LMFDB - Embedded newform 8550.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.00000 q^{8} -6.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +8.00000 q^{17} +1.00000 q^{19} -4.00000 q^{23} -6.00000 q^{26} +2.00000 q^{28} -2.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +8.00000 q^{34} +2.00000 q^{37} +1.00000 q^{38} +12.0000 q^{41} -4.00000 q^{43} -4.00000 q^{46} +12.0000 q^{47} -3.00000 q^{49} -6.00000 q^{52} +10.0000 q^{53} +2.00000 q^{56} -2.00000 q^{58} -6.00000 q^{59} -14.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +12.0000 q^{67} +8.00000 q^{68} +8.00000 q^{71} +10.0000 q^{73} +2.00000 q^{74} +1.00000 q^{76} +14.0000 q^{79} +12.0000 q^{82} +2.00000 q^{83} -4.00000 q^{86} -12.0000 q^{91} -4.00000 q^{92} +12.0000 q^{94} -2.00000 q^{97} -3.00000 q^{98} +O(q^{100})$

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
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	−6.00000	−1.17670
$27$	0	0
$28$	2.00000	0.377964
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	8.00000	1.37199
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	−6.00000	−0.832050
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	0	0
$56$	2.00000	0.267261
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	8.00000	0.970143
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	0	0
$82$	12.0000	1.32518
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	−4.00000	−0.417029
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	10.0000	0.971286
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	0	0
$112$	2.00000	0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−6.00000	−0.552345
$119$	16.0000	1.46672
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−14.0000	−1.26750
$123$	0	0
$124$	−2.00000	−0.179605
$125$	0	0
$126$	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
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	12.0000	1.03664
$135$	0	0
$136$	8.00000	0.685994
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	8.00000	0.671345
$143$	0	0
$144$	0	0
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	2.00000	0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	14.0000	1.11378
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	2.00000	0.155230
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	−12.0000	−0.889499
$183$	0	0
$184$	−4.00000	−0.294884
$185$	0	0
$186$	0	0
$187$	0	0
$188$	12.0000	0.875190
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	6.00000	0.422159
$203$	−4.00000	−0.280745
$204$	0	0
$205$	0	0
$206$	16.0000	1.11477
$207$	0	0
$208$	−6.00000	−0.416025
$209$	0	0
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	−16.0000	−1.09374
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−12.0000	−0.812743
$219$	0	0
$220$	0	0
$221$	−48.0000	−3.22883
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	−2.00000	−0.131306
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	16.0000	1.03713
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−14.0000	−0.896258
$245$	0	0
$246$	0	0
$247$	−6.00000	−0.381771
$248$	−2.00000	−0.127000
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	0	0
$266$	2.00000	0.122628
$267$	0	0
$268$	12.0000	0.733017
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	8.00000	0.485071
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−12.0000	−0.719712
$279$	0	0
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	47.0000	2.76471
$290$	0	0
$291$	0	0
$292$	10.0000	0.585206
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	−10.0000	−0.579284
$299$	24.0000	1.38796
$300$	0	0
$301$	−8.00000	−0.461112
$302$	18.0000	1.03578
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	14.0000	0.787562
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−8.00000	−0.445823
$323$	8.00000	0.445132
$324$	0	0
$325$	0	0
$326$	−8.00000	−0.443079
$327$	0	0
$328$	12.0000	0.662589
$329$	24.0000	1.32316
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	2.00000	0.109764
$333$	0	0
$334$	8.00000	0.437741
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	23.0000	1.25104
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−4.00000	−0.215666
$345$	0	0
$346$	2.00000	0.107521
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.0000	1.06449	0.532246	−	0.846590i	$-0.321348\pi$
$353$	20.0000	1.06449	0.532246	+	0.846590i	$0.321348\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−10.0000	−0.528516
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−12.0000	−0.630706
$363$	0	0
$364$	−12.0000	−0.628971
$365$	0	0
$366$	0	0
$367$	−34.0000	−1.77479	−0.887393	−	0.461014i	$-0.847486\pi$
$367$	−34.0000	−1.77479	−0.887393	+	0.461014i	$0.847486\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	20.0000	1.03835
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	12.0000	0.618031
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	0	0
$388$	−2.00000	−0.101535
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−32.0000	−1.61831
$392$	−3.00000	−0.151523
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	12.0000	0.597763
$404$	6.00000	0.298511
$405$	0	0
$406$	−4.00000	−0.198517
$407$	0	0
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	0	0
$412$	16.0000	0.788263
$413$	−12.0000	−0.590481
$414$	0	0
$415$	0	0
$416$	−6.00000	−0.294174
$417$	0	0
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	−16.0000	−0.778868
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	0	0
$427$	−28.0000	−1.35501
$428$	−16.0000	−0.773389
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−12.0000	−0.574696
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	0	0
$442$	−48.0000	−2.28313
$443$	−34.0000	−1.61539	−0.807694	−	0.589601i	$-0.799285\pi$
$443$	−34.0000	−1.61539	−0.807694	+	0.589601i	$0.799285\pi$
$444$	0	0
$445$	0	0
$446$	4.00000	0.189405
$447$	0	0
$448$	2.00000	0.0944911
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	0	0
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	24.0000	1.11178
$467$	−22.0000	−1.01804	−0.509019	−	0.860755i	$-0.669992\pi$
$467$	−22.0000	−1.01804	−0.509019	+	0.860755i	$0.669992\pi$
$468$	0	0
$469$	24.0000	1.10822
$470$	0	0
$471$	0	0
$472$	−6.00000	−0.276172
$473$	0	0
$474$	0	0
$475$	0	0
$476$	16.0000	0.733359
$477$	0	0
$478$	8.00000	0.365911
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	−14.0000	−0.637683
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	0	0
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	−14.0000	−0.633750
$489$	0	0
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	16.0000	0.717698
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	12.0000	0.535586
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.0000	−1.32324
$515$	0	0
$516$	0	0
$517$	0	0
$518$	4.00000	0.175750
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	−12.0000	−0.523225
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	2.00000	0.0867110
$533$	−72.0000	−3.11867
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	30.0000	1.29339
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−4.00000	−0.171815
$543$	0	0
$544$	8.00000	0.342997
$545$	0	0
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	28.0000	1.19068
$554$	−4.00000	−0.169944
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	20.0000	0.843649
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	8.00000	0.335673
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	0	0
$574$	24.0000	1.00174
$575$	0	0
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	47.0000	1.95494
$579$	0	0
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	0	0
$584$	10.0000	0.413803
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−26.0000	−1.07313	−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000	−1.07313	−0.536567	+	0.843857i	$0.680279\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0	0
$592$	2.00000	0.0821995
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	24.0000	0.981433
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	18.0000	0.732410
$605$	0	0
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−72.0000	−2.91281
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	0	0
$617$	20.0000	0.805170	0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000	0.805170	0.402585	+	0.915383i	$0.368112\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	−24.0000	−0.962312
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	4.00000	0.159617
$629$	16.0000	0.637962
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	14.0000	0.556890
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.0000	0.789953	0.394976	−	0.918691i	$-0.370753\pi$
$641$	20.0000	0.789953	0.394976	+	0.918691i	$0.370753\pi$
$642$	0	0
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	8.00000	0.314756
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	0	0
$656$	12.0000	0.468521
$657$	0	0
$658$	24.0000	0.935617
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	2.00000	0.0776151
$665$	0	0
$666$	0	0
$667$	8.00000	0.309761
$668$	8.00000	0.309529
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	23.0000	0.884615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	0	0
$686$	−20.0000	−0.763604
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−60.0000	−2.28582
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−10.0000	−0.379595
$695$	0	0
$696$	0	0
$697$	96.0000	3.63626
$698$	14.0000	0.529908
$699$	0	0
$700$	0	0
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	0	0
$706$	20.0000	0.752710
$707$	12.0000	0.451306
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	0	0
$716$	−10.0000	−0.373718
$717$	0	0
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	1.00000	0.0372161
$723$	0	0
$724$	−12.0000	−0.445976
$725$	0	0
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	−12.0000	−0.444750
$729$	0	0
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	−8.00000	−0.295487	−0.147743	−	0.989026i	$-0.547201\pi$
$733$	−8.00000	−0.295487	−0.147743	+	0.989026i	$0.547201\pi$
$734$	−34.0000	−1.25496
$735$	0	0
$736$	−4.00000	−0.147442
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	20.0000	0.734223
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	0	0
$746$	14.0000	0.512576
$747$	0	0
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	−40.0000	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.0000	1.29988
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	0	0
$772$	2.00000	0.0719816
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	0	0
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−18.0000	−0.645331
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	−32.0000	−1.14432
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	84.0000	2.98293
$794$	−8.00000	−0.283909
$795$	0	0
$796$	4.00000	0.141776
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	96.0000	3.39624
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	12.0000	0.422682
$807$	0	0
$808$	6.00000	0.211079
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	−4.00000	−0.140372
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.00000	−0.139942
$818$	−6.00000	−0.209785
$819$	0	0
$820$	0	0
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−12.0000	−0.417533
$827$	32.0000	1.11275	0.556375	−	0.830932i	$-0.312192\pi$
$827$	32.0000	1.11275	0.556375	+	0.830932i	$0.312192\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	0	0
$832$	−6.00000	−0.208013
$833$	−24.0000	−0.831551
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	28.0000	0.967244
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	4.00000	0.137849
$843$	0	0
$844$	−16.0000	−0.550743
$845$	0	0
$846$	0	0
$847$	−22.0000	−0.755929
$848$	10.0000	0.343401
$849$	0	0
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	−28.0000	−0.958140
$855$	0	0
$856$	−16.0000	−0.546869
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	−24.0000	−0.817443
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	0	0
$865$	0	0
$866$	34.0000	1.15537
$867$	0	0
$868$	−4.00000	−0.135769
$869$	0	0
$870$	0	0
$871$	−72.0000	−2.43963
$872$	−12.0000	−0.406371
$873$	0	0
$874$	−4.00000	−0.135302
$875$	0	0
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−10.0000	−0.337484
$879$	0	0
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	−48.0000	−1.61441
$885$	0	0
$886$	−34.0000	−1.14225
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	4.00000	0.133930
$893$	12.0000	0.401565
$894$	0	0
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	−36.0000	−1.20134
$899$	4.00000	0.133407
$900$	0	0
$901$	80.0000	2.66519
$902$	0	0
$903$	0	0
$904$	6.00000	0.199557
$905$	0	0
$906$	0	0
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	20.0000	0.663723
$909$	0	0
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	0	0
$914$	−10.0000	−0.330771
$915$	0	0
$916$	6.00000	0.198246
$917$	0	0
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	18.0000	0.592798
$923$	−48.0000	−1.57994
$924$	0	0
$925$	0	0
$926$	−22.0000	−0.722965
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	24.0000	0.786146
$933$	0	0
$934$	−22.0000	−0.719862
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	24.0000	0.783628
$939$	0	0
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	2.00000	0.0649913	0.0324956	−	0.999472i	$-0.489654\pi$
$947$	2.00000	0.0649913	0.0324956	+	0.999472i	$0.489654\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	0	0
$952$	16.0000	0.518563
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−12.0000	−0.386896
$963$	0	0
$964$	−14.0000	−0.450910
$965$	0	0
$966$	0	0
$967$	34.0000	1.09337	0.546683	−	0.837340i	$-0.315890\pi$
$967$	34.0000	1.09337	0.546683	+	0.837340i	$0.315890\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	−40.0000	−1.28168
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	16.0000	0.510581
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	−16.0000	−0.509544
$987$	0	0
$988$	−6.00000	−0.190885
$989$	16.0000	0.508770
$990$	0	0
$991$	50.0000	1.58830	0.794151	−	0.607720i	$-0.207916\pi$
$991$	50.0000	1.58830	0.794151	+	0.607720i	$0.207916\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	16.0000	0.507489
$995$	0	0
$996$	0	0
$997$	−44.0000	−1.39349	−0.696747	−	0.717317i	$-0.745370\pi$
$997$	−44.0000	−1.39349	−0.696747	+	0.717317i	$0.745370\pi$
$998$	28.0000	0.886325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.bh.1.1		1
3.2	odd	2		2850.2.a.n.1.1		1
5.4	even	2		1710.2.a.c.1.1		1
15.2	even	4		2850.2.d.e.799.1		2
15.8	even	4		2850.2.d.e.799.2		2
15.14	odd	2		570.2.a.h.1.1	✓	1
60.59	even	2		4560.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.h.1.1	✓	1	15.14	odd	2
1710.2.a.c.1.1		1	5.4	even	2
2850.2.a.n.1.1		1	3.2	odd	2
2850.2.d.e.799.1		2	15.2	even	4
2850.2.d.e.799.2		2	15.8	even	4
4560.2.a.bc.1.1		1	60.59	even	2
8550.2.a.bh.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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