LMFDB - Embedded newform 1800.4.f.f.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(649,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1800.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$106.203438010$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1800.649
Dual form			1800.4.f.f.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{7} +O(q^{10})$	$q-2.00000i q^{7} -34.0000 q^{11} -68.0000i q^{13} +38.0000i q^{17} -4.00000 q^{19} +152.000i q^{23} +46.0000 q^{29} -260.000 q^{31} +312.000i q^{37} +48.0000 q^{41} -200.000i q^{43} -104.000i q^{47} +339.000 q^{49} -414.000i q^{53} +2.00000 q^{59} -38.0000 q^{61} +244.000i q^{67} +708.000 q^{71} -378.000i q^{73} +68.0000i q^{77} +852.000 q^{79} +844.000i q^{83} +1380.00 q^{89} -136.000 q^{91} -514.000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 68 q^{11} - 8 q^{19} + 92 q^{29} - 520 q^{31} + 96 q^{41} + 678 q^{49} + 4 q^{59} - 76 q^{61} + 1416 q^{71} + 1704 q^{79} + 2760 q^{89} - 272 q^{91}+O(q^{100}) $ 2 * q - 68 * q^11 - 8 * q^19 + 92 * q^29 - 520 * q^31 + 96 * q^41 + 678 * q^49 + 4 * q^59 - 76 * q^61 + 1416 * q^71 + 1704 * q^79 + 2760 * q^89 - 272 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.107990i	−0.998541	−	0.0539949i	$-0.982805\pi$
$7$	− 2.00000i	− 0.107990i	0.998541	−	0.0539949i	$-0.0171955\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−34.0000	−0.931944	−0.465972	−	0.884799i	$-0.654295\pi$
$11$	−34.0000	−0.931944	−0.465972	+	0.884799i	$0.654295\pi$
$12$	0	0
$13$	− 68.0000i	− 1.45075i	−0.688352	−	0.725377i	$-0.741665\pi$
$13$	− 68.0000i	− 1.45075i	0.688352	−	0.725377i	$-0.258335\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	38.0000i	0.542138i	0.962560	+	0.271069i	$0.0873772\pi$
$17$	38.0000i	0.542138i	−0.962560	+	0.271069i	$0.912623\pi$
$18$	0	0
$19$	−4.00000	−0.0482980	−0.0241490	−	0.999708i	$-0.507688\pi$
$19$	−4.00000	−0.0482980	−0.0241490	+	0.999708i	$0.507688\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	152.000i	1.37801i	0.724757	+	0.689004i	$0.241952\pi$
$23$	152.000i	1.37801i	−0.724757	+	0.689004i	$0.758048\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	46.0000	0.294551	0.147276	−	0.989095i	$-0.452950\pi$
$29$	46.0000	0.294551	0.147276	+	0.989095i	$0.452950\pi$
$30$	0	0
$31$	−260.000	−1.50637	−0.753184	−	0.657810i	$-0.771483\pi$
$31$	−260.000	−1.50637	−0.753184	+	0.657810i	$0.771483\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	312.000i	1.38628i	0.720801	+	0.693142i	$0.243774\pi$
$37$	312.000i	1.38628i	−0.720801	+	0.693142i	$0.756226\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	48.0000	0.182838	0.0914188	−	0.995813i	$-0.470860\pi$
$41$	48.0000	0.182838	0.0914188	+	0.995813i	$0.470860\pi$
$42$	0	0
$43$	− 200.000i	− 0.709296i	−0.935000	−	0.354648i	$-0.884601\pi$
$43$	− 200.000i	− 0.709296i	0.935000	−	0.354648i	$-0.115399\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 104.000i	− 0.322765i	−0.986892	−	0.161383i	$-0.948405\pi$
$47$	− 104.000i	− 0.322765i	0.986892	−	0.161383i	$-0.0515953\pi$
$48$	0	0
$49$	339.000	0.988338
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 414.000i	− 1.07297i	−0.843911	−	0.536484i	$-0.819752\pi$
$53$	− 414.000i	− 1.07297i	0.843911	−	0.536484i	$-0.180248\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.00000	0.00441318	0.00220659	−	0.999998i	$-0.499298\pi$
$59$	2.00000	0.00441318	0.00220659	+	0.999998i	$0.499298\pi$
$60$	0	0
$61$	−38.0000	−0.0797607	−0.0398803	−	0.999204i	$-0.512698\pi$
$61$	−38.0000	−0.0797607	−0.0398803	+	0.999204i	$0.512698\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	244.000i	0.444916i	0.974942	+	0.222458i	$0.0714080\pi$
$67$	244.000i	0.444916i	−0.974942	+	0.222458i	$0.928592\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	708.000	1.18344	0.591719	−	0.806144i	$-0.298449\pi$
$71$	708.000	1.18344	0.591719	+	0.806144i	$0.298449\pi$
$72$	0	0
$73$	− 378.000i	− 0.606049i	−0.952983	−	0.303024i	$-0.902004\pi$
$73$	− 378.000i	− 0.606049i	0.952983	−	0.303024i	$-0.0979963\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	68.0000i	0.100641i
$78$	0	0
$79$	852.000	1.21339	0.606693	−	0.794936i	$-0.292496\pi$
$79$	852.000	1.21339	0.606693	+	0.794936i	$0.292496\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	844.000i	1.11616i	0.829788	+	0.558079i	$0.188461\pi$
$83$	844.000i	1.11616i	−0.829788	+	0.558079i	$0.811539\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1380.00	1.64359	0.821796	−	0.569782i	$-0.192972\pi$
$89$	1380.00	1.64359	0.821796	+	0.569782i	$0.192972\pi$
$90$	0	0
$91$	−136.000	−0.156667
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 514.000i	− 0.538029i	−0.963136	−	0.269014i	$-0.913302\pi$
$97$	− 514.000i	− 0.538029i	0.963136	−	0.269014i	$-0.0866979\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−702.000	−0.691600	−0.345800	−	0.938308i	$-0.612392\pi$
$101$	−702.000	−0.691600	−0.345800	+	0.938308i	$0.612392\pi$
$102$	0	0
$103$	898.000i	0.859054i	0.903054	+	0.429527i	$0.141320\pi$
$103$	898.000i	0.859054i	−0.903054	+	0.429527i	$0.858680\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 876.000i	− 0.791459i	−0.918367	−	0.395730i	$-0.870492\pi$
$107$	− 876.000i	− 0.791459i	0.918367	−	0.395730i	$-0.129508\pi$
$108$	0	0
$109$	−602.000	−0.529001	−0.264501	−	0.964386i	$-0.585207\pi$
$109$	−602.000	−0.529001	−0.264501	+	0.964386i	$0.585207\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1350.00i	1.12387i	0.827181	+	0.561935i	$0.189943\pi$
$113$	1350.00i	1.12387i	−0.827181	+	0.561935i	$0.810057\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	76.0000	0.0585455
$120$	0	0
$121$	−175.000	−0.131480
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	366.000i	0.255726i	0.991792	+	0.127863i	$0.0408118\pi$
$127$	366.000i	0.255726i	−0.991792	+	0.127863i	$0.959188\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	498.000	0.332141	0.166070	−	0.986114i	$-0.446892\pi$
$131$	498.000	0.332141	0.166070	+	0.986114i	$0.446892\pi$
$132$	0	0
$133$	8.00000i	0.00521570i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2026.00i	1.26345i	0.775192	+	0.631726i	$0.217653\pi$
$137$	2026.00i	1.26345i	−0.775192	+	0.631726i	$0.782347\pi$
$138$	0	0
$139$	−2460.00	−1.50111	−0.750556	−	0.660807i	$-0.770214\pi$
$139$	−2460.00	−1.50111	−0.750556	+	0.660807i	$0.770214\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2312.00i	1.35202i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3362.00	1.84850	0.924248	−	0.381794i	$-0.124694\pi$
$149$	3362.00	1.84850	0.924248	+	0.381794i	$0.124694\pi$
$150$	0	0
$151$	2096.00	1.12960	0.564802	−	0.825227i	$-0.308953\pi$
$151$	2096.00	1.12960	0.564802	+	0.825227i	$0.308953\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2092.00i	− 1.06344i	−0.846921	−	0.531719i	$-0.821546\pi$
$157$	− 2092.00i	− 1.06344i	0.846921	−	0.531719i	$-0.178454\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	304.000	0.148811
$162$	0	0
$163$	244.000i	0.117249i	0.998280	+	0.0586244i	$0.0186714\pi$
$163$	244.000i	0.117249i	−0.998280	+	0.0586244i	$0.981329\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2064.00i	− 0.956390i	−0.878254	−	0.478195i	$-0.841291\pi$
$167$	− 2064.00i	− 0.956390i	0.878254	−	0.478195i	$-0.158709\pi$
$168$	0	0
$169$	−2427.00	−1.10469
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1258.00i	0.552855i	0.961035	+	0.276428i	$0.0891506\pi$
$173$	1258.00i	0.552855i	−0.961035	+	0.276428i	$0.910849\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3986.00	1.66440	0.832200	−	0.554475i	$-0.187081\pi$
$179$	3986.00	1.66440	0.832200	+	0.554475i	$0.187081\pi$
$180$	0	0
$181$	2570.00	1.05540	0.527698	−	0.849432i	$-0.323055\pi$
$181$	2570.00	1.05540	0.527698	+	0.849432i	$0.323055\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1292.00i	− 0.505243i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4684.00	1.77446	0.887231	−	0.461325i	$-0.152626\pi$
$191$	4684.00	1.77446	0.887231	+	0.461325i	$0.152626\pi$
$192$	0	0
$193$	214.000i	0.0798138i	0.999203	+	0.0399069i	$0.0127061\pi$
$193$	214.000i	0.0798138i	−0.999203	+	0.0399069i	$0.987294\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3014.00i	− 1.09004i	−0.838422	−	0.545022i	$-0.816521\pi$
$197$	− 3014.00i	− 1.09004i	0.838422	−	0.545022i	$-0.183479\pi$
$198$	0	0
$199$	1792.00	0.638349	0.319175	−	0.947696i	$-0.396594\pi$
$199$	1792.00	0.638349	0.319175	+	0.947696i	$0.396594\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 92.0000i	− 0.0318085i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	136.000	0.0450111
$210$	0	0
$211$	−4540.00	−1.48126	−0.740631	−	0.671911i	$-0.765474\pi$
$211$	−4540.00	−1.48126	−0.740631	+	0.671911i	$0.765474\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	520.000i	0.162672i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2584.00	0.786510
$222$	0	0
$223$	6506.00i	1.95369i	0.213937	+	0.976847i	$0.431371\pi$
$223$	6506.00i	1.95369i	−0.213937	+	0.976847i	$0.568629\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3696.00i	1.08067i	0.841450	+	0.540335i	$0.181703\pi$
$227$	3696.00i	1.08067i	−0.841450	+	0.540335i	$0.818297\pi$
$228$	0	0
$229$	3386.00	0.977088	0.488544	−	0.872539i	$-0.337528\pi$
$229$	3386.00	0.977088	0.488544	+	0.872539i	$0.337528\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3306.00i	0.929542i	0.885431	+	0.464771i	$0.153863\pi$
$233$	3306.00i	0.929542i	−0.885431	+	0.464771i	$0.846137\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4188.00	1.13347	0.566735	−	0.823900i	$-0.308206\pi$
$239$	4188.00	1.13347	0.566735	+	0.823900i	$0.308206\pi$
$240$	0	0
$241$	5462.00	1.45991	0.729955	−	0.683495i	$-0.239541\pi$
$241$	5462.00	1.45991	0.729955	+	0.683495i	$0.239541\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	272.000i	0.0700686i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3366.00	−0.846454	−0.423227	−	0.906024i	$-0.639103\pi$
$251$	−3366.00	−0.846454	−0.423227	+	0.906024i	$0.639103\pi$
$252$	0	0
$253$	− 5168.00i	− 1.28423i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1158.00i	− 0.281066i	−0.990076	−	0.140533i	$-0.955118\pi$
$257$	− 1158.00i	− 0.281066i	0.990076	−	0.140533i	$-0.0448817\pi$
$258$	0	0
$259$	624.000	0.149705
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8304.00i	1.94695i	0.228804	+	0.973473i	$0.426519\pi$
$263$	8304.00i	1.94695i	−0.228804	+	0.973473i	$0.573481\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7478.00	1.69495	0.847475	−	0.530835i	$-0.178122\pi$
$269$	7478.00	1.69495	0.847475	+	0.530835i	$0.178122\pi$
$270$	0	0
$271$	−6792.00	−1.52245	−0.761226	−	0.648486i	$-0.775402\pi$
$271$	−6792.00	−1.52245	−0.761226	+	0.648486i	$0.775402\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2296.00i	0.498026i	0.968500	+	0.249013i	$0.0801062\pi$
$277$	2296.00i	0.498026i	−0.968500	+	0.249013i	$0.919894\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3980.00	0.844936	0.422468	−	0.906378i	$-0.361164\pi$
$281$	3980.00	0.844936	0.422468	+	0.906378i	$0.361164\pi$
$282$	0	0
$283$	− 1972.00i	− 0.414216i	−0.978318	−	0.207108i	$-0.933595\pi$
$283$	− 1972.00i	− 0.414216i	0.978318	−	0.207108i	$-0.0664052\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 96.0000i	− 0.0197446i
$288$	0	0
$289$	3469.00	0.706086
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9254.00i	1.84513i	0.385836	+	0.922567i	$0.373913\pi$
$293$	9254.00i	1.84513i	−0.385836	+	0.922567i	$0.626087\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10336.0	1.99915
$300$	0	0
$301$	−400.000	−0.0765967
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5888.00i	1.09461i	0.836933	+	0.547306i	$0.184347\pi$
$307$	5888.00i	1.09461i	−0.836933	+	0.547306i	$0.815653\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4604.00	−0.839450	−0.419725	−	0.907651i	$-0.637873\pi$
$311$	−4604.00	−0.839450	−0.419725	+	0.907651i	$0.637873\pi$
$312$	0	0
$313$	− 8026.00i	− 1.44938i	−0.689074	−	0.724691i	$-0.741983\pi$
$313$	− 8026.00i	− 1.44938i	0.689074	−	0.724691i	$-0.258017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2838.00i	− 0.502833i	−0.967879	−	0.251416i	$-0.919104\pi$
$317$	− 2838.00i	− 0.502833i	0.967879	−	0.251416i	$-0.0808963\pi$
$318$	0	0
$319$	−1564.00	−0.274505
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 152.000i	− 0.0261842i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−208.000	−0.0348554
$330$	0	0
$331$	−1020.00	−0.169378	−0.0846892	−	0.996407i	$-0.526990\pi$
$331$	−1020.00	−0.169378	−0.0846892	+	0.996407i	$0.526990\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 814.000i	− 0.131577i	−0.997834	−	0.0657884i	$-0.979044\pi$
$337$	− 814.000i	− 0.131577i	0.997834	−	0.0657884i	$-0.0209562\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8840.00	1.40385
$342$	0	0
$343$	− 1364.00i	− 0.214720i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4544.00i	0.702982i	0.936191	+	0.351491i	$0.114325\pi$
$347$	4544.00i	0.702982i	−0.936191	+	0.351491i	$0.885675\pi$
$348$	0	0
$349$	−6978.00	−1.07027	−0.535134	−	0.844767i	$-0.679739\pi$
$349$	−6978.00	−1.07027	−0.535134	+	0.844767i	$0.679739\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2818.00i	0.424892i	0.977173	+	0.212446i	$0.0681430\pi$
$353$	2818.00i	0.424892i	−0.977173	+	0.212446i	$0.931857\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−744.000	−0.109378	−0.0546892	−	0.998503i	$-0.517417\pi$
$359$	−744.000	−0.109378	−0.0546892	+	0.998503i	$0.517417\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6454.00i	0.917973i	0.888443	+	0.458986i	$0.151787\pi$
$367$	6454.00i	0.917973i	−0.888443	+	0.458986i	$0.848213\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−828.000	−0.115870
$372$	0	0
$373$	− 5900.00i	− 0.819009i	−0.912308	−	0.409505i	$-0.865702\pi$
$373$	− 5900.00i	− 0.819009i	0.912308	−	0.409505i	$-0.134298\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3128.00i	− 0.427321i
$378$	0	0
$379$	11876.0	1.60958	0.804788	−	0.593563i	$-0.202279\pi$
$379$	11876.0	1.60958	0.804788	+	0.593563i	$0.202279\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 552.000i	− 0.0736446i	−0.999322	−	0.0368223i	$-0.988276\pi$
$383$	− 552.000i	− 0.0736446i	0.999322	−	0.0368223i	$-0.0117236\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1722.00	0.224444	0.112222	−	0.993683i	$-0.464203\pi$
$389$	1722.00	0.224444	0.112222	+	0.993683i	$0.464203\pi$
$390$	0	0
$391$	−5776.00	−0.747071
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4576.00i	− 0.578496i	−0.957254	−	0.289248i	$-0.906595\pi$
$397$	− 4576.00i	− 0.578496i	0.957254	−	0.289248i	$-0.0934052\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2892.00	0.360149	0.180074	−	0.983653i	$-0.442366\pi$
$401$	2892.00	0.360149	0.180074	+	0.983653i	$0.442366\pi$
$402$	0	0
$403$	17680.0i	2.18537i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10608.0i	− 1.29194i
$408$	0	0
$409$	230.000	0.0278063	0.0139031	−	0.999903i	$-0.495574\pi$
$409$	230.000	0.0278063	0.0139031	+	0.999903i	$0.495574\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 4.00000i	0 0.000476579i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15438.0	1.79999	0.899995	−	0.435901i	$-0.143570\pi$
$419$	15438.0	1.79999	0.899995	+	0.435901i	$0.143570\pi$
$420$	0	0
$421$	12294.0	1.42321	0.711607	−	0.702578i	$-0.247968\pi$
$421$	12294.0	1.42321	0.711607	+	0.702578i	$0.247968\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	76.0000i	0.00861334i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17488.0	1.95445	0.977224	−	0.212209i	$-0.0680658\pi$
$431$	17488.0	1.95445	0.977224	+	0.212209i	$0.0680658\pi$
$432$	0	0
$433$	− 8698.00i	− 0.965356i	−0.875798	−	0.482678i	$-0.839664\pi$
$433$	− 8698.00i	− 0.965356i	0.875798	−	0.482678i	$-0.160336\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 608.000i	− 0.0665551i
$438$	0	0
$439$	−8536.00	−0.928021	−0.464010	−	0.885830i	$-0.653590\pi$
$439$	−8536.00	−0.928021	−0.464010	+	0.885830i	$0.653590\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8712.00i	− 0.934356i	−0.884163	−	0.467178i	$-0.845271\pi$
$443$	− 8712.00i	− 0.934356i	0.884163	−	0.467178i	$-0.154729\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5484.00	0.576405	0.288203	−	0.957569i	$-0.406942\pi$
$449$	5484.00	0.576405	0.288203	+	0.957569i	$0.406942\pi$
$450$	0	0
$451$	−1632.00	−0.170394
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 19402.0i	− 1.98597i	−0.118250	−	0.992984i	$-0.537728\pi$
$457$	− 19402.0i	− 1.98597i	0.118250	−	0.992984i	$-0.462272\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13578.0	−1.37178	−0.685890	−	0.727705i	$-0.740587\pi$
$461$	−13578.0	−1.37178	−0.685890	+	0.727705i	$0.740587\pi$
$462$	0	0
$463$	6222.00i	0.624537i	0.949994	+	0.312269i	$0.101089\pi$
$463$	6222.00i	0.624537i	−0.949994	+	0.312269i	$0.898911\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15260.0i	1.51210i	0.654516	+	0.756048i	$0.272872\pi$
$467$	15260.0i	1.51210i	−0.654516	+	0.756048i	$0.727128\pi$
$468$	0	0
$469$	488.000	0.0480464
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6800.00i	0.661024i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9812.00	−0.935953	−0.467977	−	0.883741i	$-0.655017\pi$
$479$	−9812.00	−0.935953	−0.467977	+	0.883741i	$0.655017\pi$
$480$	0	0
$481$	21216.0	2.01116
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7226.00i	0.672364i	0.941797	+	0.336182i	$0.109136\pi$
$487$	7226.00i	0.672364i	−0.941797	+	0.336182i	$0.890864\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6750.00	−0.620414	−0.310207	−	0.950669i	$-0.600398\pi$
$491$	−6750.00	−0.620414	−0.310207	+	0.950669i	$0.600398\pi$
$492$	0	0
$493$	1748.00i	0.159688i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1416.00i	− 0.127799i
$498$	0	0
$499$	4156.00	0.372842	0.186421	−	0.982470i	$-0.440311\pi$
$499$	4156.00	0.372842	0.186421	+	0.982470i	$0.440311\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14088.0i	1.24881i	0.781100	+	0.624406i	$0.214659\pi$
$503$	14088.0i	1.24881i	−0.781100	+	0.624406i	$0.785341\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16970.0	−1.47776	−0.738882	−	0.673835i	$-0.764646\pi$
$509$	−16970.0	−1.47776	−0.738882	+	0.673835i	$0.764646\pi$
$510$	0	0
$511$	−756.000	−0.0654471
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3536.00i	0.300799i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8500.00	−0.714763	−0.357382	−	0.933958i	$-0.616330\pi$
$521$	−8500.00	−0.714763	−0.357382	+	0.933958i	$0.616330\pi$
$522$	0	0
$523$	20620.0i	1.72400i	0.506912	+	0.861998i	$0.330787\pi$
$523$	20620.0i	1.72400i	−0.506912	+	0.861998i	$0.669213\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9880.00i	− 0.816660i
$528$	0	0
$529$	−10937.0	−0.898907
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3264.00i	− 0.265252i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−11526.0	−0.921076
$540$	0	0
$541$	5314.00	0.422304	0.211152	−	0.977453i	$-0.432278\pi$
$541$	5314.00	0.422304	0.211152	+	0.977453i	$0.432278\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 24104.0i	− 1.88412i	−0.335447	−	0.942059i	$-0.608887\pi$
$547$	− 24104.0i	− 1.88412i	0.335447	−	0.942059i	$-0.391113\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−184.000	−0.0142262
$552$	0	0
$553$	− 1704.00i	− 0.131033i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23582.0i	1.79390i	0.442134	+	0.896949i	$0.354222\pi$
$557$	23582.0i	1.79390i	−0.442134	+	0.896949i	$0.645778\pi$
$558$	0	0
$559$	−13600.0	−1.02901
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2680.00i	− 0.200619i	−0.994956	−	0.100310i	$-0.968017\pi$
$563$	− 2680.00i	− 0.200619i	0.994956	−	0.100310i	$-0.0319833\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25004.0	1.84222	0.921109	−	0.389304i	$-0.127285\pi$
$569$	25004.0	1.84222	0.921109	+	0.389304i	$0.127285\pi$
$570$	0	0
$571$	−11180.0	−0.819384	−0.409692	−	0.912224i	$-0.634364\pi$
$571$	−11180.0	−0.819384	−0.409692	+	0.912224i	$0.634364\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15862.0i	1.14444i	0.820099	+	0.572222i	$0.193918\pi$
$577$	15862.0i	1.14444i	−0.820099	+	0.572222i	$0.806082\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1688.00	0.120534
$582$	0	0
$583$	14076.0i	0.999946i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15036.0i	− 1.05724i	−0.848857	−	0.528622i	$-0.822709\pi$
$587$	− 15036.0i	− 1.05724i	0.848857	−	0.528622i	$-0.177291\pi$
$588$	0	0
$589$	1040.00	0.0727546
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 12786.0i	− 0.885427i	−0.896663	−	0.442713i	$-0.854016\pi$
$593$	− 12786.0i	− 0.885427i	0.896663	−	0.442713i	$-0.145984\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13464.0	−0.918404	−0.459202	−	0.888332i	$-0.651865\pi$
$599$	−13464.0	−0.918404	−0.459202	+	0.888332i	$0.651865\pi$
$600$	0	0
$601$	8518.00	0.578131	0.289065	−	0.957309i	$-0.406656\pi$
$601$	8518.00	0.578131	0.289065	+	0.957309i	$0.406656\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11082.0i	0.741029i	0.928827	+	0.370514i	$0.120819\pi$
$607$	11082.0i	0.741029i	−0.928827	+	0.370514i	$0.879181\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7072.00	−0.468253
$612$	0	0
$613$	− 26568.0i	− 1.75052i	−0.483649	−	0.875262i	$-0.660689\pi$
$613$	− 26568.0i	− 1.75052i	0.483649	−	0.875262i	$-0.339311\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3282.00i	− 0.214146i	−0.994251	−	0.107073i	$-0.965852\pi$
$617$	− 3282.00i	− 0.214146i	0.994251	−	0.107073i	$-0.0341479\pi$
$618$	0	0
$619$	2308.00	0.149865	0.0749324	−	0.997189i	$-0.476126\pi$
$619$	2308.00	0.149865	0.0749324	+	0.997189i	$0.476126\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 2760.00i	− 0.177491i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−11856.0	−0.751558
$630$	0	0
$631$	−24572.0	−1.55023	−0.775116	−	0.631819i	$-0.782308\pi$
$631$	−24572.0	−1.55023	−0.775116	+	0.631819i	$0.782308\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 23052.0i	− 1.43384i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2136.00	0.131618	0.0658088	−	0.997832i	$-0.479037\pi$
$641$	2136.00	0.131618	0.0658088	+	0.997832i	$0.479037\pi$
$642$	0	0
$643$	− 5508.00i	− 0.337814i	−0.985632	−	0.168907i	$-0.945976\pi$
$643$	− 5508.00i	− 0.337814i	0.985632	−	0.168907i	$-0.0540237\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4536.00i	0.275624i	0.990458	+	0.137812i	$0.0440069\pi$
$647$	4536.00i	0.275624i	−0.990458	+	0.137812i	$0.955993\pi$
$648$	0	0
$649$	−68.0000	−0.00411284
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 27914.0i	− 1.67283i	−0.548095	−	0.836416i	$-0.684647\pi$
$653$	− 27914.0i	− 1.67283i	0.548095	−	0.836416i	$-0.315353\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22842.0	−1.35022	−0.675112	−	0.737715i	$-0.735905\pi$
$659$	−22842.0	−1.35022	−0.675112	+	0.737715i	$0.735905\pi$
$660$	0	0
$661$	16458.0	0.968445	0.484222	−	0.874945i	$-0.339103\pi$
$661$	16458.0	0.968445	0.484222	+	0.874945i	$0.339103\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6992.00i	0.405894i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1292.00	0.0743325
$672$	0	0
$673$	− 16050.0i	− 0.919290i	−0.888103	−	0.459645i	$-0.847977\pi$
$673$	− 16050.0i	− 0.919290i	0.888103	−	0.459645i	$-0.152023\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5314.00i	0.301674i	0.988559	+	0.150837i	$0.0481969\pi$
$677$	5314.00i	0.301674i	−0.988559	+	0.150837i	$0.951803\pi$
$678$	0	0
$679$	−1028.00	−0.0581016
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 15876.0i	− 0.889426i	−0.895673	−	0.444713i	$-0.853306\pi$
$683$	− 15876.0i	− 0.889426i	0.895673	−	0.444713i	$-0.146694\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−28152.0	−1.55661
$690$	0	0
$691$	−13372.0	−0.736172	−0.368086	−	0.929792i	$-0.619987\pi$
$691$	−13372.0	−0.736172	−0.368086	+	0.929792i	$0.619987\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1824.00i	0.0991233i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3230.00	−0.174031	−0.0870153	−	0.996207i	$-0.527733\pi$
$701$	−3230.00	−0.174031	−0.0870153	+	0.996207i	$0.527733\pi$
$702$	0	0
$703$	− 1248.00i	− 0.0669548i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1404.00i	0.0746858i
$708$	0	0
$709$	6154.00	0.325978	0.162989	−	0.986628i	$-0.447887\pi$
$709$	6154.00	0.325978	0.162989	+	0.986628i	$0.447887\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 39520.0i	− 2.07579i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20264.0	−1.05107	−0.525535	−	0.850772i	$-0.676135\pi$
$719$	−20264.0	−1.05107	−0.525535	+	0.850772i	$0.676135\pi$
$720$	0	0
$721$	1796.00	0.0927691
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	25354.0i	1.29344i	0.762729	+	0.646718i	$0.223859\pi$
$727$	25354.0i	1.29344i	−0.762729	+	0.646718i	$0.776141\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7600.00	0.384536
$732$	0	0
$733$	− 13344.0i	− 0.672404i	−0.941790	−	0.336202i	$-0.890858\pi$
$733$	− 13344.0i	− 0.672404i	0.941790	−	0.336202i	$-0.109142\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8296.00i	− 0.414636i
$738$	0	0
$739$	28452.0	1.41627	0.708135	−	0.706077i	$-0.249537\pi$
$739$	28452.0	1.41627	0.708135	+	0.706077i	$0.249537\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 5784.00i	− 0.285591i	−0.989752	−	0.142796i	$-0.954391\pi$
$743$	− 5784.00i	− 0.285591i	0.989752	−	0.142796i	$-0.0456092\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1752.00	−0.0854695
$750$	0	0
$751$	852.000	0.0413980	0.0206990	−	0.999786i	$-0.493411\pi$
$751$	852.000	0.0413980	0.0206990	+	0.999786i	$0.493411\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5704.00i	0.273864i	0.990580	+	0.136932i	$0.0437243\pi$
$757$	5704.00i	0.273864i	−0.990580	+	0.136932i	$0.956276\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−24828.0	−1.18267	−0.591337	−	0.806425i	$-0.701400\pi$
$761$	−24828.0	−1.18267	−0.591337	+	0.806425i	$0.701400\pi$
$762$	0	0
$763$	1204.00i	0.0571268i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 136.000i	− 0.00640245i
$768$	0	0
$769$	13298.0	0.623587	0.311793	−	0.950150i	$-0.399070\pi$
$769$	13298.0	0.623587	0.311793	+	0.950150i	$0.399070\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 642.000i	− 0.0298721i	−0.999888	−	0.0149361i	$-0.995246\pi$
$773$	− 642.000i	− 0.0298721i	0.999888	−	0.0149361i	$-0.00475447\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−192.000	−0.00883070
$780$	0	0
$781$	−24072.0	−1.10290
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20236.0i	0.916564i	0.888807	+	0.458282i	$0.151535\pi$
$787$	20236.0i	0.916564i	−0.888807	+	0.458282i	$0.848465\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2700.00	0.121367
$792$	0	0
$793$	2584.00i	0.115713i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11562.0i	− 0.513861i	−0.966430	−	0.256930i	$-0.917289\pi$
$797$	− 11562.0i	− 0.513861i	0.966430	−	0.256930i	$-0.0827111\pi$
$798$	0	0
$799$	3952.00	0.174983
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12852.0i	0.564804i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18984.0	−0.825021	−0.412510	−	0.910953i	$-0.635348\pi$
$809$	−18984.0	−0.825021	−0.412510	+	0.910953i	$0.635348\pi$
$810$	0	0
$811$	−2332.00	−0.100971	−0.0504856	−	0.998725i	$-0.516077\pi$
$811$	−2332.00	−0.100971	−0.0504856	+	0.998725i	$0.516077\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	800.000i	0.0342576i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19126.0	−0.813035	−0.406518	−	0.913643i	$-0.633257\pi$
$821$	−19126.0	−0.813035	−0.406518	+	0.913643i	$0.633257\pi$
$822$	0	0
$823$	37102.0i	1.57144i	0.618583	+	0.785720i	$0.287707\pi$
$823$	37102.0i	1.57144i	−0.618583	+	0.785720i	$0.712293\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11304.0i	0.475307i	0.971350	+	0.237653i	$0.0763782\pi$
$827$	11304.0i	0.475307i	−0.971350	+	0.237653i	$0.923622\pi$
$828$	0	0
$829$	974.000	0.0408063	0.0204031	−	0.999792i	$-0.493505\pi$
$829$	974.000	0.0408063	0.0204031	+	0.999792i	$0.493505\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12882.0i	0.535816i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16480.0	−0.678132	−0.339066	−	0.940763i	$-0.610111\pi$
$839$	−16480.0	−0.678132	−0.339066	+	0.940763i	$0.610111\pi$
$840$	0	0
$841$	−22273.0	−0.913240
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	350.000i	0.0141985i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−47424.0	−1.91031
$852$	0	0
$853$	11192.0i	0.449246i	0.974446	+	0.224623i	$0.0721150\pi$
$853$	11192.0i	0.449246i	−0.974446	+	0.224623i	$0.927885\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34278.0i	1.36629i	0.730281	+	0.683147i	$0.239389\pi$
$857$	34278.0i	1.36629i	−0.730281	+	0.683147i	$0.760611\pi$
$858$	0	0
$859$	14020.0	0.556876	0.278438	−	0.960454i	$-0.410183\pi$
$859$	14020.0	0.556876	0.278438	+	0.960454i	$0.410183\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 30528.0i	− 1.20415i	−0.798438	−	0.602077i	$-0.794340\pi$
$863$	− 30528.0i	− 1.20415i	0.798438	−	0.602077i	$-0.205660\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−28968.0	−1.13081
$870$	0	0
$871$	16592.0	0.645463
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2932.00i	− 0.112892i	−0.998406	−	0.0564462i	$-0.982023\pi$
$877$	− 2932.00i	− 0.112892i	0.998406	−	0.0564462i	$-0.0179769\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7116.00	0.272127	0.136064	−	0.990700i	$-0.456555\pi$
$881$	7116.00	0.272127	0.136064	+	0.990700i	$0.456555\pi$
$882$	0	0
$883$	− 35140.0i	− 1.33925i	−0.742701	−	0.669624i	$-0.766455\pi$
$883$	− 35140.0i	− 1.33925i	0.742701	−	0.669624i	$-0.233545\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20296.0i	0.768290i	0.923273	+	0.384145i	$0.125504\pi$
$887$	20296.0i	0.768290i	−0.923273	+	0.384145i	$0.874496\pi$
$888$	0	0
$889$	732.000	0.0276159
$890$	0	0
$891$	0	0
$892$	0	0
$893$	416.000i	0.0155889i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11960.0	−0.443702
$900$	0	0
$901$	15732.0	0.581697
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 19512.0i	− 0.714317i	−0.934044	−	0.357158i	$-0.883746\pi$
$907$	− 19512.0i	− 0.714317i	0.934044	−	0.357158i	$-0.116254\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−16720.0	−0.608077	−0.304039	−	0.952660i	$-0.598335\pi$
$911$	−16720.0	−0.608077	−0.304039	+	0.952660i	$0.598335\pi$
$912$	0	0
$913$	− 28696.0i	− 1.04020i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 996.000i	− 0.0358678i
$918$	0	0
$919$	−7340.00	−0.263465	−0.131732	−	0.991285i	$-0.542054\pi$
$919$	−7340.00	−0.263465	−0.131732	+	0.991285i	$0.542054\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 48144.0i	− 1.71688i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−48932.0	−1.72810	−0.864051	−	0.503404i	$-0.832081\pi$
$929$	−48932.0	−1.72810	−0.864051	+	0.503404i	$0.832081\pi$
$930$	0	0
$931$	−1356.00	−0.0477348
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 30298.0i	− 1.05634i	−0.849138	−	0.528171i	$-0.822878\pi$
$937$	− 30298.0i	− 1.05634i	0.849138	−	0.528171i	$-0.177122\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8414.00	0.291486	0.145743	−	0.989322i	$-0.453443\pi$
$941$	8414.00	0.291486	0.145743	+	0.989322i	$0.453443\pi$
$942$	0	0
$943$	7296.00i	0.251952i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 23912.0i	− 0.820523i	−0.911968	−	0.410262i	$-0.865437\pi$
$947$	− 23912.0i	− 0.820523i	0.911968	−	0.410262i	$-0.134563\pi$
$948$	0	0
$949$	−25704.0	−0.879228
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 22866.0i	− 0.777232i	−0.921400	−	0.388616i	$-0.872953\pi$
$953$	− 22866.0i	− 0.777232i	0.921400	−	0.388616i	$-0.127047\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4052.00	0.136440
$960$	0	0
$961$	37809.0	1.26914
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 738.000i	− 0.0245424i	−0.999925	−	0.0122712i	$-0.996094\pi$
$967$	− 738.000i	− 0.0245424i	0.999925	−	0.0122712i	$-0.00390614\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44098.0	1.45744	0.728719	−	0.684813i	$-0.240116\pi$
$971$	44098.0	1.45744	0.728719	+	0.684813i	$0.240116\pi$
$972$	0	0
$973$	4920.00i	0.162105i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34426.0i	− 1.12731i	−0.826009	−	0.563657i	$-0.809394\pi$
$977$	− 34426.0i	− 1.12731i	0.826009	−	0.563657i	$-0.190606\pi$
$978$	0	0
$979$	−46920.0	−1.53174
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 30216.0i	− 0.980408i	−0.871608	−	0.490204i	$-0.836922\pi$
$983$	− 30216.0i	− 0.980408i	0.871608	−	0.490204i	$-0.163078\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30400.0	0.977415
$990$	0	0
$991$	4592.00	0.147194	0.0735972	−	0.997288i	$-0.476552\pi$
$991$	4592.00	0.147194	0.0735972	+	0.997288i	$0.476552\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24276.0i	0.771142i	0.922678	+	0.385571i	$0.125996\pi$
$997$	24276.0i	0.771142i	−0.922678	+	0.385571i	$0.874004\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.f.f.649.1		2
3.2	odd	2		1800.4.f.t.649.1		2
5.2	odd	4		360.4.a.k.1.1	yes	1
5.3	odd	4		1800.4.a.q.1.1		1
5.4	even	2	inner	1800.4.f.f.649.2		2
15.2	even	4		360.4.a.d.1.1	✓	1
15.8	even	4		1800.4.a.r.1.1		1
15.14	odd	2		1800.4.f.t.649.2		2
20.7	even	4		720.4.a.x.1.1		1
60.47	odd	4		720.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.a.d.1.1	✓	1	15.2	even	4
360.4.a.k.1.1	yes	1	5.2	odd	4
720.4.a.g.1.1		1	60.47	odd	4
720.4.a.x.1.1		1	20.7	even	4
1800.4.a.q.1.1		1	5.3	odd	4
1800.4.a.r.1.1		1	15.8	even	4
1800.4.f.f.649.1		2	1.1	even	1	trivial
1800.4.f.f.649.2		2	5.4	even	2	inner
1800.4.f.t.649.1		2	3.2	odd	2
1800.4.f.t.649.2		2	15.14	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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{7} +O(q^{10})\)	\(q-2.00000i q^{7} -34.0000 q^{11} -68.0000i q^{13} +38.0000i q^{17} -4.00000 q^{19} +152.000i q^{23} +46.0000 q^{29} -260.000 q^{31} +312.000i q^{37} +48.0000 q^{41} -200.000i q^{43} -104.000i q^{47} +339.000 q^{49} -414.000i q^{53} +2.00000 q^{59} -38.0000 q^{61} +244.000i q^{67} +708.000 q^{71} -378.000i q^{73} +68.0000i q^{77} +852.000 q^{79} +844.000i q^{83} +1380.00 q^{89} -136.000 q^{91} -514.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 68 q^{11} - 8 q^{19} + 92 q^{29} - 520 q^{31} + 96 q^{41} + 678 q^{49} + 4 q^{59} - 76 q^{61} + 1416 q^{71} + 1704 q^{79} + 2760 q^{89} - 272 q^{91}+O(q^{100}) \) 2 * q - 68 * q^11 - 8 * q^19 + 92 * q^29 - 520 * q^31 + 96 * q^41 + 678 * q^49 + 4 * q^59 - 76 * q^61 + 1416 * q^71 + 1704 * q^79 + 2760 * q^89 - 272 * q^91

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