q^{86} + 16 q^{94} + 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9 - 8 * q^13 + 2 * q^16 + 6 * q^18 - 8 * q^19 - 6 * q^25 + 8 * q^26 - 2 * q^32 - 6 * q^36 + 8 * q^38 - 8 * q^43 - 16 * q^47 - 14 * q^49 + 6 * q^50 - 8 * q^52 + 8 * q^53 - 16 * q^55 + 8 * q^59 + 2 * q^64 - 24 * q^67 + 6 * q^72 - 8 * q^76 + 18 * q^81 + 8 * q^83 + 8 * q^86 + 16 * q^94 + 14 * q^98

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